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The Non-Commutative Carathodory-Fejer Problem

Permanent Link: http://ufdc.ufl.edu/UFE0041892/00001

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Title: The Non-Commutative Carathodory-Fejer Problem
Physical Description: 1 online resource (76 p.)
Language: english
Creator: Balasubramanian, Sriram
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2010

Subjects

Subjects / Keywords: analytic, blecher, caratheodory, fejer, interpolation, matrix, noncommutative, operator, ruan, sinclair
Mathematics -- Dissertations, Academic -- UF
Genre: Mathematics thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: We pose the Caratheodory-Fejer interpolation problem for open, circled and bounded matrix-convex sets in C^d. By using the Blecher-Ruan-Sinclair characterization of an abstract unital operator algebra, we obtain a necessary and sufficient condition for the existence of a minimum-norm solution to the problem.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Sriram Balasubramanian.
Thesis: Thesis (Ph.D.)--University of Florida, 2010.
Local: Adviser: McCullough, Scott.

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Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2010
System ID: UFE0041892:00001

Permanent Link: http://ufdc.ufl.edu/UFE0041892/00001

Material Information

Title: The Non-Commutative Carathodory-Fejer Problem
Physical Description: 1 online resource (76 p.)
Language: english
Creator: Balasubramanian, Sriram
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2010

Subjects

Subjects / Keywords: analytic, blecher, caratheodory, fejer, interpolation, matrix, noncommutative, operator, ruan, sinclair
Mathematics -- Dissertations, Academic -- UF
Genre: Mathematics thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: We pose the Caratheodory-Fejer interpolation problem for open, circled and bounded matrix-convex sets in C^d. By using the Blecher-Ruan-Sinclair characterization of an abstract unital operator algebra, we obtain a necessary and sufficient condition for the existence of a minimum-norm solution to the problem.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Sriram Balasubramanian.
Thesis: Thesis (Ph.D.)--University of Florida, 2010.
Local: Adviser: McCullough, Scott.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2010
System ID: UFE0041892:00001


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THE NON-COMMUTATIVE CARATHEODORY-FEJER PROBLEM


By

SRIRAM BALASUBRAMANIAN


















A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY

UNIVERSITY OF FLORIDA

2010































2010 Sriram Balasubramanian
































I dedicate this dissertation to my family and friends.









ACKNOWLEDGMENTS

First and foremost, I would like to express my deepest gratitude to my advisor Dr.

Scott McCullough, for his ever-present guidance, encouragement and help, without

which this dissertation would never have been completed. I am forever grateful to him

for having me as his student and for inspiring me with his tremendous knowledge and

expertise in Mathematics.

I would also like to thank Dr. James Brooks, Dr. James Hobert, Dr. Michael Jury

and Dr. Li-chien Shen for serving on my doctoral committee. My special thanks to Dr.

Michael Jury for his many valuable suggestions.

Finally, I would like to thank my family and my friends especially Balaji, Krishna,

Subbu and Vijay, for their constant encouragement and support.









TABLE OF CONTENTS


ACKNOWLEDGMENTS ................... ............... 4

ABSTRACT. ..................... .................... 7

CHAPTER

1 INTRO DUCTION ..................... .............. 8

1.1 Sum m ary of Results .. .. .. .. .. .. .. 14
1.2 Organization ................... ............... 16

2 MATRIX CONVEXITY ................... ............ 18

2.1 Matrix Convex Sets in Cd ......................... .. 18
2.1.1 Exam ples . .... 19
2.1.2 Properties .................... .. ............. 19
2.2 Matricial Hahn-Banach Separation ................... 21
2.3 The Non-commutative Fock Space and Creation Operators ... 26
2.3.1 The Free Semi-group on d Letters and Intial Segments ...... .26
2.3.2 The Non-commutative Fock Space. 27
2.3.3 The Creation Operators ..... ... 27

3 ABSTRACT OPERATOR ALGEBRAS ....................... 29

3.1 Abstract Operator Algebra ................ ......... 29
3.1.1 Examples .................. ............ 31
3.1.2 The Quotient Operator Algebra ..... ... ... 31
3.2 Representations of Abstract Unital Operator Algebras .... 34

4 THE ABSTRACT OPERATOR ALGEBRAS A(/C)" & A(C)"/I(/) ..... ..35

4.1 The Algebra A(KC)" of Scalar Formal Power Series .. 35
4.1.1 Formal Power Series ... 35
4.1.2 The Vector Space A(KC)" .............. ......... .. 36
4.1.3 Matrix Norms on A(/C)" ........................ 37
4.1.4 The Algebra A(KC) ........................... 39
4.2 Weak Compactness and A(KC)" ....................... 44
4.3 The Abstract Operator Algebra A(C) .. 47
4.4 Completely Contractive Representations of A(C) ... 48
4.5 The Abstract Operator Algebra A(C) / (C) .. 53
4.6 Attainment of Norms of Classes in Mq(A(1C)")/Mq(I(1C)) ... 54

5 THE NON-COMMUTATIVE CARATHEODORY-FEJER PROBLEM ...... ..56

5.1 The Caratheodory-Fej6r Interpolation Problem (CFP) .... 56
5.2 The Matrix Version .......... ........ ............. 56









5.3 The Operator Version ............... .......... .. 58

6 INFINITE INITIAL SEGMENTS ............... .......... 62

6.1 Examples of Non-commutative Operator Domains .... 62
6.2 The d-dimensional Non-commutative Polydisc ... 64
6.3 The d x d Non-commutative Mixed Ball ... 69

7 FUTURE RESEARCH ................... ............. 72

R EFER EN C ES . . 73

BIOGRAPHICAL SKETCH ................... ............. 76









Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy

THE NON-COMMUTATIVE CARATHEODORY-FEJER PROBLEM

By

Sriram Balasubramanian

August 2010

Chair: Scott McCullough
Major: Mathematics

We pose the Caratheodory-Fej6r interpolation problem for open, circled and

bounded matrix-convex sets in Cd. By using the Blecher-Ruan-Sinclair characterization

of an abstract unital operator algebra, we obtain a necessary and sufficient condition for

the existence of a minimum-norm solution to the problem.









CHAPTER 1
INTRODUCTION

A classical interpolation problem in function theory is the Caratheodory-Fej6r

interpolation problem (CFP): Given n + 1 complex numbers co, cl,..., c, does there exist

a complex valued analytic function f(z) = Yo fjzJ defined on the open unit disc D c C

such that j = cj for all 0
The problem and some of its variants were studied by Caratheodory, Fejer and

Schur during the early 20th century in [30], [31] and [8] A necessary and sufficient

condition for the solvability of the CFP, which is commonly referred to as the Schur

Criterion, is that the Toeplitz matrix

co 0 ... 0

(1.1)
0

Cn ... Ci Co

is a contraction.

An operator theoretic view of the CFP was first presented by Sarason in his

pioneering work [29]. His formulation has had a major impact not only on the CFP

and the related Pick interpolation problem but the development of operator theory and

the study of non-self adjoint operator algebras generally. We now present a proof of

the equivalence between the solvability of the CFP and the Schur Criterion that uses

Sarason's ideas. We will begin with some definitions and state some well-known facts

(without proofs).

Let H2(D) denote the Hardy Hilbert Space defined by

H2(D) = f : D C : f(z) = az" and I a,,2 < co .
n=O nO=









The inner product on H2(D) is given by


00
(f, g) = anbn
n=0

where f(z) = anz" and g(z) = bnz.
n=O n=O
Let L2 denote the Hilbert space of square-integrable functions on T with respect to

the normalized Lebesgue measure. The inner product is defined as:
1 027T
(f g) = 2 j f(e')g(e'o) dO

where dO denotes the Lebesgue measure on [0, 27].

As is often done, we will view L2 as a space of functions rather than as a space

of equivalence classes of functions, by identifying two functions to be equal if they are

equal a.e. with respect to the normalized Lebesgue measure. i.e.

L = f : T -- C : I f(e0) 2d < oI .

The Hilbert space L2 is separable with orthonormal basis {en}nez, where en : T -I C

is defined by en(ei') = ei'0. For a proof of this fact see [7]. The expansion of the function

f e L2 with respect to this orthonormal basis is called the Fourier Series expansion of f.

We write

f(e'0) > Y f(n)e'"
n=-oo
where the Fourier coefficients f(n) are given by

1 2 (eifle ii
f(n) = (f, en) = OdO

and they satisfy
>I(n)12 < cc.
n--oo
The Fourier Transform produces a canonical unitary equivalence between H2(D)

and a subspace of L2 as described below. It is a consequence of Fatou's Theorem









(see pp 15 20, [22]) that for f c H2(D), the radial limit f(e'0) = lim f(re') exists
r--- l
for almost all 0. We will call f the boundary function of f. An important property of f is

that f c L2. The linear mapping that takes f c H2(D) to f e L2 is an isometry from

H2(D) onto a closed subspace of L2 which we will denote by H2. If g e L2 is given by
00
g(e'0) a g(n)e'"0, then it is well known that the function f defined on the unit disc
n=0
by f(z) = g(n)z" is in H2(D) and in addition, f = g. Conversely, if f e H2(D) is
n=0
given by f(z) = fz" then, the Fourier series expansion of f is f(e'") f ,e'"0.
n=O n=0
This equivalence gives us the following way to view the space H2 in terms of Fourier

coefficients, namely,

H2 = {g L2: g(n) = 0 for n < 0}.

Let f be a Lebesgue measurable function defined on 'T. The essential supremum of

f is defined by II||f| = inf{M > 0 : |f(z)| < M a.e.}. Let L" denote the Banach space

L" = {f : T I C : fis measurable and |lfll < oo}


and let H" denote H2 q L0.

Remark 1.0.1. If f is a bounded analytic function defined on the unit disc, then f e H"

and 1||f || < sup{|f(z)| : z e D}. Conversely, iff c H", then the function f c H2(ID) is

bounded by I f | .

For ce LPc, the Toeplitz Operator with symbol 4 is defined by


Tof = P(Of)


for each f e H2, where P is the orthogonal projection of L2 onto H2. If ec H00, then

Tof = P(Of) = Of.









In this case the Toeplitz operator is said to be analytic and its matrix with respect to the
basis {ei 0} is given by


An important property of the Toeplitz operator To is that


11 Tll = 11 110


(1.2)


Lemma 1.0.1 (Sarason). Let


(i) U represent the Unilateral shift on H2, i.e. (Uf)(z) = zf(z) for all f c H2

(ii) K = H2 Q H2 where b is an inner function, i.e. b e H" and I (z) = 1 a.e.

(iii) S = PU K where P denotes the orthogonal projection of H2 onto K.

(iv) T be a (bounded) operator on K.
If TS = ST, then there exists a function ce H" such that T = PT| K where To is the
analytic Toeplitz operator with symbol 0 and I1011, = II TII.
A proof of the above lemma can be found in [29].
Theorem 1.0.1. The CFP has a solution if and only if the matrix in equation (1.1) has
norm at most one.

Proof. Suppose that there exists a solution f to the CFP. It follows from remark 1.0.1
and equation (1.2) that f e H" and,

1 > sup{|f(z)l :z C D} > ||f || = II T II|.









Since the matrix in equation (1.1) is the compression of the analytic Toeplitz operator TT
to the subspace spanned by the orthonormal set (l, z, ..., z") of H2 (and computed
with respect to this basis), it follows that the norm of that matrix is at most 1.
For the converse, we apply Lemma 1.0.1 with b(z) = z"+1, K = (l, z, z ..., z")
and T : K K being the matrix in (1.1). Suppose that II TI| < 1. Let S and P be the

operators that appear in the hypothesis of Lemma 1.0.1. The matrix of S with respect to
the basis {l, z, z ... z"} is given by

0

S = (1.3)

1 0

Thus TS = ST and Lemma 1.0.1 implies that there exists c e H" such that T = PTI K
and

11 11. = II Tll.

Since ce H", = f for some f c H2(D). It follows from Remark 1.0.1 that

sup{|f(z)l : z eD} < 11 .11

Hence this f is a solution to the CFP. D

From the operator theory/algebra point of view, the CFP is essentially unchanged
if the coefficients co, c, ..., cn are taken to be elements of B(U) for some separable
Hilbert space U. Indeed, even in this case, a necessary and sufficient condition for the

solvability of the CFP is the same as before, only now the entries of the Toeplitz matrix in
(1.1) are bounded operators in B(U) and the matrix itself is an element of B(( n U). An

alternate way of viewing the Schur Criterion which is more convenient for our purposes

is the following.










Lemma 1.0.2. Let p(z) = YJno cjzJ e B() and d-i be an arbitrary Hilbert space. The

Schur Criterion is equivalent to the contractivity of the operator p( T) = Y:n cj 0 TJ

B(U H) for every contraction T E B(H) which is nilpotent of order n + 1; i.e., T"+ = 0.

Proof. (=) Let S be the matrix in equation (1.3). Since S is a contraction and S"+1 = 0

we have I|p(S)l| < 1. The fact that the norm of the Toeplitz matrix in equation (1.1) is

equal to the norm of the operator p(S) completes the argument.

(=) Fix a Hilbert space -H and a contraction T e B((-() which satisfies T"+1 = 0.

Let DTr = (I TT*)2. DTr is called the defect operatorof T*. Let {e}j}jo denote the

standard orthonormal basis of C"+1. Define the operator V: H C"+1 '0 by

n
Vh= ej e DT.(T*yh.
j=o

Then

n n
(Vh, Vh)= ( ej DT*(T*yh, ek DT*(T*)kh)
j= 0 k= 0
n
= Z(DT.(T*yh, DT*(T*)Jh)
j=o
n
= {(TJ(/ TT*)(T*)Jh, h)
j=o

= (h, h).


Thus V is an isometry. Moreover, For each k = 0, 1,2,... we have,

n
V(T*)kh = Y e, DTr(T*y+kh
J=o
n-k
= e, DTr(T*Y kh
j=o
n
= ((S*)k I)( eg j DT*(T*)Jh)
j=o

((S*)k l) Vh.









This implies that for each k = 0, 1, 2,...,


Tk =V*(Sk l)V. (1.4)


Using equation (1.4) we get,
n
||p(T)||= II c, : V*(S /H)V|
j= 0
n
= I1(G/0 v*)(Y- C 0 (Sj 0/H)) (G V)ll

=o0
j 0

= 1|p(S)l|

<1.




1.1 Summary of Results

Several commutative multi-variable generalizations of the CFP have been obtained

for different domains the polydisc Dd c Cd for example and for different interpolating

classes of functions, for example the Schur-Agler class of analytic functions that

take contractive operator values on any d-tuple of commuting strict contractions in a

manner discussed in [1]. For more details see [14], [6]. Some results on the problem

for bounded circular domains in Cd can also be found in [11]. Some non-commutative

generalizations of the CFP have also been studied in [25], [26], [9], [20], [4].

In this thesis, some of the existing results on the CFP have been extended to

the non-commutative setting of the free algebra on a finite number of generators. An

example of a domain we consider here is the d x d non-commutative matrix mixed ball

defined by


Ddd = U{X = (X1, X2.... Xd) : Xi are n x n matrices and |X||op < 1}.
nEN









where I||X|op is the norm of the operator X = (Xy)d d1 (Cn)ad (C)d

What follows are some definitions which will lead us to the statement of the CFP for

this particular domain.

* Let FdT denote the semigroup of words generated by the symbols {g,~})jd1

* A set A c TFd is said to be an initial segment if wg,, gw A for all w A,
1 < i < d, 1
For X e Ddd and w = gilg,... gij, c Fda, the evaluation of X at w is defined by
XW = X"l, X"j,2 ... Xik .

X e Ddd is said to be A-nilpotent if Xw = 0 for all w A.

A formal power series f is an expression of the form f = C,, fww where the
coefficients fw are complex numbers

For X C Ddd we define f(X) = J H0 C wl= fwXW whenever the series converges
converges (in the operator norm) in the indicated order.

The CFP for the d x d non-commutative matrix mixed ball is the following: Let A, a

finite initial segment, and

P = Pww
wEA
be given. Does there exist a formal power series f that fw = pw for w e A and
sup{||f(X)|| : X C d} < 1?
(A special case of) Our main result is the following.
Theorem 1.1.1. There exists a (minimum-norm) solution f to the above problem if and

only if

sup{||p(X) | : X e Ddd, X is A nilpotent} < 1.

In the body of the thesis we actually pose the CFP for more general domains that

are matrix convex sets in Cd, and using the Blecher-Ruan-Sinclair characterization of

abstract operator algebras, prove a generalization of Theorem 1.1.1 which allows for

operator-valued coefficients pw and fw.









Throughout this thesis, we will be working with non-commutative analytic functions,

i.e. formal power series with matrix or operator coefficients that converge on some

non-commutative neighborhood of the origin (see [32], [33], [34], [24], [25], [26], [21],

[19]). These functions are not only objects of great mathematical interest, but have
applications in areas such as control theory and optimization. Non-commutative

polynomials, in particular, are of special interest since non-commutative polynomial

inequalities (matrix inequalities where the unknowns are matrices too), occur naturally

in the context of dimension-free linear systems. Recent advances in the study of

non-commutative linear matrix inequalities can be found in [12], [18].

1.2 Organization

This thesis is organized as follows: In Chapter 2, the definition of a matrix

convex set in Cd is given along with some examples and properties and a proof of

the Effros-Winkler matricial Hahn-Banach Separation Theorem. The chapter ends with a

discussion on the Non-commutative Fock Space and the associated Creation Operators.

In Chapter 3, the definition an abstract operator algebra is introduced along with

some examples. It is also shown that the quotient of an abstract operator algebra by

a closed two-sided ideal is an abstract operator algebra. The chapter ends with the

statement of the Blecher-Ruan-Sinclair Theorem for abstract unital operator algebras.

Chapter 4 is where the interpolating class A(IC)" and the ideal Z(IC) are introduced.

It is shown that A(/C)" and the quotient A(/C)/IZ(/C) are abstract operator algebras.

