University of Pretoria etd – Lee, W-S (2004) Ideal perturbation of elements in C*-algebras by Wha-Suck Lee Submitted in partial fulfillment of the requirements for the degree MSc in the Faculty of Natural and Agricultural Sciences Department of Mathematics and Applied Mathematics University of Pretoria Pretoria June 2004. i University of Pretoria etd – Lee, W-S (2004) Contents Acknowledgements ........................... v Preface ................................... vi 1 Introduction 1 1.1 C*-algebra : Preliminaries . 1 1.2 C*-algebra : Global Representations . 9 1.2.1 The Concept of a *-Representation . 9 1.2.1.1 Irreducible *-representation . 13 1.2.1.2 Non-degenerate *-representation . 15 1.2.1.3 Irreducible Cyclic *-representations . 18 1.2.2 The Universal Representation : A Non-Degenerate *-Representation . 21 1.2.3 The Double Centralizer Algebra Representation (DCAR) . 25 1.2.3.1 Left Regular Representation : Category of Banach Spaces 25 1.2.3.2 Inadequacy of LRR in Category of C*-algebras . 25 1.2.3.3 DCAR : Category of Semigroups . 26 1.2.3.4 Embedding Theorem I : Category of Semigroups . 28 1.2.3.5 Embedding Theorem II : Category of Semigroups . 29 1.2.3.6 DCAR : Category of Rings . 31 1.2.3.7 DCAR : Category of C*-algebras . 35 1.2.4 The DCAR and the Universal Representation . 38 1.2.4.1 Closed 2-sided Ideals in C*-algebras . 38 1.3 C*-algebra : Local Representations . 43 1.3.1 Local Representation Theory I: The Functional Calculus for Normal Elements . 43 1.3.2 Local Representation II: The Polar Decomposition . 47 1.3.3 The Functional Calculus [Local Representation Theorem I] and The Polar Decomposition Theorem [Local Repre- sentation Theorem II] . 51 1.3.4 Local Representations in the Quotient C*-algebra . 52 2 Lifting: Zero Divisors 54 2.1 Lifting Problem . 54 2.2 Lifting Zero Divisors . 56 ii University of Pretoria etd – Lee, W-S (2004) 2.3 Lifting n-zero divisors : Abelian C*-algebra . 62 2.3.1 Lifting n-zero divisors : Statement of Problem . 62 2.3.2 Lifting n-zero divisors : Commutative C*-algebras . 62 2.4 Lifting n-zero divisors : Von Neumann C*-algebra . 64 2.4.1 Definition of Von Neumann C*-algebra . 64 2.4.2 Property of Von Neumann C*-algebra : Closure Under Range Projection Of Operator . 66 2.4.3 Lifting n-zero divisors : Von Neumann C*-algebra . 70 2.5 SAW*-algebra : Corona C*-algebra . 72 2.5.1 The Paradigm of Non Commutative Topology . 72 2.5.2 Non Commutative Topology : Sub-Stonean Spaces and Corona Sets . 76 2.5.3 Application of Non-Commutative Topology : SAW*-algebra, Corona C*-algebra . 84 2.5.3.1 Local Corona Properties are Global Multiplier Properties 84 2.5.3.2 Two Fundamental Results For Lifting the Property of n-zero divisors from the corona C(A) onto M(A). 87 2.6 Lifting n-zero divisors : Corona C*-algebra . 89 2.7 Lifting n-zero divisors : The General Case . 93 2.7.1 Essential Ideals From Ideals : Annihilators . 93 2.7.2 Lifting n-zero divisors . 96 3 Lifting: Nilpotent Elements 99 3.1 Lifting nilpotent elements : Statement of Problem . 99 3.2 Lifting nilpotent elements: Degree of nilpotency 2 . 101 3.3 Lifting nilpotent elements: Preliminary Results For the General Case . 103 3.4 Lifting nilpotent elements: The Corona . 106 3.4.1 The General Overview . 106 3.4.2 Triangular Form: The Corona . 107 2.4.2.1 Corollaries of the Triangular Form for Nilpotent Elements108 2.4.2.2 Proof of The Triangular Form For Nilpotent Elements . 109 3.4.3 Lifting the Triangular Form : The Double Centralizer Al- gebra. 113 3.4.4 Lifting Nilpotent Elements : The Corona. 117 3.5 Lifting nilpotent elements : The General Case . 122 4 Lifting Polynomially Ideal Elements : A Criteria 124 4.1 Lifting Polynomially Ideal Elements : A Counter Example . 124 4.2 Lifting Polynomially Ideal Elements : Preliminary Results . 127 4.2.1 Linear Algebra Preliminaries . 127 4.2.2 C*-algebra Preliminaries I: Construction of Positive In- vertible Elements . 131 4.2.3 C*-algebra Preliminaries II: Lifting Positive Invertible El- ements . 133 4.2.4 Lifting Polynomially Ideal Elements : A Criteria . 137 iii University of Pretoria etd – Lee, W-S (2004) A Computing Double Centralizers 146 N A.1 The Hilbert Tensor Product H h H = K(H) . 146 A.2 Computing The Double Centralizer Algebra of K(H) . 151 A.3 Computing The Double Centralizer Algebra of C0(Ω) . 154 B Anti-Unitization 159 B.1 The Holomorphic Functional Calculus : Every C*-algebra is a Local Banach Algebra . 159 B.1.1 Preliminary Result I : Cauchy Integral Formula for C*- algebra valued analytic functions of a complex variable . 160 B.1.2 Preliminary Result II : Analyticity of an important C*- algebra valued function . 162 B.1.3 The Holomorphic Functional Calculus . 162 N B.2 The Spatial Tensor Product A Mn(C) . 165 B.2.1 The Spatial Tensor Product : The General Case . 