Several key properties including a weak-compactness type property of the algebra

A(KC)" and the norm attainment property of the algebra A(/C)"/IZ(/) are established.

The chapter ends with a discussion on completely contractive representations of the

algebra A(KC)", where it is shown that tuples of finite-dimensional compressions of

operators that give rise to completely contractive representations of A(KC)" lie on the

boundary of the underlying matrix convex set.









In Chapter 5, the matrix and the operator versions of the CFP (with a finite

initial segment A) for a class of matrix-convex sets in Cd are posed and using the

results from Chapters 2 4, a necessary and sufficient condition for the existence of a

minimum-norm solution is obtained.

In Chapter 6, a version of Theorem 1.1.1 where it is assumed that the initial

segment A is an infinite set, is proved for two special non-commutative domains namely,

the d-dimensional non-commutative (operator) polydisc and the d x d non-commutative

(operator) mixed ball.

In Chapter 7, two important and very interesting questions that came up while this

work was in progress, are posed.









CHAPTER 2
MATRIX CONVEXITY

A basic object of study in this thesis is a quantized, or non-commutative, version of

a convex set. While the definitions easily extend to convex subsets of arbitrary vector

spaces, here the focus is on subsets of Cd, the complex d-dimensional space. In this

chapter we present the definition of a matrix convex subset of Cd and introduce our

standard assumptions regarding these sets. Some properties and examples of such

sets and a matricial Hahn-Banach separation result are also presented. The chapter

ends with a discussion of the Creation Operators on the Non-commutative Fock Space.

2.1 Matrix Convex Sets in Cd

Let Mm,n = Mm,n(C) denote the m x n matrices over C. In the case that m = n,

we write Mn instead of Mn,,. Let Mn(Cd) denote d-tuples with entries from Mn. Thus, an

X e Mn(Cd) has the form X = (X1,..., Xd) where each X, e Mn.

A non-commutative set is a sequence ((n)) where, for n e N, (n) c Mn(Cd),

which is closed with respect to direct sums; i.e., if X e (n) and Y c L(m), then

X Y = (X1 Y1,..., Xd Yd) E (n + m) (2.1)

where

xi (= x

A non-commutative set = ((n)) is open (closed) if each (n) is open (closed).

A matrix convex set C = (IC(n)) is a non-commutative set which is closed with

respect to conjugation by an isometry; i.e., if a c Mm,n and a*a = In, and if X =

(X ..., Xd) c C(m), then

a*Xa = (a**Xia, ..., a*Xda) e /C(n). (2.2)

A subset U of Mn(Cd) is circled if eieU c U for all 0 e R. A matrix convex set CK is

circled if each IC(n) is circled. As a canonical example of a circled matrix convex set,









suppose 7 > 0 and consider the non-commutative y-neighborhoodC, = (C,(n)) of

0 e Cd defined by
d
C(n) = {X e Mn(Cd) XjX 72}.
j-1
A matrix convex set /C = (C(n)), is said to be bounded if there exists 7 < 1, F > 1

such that, for each n c N,

C(n) C /C(n)C Cr(n), (2.3)

Assumption 2.1.1. Here, it is typically assumed that the matrix convex set /C

(a) is open;

(b) is bounded; and

(c) is circled;
2.1.1 Examples

The following are some examples of matrix convex sets in Cd that satisfy the

conditions of Assumption 1.

(i) Let /C(n) = {(X,,..., Xd) : I||XJ < 1} with 7 = 1 and F = vd. /C = (/C(n)) is the
d-dimensional non-commutative matrix polydisc.

(ii) Let /C(n) = {X = (X11, X12..., XdX) : I|X|op < 1}, where ||X||op is the norm of the
operator X = (X,)d 1 (Cn) (C)d, with 7 and F /C (n) is
the d x d non-commutative matrix mixed ball.

(iii) Let C(n) = {(X ,...,Xd) : sup{ d (X/y, y)| : ||y| 1} < 1} with 7= and
F = 2 d. If d = 1, then C is the collection of all strict numerical radius contractions.

2.1.2 Properties

Below we present three important properties of a matrix convex set = ((n)) in

Cd which we will use in the forthcoming chapters.

(i) (n) is convex for each n c N.

(ii) If 0 e L, then L is closed with respect to conjugation by a contraction; i.e. a in
equation (2.2) can be assumed to be a contraction.

(ii) The closure of L namely = ((n)) is matrix convex.









Proof. (i) Let X, Y e IC(n) and 0 < t < 1. Consider the the block matrix a* =

( tin 1 t) e Mn,2,. Since a is an isometry it follows that,

tX +(1 t)Y = *(X E Y)a IC(n).

(ii) Let a e Mn,, be such that I||j| < 1 and X e (n). We first observe that

a = (a 0) e Mmn,.m,, is a contraction. Let

X = (X E 0) e C(n + m)


and


X = (X e 0) /C(2(n


m)).


Let Da denote the defect operator of 5. i.e.,


D = (Im +n )'.

Then the Julia matrix of a namely,

d Dj
Ji(s =Tr e
Da -d*

is unitary. Therefore,


d*Xa = lm+n Om-


It follows that


a*Xa =X Inm Omn)


(iii) Consider Mn(Cd) with the topology defined by the norm


11 l(X ... X ) d) |l


(2.4)


mn /C(m
Om nJ


j=1


-) J(5)*kXJ( 5)


X5 O e /C(m).
0n,m









where the norm on the RHS is the usual operator norm.

Let X = (X, ..., Xd) e (p) and Y = (Y1, ..., Yd) e (q). Choose sequences

Xm = (Xmi ...- Xmd) and Y, = (Y,,1..., Ymd) from (p) and (q) respectively such that

Xm X and Y,+ Y.

We have,
d
II| ( ,X Y)||| = || (X,, Y,,) (Xj Y Y,)||
j= 1
d
S-(Xr Xj) E (Yrj Yj)
j 1
d
= max{||X,,j Xj ||, || Yj | }
j 1

Since X, X and Yr, Y, the sum on the RHS above can be made arbitrarily

small for all large values m. Thus X Y c (p + q).

If a e Mp is an isometry, then we have

d
|| |aX,o a*X(, ||| = oa*(X,, X))o||

d

j 1

Since X, X, it follows that a*Xoa (e).



2.2 Matricial Hahn-Banach Separation

In this section we present a Hahn-Banach separation theorem for matrix convex

sets in Cd due to Effros and Winkler (See [13]). The following contents, are minor

variants of lemmas and theorems from [16].

Given a positive integer n, let E, denote the collection of all positive semi-definite

n x n complex matrices of trace one. A T e E corresponds to a state on M,, via the









trace,


Mn A 3A tr(AT).

An affine linear mapping f : T IR is a function of the form f(x) = af Af+(x),

where Af is linear and af e R.

Lemma 2.2.1. Suppose F is a cone of affine linear mappings f : T, R. If for each

f e F there is a T E 7E such that f(T) > 0, then there is an S E 7E such that f(S) > 0

for every f c F.

Proof. For f e F, let

Bf = {S T: f(S) > 0}.

By hypothesis each Bf is non-empty and it suffices to prove that

nfEBf # 0.


Since each Bf is compact, it suffices to prove that the collection {Bf : f e F} has

the finite intersection property. Accordingly, let f, ...f, f F be given. Arguing by

contradiction, suppose

qn if 0.

In this case, the range F(T7) of the mapping F : T RW" defined by


F(S)= (fl(S),..., fm(S))


is both convex and compact because En is both convex and compact. Moreover, it does

not intersect
R" { = {x = (x, ..., x) : xj > 0 for each j}.

Hence there is a linear functional A : R"' R such that A(F(E)) < 0 and A(R"n) > 0.

There exists Aj such that
m
S(x) Ajxj
y 1









Since A(Rm) > 0 it follows that each Aj > 0 and since A / 0, for at least one k, Ak > 0.

Let
m

j= 1
Since F is a cone and Aj > 0, we have f e F. On the other hand, if T e En, then

f(T) < 0. Hence for this f there does not exist a T e n such that f(T) > 0, a

contradiction which completes the proof. O

Lemma 2.2.2. Let C = (C(n)) denote a matrix convex set in Cd which contains 0 e Cd.

Let n c N and a linear functional F : Mn(Cd) C be given. If


Re F(C(n)) < 1,


then there exists S E En such that for each m c N, Y e C(m) and C c Mm,n


Re F(C*YC) tr(CSC*).


Proof. For each m c N, Y e C(m) and C c M,,n, we define the affine linear map

fyc : n -IR by

fyc(T) = tr(CTC*) Re F(C*YC).

Let Fn = {fy,c : Y C(m), C e M,,n, m e N}. The set Fn is a cone since


f3y,c = fy, c

and

fry,C, + fY2,2 = fZ,D

where Z = Yi Y2, D* is the block matrix (CQ C) and 8 > 0 is arbitrary.

Choosing T = aa* where a is a unit vector such that


II Ca|l = | C||,









it follows that


fyc(7) = IC Re F(C*YC).

If II CII = 1, then by property (iii) from Subsection 2.1.2, C* YC e C(n) and so by the
hypothesis of the lemma, the right hand side of the above equation is non-negative. If
C does not have norm 1, but is not zero, then a simple scaling argument shows that

f,c(T)> 0.
Hence by Lemma 2.2.1, there exists an S e Tn such that fy,c(S) > 0 for every m,
Y C(m) and C e Mm,n. O

A linear pencil L of size n is a formal expression of the form yL 1 Lege where
L, e Mn. For a d-tuple T = (T,,..., Td) of bounded operators on a Hilbert Space TH, the
evaluation of L at T is defined as the operator L( T) = e Le Te.
Theorem 2.2.1 (Matricial Hahn-Banach Separation). LetC = (C(n)) denote a closed
matrix convex set in Cd which contains a non-commutative neighborhood of 0 e Cd. If

X C(n), then there is a linear pencil L (of size n) that satisfies the following conditions.

(i) 2 L(Y) L(Y)* > 0 for all m E N and Y e C(m)

(ii) 2 L(X) L(X)* E 0.

Proof. By applying the usual Hahn-Banach separation theorem on the closed convex
subset C(n) of Mn(Cd) and using the assumption that C(n) contains a non-commutative
neighborhood of 0, we obtain a linear functional F : Mn(Cd) -+ C such that

Re F(C(n)) < 1 < Re F(X).

Choose 0 < c < 1 sufficiently small such that G = (1 c)F satisfies

Re G(C(n)) < 1 < Re G(X). (2.5)

From Lemma 2.2.2 there exists S e Tn such that

Re F(C*YC) < tr(CSC*)









for each m c N, Y c C(m) and C c Mm,n.

If we let R = (1 e)S + then R Ec T, it is invertible and
If w letR = L )5 +n n


Re G(C*YC) < tr(CRC*).


(2.6)


Let {el,..., ed} denote the standard orthonormal basis for Cd. Given 1 < < d, and

column vectors c, d e Cn, define a bounded sesquilinear form on Cn by


B,(c, d) = G(R- dcTR-& e)


where cT denotes the transpose of c.

There exists a unique matrix Be e Mn such that


B(c, d) = (Bec, d).


Define the linear pencil L by dE BEg.

Fix a positive integer m. Let Y =(Yi,..., Yd) e C(m) be given and consider L(Y),

the evaluation of L at Y. Let {e, ..., em} denote the standard orthonormal basis of Cm.

For 6 = Ej1 J 0 ej e C" C"m, we have


m d
(L(Y)6,} = ~Bj, {Yej, e
ij=1 = 1
m d
1--
= G(R-i'6JR- ,
ij= 1 =i 1
m d
= G((R- 2j) (R-P
ij=1 = 1
d m
= G( (C(R-i)(Ye4,
e=1 ij=1
i G 1 -
= G((R-6)Y(R-6)*)


ee)(Y ej, ei)


)0e* e)<(Y ej, ei)


ei)(R-'6j)*) & e)









where J6 is the column vector whose entries are complex conjugates of the column

vector 6, and 6 is the n x m matrix with j-th column 6j. Using equation (2.6) we get,

Re (L(Y)6, ) = Re G((R-'6)Y(R-6)*)

< tr((R-'6)* R(R-2)6))
m
11
= 'R(R-6_), (R-'6j))
j= 1
m m
=<( ;& e,, 0& ej)
i=1 j=1

= 116112.

On the other hand, computing as above and using equation (2.5) we get,
n n
Re (L(X) Rei, e,, Rej 0 ej) = Re G(InXIn)
i=1 j=1

>1
n
= IRei ei, 2.
i= 1



2.3 The Non-commutative Fock Space and Creation Operators

The Fock space and the Creation Operators that act on it play a central role in the

analysis to follow in the forthcoming chapters. One of the key properties is that tuples

of finite-dimensional compressions of the creation operators lie in the underlying (open,

circled and bounded) matrix convex set IC. We provide a proof of this fact in this section.

2.3.1 The Free Semi-group on d Letters and Intial Segments

The Fock space is defined in terms of the free semi-group on d letters.

Let Fd denote the set of all words generated by d symbols {g, ..., gd}. Define the

product on Fd by concatenation. i.e., if w = gi...gi and v = gj1...gj,, then the product

wv is given by g,i...ggj,...gj,. Fd is a semi-group with respect to this product, with the

empty word 0 acting as the identity element, i.e. w0 = w = Ow for all w e Fd.









The length of the word w = gi,...gi, is declared to be m (where it is assumed that

g, / 0) and is denoted I w. The length of 0 is zero.
A set A c Fd is an initial segment if its complement is an ideal in the semi-group Fd;

i.e., if both gjw, wgj e Fd \ A (1 < j < d), whenever we Fd \ A. In the case that d = 1 an

initial segment is thus a set of the form {0, g,, g',..., gm} for some m.

2.3.2 The Non-commutative Fock Space

Let C(g) C(g, ..., gd) denote the algebra of non-commuting polynomials in the

variables {g, ..., gd}. Thus elements of C(g) are linear combinations of elements of Fd;

i.e., an element of C(g) of degree (at most) k has the form
k

j=0 Iwl=j

where the pw are complex numbers.

To construct the Fock space, F2, define an inner product on C(g) by defining

0 if w / v
(w, v} = : (2.7)
1 if w = v

for w, v e Fd and extending it by linearity to all of C(g). The completion of C(g) in this

inner product is then the Hilbert space F2.

2.3.3 The Creation Operators

There are natural isometric operators on F2 called the creation operators which

have been studied intensely in part because of their connection to the Cuntz algebra [7].

Given 1 < j < d, define Sj : F2 F2 by Sjv = gjv for a word v e d and extend Sj

by linearity to all of C(g). It is readily verified that Sj is an isometric mapping of C(g) into

itself and it thus follows that Sj extends to an isometry on all of F2. In particular S Sj = I,

the identity on F2. Also of note is the identity,
d
i~S SjS = P, (2.8)
j= 1








where P is the projection onto the orthogonal complement of the one-dimensional
subspace of F2 spanned by 0, which follows by observing, for a word w e Fd and
1
S*(w) if w = gjv
0 otherwise.

Of course, as it stands the tuple S =(S1, ...,Sd) acts on the infinite dimensional
Hilbert space F2. There are however, finite dimensional subspaces which are essentially
determined by ideals in Fd and which are invariant for each S*.
The subset A(f) = {w : wl < } ofFd is a canonical example of a finite
initial segment. And the subspace F(T)2 of F2 spanned by A(f) is invariant for S5,
1 < j < d. Let V(f) denote the inclusion of F(f)2 into F2 and let 5() denote the operator
V(e)*SV(e). Thus, 5() = ((S())1,..., (S(f))d) where (S(C))j = V()*SjV(C).
Recall 7 from the definition of the (open, bounded and circled) matrix-convex set /C.
Lemma 2.3.1. If t < 7, then tS(f) e IC(n) for some n c N.

Proof. Let P denote both the projection of F2 and F(C)2 onto the orthogonal complement
of the span of 0 in F2 and F(C)2 respectively. It follows from equation (2.8) that

P =V()*PV()

=v(ef*Y Sisj* v(

d
Y= (Sv))(Sv));.
j=1

Thus for t < 7, that tS() e C,(n) c /C(n), where n = o= di is the dimension of
F()e2. D

Remark 2.3.1. S()W = 0 for all w A(C).









CHAPTER 3
ABSTRACT OPERATOR ALGEBRAS
We begin this chapter with the definition of an abstract operator algebra. Following
that we present some examples and the proof of the fact that the quotient of an
abstract operator algebra by a closed two-sided ideal is an abstract operator algebra.
Furthermore, we present a characterization of an abstract unital operator algebra due to
Blecher, Ruan and Sinclair.
3.1 Abstract Operator Algebra

Let V be a complex vector space and Mp,q(V) denote the set of all p x q matrices
with entries from V. V is said to be a matrix normed space provided that there exist
norms I| I|p,q on Mpq(V) that satisfy

IIA X BI,,r < IIAIIIIX IpqII BI

for all A Mc,p, X Mp,q(V), B Mq,r.
A matrix normed space V is said to be an abstract operator space if

IIX E Y||p+e,4,, = max{||Xll p,q,, I| Y|j,


where Xe Mp,,(V) and Y e M ,(V) and X Y =
v0 Y
It is important to note that, without loss of generality, we can replace the rectangular
matrices in the above definitions with square matrices.
V is an abstract operator algebra if V is an algebra, an abstract operator space
and if the product on V is completely contractive i.e. ||XY||p < 1 whenever ||X|lp < 1
and I| Yllp < 1 for all X, Y e Mp(V) and for all p. We say V is unitalif V contains a
multiplicative unit.
Before we look at examples of abstract operator algebras, we present, as a remark,
an interesting fact about the abstract operator space Cd.








Remark 3.1.1. The closed unit balls of the abstract operator space Cd form a matrix
convex set.
A partial converse to the above remark is the following.
Lemma 3.1.1. Let L = ((n)) be a closed, bounded, absorbing and circled matrix
convex set in Cd. If L is strongly circled, i.e. UL(n) c (n) for all n e N and unitary
matrices U e Mn, then there exists a sequence of norms I|I |, such that L(n) is the
closed unit ball of Mn(Cd) with respect to II| ||n, and Cd together with the sequence of
norms II| ||n is an abstract operator space.