165 B.2.2 The Spatial Tensor Product : Explicit Description . 166 B.3 The Normed Inductive Direct Limit M∞(A) . 169 B.4 The stable algebra A J K(H) . 173 Bibliography ...............................175 Summary .................................178 iv University of Pretoria etd – Lee, W-S (2004) ACKNOWLEDGEMENTS The list of people I would like to thank would add another chapter to this thesis : the road to a Masters Degree was far from straight. I would therefore like to give special thanks to a special sublist. To God, my rock and refuge. To my father and my mother, my origin. To my brothers, Wha Joon, Wha Yong and Wha Choul, my joy. To Professor Stroh, my navigator and advisor; the teaching assistantships, the technical assistance, gentle guidance, quality lectures on C*-algebra, and the topic of the thesis was a defining moment in my mathematical education. To Mrs McDermot, forever helpful, my warmest thanks. To Professor Swart, for the technical extras and light hearted humour. To Professor Rossinger and Professor Lubuma. To my colleague and friend Conrad Beyers, a special thanks from the bottom of my heart. To Rocco Duvenhage, for a definition and sound advise. To Tannie Willie, my role model in life. To Conrad Pienaar, my best friend : thanks for the warm unique intellectual friendship. To Naomi Pienaar, for the fun times and example of non-conformity. Michael Brink, 26th avenue, House 434, Villeria will forever be etched in my heart; it is a house of warmth and Godliness. To the soccer playing theologicians: John Mokoena, a Rivaldo in the making, and Mzwamadoda Mtyhobile. To Reverend Shin and his wife, Rev- erend Yoo and his wife and the Korean Glory Church. To Siegfied Masokameng, Lucky Ntsangwane, and Graham Litchfield for the spiritual encouragements. To the Swaziland Government for funding my undergraduate and Honours stud- ies at UCT. To Dinah Paulse, Zeki Mushaandja and Johan Vogiatzis. To Keyur, a poet in the making, Lass, Vee and the gang for the warm friendship in U.C.T . To Professor Brink, Professor Barashenkov and the late Dr Vermeulen of UCT, may I write more to vindicate your faith in me. Last but not least, to Lisa Wray, my deepest respect. v University of Pretoria etd – Lee, W-S (2004) PREFACE The examples in this thesis serve not only to illustrate the abstract ideas of the thesis but to introduce notation that shall be used consistently throughout the thesis. The index of symbols follows after this preface. Hence many concrete examples of C*-algebras will be given. The first chapter of the thesis sets the ground work for the thesis and has three dominant themes. The first theme (Chapter 1, section 1) is the anti-unitization of a C*-algebra, where we embed a C*-algebra as a closed 2-sided ideal of another C*-algebra which lacks an identity. This embedding gives us licence to treat any C*-algebra as a C*-algebra without an identity for our purposes. This of fundamental im- portance in the thesis : it is an appropriate first theme. The second theme (Chapter 1, section 2) is the representation of an abstract C*- algebra in terms of more concrete C*-algebras : the C*-algebra of all continuous complex valued functions which vanish at infinity on a locally compact Hausdorff space, the C*-algebra of all bounded operators on a Hilbert space and the Double Centralizer Algebra. These representations formed the frameworks in which the problems of the thesis was solved. The bulk of Chapter 1, section 2 focusses on the latter two representation theories. The second representation theory which is well established is attacked from the viewpoint of theory of a *-representation, a *-algebraic concept void of the norm. In particular for C*-algebras, we look at irreducible and non-degenerate *-representations, the former being the stronger condition. The well known second representation theory is a non-degenerate *-representation which uses irreducible cyclic *-representations in its construc- tion : the crux is the bijective correspondence between irreducible cyclic *- representations and pure states. We furnish amongst the concrete examples, an example of a *-representation which is non-degenerate but far from the being irreducible. The Double Centralizer Algebra Representation theory is the least known. We introduce it as an improved left regular representation by showing the inade- quacy of the left regular representation. Indeed, the solution to this inadequacy is a triumph of the school of solving the problem by looking at it in a more abstract setting. Hence, we build up the theory from the very general theory of the category of semigroups which will provide reasons for the definitions of a double centralizer, which otherwise would appear as if it were plucked out of the air.