Proof. It follows from the hypothesis and the definition of matrix convexity that for any
unitary matrix U e Mn,
(n)U = U*(UL(n))U C (n).

This implies that for unitaries U, V e M,

ULn)V C (n). (3.1)

Since (n) is convex, closed, bounded, absorbing and circled, it is the closed unit
ball of M,(Cd) with respect to some norm, which we will denote I| |nII. We need to show
that Cd together with the sequence of norms II| ||n is an abstract operator space.
Let X e M,(Cd) be such that I||X|n = 1 and A, B e M, be of unit norm. Consider
the Julia matrices (see (2.4)) J(A), J(B) of A and B. Since J(A) and J(B) are unitary, it
follows from equation (3.1) that,

I|J(A)(X O0)J(B)||2n < 1.

Hence,

IIAXBEI = || In On J(A)(X 0)J(B) < 1.

If any of A, B c M, and X e M,(Cd) are not of unit norm, then a simple scaling
argument shows that IIAXBII < I|AIIIIXlI B|| |. Thus (Cd, 1 I n) is a matrix-normed








space. Next we show that if X e Mn(Cd) and Y e Mm(Cd) then, IIX YI||In =
max{||X|| |I Y m}.
Since (Cd, I In) is a matrix normed space it follows that,

IIX Y||n m > max{||lX||, | Y|m}.

To prove the reverse inequality, observe that II x IIn and YaxImII are
max{xll X ljlYll,.} max{f llXjll, Y }ma}
at most one. Hence, by the matrix convexity of L, it follows that,

X Y
max{llXlln, IIYI i, max{||X||n, I IYIm})



3.1.1 Examples

(i) Let R- be a Hilbert space and V = B(H-). Define matrix norms on V by

i( (T ) p,q= IITII
where T is the operator (T,,)p, 1 : (q'T i c ) and I| TI| is its operator norm.
Then V together with the sequence of norms I|| I|p,q is an abstract unital operator
algebra.

(ii) Let R- be an arbitrary separable Hilbert space, and V denote the algebra of
polynomials in d variables. Define matrix norms on V by

II||(x,)||lp,q = sup{||(x, (T))ll}
where the supremum is taken over all d-tuples T = (T, ... Td), where { Tk} = C
B(-H) is a set of commuting contractions. Then V together with the sequence of
norms I|| IIp,q is an abstract unital operator algebra.
3.1.2 The Quotient Operator Algebra
Let V be an abstract operator space with the sequence of norms I| I|p,q, and let W
be a closed subspace. Let r : V V/W denote the quotient map rT(x) = x + W. By
identifying Mp,q(V/W) with Mp,q(V)/Mp,q(W) we get a sequence of norms 1||| IIp,q on

Mp,q(V/W) defined by

IIl(r(x,))ll l,q = inf{||(xj + Yij)||, : yY E W }








Lemma 3.1.2. V/W with the sequence of norms | | || p,q defined as above is an abstract
operator space.

Proof. Let A c MKp, X = ( (xy)) e Mp,q(V/W) and B e Mqr. Choose (yi) E Mp,q(W)
such that
IIl( x ))lll p,-q + > II(x, + y )lp,q. (3.2)

Observe that

AXB = A((x-)) B = I(A(x)-)B) = T(A(x + y-)B). (3.3)

Using equations (3.2) and (3.3) gives,

IIIAXBIII ,q = I I(A(x + y,)B)) | Ip,q

< IIA(x + yj)BIp,q

< IIAII (x,+ Y )| p,q)II BI


= IIA II(IIeIX II l,, c)IIB II.

By letting c 0, it follows that V/W is a matrix-normed space.
Let X = (q(x,)) e Mp,q(V/W) and Y = (q(ymn)) E Mt.,(V/W). Since V/W is a
matrix-normed space, it can be seen that

IIIX E Y|||lp+,q > > max{||Xl|p,q, II Y|l,,}.

To prove the reverse inequality, choose (a,) e Mp,q(W) and (bmn) e Mr.,(W) such that

11l(Tx( ))lll,. + > II(xd 4+ ad)|l|p,q (3.4)

IIl(T (ym))ll + > |(ymn bmn)|,, (3.5)

and observe that
X Y Y = n((x, ay a) (ynn + bmn)). (3.6)








Using equations (3.4), (3.5) and (3.6) yields,


|||X E Y||l ,p+t, = I|I Ir ((xj + ay) ( (y n + bmn))l Ip+,q+r

< |I(xy + ay) (ymn + bmn) p+,q+ r

= max{||(x + ay)| p,q, I(Ymn + bmn).e),r}

< max{|||X |||p, III Ylll ,} + .

Letting c 0 completes the proof. O

Corollary 3.1.1. Let V be an abstract unital operator algebra and W be a closed
two-sided ideal in V. Then, V/ W is an abstract unital operator algebra.

Proof. From Lemma 3.1.2 we know that V/W is an abstract operator space. Moreover,
1 + W is the unit of V/W. It remains to show that the product on V/W is completely
contractive. For that purpose, let X = (r(x,()) and Y = (r(yy)) e Mp(V/W). Choose
(ay), (bu) e Mp(W) such that

IiI l(s xj))lll p + > I(x o+V adt)l op (3.7)

|||l(T(y ))lllp + e > ||(yu + b )ll|p (3.8)

Since W is a two-sided ideal of V, it follows that

XY = (Tl(xy))(Tl(yy)) = Tl((X-)(y)) = l((xu + a)(yyi + by)) (3.9)









Using equations (3.7), (3.8), (3.9) and the fact that multiplication in V is completely
contractive yields,

IIIXYlllp = I|I((xy ay)(yy + b,))| lp

II(xdj + ad)(yd + b,)llp

"< I(x -+ a,jlp\l(y -+ b,)llp


< (lll(x ))lllp + )(Y IIIP + lll( Y-))lll )


The corollary follows by letting c 0. O

3.2 Representations of Abstract Unital Operator Algebras
The following is a characterization of abstract unital operator algebras due to
Blecher, Ruan and Sinclair.
Let V and W be abstract operator spaces and : V V W be a linear map. Define
Oq: Mq 0 V -- Mq 0 W by Oq = Iq & where Iq is the q x q identity matrix.
The map 0 is said to be completely contractive (isometric) if Oq is a contraction
isometryy) for each q e N.
A completely contractive (isometric) representation of an algebra A is a completely
contractive (isometric) algebra homomorphism 0 : A -, B(M) for some Hilbert space
M.
Theorem 3.2.1. (Blecher-Ruan-Sinclair) Every abstract unital operator algebra A
admits a completely isometric representation. i.e. there exists a Hilbert space M and a
unital completely isometric algebra homomorphism 0 : A B(M)).









CHAPTER 4
THE ABSTRACT OPERATOR ALGEBRAS A(/C)" & A(1C)"/I(kC)

Recall IC, the matrix convex set satisfying the conditions of Assumption 2.1.1. In

particular, IC(1) is an circled open convex subset of Cd. This chapter is divided into two

parts. In the first part, which consists of four subsections, we construct an abstract unital

operator algebra which is a natural non-commutative analog of the Banach algebra
H-(IC(1)). We also present a few lemmas on completely contractive representations of

this algebra.

In the second part, Sections 4.5 and 4.6, we consider the ideal Z(/C) of the algebra

A(KC)" determined by a finite initial segment A. We show that the quotient algebra

A(IC)"/Z(/C) determined by the ideal is an abstract unital operator algebra. We also

show that norms of classes in the quotient algebra are attained.

4.1 The Algebra A(KC)" of Scalar Formal Power Series

In this section we establish that the collection of scalar formal power series in

non-commuting variables which converge uniformly on K is an algebra. We begin with

the definition of a formal power series.

4.1.1 Formal Power Series

Let U and U' denote separable Hilbert spaces. A formal power series with

coefficients from B(U, U') is an expression of the form


S fw (4.1)
wEEd

where fw e B(U,U'). It is convenient to sum f according to its homogeneous of degree

terms; i.e.,

f = fww = (4.2)
j=0 Iw=j j=0
Recall, for a d-tuple T = (TT,..., Td) of operators on a common separable Hilbert

space TH and a word w = ggg,... d, i {1, 2 ..., d}, the evaluation of w at T









is defined as


T = T T, ... -T .

Given a formal power series f as above, define
00
f(T)= fw DTW (4.3)
j=0 Iwl=j

provided the sum converges in the operator norm in B(U X-, U' 'H) in the indicated

order. We note for clarity that if U = U' = C, then 0 in (4.3) is the usual scalar product

and if f(T) converges, it is an element of B(-).

Recall the matrix convex set K = (IC(n)) which satisfies the conditions of

Assumption 2.1.1. We will write X e KC to denote X e UnN I(n). For the formal

power series f as above, we define


I|f l = sup{| f(X) | : X /C}. (4.4)

4.1.2 The Vector Space A(/C)"

As it stands, the supremum in equation (4.4) can be infinite. We are only interested

in those formal power series f for which this is not the case. Let

A(IC) = f f= W:f fw e C, ||f| < oo.
S wEFd

Thus, elements of A(/C)" are in some sense analogous to elements of the classical

commutativee) Hardy space, H"(/C(1)) of bounded analytic functions on /(1). It is not

hard to see that A(/C)") is a complex vector space with respect to term-wise addition

and scalar multiplication.

Lemma 4.1.1. 1|| || defines a norm on A(1C)".

Proof. It follows from the definition that I|| || is a seminorm. Thus it suffices to show that

|| fl = 0 implies f = 0. Let e { 0, 1, 2,... } and 5(f) be as in Subsection 2.3.3. For









0 < t < 7, using Lemma 2.3.1 and Remark 2.3.1, we get


0 = I|lf2 (4.5)

> ||f(tS( ))(0) |2 (4.6)

= tj fW2 (4.7)
j=0 Iwl=j

= t2 f .2 (4.8)
j=0 w =j

Thus fw =0 for all w such that Iwl < Since f is arbitrary, the lemma follows. O

4.1.3 Matrix Norms on A(KC)"

Since it will be necessary to consider, in the sequel, matrices with entries from
A(KC)", we define them here. Let


Mp,q(A(C)")= f fww: fw Mpq, ,,|f||p.,<
wEFd

where the norm I|| I|p,q is given by


I|fllp, = sup{||f(X)| : X e KC}. (4.9)

The following Lemmas plays an important role in the analysis to follow generally,
and in proving that A(KC)" is an algebra, in particular.
00
Lemma 4.1.2. Iff = fww e Mp,q(A(/C)0), then ||2 If|2 converges.
wEFd j=0 Iw=j
Proof. Recall 7 from equation (2.3) and S() from Subsection 2.3.3. Fix 0 < t < 7 and a

unit vector x e Cq. Using Lemma 2.3.1 and Remark 2.3.1, we get













j=0 wl=j


J=0

> | t
j=o

= t
j-0 I


0)112


S()w (x 0)112


) w112


Since E is arbitrary, allowing t T 7 yields


(4.10)


|j= fwx2 < j=o Iwl=


Let {el, e2...., eq} denote the standard orthonormal basis for Cq. We know that for


each we cFd


fI < I fee .2
i 1=


(4.11)


Moreover, for each 1 < i < q, equation (4.10) implies that


Using equations (4.11) and (4.12), we get


o
2= j 2 i IfWI2
j=0o Iwl=j


oo q
< 72i Y'(Y Ifweell2)


= ( | fe, l2)
< qlwfl i2.
< q f 2


5fw2
WI j


WI j


wl=j


2 | fwee 2 < f 2.
j=0 wl=j


(4.12)


(ts(V))w(x









Lemma 4.1.3. Suppose that f = fww e Mp,q(A(KC)") and X e IC(n). Let
weE

A,= f XW.


If < r < sup{s > 0 : sX e /C(n)}, then


r |llA l < I fll .

In particular, there is a p < 1 such that IIAj II < pl I f l.

Proof. Because IC(n) is open, convex, and circled, the function F(z) = f(zX) is defined

on a neighborhood of D; i.e. there exists a 6 > 1 for which the series,
00
F(z) = Ajz
j=0

converges for all z such that |z < 6. Using the fact that the series F(z) converges

uniformly on the closed disc {z : zl < (} for every ( < 6, we get, choosing 1 = 1, for

each j that,
1 02T
271 0
It follows that
A < 2F(eit) dt.

Since I F(e't) = | f(e'tX)|| and eitX e IC(n), it follows that IF(e't)l < f| f and the

lemma follows. O

4.1.4 The Algebra A(/C)"

There is a natural multiplication on A(/C)" which turns it into an algebra over C.

Given f = wE,, fww e Mp,(A(/C)00) and g = wE gww e Mq,,(A(kC)"), define

the product fg of f and g as the convolution product; i.e.,


fg9 = ( fu"v ) W.
wE)7d \uvI w /








The remainder of this subsection is devoted to demonstrating that this convolution
product corresponds to pointwise product, extends the natural product of non-commutative
polynomials (formal power series with only finitely many non-zero coefficients), and
makes A(KC)" an algebra with unit 0.
Lemma 4.1.4. Iff C Mp,4(A(/C)") and g e M,,(A(/C)") and X e /C, then

(i) fg(X) converges;

(ii) fg(X) = f(X)g(X);

(iii) fg is in Mp,r(A(KC)"); and

(iv) I|fg|| < f Ifl lgll.

Proof. Fix X e IC(a) c KC.
(i) As in the proof of Lemma 4.1.3, let

A,= f, X",
Iwl=j






Observe that C = ECo AkBj-k-

Let F(z) = f(zX) and G(z) = g(zX), both of which are defined in a neighborhood
of D. From Lemma 4.1.3, there is a p < 1 such that |iAmr\ < prllfll and II||Bk < pkllgll.
Hence






j=0 k=0

converges. In particular fg(X) = C0o Q converges in norm.









(ii) Consider the function FG(z) = fg(zX). From the proof of (i) we know that FG(z)
is defined whenever z| < We will prove the more general fact that FG(z) = F(z)G(z)

whenever |z < from which the claim will follow by setting z = 1. Recall 7, F from the

definition of the matrix convex set IC. Let z be such that |z < 2. Observe that for any k,


I C zj A,,zm Bz" | < C z Azm B zn
j= 0 m= 0 /n=O j=0 m= 0 n=0O
k oo
|| Amzm 1111 Bnz |
m=0 n=k+l
oo k
|II 1 Amzm llll 11 B
m=k+l n=0
|| amzmllll Bz"11
1 Amz|||| BnA z n
m=k+l n=k+l
(4.13)

We claim that the LHS of equation (4.13) converges to zero as k oo. It suffices

to show that the first term on the RHS of equation (4.13) converges to zero, in view of

the convergence of the second, third and the fourth terms on the RHS to zero due to the

following reasons:

(a) both k AmZ'm and I EC 0 Bnz are finite.

(b) both 1Ek+1 ArmZm and k+ 1 BZn, being tails of the convergent series f(zX)
and g(zX) respectively, converge to zero as k oo.

Consider the first term on the RHS of equation (4.13).










k k ( z\
mz"' Bnzn
m-0 (n-0


(fgv XU V)
-k

(fugv XUV)
k
, (f gv)
k,lvl=k


+
ul= 2,|vl k-1

+
|ul=3,|vl=k-1

x"u) IZ2k


(fgv XUV) ...

(f.gv0 XV)+...


S(fgv XuV) I Zlk 1
|u|I k,lvl 1

S(f9gv XUV)|| Izlk+2
|u|l k,lvl 2


< ( Ifugv11
Jul 1,vl =k


lul=2,|vl=k-1


( 5 fgvll- + I f 1gv
|ul=2,|vl=k |ul=3,vl= k-1

..-- ( I fgv (Fz)2k
|ul|-k,lvl =k


...+ f vf l (Flz)k 1
u =k, lv=1

S...+ I fgv (Fz)k+ 2
u|-k,lvl|-2


<( u I 2 kg 1 ,2 )(d k 2
Iwlk +luv w w 2uv=w

+ .... || If 11 v 1l
|w|=2k uv=w


< ( 4(k+1) iif. (kd +2)dk+ 1
11 2 2dk+l +
ww =k+1 uv=w


+ .4(2k) iif. fv 2(2k )d2k
2"d2k
Iw\=1k uv w


S ,4(k 2) I fgv) 2 (kdk +2
w k 2 uvw+2
|wl|k+2 uI w


=o
j=


+ 111
|u =1 v|=


ul =2, 1v

+.. + 11
IL11


IIfu9gv














k uv=w
(2 d2k
7+ 2m + 4 k+2) f

J dk 1 d d1k








Iwl|= uv=w
Since the series d ... d are convergent,the
k=0 k=0 k0
corresponding kth term sequences converge to zero, and so does the sum of the

sequences namely ... / Moreover Lemma 4.1.2 implies
that e g2m ro fie2 and t t2n" 5 gvl 2 are finite. These facts together imply the

desired convergence. Thus fg(zX) = f(zX)g(zX) whenever z <
Fix x e eris and y e CP Ca and consider the complex valued functions
A(z) = (cG(z)x, y) and B(z) = (F(z)G(z)x, y). Observe that A(z) and B(z) are

analytic on the disc {z : Izl < } and A(z)= B(z) on the subdisc {z : zl < }. Hence,
A(z) = B(z) on {z : Izl < ,}. Since x and y are arbitrary, we have FG(z) = F(z) G(z)
whenever znl < Choosing z = 1 gives fg(X) = f( X)g(X).
(iii) & (iv) Since, for each X E C, fg(X) = f(X)g(X) it follows that fg(X) <

||fl I|g||. Thus I|fgl| < ||fl I|g|l and fg e Mp,r(A(C)"). D








Corollary 4.1.1. A(/C)" is an algebra.


Proof. Take p = q = r = 1 in the above lemma. O

4.2 Weak Compactness and A(KC)"
In this section it is shown that every bounded sequence in A(KC)" has a pointwise
convergent subsequence. Indeed, A(KC)" has weak compactness properties with
respect to bounded pointwise convergence mirroring those for H"(D), the usual space
of bounded analytic functions on the unit disk D.
Proposition 4.1. Suppose that fm = wCe (fm)ww is a Mp,,(A(/C)) sequence. If,
for each X e 1C the sequence (fm(X)) converges or if for each w E Fd the sequence

(fm)w converges and if (fm) is a bounded sequence (so there is a constant c such that
II fml < c for all m), then there is an f e Mp,,(A(/C)") such that fm(X) converges to f(X)
for each X e 1C and moreover I fll < c.

Proof. Recall S(f) defined in Subsection 2.3.3. Fix 0 < t < 7. If fm converges pointwise,
then the sequence (fm(tS(f))) is Cauchy. Given e > 0, there exists N E N such that

Ilfm(tS()) f(tS(t))l <

for all m, n > N. Thus, if x e Cq be a unit vector, then

e2 > Ifm(tS()) f,(ts())l2

> |I f(tS(f)) f(tS(f))(x 0) 2|

= t2J ((fm)w- (fn)w) (x) 2
= o Iwl=J
> t2ll((fm)w (f)w) (x) 11

for each word w of length at most Since f and the unit vector x e C are arbitrary, it
follows that, the sequence ((fm)w) c B(Cq, CP) is Cauchy for each word we Fd. Thus

(fm)w converges to some fw for each w.









Hence, to prove the Proposition it suffices to prove that, if (f,), converges to f, for

each w and IIfmll < c for each m, then for each X e /C, the series
00
f(X)= f, O XW
j= o w =

converges and (f,(X)) converges to f(X).

For each j and X e C,


S (fm)w XW fw XW. (4.14)
Iwl=j Iwl=j

From Lemma 4.1.3, there is a p < 1 such that for each j,


II (fm)w, XW

Iwl=J

From equations (4.14) and (4.15), it follows for each j that,


II fw, XWl <- c,
Iwl=J

an estimate which implies that the series f(X) converges.

Fix J e N such that

<- (4.16)
4c
j=J+1
Recall F from the definition of the matrix convex set /C. Choose K e N such that for

all m > K,

ll( ) fwll < 2(J(4.17)
2(+ J 1)J +

for each 0








Thus for all m > K, from equations (4.16) and (4.17), it follows that


Ilfm(X)- f(X)ll

< | 1 (( M)w
j=0o Iwl=


j=0 Iwl=j


o0
fw) XW"|| 1
j=J+1


fl r ~j I |(|m)w
j J+1 Iwl
_/^j+i |M/|^_


J 00

2(J 1) 2c p
j=0 JJ1
S.

Thus fm(X) f(X) for all X c /C. Since |fm(X)ll < c, we have If(X)ll < c. This


implies that Ilfll < c.


Lemma 4.2.1. If fm


Ewe (fm,)wW C Mp,q(A(/C)") satisfies I||lfm < c for allm c N


then,


(i) I(fm)wl < c forallw e Fd and forallme N;

(ii) There exists a subsequence {f,,} of{f,} and fw e Mp,q such that (fm,)w f for
all w;

(iii) Let f = f w. For each X c IC the sequence (fi, (X)) converges to f(X) and
wMeE
moreover l f(X)II < c.

Proof. To prove item (i), Recall 7 from the definition of /C. Fix 0 < t < 7 and a unit vector


x e Cq. Forj


0, 1, 2, ..., the hypothesis I| fmll < c together with the conclusion of


Lemma 4.1.3 for X = tS(f) imply that


|| t (f=)j
|w|=j


II ((fm)
|wl=j


f,) XWII


xWII


xWII


JJ1


II fw
I j


s()w|| < c.









Hence


c2 > t (fm,),x S()wO 12
Iwlj
St2J I (fm)wX112
Iwlj
> t2, Jll(fm)wX 112

Since x and f are arbitrary, letting t 1 7 it follows that ||(fm)wl| < < for all m e N.
The proof of item (ii) uses a standard diagonal argument. Let {wl, w2, ...} be an
enumeration of words in d which respects length (i.e., if v < w, then vl < Iwl). Since

I(f,)w, 1 < c there exists a subsequence say, {f,,m} of {f,,} such that (fi,m)w, fw,.
Since 1|(fim)w2|| < c, there exists a subsequence say, {f2,m} of {fi,m} and thereby of

{f,}, such that (f2,m)w2 f2. Continue this procedure to obtain a subsequence {fk,m} of
{fk-1,m} and thereby of {fm} such that for all k e N,

(fk,m)w fwk

Now consider the diagonal sequence {fm,m}. It follows that {fm,m} is a subsequence
of {f,} and satisfies (f,,m)w fw for all we c d.
In view of what has already been proved, an application of Proposition 4.1 proves
item (iii). D

4.3 The Abstract Operator Algebra A(KC)"

Consider A(KC)" with matrix norms I| |p,q on M, q(A(/C)") as defined in (4.9).
Theorem 4.3.1. A(KC)" with the family of norms II IIp,q, is an abstract unital operator
algebra.

Proof. Let A c M.,, F c Mp,(A"), B e Mq,,. Interpret A and B as AO e M.,p(A(/C)")
and BO e Mq,r(A(KC)") respectively. As a notational convenience we will drop the








subscripts that go with the norms. It follows from Lemma 4.1.4 (ii) that for all X e IC(n),

IIAFB(X)II = IIA(X)F(X)B(X)II < IIA & I, ||F(X)| ||IB 0 Ini < I||A|| IFI| |BI|.

Thus,
IIAFBI| < I||A|| IFI| |BI|. (4.18)

Let F e Me ,(A(/C)-), G e Mp,q(A(/C)"), X e IC(n). Observe that

(F(X) 0N^
||F G (X)|| = F(X) 0(X) < max{||F(X)||, || G(X)||} < max{|| F||II, ||GII||}.
0 G(X)

Thus,
|IF GI| < max{||F||, I|G||} (4.19)

Let c > 0 be given. Without loss of generality assume that I|F|| > I| GI. Choose m e N
and R e IC(m) such that |IF(R)II > |IFII c. Therefore


||F GI| > ( ) ) > I|F(R)I| > |F| -e. (4.20)
S0 G(R)

Letting c 0 in the inequality (4.20) and from the inequality (4.19) it follows that,

I||F G | = max{||F||, I||G|}. (4.21)

Lastly, complete contractivity of multiplication in M,(A(KC)") follows directly from Lemma
4.1.4 (iv). Thus A(KC)" is an abstract operator algebra. O

4.4 Completely Contractive Representations of A(KC)"
Recall the definitions of a completely contractive and completely isometric
representation from Section 3.2. Theorem 3.2.1 guarantees the existence of a
completely isometric representation for the abstract unital operator algebra A(C)".









Let : A(/C)" -- B(M) be a completely contractive unital representation and let
T = (T, ..., Td) where T, = (gj). As a notional device, we will write 7T for T. Further,
we will also use 7, to denote the map Iq 7 : Mq(A(lC)") Mq 0 B(M).
In this section, we prove that for a completely contractive representation 7r of

A(/C)"), for any n e N and finite dimensional subspace W of M of dimension n and
0 < t < 1 the tuple
tZ = tV*TV = (tV* T V,..., tV*TdV)

is in IC(n). The proof begins with a couple of lemmas. Given f e Mq(A()C)") and
0 < r < 1, define fr as follows. If

f= EEfw = (4.22)
j 0 Iwl= j=0
then
00 00
fr= ri fww = rj.
j=0 Iw=j j=0
Lemma 4.4.1. If 7T is a completely contractive representation of A(1C)0 and f e

Mq(A(1C)"), then fr(T) converges in operator norm. Moreover r(fr) = fr(T) and

IIf,(T)II < fIIrll < If ll. If in addition 7T is completely isometric, then limr fr(T) || =



Proof. Write f as in equation (4.22). Lemma 4.1.3 implies that 1~ fj1 < I f l. Because T7-

is completely contractive I||(T)|| < I|f l. It follows that fr(T) converges in norm. Since
also the partial sums of fr converge (to fr) in the norm of Mq(A()C)"), it follows that

ir(fr) = fr(T) and so I|f,(T)|| < I|fr|.
The inequality I|frl| < lf ll is straightforward because rC c /C.
Now suppose that 7T is completely isometric. In this case | fr(T) | = I| fr,. On the
other hand lim I fr,| = ||Ilf D









Lemma 4.4.2. Given k x k matrices A1,..., Ad, let


d

j=1

Suppose

2 L(X) L(X)* >- 0

for all X e IC(T) and for all e N. Let OL denote the formal power series,


OL = L(2 L)-1 LJ1
2J1
j=0

(a) 2 L(X) L(X)* >- 0 for all X e C if and only if 2 L(U) L(U)* > 0 forall

U eC.

(b) If X e IC(f), then OL(X) converges in norm; i.e., the series

L(X)+ 1
2j+1
j 0
converges.

(c) II1L(X)II < 1 and hence OL is in Mk (A(C)00) and has norm at most one.
(d) If ,T is a completely contractive representation of A(IC)0, then 2 (L(T) +

L(T)*) 0.

Proof. To prove part (a), suppose that 2 L(X) L(X)* > 0 for all X e /C. Since K

is the closure of KC it follows that 2 L(U) L(U)* > 0 for all U e kC. To prove the

converse, assume the contrary, i.e. suppose there exists X e K and a unit vector v such

that ((2 L(X) L(X)*)v, v) = 0. The following argument has been adapted from [16].

Define the map q : R -- R by


q(t) = ((2 L(tX)- L(tX)*)v, v).

Observe that the q is an affine map that satisfies q(0) = 2 and q(1) = 0. Hence q(t) < 0

for all t > 1. Choose s > 1 such that sX e /C. Such an s exists because K/ is open. For









this s, we get

q(s) = ((2 L(sX) L(sX)*)v, v) < 0

which is a contradiction to the hypothesis.

To prove part (b) of the lemma, let X e /C(f) be given. Because IC(f) is circled, it
follows that e'eX e /C() for each 0. Hence,

2 e'oL(X) e-L(X)* >- 0 (4.23)

for each 0. For notational ease, let Y = L(X). Thus Y is a k x kt matrix and equation

(4.23) implies that the spectrum of Y lies strictly within the disc; i.e., each eigenvalue of
Y has absolute value less than one. Thus,

2 = (2 Y)
j= 0
converges in norm. It follows that

) 2Jy+1
L(X)- = Y(2- Y)-= 2j+1
J=0
converges.

To prove (c) observe that I| Y(2 Y)-l11 < 1 if and only if

(2 Y)*(2 Y) > Y*Y

which is equivalent to 2 (Y + Y*) > 0. Thus I|IL(X)|| < 1 which implies that

OL E Mk(A(/C)0) with 1||L| < 1. This completes the proof of (c).
To prove part (d), observe, Since ,7 is completely contractive and OL E Mk(A(/C))
with norm at most one, an application of Lemma 4.4.1 yields, I||L(rT)II < 1. Arguing as
in the proof of part (b), it follows that 2 (L(rT) + L(rT)*) > 0. This inequality holds for

all 0 < r < 1 and thus the conclusion of part (c) follows. D









Proposition 4.2. If T = (T,,..., Td), and j B( 4M) for some Hilbert space M, and

r(gj) = T determines a completely contractive representation of A(1C), then, for each
positive integer n and finite dimensional subspace W of M of dimension n and each
0 < t < 1 the tuple
tZ = tV*TV = (tV* T V,..., tV*TdV)

is in 1C(n), where V : W M is the inclusion map.

Proof. Let n and H- be given and define Z as in the statement of the proposition.
Suppose that L is as in the statement of Lemma 4.4.2. From part (d) of the previous
lemma, it follows that 2 (L(T)+ L(T)*) > 0. Applying Ik 0 V* on the left and Ik 0 V on
the right of this inequality gives,

2 (L(Z)+ L(Z)*) = (k 0 V*)(2 (L(T) + L(T)*)(Ik 0 V) > 0.

Part (a) of Lemma 4.4.2 and an application of Theorem 2.2.1 imply that Z e /C(n).

Hence tZ e 1C(n) for all 0 < t < 1. D

Lemma 4.4.3. Let A c d be a finite initial segment, f C Mq(A(/C)") be as in equation
(4.22) and suppose that T is a completely contractive representation of A(1C)" into

B(M) and T is A nilpotent. Then I fr(T)l < sup{| f(X)| : X e /C, X is A nilpotent}
for all 0 < r < 1. Moreover if f = 0 for all w A, then I|f(T) | < sup{| f(X) | : X e

C, X is A nilpotent}.

Proof. Since A is finite and T is A nilpotent, fr(T) = wA f, 0 (rT)w. Let {ej}j 1

denote the standard basis of Cq. Given c > 0, choose a unit vector y = e; 0 hj e
Cq M A such that

IIf,(T)|| < I|f,(T)yll + .

Let -H denote the finite-dimensional subspace of M spanned by the vectors
{T"(h) w e A, 1 < j < q} and V : H M4 be the inclusion map. Then Z = V* TV is








A-nilpotent and
V* TWV if w A
Zw ={
0 otherwise.
Proposition 4.2 implies that rZ e 1C. Thus,

lfr(T)ll < ( fw rlwI T" yll+

= fII(Z)yll+

< fr(Z)| +C
< sup{|| f(X) | : X e /C, X is A nilpotent} + .

Letting c 0 yields the desired inequality. If fw = 0 for all w A, then f = weA fwW is
a non-commutative polynomial in which case we have limr,- Ifr(T)I| = 11 f(T)II and this
completes the proof. D

4.5 The Abstract Operator Algebra A(kC)/I-Z(C)
Recall the definition of an initial segment from Subsection 2.3.1. Fix a finite initial
segment A c Fd and let

Z(/C) = f= ffww : ||f < 0 c A(/C)"

Observe that Z(IC) is a two-sided ideal in the operator algebra A(C)".
Lemma 4.5.1. Mp,q(Z(lC)) is closed in Mp,q(A(IC)").

Proof. Let fm = CwEA(fm)wW be a sequence in Mp,q(I(IC)) be such that fm f =

EW7dw fww e M,,q(A(C)"). We need to show that f = 0 for all w e A. Let f be the least
integer such that A c A(.). Consider tS(f) where 0 < t < 7. Given C > 0, there exists
N e N such that

|fm(tS()) f(ts())| <








for all m > N. Let x E Cq be an arbitrary unit vector. We have


> I fm((tS()) f(tS(f))l

> |(fm( tS()) f(tS ()))(x 0)) 2

-IS t W((fm)w fw)X W W + WfWX DW 12
wEA()\A wEA
= | t2wI l((f,), f )xl 2 w t2wl 1 fwx 2
wEA()\A wEA
> t21WI fwx 12

for all w A and m> N. Hence fw =0 for all w A.

By identifying Mp,q(A((C)"/I(lC)) with Mp,q(A(1C)")/Mp,q(I(1C)), Corollary 3.1.1
implies that the quotient A(KC)"I/(/C) is an abstract operator algebra. We formally
record this fact.
Theorem 4.5.1. A(IC)"/I(/C) is an abstract unital operator algebra.
4.6 Attainment of Norms of Classes in Mq(A(C)C")/Mq(Z(/C))
Let p e Mq(A(/C)"). In this subsection it is shown that there exists f e Mq(I(/C))
such that
|p + f| = l|p + M,(Z(IC))I| = inf{||p + gll g e M,(I(/C))}.

Let {fm} be a sequence in Mq(I(/C)) such that

p + M,(Z(IC))|| < lp + fml < ||p+ M,(Z(IC))I| +

It follows that the sequence {fm} is bounded and that ||p + fmll lip + Mq(Z(C))||. An
application of Lemma 4.2.1 yields a subsequence {fmk} of {fm} and f e Mq(I(IC)) such
that

(p+ f)(X) (p + f)(X)

for all X E /C.








Proposition 4.3. If p, {fmk}, f are as above, then lip + f| = lip + Mq(I(IC))||.

Proof. Let e > 0 be given. Choose R /C such that

l|p f|| < I|(p+ f)(R)|| + (4.24)
4

Since I|(p + f)(R)|| I(p+ f)(R) |, there exists K1 e N such that,

I|(p+ f)(R)ll < I|(p + fm)(R)|| + (4.25)

for all k > K1. Combining the inequalities from equations (4.24) and (4.25), implies that,
for all k > K1,

lip + fl < lip fm, + (4.26)

Since lip + fm, |II p + Mq(I(/C)) |, there exists a Natural number K2 such that for
all k > K2,

lip + f, < lip + Mq(I(C))I| + (4.27)

Setting k = max{Ki, K2} in equations (4.26) and (4.27), and letting C 0 yields


lip+ fll < lp+ M,(I(/C))I|.

On the other hand, since f e Mq(Z(IC)),


lip+ fll > lp+ M,(Z(/C))I|.

E









CHAPTER 5
THE NON-COMMUTATIVE CARATHEODORY-FEJER PROBLEM

In this chapter we pose the Caratheodory-Fej6r Interpolation problem (CFP) for our

open, bounded and circled matrix convex set /C. Using the results from Chapters 2 4,

we prove a necessary and sufficient condition for the solvability of the problem.
5.1 The Caratheodory-Fejer Interpolation Problem (CFP)

The statement of the CFP is as follows: Fix a matrix convex set K/ satisfying the

conditions of Assumption 2.1.1. Let A c 'd be a finite initial segment, and

P = p,w A(/C)0
wEA
be given. Does there exist x E A(/C)- such that xw = pw for w e A and I||1| < 1?

Theorem 5.1.1. There exists a (minimum-norm) solution x to the above problem if and

only if

sup{||p(X)l| : Xe c X is A nilpotent} < 1.

The generalization of Theorem 5.1.1 allowing for operator coefficients is proved in

this chapter.

The strategy is to first prove the result for matrix coefficients. This is done in Section

5.2 below. Passing from matrix to operator coefficients is then accomplished using

well-known facts about the Weak Operator Topology (WOT) and the Strong Operator

Topology (SOT) on the space of bounded operators on a separable Hilbert space. The

details are in Section 5.3.

5.2 The Matrix Version

Fix A c Fd, a finite initial segment, and a polynomial p = EwC, pww e Mq(A(/C)").

Proposition 5.1. There exists f e Mq(A(KC)") such that lip + fll = l|p + Mq(I(/C))| =

sup{||p(X) | : X IC, X is A nilpotent}.

Proof. From Theorems 4.5.1 and 3.2.1 it follows that there exists a Hilbert space

M and a completely isometric homomorphism 0 : A(C)"/IZ(KC) B(M4). As








before, identify Mq(A(/C)/Z(K/C)) with Mq(A(lC)")/Mq(IZ(l)). Let 0q denote the map
,Iq 0 : Mq(A(IC)f)/Mq(I(1C)) M, B(M.A). Let R be the d-tuple (R1, R,,..., Rd),
where R, = 0(gj + Z(IC)) e B(M), for 1 < j < d. Observe that R is A-nilpotent. Let
l : A(KC)" A(KC)"/IZ(K) be the quotient map. The composition 7 = 0 o r : A(C) -)
B(M) is a completely contractive representation of A(KC)". Since r(gj) = Rj, consistent
with the notation introduced in Section 4.4, we will use rR to denote the map r.
It follows from Theorem 4.3 that there exists f e Mq(I(IC)) such that

l|p+ fl| = l p+ M,(I(/C))||. (5.1)

The fact that 0 is completely isometric implies that

|p + M,((I(C)) I = I|l(P + M,(I(/C))|| = I|p(R)II. (5.2)

Since 7R is a completely contractive representation of A(/C)", Lemma 4.4.3 implies that

Ilp(R)II < sup{||p(X)|| : X e /C, X is A nilpotent}. (5.3)

Combining the equations (5.1), (5.2) and (5.3), it follows that

lp + fl < sup{||p(X)l| X e /, X is A nilpotent}.

But the definition of l|p + f|| implies that

lip f+ > sup{||p(X) : X e IC, X is A nilpotent}

and this completes the proof. D

(The matrix version of) Theorem 5.1.1 follows from the above proposition by setting
S= p f.









5.3 The Operator Version

As before, let A c Fd be a finite initial segment. Departing from the previous

section, let U be an infinite dimensional separable Hilbert space and let the polynomial

P = YW,, pw, where now {pw}wWE c B(U), be given.
Theorem 5.3.1. There exists a formal power series x = Cv,,e d w such that w = pw

for all w e A and Ix\11 = sup{|lp(X) | : X e IC, X is A nilpotent}.

Proof. Let {ul, u2,...} denote an orthonormal basis for the separable Hilbert space U

and let Urn be the subspace of U spanned by the vectors {uj} 1. For notation ease, let
C = sup{|lp(X)l| : X e X is A nilpotent}. Observe that C = 0 if and only if p = 0.

For w c A, define Mm 3 (pm)w = V,~PwV where Vm : Um -- U is the inclusion map.

Let pm denote the formal power series


Pm = Y(Pm)wW
wEA

For each X e IC, Observe that Ilpm(X)II < Ilp(X)II < lpll. Thus Ilpll < lpll and

pm E Mm(A(1C)) for all m c N. From Proposition 5.1, there exists fr, e M(Z(I(C)) such
that xr = Pm + fr, Mr(A(C)) and

||xrl| = sup{l|lp(X)| : X e IC, X is A nilpotent}.

For w c Fd, define B(U) D (5rm)w = Vr.(x)wV,. Let mr denote the formal power

series Yv(xr)ww. For X KC andj = 0, 1, 2,..., it follows from From Lemma 4.1.3

that there exists 0 < p < 1 such that

II 3Y, ()w XWI <- (Xm)w XWI
Iwl=j Iwl=J
< ,llXrn (5.4)

< Cpi








This implies that the series for x,(X) converges for each X e /C and moreover we have


IIXmll < IlXmll < C.


(5.5)


Recall 7 and 5() from Subsection 2.3.3. Fox 0 < t < 7 and a unit vector u c U. For
each 0 < j < and Xe IC, it follows that



c2 > It (X)w S(C)w(U 0)112
Iwl~
> tJ C (Xt)wu G wll2
Iwl j

Iwl=J
> t2|| (xm)wU ||2



Since f is arbitrary, letting t 1 7 implies that I||(m)wll < c for all w e d and m e N.
Since U is a separable Hilbert space and the sequence {(,m)w}- 1 is bounded (by
C ), for each w e Fd, there exists a subsequence of {(Im)w}L1 that converges with

respect to the WOT on B(U). By a diagonal argument similar to the one in Lemma 4.2.1,
it follows that there exists a subsequence {m,,} of {xm,} and {2w}wj c B(U) such that
for each w e Fd

(5m,)w 5w

with respect to the WOT on B(U).
Let X e IC(n) for some n. Since Hlwl(mk)w & Xw lwl| w 0& X" with respect
to the WOT on B(U 0 Cn), it follows from equation (5.4) that I|| EY:W, XW"l < Co.
Hence the series for x(X) converges in norm.
Let e > 0 be given. Choose Lx E N such that for all k > Lx,


xj=k w= <. (5.6)
jIk wlj









Let Nx E N be such that for all k > Nx,


(5.7)


J <
j=k


Thus for all k > Nx it follows from equations (5.4) and (5.7) that,


j (-k)w Xw <
j=k w|=j


(5.8)


Let Mx = max{Lx, Nx}, h e U 0 Cn be a unit vector and y

From equations (5.5) and (5.8) it follows that


X")


(MxjIw


(jxw
J a -W


j MI


< Iyl(C + )


Since EYM (lwl~( ),w

B(U & Cn), it follows that


x -w Z--" :- :


2 x, Xw with respect to the WOT on


xw)


, ( r, )w
w=j


Equations (5.10) and (5.11) together imply


lyll| < c +

Since h in the definition of y is arbitrary, it follows that


Mx

j=0


SIw
Iw j


XWl < c


Thus from equations (5.6) and (5.13) it follows that


X") h.


lIwl=j
5c^l-mf
^1~ | j|^/


(5.9)


(5.10)


h, Y (y, y).


(5.11)


(5.12)


(5.13)


h, Y









S0 Mx o0
II^W(X)ll=II EE ^w XWll<-II IIE^w IXWll-+II Y, E^0w XWll
j o Iwl= jo Iwl=j j=Mx 1l wl=J
< C+2e.

Letting c 0 implies that I||(X)|| < C. Since X e IC was arbitrary, it follows that

I1x|| < C. (5.14)

To prove the reverse inequality, observe that for w e A, (,m)w = V VmVPw VmV and

Vm V, pwV V, V- pw with respect to the WOT on B(U). This implies that xw = pw for all
w e A and so by definition, we get I1x|| > C. Thus


I||x| = C. (5.15)

[]









CHAPTER 6
INFINITE INITIAL SEGMENTS

In this chapter we present two examples of non-commutative operator domains,

and consider the Caratheodory-Fej6r Interpolation problem (CFP) for these domains

under the assumption that the initial segment A is an infinite set. The examples we will

consider here will be the operator (as opposed to matrix) versions of those presented

in Subsection 2.1.1. So naturally, most of the definitions including non-commutative

neighborhood of zero, circled etc. extend analogously. A slight modification in the notion

of the 7 > 0 neighborhood of 0 is necessary. Given an operator A on a Hilbert space

-H, write A > 0 if there is an c > 0 such that A > lc; i.e., A = A* and for all vectors

h e -H, the inequality (Ah, h) > c(h, h) holds. In the operator version, a non- commutative

7-neighborhood of zero, is the set of T = (T, ..., Td) acting on -H such that
d
7 2/ Y, Tj Tj*
j= 1

6.1 Examples of Non-commutative Operator Domains

Let 'H be a separable infinite dimensional Hilbert Space.

The d-dimensional Non-commutative Polydisc is defined by

Cd= (T-...,. Td) : T B() T 11 1<1}

Just as for the non-commutative matrix polydisc, = 1 and F = Vd.

The d x d-dimensional Non-commutative Mixed Ball is defined by,

Ddd {T =(T11, T12, B Td) T E() TI o < 1}


where I| TI|op is the norm of the operator (Ty) : (d) ( B(d). As expected

7 = and rF = dd.
\Idd









We will demonstrate that the operator algebra approach that we used in Chapter 5

(for the finite A case) can also be applied here (to handle the infinite A case) and that it

leads to a similar necessary and sufficient condition for the solvability of the CFP.

Fix an infinite initial segment A c Td. In order to make the proofs from the Chapters

2 5 work for this setting, some minor modifications need to be made.

Recall the Non-commutative Fock Space and the d-tuple of Creation Operators

S = (S1,..., Sd) from Section 2.3. A more general property of S is the following.
Let F2(A) denote the completion of the linear span of A with respect to the inner

product defined in (2.7).
Lemma 6.1.1. If V : F2(A) F2 denotes the inclusion map, then

(i) (V*SV)" = V*SV for all w Fd.

(ii) V*SV is A-nilpotent, i.e. (V*SV)" = 0 for all w i A.
Here V*SV = (V*51V,..., V*SdV).

Proof. We prove item (i) by induction. When Iwl = 0 or 1, the statement is true. Assume

the statement is true for all words of length at most n. Let w be a word of length n + 1.
Then w = vwg for some word w of length n and some j such that 1 < j < d. Let u e A.

(V*SV)w(u) = (V*sV)w(V*SV)s(u)

= (V*SWV)(V*S V)(u)

SV*S(gju) if gju E A

0 if gu A

Swu if wu A

0 if wu A

=(V*SWV)(u).









Thus we get (V*SV) = V*SwV for all w such that Iwl = n + 1, and this completes

the proof.

To prove item (ii), fix w ( A. In lieu of Part (i), it suffices to show that V*SwV = 0.

Let u e A. Since w A and A is an initial segment, we get wu A. It follows that
V*SwV(u) = V*(wu) = 0.

D

6.2 The d-dimensional Non-commutative Polydisc

In this section we pose the CFP for Cd, the d-dimensional Non-commutative

Polydisc, with the infinite initial segment A c Fd and give our main result. We begin with

the following remark. Let V be as in Lemma 6.1.1.
Remark 6.2.1. If 0 < t < 1, then

(i) tS = (tS1,..., tSd) Cd

(ii) tV*SV e Cd.
We define A(Cd)- and Z(Cd) as before (See Sections 4.1 and 4.5). The proofs

from Sections 4.1, 4.2 and 4.3, can be generalized to this current setting (A is infinite) by

replacing S(f) by S.
In order to generalize the proof of Lemma 4.5.1, to the current setting, we replace

S() with V*SV. For clarity, we present the modified argument here.

Let 0 < t < 1 be given. Since fm e Mq(Z(Cd)), using Lemma 6.1.1 (ii), it follows that

fm(tV*SV) = 0. Since fm f in norm, we have f(tV*SV) = 0. If x e Cq is a unit vector,
then

0 = If(tV*SV)|12

> IIf(tV*SV)(x 08)12

= ti IY fx l12
j=0 Iwl=j,w6A
> t21wl11 fxll2








for all w e A. Hence f = 0 for all w e A.
We now state the CFP and our main result.
Let U be an infinite dimensional separable Hilbert space, A c Fd be an infinite initial
segment and p = EwA pwW be a formal power series such that pw e B(L) and

|p||l = sup{||p(T)| : T C Cd < .

Does there exist x = Cv,,e ww such that x2 = pw for all w c A and 112ll < 1?
Theorem 6.2.1. There exists a solution x to the above problem if and only if

sup{||p(T)I| : T Cd, T is A nilpotent} < 1.

As before, we will prove Theorem 6.2.1 by first proving it for matrix coefficients, i.e.
pw e Mq and then by following that with a WOT approximation argument.
Let 7 : A(Cd)/00(Cd) B(MA) denote a completely isometric algebra
homomorphism obtained by applying Theorem 3.2.1 to the abstract unital operator
algebra A(Cd)/Z(Cd). For 1 < j < d, let R = r(gj +Z(Cd)), and R = (R1, R,,..., Rd).
Since Ig + (Cd)ll < I 1 and 7 is isometric, we have IIRjI < 1. Moreover R is
A-nilpotent.
In Section 5.1, p was a polynomial, which automatically gave us the finiteness
of ||p(R)II. But here, since A is infinite, we will need an approximation argument to
generalize Proposition 5.1. We present the generalization below.
Proposition 6.1. There exists f e Mq(I(Cd)) such that lip + f l = lip + M(I(Cd))l =
sup{|lp(T)|| : T Cd, T is A nilpotent}.

Proof. We know from the generalization of Proposition 4.3 to the current setting of
infinite A, that there exists an f e Mq(I(Cd)) such that

lp+ fl = lp+ Mq,((Cd))l.









Let 0 < t < 1. Define the formal power series pt by


Pt = tlPww.
wEA

Since for each T c Cd we have pt(T) = p(tT), it follows that l|ptll < |lpll. It is also

true that if 0 < t1 < t2 < 1, then IlptllI< Ilpt21.

For 0 < t < 1, define p = wEA,.Iwlk tlwpww and pk = wEA,.wl
denote the map Iq & : Mq(A(Cd)d)/Mq(Z(Cd)) M, B(A4).

We will first prove that I|,(pt + Mq ((Cd))l = Ipt(R)II = IIp(tR)ll.

We know that

(p + MQ(Z(Cd)) = pk(R) = pk(tR)

for each k. Moreover, for each T E Cd and each j, we have


II Pw Twl < Ilpll.
Iwlj,wEA

Let e > 0 be given. Choose a natural number N such that


t < li-pi
j= N+ 1

Let T e Cd be arbitrary. For k > N we have,


( -p- )(T)|II = 1 t P., 0 T II
j= k 1 wjwEA

< till Pw & TWII
j k 1 Iwj,wEA
00
< HIpil Y tj < C.


This implies that |ip P ptll < c for all k > N; i.e. the sequence of partial sums of Pt

converges to pt in norm.








Since (p Mq(I(Cd))) (Pt + Mq((Cd)))ll I pPt we get that for all k > N,

||(p M(I(Cd)))- (Pt + M((C)))d < .

Since 7q is continuous (it is an isometry), we have

Pt (R) = (p + Mq(I(Cd))) 7r,(pt + MZq(dC)).

Thus ~q(Pt + Mq((Cd))) = pt(R) which in turn implies that

q(Pt + Mq(I(Cd))) = IPt(R)II. (6.1)

Let {t,,} be an increasing positive sequence that converges to 1. Choose h, e
Mq(I(Cd)) such that
Ipt, + hm,= IIPt, + Mq((Cd)) 1 (6.2)

for each m.
We have
IPt, hmll = Ipt -+ Mq(Z(Cd))ll < Ptjl < IIP

It follows that the sequence {h,} is bounded (by 2 l||p|). Therefore, by the weak
compactness property of the algebra A(Cd) we get a subsequence of {h,} which we will
again denote by {h,} and g e Mq(I(Cd)) such that

h( T) g(T)

for each T e Cd.
Moreover, since pt (T) p(T) it follows that

II(pt + hm)(T)| -i (p + g)(T)II

for each T e Cd.








Given e > 0, choose H e Cd such that

lp + gl < I|(p g)(H)|| + (6.3)

Choose a natural number N such that for all m > N we have

I|(p g)(H)|| < |(t, + hm)(H)ll + (6.4)

From equations (6.3) and (6.4), we have,

ip + g (l < I|(pt, + hm)(H)I| + < IPt, + hmll + (6.5)

for all m > N. On the other hand,

lip + g>l > IIpt + g9t, > I|t, + hml > IIP + hm| c (6.6)

for all m > N, Combining equations (6.5) and (6.6), we get

I Ipt, + hmll p + 9gl I <

for all m > N. i.e. ||pt + hml liP + gll.
Combining equations (6.1) and (6.2), and using the fact that 7q is isometric and
yields,

Ipt, + hmll = II(tmR)ll.

Thus IIp(tmR)II lip + g||. Let C = sup{||p(T)|| : T E Cd, T is A-nilpotent}. Since
I|p(tmR)|| < C, it follows that

lip + gl C.

Hence,

llp+ fll = lp+ Mq((Cd)) < p+ gl
On the other hand, by definition,

ip + fl > sup{||(p + f)(T) : T E Cd, T is A-nilpotent} = C









and this completes the proof.


The matrix version of Theorem 6.2.1 follows as a consequence by setting x = p + f

in the above proposition.

To pass from the case of matrix coefficients to the case of operator coefficients, i.e.
to prove Theorem 6.2.1, we can imitate the proof of Theorem 5.3.1 by using the d-tuple

S in place of 5(.).
6.3 The d x d Non-commutative Mixed Ball

In this section we pose the CFP for Ddd, the d x d Non-commutative Mixed Ball, with
the infinite initial segment A and give our main result.

Let Fd be the semi-group generated by the dd symbols {g}j 1. Let F2 denote the
corresponding Non-commutative Fock Space and S = (51, S12,..., Sd), the dd-tuple of

Creation Operators. Fix an infinite initial segment A c Fda.
As in Section 6.2, we begin with the following remark. Let V : F2(A) IF2 be the
inclusion map.

Remark 6.3.1. If 0 < t < -, then
,dd

(i) tS= (tS11, 512, .. tSd. ) Edd

(ii) tV*SV e d.
We define A(Ddd)a and _(VDdd) as before (See Sections 4.1 and 4.5). The proofs
from Sections 4.1, 4.2 and 4.3, can be generalized to this current setting (A is infinite) by

replacing 5(f) by the dd-tuple S.
In order to generalize the proof of Lemma 4.5.1, to the current setting, we replace

S() with the dd-tuple V*SV, and modify the proof as we did in Section 6.2.
We now state the CFP and give our main result.
Let U be an infinite dimensional separable Hilbert space, A c Fd be an infinite
initial segment and p = Y^pw, w be a formal power series such that pw e B(U) and

l||p| = sup{|lp(T)|| : T EC dd} < o.









Does there exist x = Y,,d xRw such that x2 = pw for all w e A and 1||1| < 1?

Theorem 6.3.1. There exists a solution x to the above problem if and only if


sup{||p(T)|| : T cDdd, T is A- nilpotent} < 1.

The same strategy that we used to prove Theorem 6.2.1 can be used to Theorem

6.3.1 as well.

As before, let 6.1, let 7 : A(Ddd)-/I(dd) B(M) denote a completely isometric

homomorphism obtained by applying Theorem 3.2.1 to the abstract unital operator

algebra A(Ddd)/Z00 (d). And for 1 < i < d and 1
R = (R11, R12, .., Rd). It follows that R is A-nilpotent.

To prove the matrix version of Theorem 6.3.1, we can imitate the proof of Proposition

6.1. The only point that needs clarification is that the dd-tuple tR for 0 < t < 1 lies in

Dda, i.e. I||tR||op < 1.

To see this we first observe that the formal power series

d, d
E EgU E M1(A(DddY))
i,j= 1

has norm at most one, where = max{d, d}, and Ey is the x matrix whose (i,j)-th

entry is 1 and other entries are 0; 1 < i < d, 1 < j < d.

The fact that the map F : M P(A(Ddda))/M(Z(Pdd)) Me BA(M) is isometric

implies that

d,d


d,d
RE= g|| E,, M (
i,j= 1
ij=i









d, d
= Eg+ M(I()Ddd))

d,d

ij= 1

< 1.


To pass from the matrix version to the operator version, i.e. to prove Theorem 6.3.1,

we can imitate the techniques in the proof of Theorem 5.3.1 by using the dd-tuple S in

place of 5(f).









CHAPTER 7
FUTURE RESEARCH

The following questions came up while this work was in progress: Could the

operator algebras approach used in the examples discussed in Chapter 6 be generalized

to handle non-commutative domains that are defined by a possibly infinite collection of

Linear Matrix Inequalities? Is the algebra A(IC)" a dual algebra?

It would be interesting to know the answers to these questions.









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BIOGRAPHICAL SKETCH

Sriram Balasubramanian was born in Chennai, India. He obtained his bachelor's

degree from the University of Madras and a master's degree in mathematics from the

Indian Institute of Technology Madras, before coming to the University of Florida for

doctoral study. His interests other than mathematics are the game of Cricket and South

Indian Classical and Film Music.





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Firstandforemost,IwouldliketoexpressmydeepestgratitudetomyadvisorDr.ScottMcCullough,forhisever-presentguidance,encouragementandhelp,withoutwhichthisdissertationwouldneverhavebeencompleted.IamforevergratefultohimforhavingmeashisstudentandforinspiringmewithhistremendousknowledgeandexpertiseinMathematics.IwouldalsoliketothankDr.JamesBrooks,Dr.JamesHobert,Dr.MichaelJuryandDr.Li-chienShenforservingonmydoctoralcommittee.MyspecialthankstoDr.MichaelJuryforhismanyvaluablesuggestions.Finally,IwouldliketothankmyfamilyandmyfriendsespeciallyBalaji,Krishna,SubbuandVijay,fortheirconstantencouragementandsupport. 4

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page ACKNOWLEDGMENTS .................................. 4 ABSTRACT ......................................... 7 CHAPTER 1INTRODUCTION ................................... 8 1.1SummaryofResults .............................. 14 1.2Organization .................................. 16 2MATRIXCONVEXITY ................................ 18 2.1MatrixConvexSetsinCd 18 2.1.1Examples ................................ 19 2.1.2Properties ................................ 19 2.2MatricialHahn-BanachSeparation ...................... 21 2.3TheNon-commutativeFockSpaceandCreationOperators ........ 26 2.3.1TheFreeSemi-groupondLettersandIntialSegments ....... 26 2.3.2TheNon-commutativeFockSpace .................. 27 2.3.3TheCreationOperators ........................ 27 3ABSTRACTOPERATORALGEBRAS ....................... 29 3.1AbstractOperatorAlgebra ........................... 29 3.1.1Examples ................................ 31 3.1.2TheQuotientOperatorAlgebra .................... 31 3.2RepresentationsofAbstractUnitalOperatorAlgebras ........... 34 4THEABSTRACTOPERATORALGEBRASA(K)1&A(K)1=I(K) 35 4.1TheAlgebraA(K)1ofScalarFormalPowerSeries ............. 35 4.1.1FormalPowerSeries .......................... 35 4.1.2TheVectorSpaceA(K)1 36 4.1.3MatrixNormsonA(K)1 37 4.1.4TheAlgebraA(K)1 39 4.2WeakCompactnessandA(K)1 44 4.3TheAbstractOperatorAlgebraA(K)1 47 4.4CompletelyContractiveRepresentationsofA(K)1 48 4.5TheAbstractOperatorAlgebraA(K)1=I(K) 53 4.6AttainmentofNormsofClassesinMq(A(K)1)=Mq(I(K)) 54 5THENON-COMMUTATIVECARATHEODORY-FEJERPROBLEM ....... 56 5.1TheCaratheodory-FejerInterpolationProblem(CFP) ............ 56 5.2TheMatrixVersion ............................... 56 5

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............................. 58 6INFINITEINITIALSEGMENTS ........................... 62 6.1ExamplesofNon-commutativeOperatorDomains ............. 62 6.2Thed-dimensionalNon-commutativePolydisc ............... 64 6.3Thed~dNon-commutativeMixedBall ................... 69 7FUTURERESEARCH ................................ 72 REFERENCES ....................................... 73 BIOGRAPHICALSKETCH ................................ 76 6

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WeposetheCaratheodory-Fejerinterpolationproblemforopen,circledandboundedmatrix-convexsetsinCd.ByusingtheBlecher-Ruan-Sinclaircharacterizationofanabstractunitaloperatoralgebra,weobtainanecessaryandsufcientconditionfortheexistenceofaminimum-normsolutiontotheproblem. 7

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AclassicalinterpolationprobleminfunctiontheoryistheCaratheodory-Fejerinterpolationproblem(CFP):Givenn+1complexnumbersc0,c1,...,cndoesthereexistacomplexvaluedanalyticfunctionf(z)=P1j=0fjzjdenedontheopenunitdiscDCsuchthatfj=cjforall0jnandjf(z)j1forallz2D? TheproblemandsomeofitsvariantswerestudiedbyCaratheodory,FejerandSchurduringtheearly20thcenturyin[ 30 ],[ 31 ]and[ 8 ].AnecessaryandsufcientconditionforthesolvabilityoftheCFP,whichiscommonlyreferredtoastheSchurCriterion,isthattheToeplitzmatrix isacontraction. AnoperatortheoreticviewoftheCFPwasrstpresentedbySarasoninhispioneeringwork[ 29 ].HisformulationhashadamajorimpactnotonlyontheCFPandtherelatedPickinterpolationproblembutthedevelopmentofoperatortheoryandthestudyofnon-selfadjointoperatoralgebrasgenerally.WenowpresentaproofoftheequivalencebetweenthesolvabilityoftheCFPandtheSchurCriterionthatusesSarason'sideas.Wewillbeginwithsomedenitionsandstatesomewell-knownfacts(withoutproofs). LetH2(D)denotetheHardyHilbertSpacedenedbyH2(D)=(f:D!C:f(z)=1Xn=0anznand1Xn=0janj2<1).

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LetL2denotetheHilbertspaceofsquare-integrablefunctionsonTwithrespecttothenormalizedLebesguemeasure.Theinnerproductisdenedas:hf,gi=1 2Z20f(ei) Asisoftendone,wewillviewL2asaspaceoffunctionsratherthanasaspaceofequivalenceclassesoffunctions,byidentifyingtwofunctionstobeequaliftheyareequala.e.withrespecttothenormalizedLebesguemeasure.i.e.L2=f:T!C:1 2Z20jf(ei)j2d<1. 7 ].Theexpansionofthefunctionf2L2withrespecttothisorthonormalbasisiscalledtheFourierSeriesexpansionoff.Wewritef(ei)1Xn=bf(n)ein 2Z20f(ei)eind 9

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22 ])thatforf2H2(D),theradiallimit~f(ei)=limr!1f(rei)existsforalmostall.Wewillcall~ftheboundaryfunctionoff.Animportantpropertyof~fisthat~f2L2.Thelinearmappingthattakesf2H2(D)to~f2L2isanisometryfromH2(D)ontoaclosedsubspaceofL2whichwewilldenotebyH2.Ifg2L2isgivenbyg(ei)1Xn=0bg(n)ein,thenitiswellknownthatthefunctionfdenedontheunitdiscbyf(z)=1Xn=0bg(n)znisinH2(D)andinaddition,~f=g.Conversely,iff2H2(D)isgivenbyf(z)=1Xn=0fnznthen,theFourierseriesexpansionof~fis~f(ei)1Xn=0fnein.ThisequivalencegivesusthefollowingwaytoviewthespaceH2intermsofFouriercoefcients,namely,H2=fg2L2:bg(n)=0forn<0g.

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(i) (iii) (iv) IfTS=ST,thenthereexistsafunction2H1suchthatT=PTjKwhereTistheanalyticToeplitzoperatorwithsymbolandkk1=kTk. 29 ]. 1.1 )hasnormatmostone. Proof. 1.0.1 andequation( 1.2 )that~f2H1and,1supfjf(z)j:z2Dgk~fk1=kT~fk.

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1.1 )isthecompressionoftheanalyticToeplitzoperatorT~ftothesubspacespannedbytheorthonormalseth1,z,z2,...,zniofH2(andcomputedwithrespecttothisbasis),itfollowsthatthenormofthatmatrixisatmost1. Fortheconverse,weapplyLemma 1.0.1 with(z)=zn+1,K=h1,z,z2,...,zniandT:K!Kbeingthematrixin( 1.1 ).SupposethatkTk1.LetSandPbetheoperatorsthatappearinthehypothesisofLemma 1.0.1 .ThematrixofSwithrespecttothebasisf1,z,z2,...,zngisgivenby ThusTS=STandLemma 1.0.1 impliesthatthereexists2H1suchthatT=PTjKandkk1=kTk. 1.0.1 thatsupfjf(z)j:z2Dgkk1. Fromtheoperatortheory/algebrapointofview,theCFPisessentiallyunchangedifthecoefcientsc0,c1,...,cnaretakentobeelementsofB(U)forsomeseparableHilbertspaceU.Indeed,eveninthiscase,anecessaryandsufcientconditionforthesolvabilityoftheCFPisthesameasbefore,onlynowtheentriesoftheToeplitzmatrixin( 1.1 )areboundedoperatorsinB(U)andthematrixitselfisanelementofB(n+11U).AnalternatewayofviewingtheSchurCriterionwhichismoreconvenientforourpurposesisthefollowing. 12

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Proof. 1.3 ).SinceSisacontractionandSn+1=0wehavekp(S)k1.ThefactthatthenormoftheToeplitzmatrixinequation( 1.1 )isequaltothenormoftheoperatorp(S)completestheargument. 2.DTiscalledthedefectoperatorofT.Letfejgnj=0denotethestandardorthonormalbasisofCn+1.DenetheoperatorV:H!Cn+1HbyVh=nXj=0ejDT(T)jh.

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Usingequation( 1.4 )weget,kp(T)k=knXj=0cjV(SjIH)Vk=k(IUV)(nXj=0cj(SjIH))(IUV)kknXj=0cj(SjIH)k=kp(S)k1. 1 ].Formoredetailssee[ 14 ],[ 6 ].SomeresultsontheproblemforboundedcirculardomainsinCdcanalsobefoundin[ 11 ].Somenon-commutativegeneralizationsoftheCFPhavealsobeenstudiedin[ 25 ],[ 26 ],[ 9 ],[ 20 ],[ 4 ]. Inthisthesis,someoftheexistingresultsontheCFPhavebeenextendedtothenon-commutativesettingofthefreealgebraonanitenumberofgenerators.Anexampleofadomainweconsiderhereisthed~dnon-commutativematrixmixedballdenedbyDd~d=[n2NfX=(X11,X12,...,Xd~d):XijarennmatricesandkXkop<1g.

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WhatfollowsaresomedenitionswhichwillleadustothestatementoftheCFPforthisparticulardomain. TheCFPforthed~dnon-commutativematrixmixedballisthefollowing:Let,aniteinitialsegment,andp=Xw2pww (Aspecialcaseof)Ourmainresultisthefollowing. 1.1.1 whichallowsforoperator-valuedcoefcientspwandfw. 15

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32 ],[ 33 ],[ 34 ],[ 24 ],[ 25 ],[ 26 ],[ 21 ],[ 19 ]).Thesefunctionsarenotonlyobjectsofgreatmathematicalinterest,buthaveapplicationsinareassuchascontroltheoryandoptimization.Non-commutativepolynomials,inparticular,areofspecialinterestsincenon-commutativepolynomialinequalities(matrixinequalitieswheretheunknownsarematricestoo),occurnaturallyinthecontextofdimension-freelinearsystems.Recentadvancesinthestudyofnon-commutativelinearmatrixinequalitiescanbefoundin[ 12 ],[ 18 ]. InChapter3,thedenitionanabstractoperatoralgebraisintroducedalongwithsomeexamples.Itisalsoshownthatthequotientofanabstractoperatoralgebrabyaclosedtwo-sidedidealisanabstractoperatoralgebra.ThechapterendswiththestatementoftheBlecher-Ruan-SinclairTheoremforabstractunitaloperatoralgebras. Chapter4iswheretheinterpolatingclassA(K)1andtheidealI(K)areintroduced.ItisshownthatA(K)1andthequotientA(K)1=I(K)areabstractoperatoralgebras.Severalkeypropertiesincludingaweak-compactnesstypepropertyofthealgebraA(K)1andthenormattainmentpropertyofthealgebraA(K)1=I(K)areestablished.ThechapterendswithadiscussiononcompletelycontractiverepresentationsofthealgebraA(K)1,whereitisshownthattuplesofnite-dimensionalcompressionsofoperatorsthatgiverisetocompletelycontractiverepresentationsofA(K)1lieontheboundaryoftheunderlyingmatrixconvexset. 16

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InChapter6,aversionofTheorem 1.1.1 whereitisassumedthattheinitialsegmentisaninniteset,isprovedfortwospecialnon-commutativedomainsnamely,thed-dimensionalnon-commutative(operator)polydiscandthed~dnon-commutative(operator)mixedball. InChapter7,twoimportantandveryinterestingquestionsthatcameupwhilethisworkwasinprogress,areposed. 17

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Abasicobjectofstudyinthisthesisisaquantized,ornon-commutative,versionofaconvexset.Whilethedenitionseasilyextendtoconvexsubsetsofarbitraryvectorspaces,herethefocusisonsubsetsofCd,thecomplexd-dimensionalspace.InthischapterwepresentthedenitionofamatrixconvexsubsetofCdandintroduceourstandardassumptionsregardingthesesets.SomepropertiesandexamplesofsuchsetsandamatricialHahn-Banachseparationresultarealsopresented.ThechapterendswithadiscussionoftheCreationOperatorsontheNon-commutativeFockSpace. Anon-commutativesetLisasequence(L(n))where,forn2N,L(n)Mn(Cd),whichisclosedwithrespecttodirectsums;i.e.,ifX2L(n)andY2L(m),then whereXjYj=0B@Xj00Yj1CA. AmatrixconvexsetK=(K(n))isanon-commutativesetwhichisclosedwithrespecttoconjugationbyanisometry;i.e.,if2Mm,nand=In,andifX=(X1,...,Xd)2K(m),then AsubsetUofMn(Cd)iscircledifeiUUforall2R.AmatrixconvexsetKiscircledifeachK(n)iscircled.Asacanonicalexampleofacircledmatrixconvexset, 18

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isopen; (b) isbounded;and (c) iscircled; (i) LetK(n)=f(X1,...,Xd):kXjk<1gwith=1and=p (ii) LetK(n)=fX=(X11,X12,...,Xd~d):kXkop<1g,wherekXkopisthenormoftheoperatorX=(Xij)d,~di,j=1:(Cn)~d!(Cn)d,with=1 (iii) LetK(n)=f(X1,...,Xd):supfPdj=1jhXjy,yij:kyk=1g<1gwith=1 (i) (ii) If02L,thenLisclosedwithrespecttoconjugationbyacontraction;i.e.inequation( 2.2 )canbeassumedtobeacontraction. (ii) TheclosureofLnamely 19

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2. isunitary.Therefore,~~X~=Im+n0m+nJ(~)bXJ(~)0B@Im+n0m+n1CA2K(m+n).

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LetX=(X1,...,Xd)2 L(p)andY=(Y1,...,Yd)2 L(q).ChoosesequencesXm=(Xm1,...,Xmd)andYm=(Ym1,...,Ymd)fromL(p)andL(q)respectivelysuchthatXm!XandYm!Y. Wehave,kj(XmYm)(XY)kj=dXj=1k(XmjYmj)(XjYj)k=dXj=1k(XmjXj)(YmjYj)k=dXj=1maxfkXmjXjk,kYmjYjkg L(p+q). If2Mp,`isanisometry,thenwehavekjXmXkj=dXj=1k(XmjXj)kdXj=1kXmjXjk. L(`). 13 ]).Thefollowingcontents,areminorvariantsoflemmasandtheoremsfrom[ 16 ]. Givenapositiveintegern,letTndenotethecollectionofallpositivesemi-denitenncomplexmatricesoftraceone.AT2TncorrespondstoastateonMn,viathe 21

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Proof.

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Choosing~T=whereisaunitvectorsuchthatkCk=kCk,

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2.1.2 ,CYC2C(n)andsobythehypothesisofthelemma,therighthandsideoftheaboveequationisnon-negative.IfCdoesnothavenorm1,butisnotzero,thenasimplescalingargumentshowsthatfY,C(~T)0. HencebyLemma 2.2.1 ,thereexistsanS2TnsuchthatfY,C(S)0foreverym,Y2C(m)andC2Mm,n. AlinearpencilLofsizenisaformalexpressionoftheformPd`=1L`g`whereL`2Mn.Forad-tupleT=(T1,...,Td)ofboundedoperatorsonaHilbertSpaceH,theevaluationofLatTisdenedastheoperatorL(T)=Pd`=1L`T`. (i) Proof. ReG(C(n))1
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IfweletR=(1)S+ ReG(CYC)tr(CRC).(2.6) Letfe1,...,edgdenotethestandardorthonormalbasisforCd.Given1`d,andcolumnvectorsc,d2Cn,deneaboundedsesquilinearformonCnbyB`(c,d)=G(R1 2 2e`) ThereexistsauniquematrixB`2MnsuchthatB`(c,d)=hB`c,di. Fixapositiveintegerm.LetY=(Y1,...,Yd)2C(m)begivenandconsiderL(Y),theevaluationofLatY.Letfe1,...,emgdenotethestandardorthonormalbasisofCm.For=Pmj=1jej2CnCm,wehavehL(Y),i=mXi,j=1dX`=1hB`j,iihY`ej,eii=mXi,j=1dX`=1G(R1 2 2e`)hY`ej,eii=mXi,j=1dX`=1G((R1 2 2 2 2 2 2

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2.6 )weget,RehL(Y),i=ReG((R1 2 2 2 2 2 2 2.5 )weget,RehL(X)nXi=1 2eiei,nXj=1 2ejeji=ReG(InXIn)>1=knXi=1 2eieik2. LetFddenotethesetofallwordsgeneratedbydsymbolsfg1,...,gdg.DenetheproductonFdbyconcatenation.i.e.,ifw=gi1...gimandv=gj1...gjn,thentheproductwvisgivenbygi1...gimgj1...gjn.Fdisasemi-groupwithrespecttothisproduct,withtheemptyword;actingastheidentityelement,i.e.w;=w=;wforallw2Fd. 26

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AsetFdisaninitialsegmentifitscomplementisanidealinthesemi-groupFd;i.e.,ifbothgjw,wgj2Fdn(1jd),wheneverw2Fdn.Inthecasethatd=1aninitialsegmentisthusasetoftheformf;,g1,g21,...,gm1gforsomem. ToconstructtheFockspace,F2,deneaninnerproductonChgibydening forw,v2FdandextendingitbylinearitytoallofChgi.ThecompletionofChgiinthisinnerproductisthentheHilbertspaceF2. 7 ].Given1jd,deneSj:F2!F2bySjv=gjvforawordv2FdandextendSjbylinearitytoallofChgi.ItisreadilyveriedthatSjisanisometricmappingofChgiintoitselfanditthusfollowsthatSjextendstoanisometryonallofF2.InparticularSjSj=I,theidentityonF2.Alsoofnoteistheidentity, 27

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Ofcourse,asitstandsthetupleS=(S1,...,Sd)actsontheinnitedimensionalHilbertspaceF2.Therearehowever,nitedimensionalsubspaceswhichareessentiallydeterminedbyidealsinFdandwhichareinvariantforeachSj. Thesubset(`)=fw:jwj`gofFdisacanonicalexampleofaniteinitialsegment.AndthesubspaceF(`)2ofF2spannedby(`)isinvariantforSj,1jd.LetV(`)denotetheinclusionofF(`)2intoF2andletS(`)denotetheoperatorV(`)SV(`).Thus,S(`)=((S(`))1,...,(S(`))d)where(S(`))j=V(`)SjV(`). Recallfromthedenitionofthe(open,boundedandcircled)matrix-convexsetK. Proof. 2.8 )thatP=V(`)PV(`)=V(`)dXj=1SjSj!V(`)=dXj=1(S(`))j(S(`))j.

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Webeginthischapterwiththedenitionofanabstractoperatoralgebra.Followingthatwepresentsomeexamplesandtheproofofthefactthatthequotientofanabstractoperatoralgebrabyaclosedtwo-sidedidealisanabstractoperatoralgebra.Furthermore,wepresentacharacterizationofanabstractunitaloperatoralgebraduetoBlecher,RuanandSinclair. AmatrixnormedspaceVissaidtobeanabstractoperatorspaceifkXYkp+`,q+r=maxfkXkp,q,kYk`,rg Itisimportanttonotethat,withoutlossofgenerality,wecanreplacetherectangularmatricesintheabovedenitionswithsquarematrices. Beforewelookatexamplesofabstractoperatoralgebras,wepresent,asaremark,aninterestingfactabouttheabstractoperatorspaceCd. 29

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Proof. SinceL(n)isconvex,closed,bounded,absorbingandcircled,itistheclosedunitballofMn(Cd)withrespecttosomenorm,whichwewilldenotekkn.WeneedtoshowthatCdtogetherwiththesequenceofnormskknisanabstractoperatorspace. LetX2Mn(Cd)besuchthatkXkn=1andA,B2Mnbeofunitnorm.ConsidertheJuliamatrices(see( 2.4 ))J(A),J(B)ofAandB.SinceJ(A)andJ(B)areunitary,itfollowsfromequation( 3.1 )that,kJ(A)(X0)J(B)k2n1. 30

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Since(Cd,kkn)isamatrixnormedspaceitfollowsthat,kXYkn+mmaxfkXkn,kYkmg. LetHbeaHilbertspaceandV=B(H).DenematrixnormsonVbyk(Tij)kp,q=kTk (ii) LetHbeanarbitraryseparableHilbertspace,andVdenotethealgebraofpolynomialsindvariables.DenematrixnormsonVbyk(xi,j)kp,q=supfk(xi,j(T))kg

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Proof. Observethat Usingequations( 3.2 )and( 3.3 )gives,kjAXBkjp,q=kj(A(xij+yij)B)kjp,qkA(xij+yij)Bkp,qkAkk(xij+yij)kp,qkBk
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3.4 ),( 3.5 )and( 3.6 )yields, Proof. 3.1.2 weknowthatV=Wisanabstractoperatorspace.Moreover,1+WistheunitofV=W.ItremainstoshowthattheproductonV=Wiscompletelycontractive.Forthatpurpose,letX=((xij))andY=((yij))2Mp(V=W).Choose(aij),(bij)2Mp(W)suchthatkj((xij))kjp+>k(xij+aij)kp SinceWisatwo-sidedidealofV,itfollowsthat 33

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3.7 ),( 3.8 ),( 3.9 )andthefactthatmultiplicationinViscompletelycontractiveyields,kjXYkjp=kj((xij+aij)(yij+bij))kjpk(xij+aij)(yij+bij)kpk(xij+aij)kpk(yij+bij)kp<(kj((xij))kjp+)(kj((yij))kjp+)=(kjXkjp+)(kjYkjp+). LetVandWbeabstractoperatorspacesand:V!Wbealinearmap.Deneq:MqV!MqWbyq=Iq,whereIqistheqqidentitymatrix. Themapissaidtobecompletelycontractive(isometric)ifqisacontraction(isometry)foreachq2N. Acompletelycontractive(isometric)representationofanalgebraAisacompletelycontractive(isometric)algebrahomomorphism:A!B(M)forsomeHilbertspaceM.

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2.1.1 .Inparticular,K(1)isancircledopenconvexsubsetofCd.Thischapterisdividedintotwoparts.Intherstpart,whichconsistsoffoursubsections,weconstructanabstractunitaloperatoralgebrawhichisanaturalnon-commutativeanalogoftheBanachalgebraH1(K(1)).Wealsopresentafewlemmasoncompletelycontractiverepresentationsofthisalgebra. Inthesecondpart,Sections 4.5 and 4.6 ,weconsidertheidealI(K)ofthealgebraA(K)1determinedbyaniteinitalsegment.WeshowthatthequotientalgebraA(K)1=I(K)determinedbytheidealisanabstractunitaloperatoralgebra.Wealsoshowthatnormsofclassesinthequotientalgebraareattained. wherefw2B(U,U0).Itisconvenienttosumfaccordingtoitshomogeneousofdegreejterms;i.e., Recall,forad-tupleT=(T1,...,Td)ofoperatorsonacommonseparableHilbertspaceHandawordw=gi1gi2...gin2Fd,i1,...,in2f1,2,...,dg,theevaluationofwatT

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providedthesumconvergesintheoperatornorminB(UH,U0H)intheindicatedorder.WenoteforclaritythatifU=U0=C,thenin( 4.3 )istheusualscalarproductandiff(T)converges,itisanelementofB(H). RecallthematrixconvexsetK=(K(n))whichsatisestheconditionsofAssumption 2.1.1 .WewillwriteX2KtodenoteX2Sn2NK(n).Fortheformalpowerseriesfasabove,wedene 4.4 )canbeinnite.Weareonlyinterestedinthoseformalpowerseriesfforwhichthisisnotthecase.LetA(K)1=(f=Xw2Fdfww:fw2C,kfk<1). Proof. 2.3.3 .For 36

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2.3.1 andRemark 2.3.1 ,weget0=kfk2 Thusfw=0forallwsuchthatjwj`.Since`isarbitrary,thelemmafollows. ThefollowingLemmasplaysanimportantroleintheanalysistofollowgenerally,andinprovingthatA(K)1isanalgebra,inparticular. Proof. 2.3 )andS(`)fromSubsection 2.3.3 .Fix0t
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Letfe1,e2....,eqgdenotethestandardorthonormalbasisforCq.Weknowthatforeachw2Fd, 2.(4.11) Moreover,foreach1iq,equation( 4.10 )impliesthat Usingequations( 4.11 )and( 4.12 ),weget1Xj=02jXjwj=jkfwk21Xj=02jXjwj=j(qXi=1kfweik2)=qXi=1(1Xj=02jXjwj=jkfweik2)qkfk2.

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Proof. 2Z20F(eit)eijtdt. 2Z20kF(eit)kdt. Givenf=Pw2Fdfww2Mp,q(A(K)1)andg=Pw2Fdgww2Mq,r(A(K)1),denetheproductfgoffandgastheconvolutionproduct;i.e.,fg=Xw2FdXuv=wfugv!w.

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(i) (ii) (iii) (iv) Proof. (i)AsintheproofofLemma 4.1.3 ,letAj=Xjwj=jfwXw,Bj=Xjwj=jgwXwCj=Xjwj=j(Xuv=wfugv)Xw. LetF(z)=f(zX)andG(z)=g(zX),bothofwhicharedenedinaneighborhoodof 4.1.3 ,thereisa<1suchthatkAmkmkfkandkBkkkkgk.HencekCjk(j+1)kfkkgkj. 40

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WeclaimthattheLHSofequation( 4.13 )convergestozeroask!1.ItsufcestoshowthatthersttermontheRHSofequation( 4.13 )convergestozero,inviewoftheconvergenceofthesecond,thirdandthefourthtermsontheRHStozeroduetothefollowingreasons: (a) bothkPkm=0AmzmkandkPkn=0Bnznkarenite. (b) bothP1m=k+1AmzmandP1n=k+1Bnzn,beingtailsoftheconvergentseriesf(zX)andg(zX)respectively,convergetozeroask!1. 4.13 ). 41

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2p 2p 2p

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2r 2r 2r 20@1Xn=02nXjvj=nkgvk21A1 2r Sincetheseries1Xk=0r 4.1.2 impliesthat1Xm=02mXjuj=mkfuk2and1Xn=02nXjvj=nkgvk2arenite.Thesefactstogetherimplythedesiredconvergence.Thusfg(zX)=f(zX)g(zX)wheneverjzj<2 Fixx2CrCaandy2CpCaandconsiderthecomplexvaluedfunctionsA(z)=hFG(z)x,yiandB(z)=hF(z)G(z)x,yi.ObservethatA(z)andB(z)areanalyticonthediscfz:jzj<1 (iii)&(iv)Since,foreachX2K,fg(X)=f(X)g(X)itfollowsthatkfg(X)kkfkkgk.Thuskfgkkfkkgkandfg2Mp,r(A(K)1). 43

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Proof. Proof. 2.3.3 .Fix00,thereexistsN2Nsuchthatkfm(tS(`))fn(tS(`))k< 44

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ForeachjandX2K, FromLemma 4.1.3 ,thereisa<1suchthatforeachj, Fromequations( 4.14 )and( 4.15 ),itfollowsforeachjthat,kXjwj=jfwXwkjc, FixJ2Nsuchthat RecallfromthedenitionofthematrixconvexsetK.ChooseK2NsuchthatforallmK, foreach0jJ. 45

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4.16 )and( 4.17 ),itfollowsthatkfm(X)f(X)kJXj=0kXjwj=j((fm)wfw)Xwk+1Xj=J+1kXjwj=j((fm)wfw)XwkJXj=0Xjwj=jk(fm)wfwkj+1Xj=J+1kXjwj=j(fm)wXwk+1Xj=J+1kXjwj=jfwXwkJXj=0 (i) (ii) Thereexistsasubsequenceffmkgofffmgandfw2Mp,qsuchthat(fmk)w!fwforallw; (iii) Letf=Xw2Fdfww.ForeachX2Kthesequence(fmk(X))convergestof(X)andmoreoverkf(X)kc. Proof. 4.1.3 forX=tS(`)implythat

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Theproofofitem(ii)usesastandarddiagonalargument.Letfw1,w2,...gbeanenumerationofwordsinFdwhichrespectslength(i.e.,ifvw,thenjvjjwj).Sincek(fm)w1kc Inviewofwhathasalreadybeenproved,anapplicationofProposition 4.1 provesitem(iii). 4.9 ). Proof. 47

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4.1.4 (ii)thatforallX2K(n),kAFB(X)k=kA(X)F(X)B(X)kkAInkkF(X)kkBInkkAkkFkkBk. LetF2M`,r(A(K)1),G2Mp,q(A(K)1),X2K(n).ObservethatkFG(X)k=0B@F(X)00G(X)1CAmaxfkF(X)k,kG(X)kgmaxfkFk,kGkg. Let>0begiven.WithoutlossofgeneralityassumethatkFkkGk.Choosem2NandR2K(m)suchthatkF(R)k>kFk.Therefore Letting!0intheinequality( 4.20 )andfromtheinequality( 4.19 )itfollowsthat, Lastly,completecontractivityofmultiplicationinMp(A(K)1)followsdirectlyfromLemma 4.1.4 (iv).ThusA(K)1isanabstractoperatoralgebra. 3.2 .Theorem 3.2.1 guaranteestheexistenceofacompletelyisometricrepresentationfortheabstractunitaloperatoralgebraA(K)1. 48

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Inthissection,weprovethatforacompletelycontractiverepresentationTofA(K)1,foranyn2NandnitedimensionalsubspaceWofMofdimensionnand0t<1thetupletZ=tVTV=(tVT1V,...,tVTdV) thenfr=1Xj=0rjXjwj=jfww=1Xj=0rjfj. Proof. 4.22 ).Lemma 4.1.3 impliesthatkfjkkfk.BecauseTiscompletelycontractivekfj(T)kkfk.Itfollowsthatfr(T)convergesinnorm.Sincealsothepartialsumsoffrconverge(tofr)inthenormofMq(A(K)1),itfollowsthatT(fr)=fr(T)andsokfr(T)kkfrk. TheinequalitykfrkkfkisstraightforwardbecauserKK. NowsupposethatTiscompletelyisometric.Inthiscasekfr(T)k=kfrk.Ontheotherhandlimr!1kfrk=kfk. 49

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K. (b)IfX2K(`),thenL(X)convergesinnorm;i.e.,theseries1Xj=0L(X)j+1 (c)kL(X)k<1andhenceLisinMk(A(K)1)andhasnormatmostone. (d)IfTisacompletelycontractiverepresentationofA(K)1,then2(L(T)+L(T))0. Proof. K.Toprovetheconverse,assumethecontrary,i.e.supposethereexists~X2Kandaunitvectorvsuchthath(2L(~X)L(~X))v,vi=0.Thefollowingargumenthasbeenadaptedfrom[ 16 ].Denethemapq:R!Rbyq(t)=h(2L(t~X)L(t~X))v,vi. 50

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Toprovepart(b)ofthelemma,letX2K(`)begiven.BecauseK(`)iscircled,itfollowsthateiX2K(`)foreach.Hence, foreach.Fornotationalease,letY=L(X).ThusYisak`k`matrixandequation( 4.23 )impliesthatthespectrumofYliesstrictlywithinthedisc;i.e.,eacheigenvalueofYhasabsolutevaluelessthanone.Thus,1 21Xj=0Y Toprove(c)observethatkY(2Y)1k<1ifandonlyif(2Y)(2Y)YY Toprovepart(d),observe,SinceTiscompletelycontractiveandL2Mk(A(K)1)withnormatmostone,anapplicationofLemma 4.4.1 yields,kL(rT)k1.Arguingasintheproofofpart(b),itfollowsthat2(L(rT)+L(rT))0.Thisinequalityholdsforall0r<1andthustheconclusionofpart(c)follows. 51

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Proof. 4.4.2 .Frompart(d)ofthepreviouslemma,itfollowsthat2(L(T)+L(T))0.ApplyingIkVontheleftandIkVontherightofthisinequalitygives,2(L(Z)+L(Z))=(IkV)(2(L(T)+L(T))(IkV)0. 4.4.2 andanapplicationofTheorem 2.2.1 implythatZ2 K(n).HencetZ2K(n)forall0t<1. 4.22 )andsupposethatTisacompletelycontractiverepresentationofA(K)1intoB(M)andTisnilpotent.Thenkfr(T)ksupfkf(X)k:X2K,Xisnilpotentgforall0r<1.Moreoveriffw=0forallw62,thenkf(T)ksupfkf(X)k:X2K,Xisnilpotentg. Proof. 52

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Proposition 4.2 impliesthatrZ2K.Thus,kfr(T)k
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ByidentifyingMp,q(A(K)1=I(K))withMp,q(A(K)1)=Mp,q(I(K)),Corollary 3.1.1 impliesthatthequotientA(K)1=I(K)isanabstractoperatoralgebra.Weformallyrecordthisfact. 4.2.1 yieldsasubsequenceffmkgofffmgandf2Mq(I(K))suchthat(p+fmk)(X)!(p+f)(X) 54

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Proof. Sincek(p+fmk)(R)k!k(p+f)(R)k,thereexistsK12Nsuchthat, forallkK1.Combiningtheinequalitiesfromequations( 4.24 )and( 4.25 ),impliesthat,forallkK1, Sincekp+fmkk!kp+Mq(I(K))k,thereexistsaNaturalnumberK2suchthatforallkK2, Settingk=maxfK1,K2ginequations( 4.26 )and( 4.27 ),andletting!0yields

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InthischapterweposetheCaratheodory-FejerInterpolationproblem(CFP)forouropen,boundedandcircledmatrixconvexsetK.UsingtheresultsfromChapters2-4,weproveanecessaryandsufcientconditionforthesolvabilityoftheproblem. 2.1.1 .LetFdbeaniteinitialsegment,andp=Xw2pww2A(K)1 5.1.1 allowingforoperatorcoefcientsisprovedinthischapter. Thestrategyistorstprovetheresultformatrixcoefcients.ThisisdoneinSection 5.2 below.Passingfrommatrixtooperatorcoefcientsisthenaccomplishedusingwell-knownfactsabouttheWeakOperatorTopology(WOT)andtheStrongOperatorTopology(SOT)onthespaceofboundedoperatorsonaseparableHilbertspace.ThedetailsareinSection 5.3 Proof. 4.5.1 and 3.2.1 itfollowsthatthereexistsaHilbertspaceMandacompletelyisometrichomomorphism:A(K)1=I(K)!B(M).As 56

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4.4 ,wewilluseRtodenotethemap. ItfollowsfromTheorem 4.3 thatthereexistsf2Mq(I(K))suchthat Thefactthatiscompletelyisometricimpliesthat SinceRisacompletelycontractiverepresentationofA(K)1,Lemma 4.4.3 impliesthat Combiningtheequations( 5.1 ),( 5.2 )and( 5.3 ),itfollowsthatkp+fksupfkp(X)k:X2K,Xisnilpotentg. (Thematrixversionof)Theorem 5.1.1 followsfromtheabovepropositionbysetting~x=p+f. 57

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Proof. Forw2,deneMm3(pm)w=VmpwVmwhereVm:Um!Uistheinclusionmap.Letpmdenotetheformalpowerseriespm=Xw2(pm)ww 5.1 ,thereexistsfm2Mm(I(K))suchthatxm=pm+fm2Mm(A(K)1)andkxmk=supfkpm(X)k:X2K,Xisnilpotentg. 4.1.3 thatthereexists0<1suchthat 58

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RecallandS(`)fromSubsection 2.3.3 .Fox0t0begiven.ChooseLX2Nsuchthatforallk>LX, 59

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Thusforallk>NXitfollowsfromequations( 5.4 )and( 5.7 )that, LetMX=maxfLX,NXg,h2UCnbeaunitvectorandy=0@MXXj=0Xjwj=j~xwXw1Ah. Fromequations( 5.5 )and( 5.8 )itfollowsthat*0@MXXj=0Xjwj=j(~xmk)wXw1Ah,y+kyk0@k~xmk(X)k+k1Xj=MX+1Xjwj=j(~xmk)wXwk1A SincePMXj=0Pjwj=j(~xmk)wXw!PMXj=0Pjwj=j~xwXwwithrespecttotheWOTonB(UCn),itfollowsthat Equations( 5.10 )and( 5.11 )togetherimply Sincehinthedenitionofyisarbitrary,itfollowsthat Thusfromequations( 5.6 )and( 5.13 )itfollowsthat 60

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Toprovethereverseinequality,observethatforw2,(~xm)w=VmVmpwVmVmandVmVmpwVmVm!pwwithrespecttotheWOTonB(U).Thisimpliesthat~xw=pwforallw2andsobydenition,wegetk~xkC.Thus 61

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Inthischapterwepresenttwoexamplesofnon-commutativeoperatordomains,andconsidertheCaratheodory-FejerInterpolationproblem(CFP)forthesedomainsundertheassumptionthattheinitialsegmentisaninniteset.Theexampleswewillconsiderherewillbetheoperator(asopposedtomatrix)versionsofthosepresentedinSubsection 2.1.1 .Sonaturally,mostofthedenitionsincludingnon-commutativeneighborhoodofzero,circledetc.extendanalogously.Aslightmodicationinthenotionofthe>0neighborhoodof0isnecessary.GivenanoperatorAonaHilbertspaceH,writeA0ifthereisan>0suchthatAI;i.e.,A=Aandforallvectorsh2H,theinequalityhAh,hihh,hiholds.Intheoperatorversion,anon-commutative-neighborhoodofzero,isthesetofT=(T1,...,Td)actingonHsuchthat2IdXj=1TjTj. Thed-dimensionalNon-commutativePolydiscisdenedbyCd=f(T1,...,Td):Tj2B(H)&kTjk<1g Thed~d-dimensionalNon-commutativeMixedBallisdenedby,Dd~d=fT=(T11,T12,...,Td~d):Tij2B(H)&kTkop<1g 62

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5 (forthenitecase)canalsobeappliedhere(tohandletheinnitecase)andthatitleadstoasimilarnecessaryandsufcientconditionforthesolvabilityoftheCFP. FixaninniteinitialsegmentFd.InordertomaketheproofsfromtheChapters2-5workforthissetting,someminormodicationsneedtobemade. RecalltheNon-commutativeFockSpaceandthed-tupleofCreationOperatorsS=(S1,...,Sd)fromSection 2.3 .AmoregeneralpropertyofSisthefollowing. LetF2()denotethecompletionofthelinearspanofwithrespecttotheinnerproductdenedin( 2.7 ). (i) (ii) HereVSV=(VS1V,...,VSdV). Proof.

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Toproveitem(ii),xw62.InlieuofPart(i),itsufcestoshowthatVSwV=0.Letu2.Sincew62andisaninitialsegment,wegetwu62.ItfollowsthatVSwV(u)=V(wu)=0. 6.1.1 (i) 4.1 and 4.5 ).TheproofsfromSections 4.1 4.2 and 4.3 ,canbegeneralizedtothiscurrentsetting(isinnite)byreplacingS(`)byS. InordertogeneralizetheproofofLemma 4.5.1 ,tothecurrentsetting,wereplaceS(`)withVSV.Forclarity,wepresentthemodiedargumenthere. Let0
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WenowstatetheCFPandourmainresult. LetUbeaninnitedimensionalseparableHilbertspace,Fdbeaninniteinitialsegmentandp=Pw2pwwbeaformalpowerseriessuchthatpw2B(U)andkpk=supfkp(T)k:T2Cdg<1. 6.2.1 byrstprovingitformatrixcoefcients,i.e.pw2MqandthenbyfollowingthatwithaWOTapproximationargument. Let:A(Cd)1=I(Cd)!B(M)denoteacompletelyisometricalgebrahomomorphismobtainedbyapplyingTheorem 3.2.1 totheabstractunitaloperatoralgebraA(Cd)1=I(Cd).For1jd,letRj=(gj+I(Cd)),andR=(R1,R2,...,Rd).Sincekgj+I(Cd)kkgjk1andisisometric,wehavekRjk1.MoreoverRis-nilpotent. InSection 5.1 ,pwasapolynomial,whichautomaticallygaveusthenitenessofkp(R)k.Buthere,sinceisinnite,wewillneedanapproximationargumenttogeneralizeProposition 5.1 .Wepresentthegeneralizationbelow. Proof. 4.3 tothecurrentsettingofinnite,thatthereexistsanf2Mq(I(Cd))suchthatkp+fk=kp+Mq(I(Cd))k.

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For0t<1,denepkt=Pw2,jwjktjwjpwwandpk=Pw2,jwjktjwjpww.LetqdenotethemapIq:Mq(A(Cd)1)=Mq(I(Cd))!MqB(M). Wewillrstprovethatkq(pt+Mq(I(Cd))k=kpt(R)k=kp(tR)k. Weknowthatq(pkt+Mq(I(Cd))=pkt(R)=pk(tR) 66

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Letftmgbeanincreasingpositivesequencethatconvergesto1.Choosehm2Mq(I(Cd))suchthat foreachm. Wehavekptm+hmk=kptm+Mq(I(Cd))kkptmkkpk. Moreover,sinceptm(T)!p(T)itfollowsthatk(ptm+hm)(T)k!k(p+g)(T)k 67

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ChooseanaturalnumberNsuchthatforallmNwehave Fromequations( 6.3 )and( 6.4 ),wehave, forallmN.Ontheotherhand, forallmN,Combiningequations( 6.5 )and( 6.6 ),wegetjkptm+hmkkp+gkj< Combiningequations( 6.1 )and( 6.2 ),andusingthefactthatqisisometricandyields,kptm+hmk=kp(tmR)k.

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ThematrixversionofTheorem 6.2.1 followsasaconsequencebysetting~x=p+fintheaboveproposition. Topassfromthecaseofmatrixcoefcientstothecaseofoperatorcoefcients,i.e.toproveTheorem 6.2.1 ,wecanimitatetheproofofTheorem 5.3.1 byusingthed-tupleSinplaceofS(`). LetFd~dbethesemi-groupgeneratedbythed~dsymbolsfgijgd,~di,j=1.LetF2denotethecorrespondingNon-commutativeFockSpaceandS=(S11,S12,...,Sd~d),thed~d-tupleofCreationOperators.FixaninniteinitialsegmentFd~d. AsinSection 6.2 ,webeginwiththefollowingremark.LetV:F2()!F2betheinclusionmap. (i) 4.1 and 4.5 ).TheproofsfromSections 4.1 4.2 and 4.3 ,canbegeneralizedtothiscurrentsetting(isinnite)byreplacingS(`)bythed~d-tupleS. InordertogeneralizetheproofofLemma 4.5.1 ,tothecurrentsetting,wereplaceS(`)withthed~d-tupleVSV,andmodifytheproofaswedidinSection 6.2 WenowstatetheCFPandgiveourmainresult. LetUbeaninnitedimensionalseparableHilbertspace,Fd~dbeaninniteinitialsegmentandp=Pw2pwwbeaformalpowerseriessuchthatpw2B(U)andkpk=supfkp(T)k:T2Dd~dg<1.

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6.2.1 canbeusedtoTheorem 6.3.1 aswell. Asbefore,let 6.1 ,let:A(Dd~d)1=I(Dd~d)!B(M)denoteacompletelyisometrichomomorphismobtainedbyapplyingTheorem 3.2.1 totheabstractunitaloperatoralgebraA(Dd~d)1=I(Dd~d).Andfor1idand1j~d,letRij=(gij+I(Dd~d)),andR=(R11,R12,...,Rd~d).ItfollowsthatRis-nilpotent. ToprovethematrixversionofTheorem 6.3.1 ,wecanimitatetheproofofProposition 6.1 .Theonlypointthatneedsclaricationisthatthed~d-tupletRfor0t<1liesinDd~d,i.e.ktRkop<1. Toseethiswerstobservethattheformalpowerseriesd,~dXi,j=1Eijgij2M`(A(Dd~d)1) Thefactthatthemap`:M`(A(Dd~d)1)=M`(I(Dd~d))!M`B(M)isisometricimpliesthatkRkop=kd,~dXi,j=1EijRijk=k`(d,~dXi,j=1Eijgij+M`(I(Dd~d))k

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6.3.1 ,wecanimitatethetechniquesintheproofofTheorem 5.3.1 byusingthed~d-tupleSinplaceofS(`). 71

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Thefollowingquestionscameupwhilethisworkwasinprogress:CouldtheoperatoralgebrasapproachusedintheexamplesdiscussedinChapter6begeneralizedtohandlenon-commutativedomainsthataredenedbyapossiblyinnitecollectionofLinearMatrixInequalities?IsthealgebraA(K)1adualalgebra? Itwouldbeinterestingtoknowtheanswerstothesequestions. 72

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[1] J.Agler,Ontherepresentationofcertainholomorphicfunctionsdenedonapolydisc,Topicsinoperatortheory:ErnstD.Hellingermemorialvolume,volume48ofOper.TheoryAdv.Appl.,pp47-66.Birkhauser,Basel,1990. [2] J.Agler,J.E.McCarthy,PickInterpolationandHilbertFunctionSpaces,AMSPublication,2002. [3] J.A.Ball,V.Bolotnikov,InterpolationinthenoncommutativeSchur-Aglerclass,J.OperatorTheory58(2007),no.1,83. [4] J.A.Ball,G.Groenewald,T.Malakorn,Conservativestructurednoncommutativemultidimensionallinearsystems,Thestatespacemethodgeneralizationsandapplications,179,Oper.TheoryAdv.Appl.,161,Birkhuser,Basel,2006. [5] J.A.Ball,D.Kalyuzhnyi-Verbovetzkii,Conservativedilationsofdissipativemultidimensionalsystems:thecommutativeandnon-commutativesettings.Multidimens.Syst.SignalProcess.19(2008),no.1,79. [6] J.A.Ball,W.S.Li,D.Timotin,T.T.Trent,Acommutantliftinttheoremonthepolydisc,IndianaUniv.Math.J.,48(2):653-675,1999. [7] J.Cuntz,SimpleC*-algebrasgeneratedbyisometries,Comm.Math.Phys.57,173-185(1977). [8] C.Caratheodory,L.Fejer,UberdenZusammenhangderExtremenvonharmonischenFunktionenmitihrenKoefzientenunduberdenPicardLandauschenSatz,Rend.Circ.Mat.Palermo,32:pp218-239,1911. [9] T.Constantinescu,J.L.Johnson,Anoteonnoncommutativeinterpolation.Canad.Math.Bull.,46(1):5970,2003. [10] J.B.Conway,ACourseinFunctionalAnalysis,Springer-VerlagPublications,1985 [11] Sh.A.Dautov,G.Khudaiberganov,TheCaratheodory-Fejerprobleminhigher-dimensionalcomplexanalysis,Sibirsk.Mat.Zh.23(2)(1982)58.64,215. [12] M.DeOliviera,J.W.Helton,S.McCullough,M.Putinar,EngineeringSystemsandFreeRealAlgebraicGeometry,EmergingApplicationsofAlgebraicGeometry,IMAVol.Math.Appl.149(2009)47-87. [13] E.G.Effros,S.Winkler,Matrixconvexity:operatoranaloguesofthebipolarandHahn-Banachtheorems,J.Funct.Anal.144(1997),no.1,117. [14] J.Eschmeier,L.Patton,M.Putinar,Caratheodory-Fejerinterpolationonpolydisks,Math.Res.Lett.7(2000),no.1,25. 73

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C.Foias,A.E.Frazho,Thecommutantliftingapproachtointerpolationproblems,OperatorTheory:AdvancesandApplications,vol.44,Birkhuser,Verlag,Basel,1990. [16] J.W.Helton,S.McCullough,Everyfreebasicsemi-algebraicsethasanLMIrepresentation,arXiv:0908.4352v2 [17] J.W.Helton,S.McCullough,V.Vinnikov,NoncommutativeconvexityarisesfromLinearMatrixInequalities,J.Funct.Anal240(2006),no.1,pp105-191. [18] J.W.Helton;S.A.McCullough;M.Putinar;V.Vinnikov:ConvexMatrixInequalitiesVersusLinearMatrixInequalities,IEEETrans.Automat.Control54(2009)952-964. [19] J.W.Helton,I.Klep,S.McCullough,N.Slinglend,Noncommutativeballmaps,J.Funct.Anal.257(2009),no.1,47. [20] D.Kalyuzhnyi-Verbovetzkii:CaratheodoryInterpolationontheNoncommutativePolydisk,J.Funct.Anal.,229(2005),pp.241-276. [21] D.Kalyuzhnyi-Verbovetzkii,V.Vinnikov,Foundationsofnoncommutativefunctiontheory,inpreparation. [22] R.A.Martinez-Avendano,P.Rosenthal,AnIntroductiontoOperatorsontheHardy-HilbertSpace,SpringerPublications,2007. [23] V.Paulsen,CompletelyBoundedMapsandOperatorAlgebras,CambridgeUniversityPress,1stedition,Jan2003. [24] G.Popescu,FreeholomorphicfunctionsontheunitballofB(H)n,J.Funct.Anal.241(2006),pp268-333. [25] G.Popescu,Freeholomorphicfunctionsandinterpolation,Math.Ann.342(2008)130. [26] G.Popescu,Interpolationproblemsinseveralvariables,J.Math.Anal.Appl.,227(1):227250,1998. [27] W.Rudin,PrinciplesofMathematicalAnalysis,McGraw-HillScience/Engineering/Math;3rdedition,Jan1976. [28] M.Rosenblum,J.Rovnyak,HardyClassesandOperatorTheory,DoverPublicationsInc.,NewEdedition,Jun1997. [29] D.Sarason,GeneralizedinterpolationinH1,Trans.Amer.Math.Soc.127(1967),pp.179-203. [30] I.Schur,UberPotenzreihendieimInnerndesEinheitskreisesbeschranktsind,J.ReineAngew.Math.,147:pp205-232,1917. 74

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O.Toeplitz,berdieFourierscheEntwickelungpositiverFunktionen,Rend.Circ.Mat.Palermo32(1911)191192. [32] D.V.Voiculescu,FreeProbabilityTheory,AmericanMathematicalSociety,1997. [33] D.V.Voiculescu,FreeanalysisquestionsI:DualitytransformforthecoalgebraofX:B,Int.Math.Res.Not.16(2004)793.822. [34] D.V.Voiculescu,K.J.Dykema,A.Nica,FreeRandomVariables:anoncommutativeprobabilityapproachtofreeproductswithapplicationstorandommatrices,operatoralgebras,andharmonicanalysisonfreegroups,AmericanMathematicalSociety,1992. 75

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SriramBalasubramanianwasborninChennai,India.Heobtainedhisbachelor'sdegreefromtheUniversityofMadrasandamaster'sdegreeinmathematicsfromtheIndianInstituteofTechnologyMadras,beforecomingtotheUniversityofFloridafordoctoralstudy.HisinterestsotherthanmathematicsarethegameofCricketandSouthIndianClassicalandFilmMusic. 76