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Lectures on C∗-algebras Vahid Shirbisheh c Draft date November 15, 2012 arXiv:1211.3404v1 [math.OA] 14 Nov 2012 Contents Contents i 1 Introduction 1 2 Banach algebras and spectral theory 3 2.1 BasicsofBanachalgebras ........................ 5 2.2 L1(G) ................................... 16 2.3 ThespectrumofelementsofaBanachalgebra . 31 2.4 Thespectraltheoryofcompactoperators. .. 34 2.5 The holomorphic functional calculus . 46 2.6 Problems.................................. 51 3 The Gelfand duality 55 3.1 TheGelfandtransform .......................... 56 3.2 Thecontinuousfunctionalcalculus . 70 3.3 TheGelfandduality ........................... 77 3.4 Problems.................................. 84 4 Basics of the theory of C∗-algebras 87 4.1 Positivity ................................. 88 4.2 Approximateunits ............................ 97 4.3 Idealsandhomomorphisms. .102 4.4 Problems..................................114 i ii CONTENTS 5 Bounded operators on Hilbert spaces 117 5.1 Hilbertspaces...............................118 5.2 BoundedoperatorsonHilbertspaces . .133 5.3 Concrete examples of C∗-algebras ....................142 5.4 Locally convex topologies on B(H) ...................147 5.5 TheBorelfunctionalcalculus . .155 5.6 Projections and the polar decomposition . 159 5.7 Compactoperators ............................168 5.8 ElementsofvonNeumannalgebras . .170 5.9 Problems..................................172 Bibliography 174 Chapter 1 Introduction These notes are mainly based on a course given by the author in Fall 2008. The title of the course was “topics in functional analysis”, but with a very flexible syllabus mainly about operator algebras. Therefore at the time, we decided to focus only on one topic which was “C∗-algebras”. We mainly followed Bruce Blackadar’s book [6] in the course. Meanwhile, we had to refer to other books on C∗-algebras and operator algebras for more details. Therefore we also added many topics, results, examples, details and exercises from other sources. These additional sources are mentioned in these lectures from time to time, but to do them justice we have to name a few of the most important of them; [27, 29, 32, 33, 34, 41, 43]. This mixture of sources for the course made us to design the order and depth of the topics differently than other books. Besides, since students attending the course had different background, we had to give full proofs for every statement and explain many details from measure theory and functional analysis as well as the theory of C∗-algebras itself. So, the result was a very self contained series of lectures on C∗- algebra. Hoping that this level of details would help beginners, we decided to prepare these notes in an organized and standard form. During rewriting these notes, we frequently were tempted to add more materials to the original lectures. Although most of the time, we managed to control this temptation, we have added some new topics in order to make the whole notes more consistent and useful. For instance, Sections 2.2, 2.4, 3.3, 5.7, and 5.8 were not part of the original course. On the other hand, we presented GNS construction fully in the course, but it is not given in these notes. Hopefully, a chapter on states, representations and GNS construction will be added to the present notes in the near future. The order and list of the topics covered in these notes are as follows: Chapter 2 begins with elements of Banach algebras and some examples. We also devote a section to detailed study of Banach algebras of the form L1(G), where G is a locally compact group. Afterwards, we discus spectrum of elements of Banach algebras. In 1 2 CHAPTER 1. INTRODUCTION Section 2.4, we study basics of the spectral theory of compact operators on Banach spaces. The first chapter is concluded with a section on the holomorphic functional calculus in Banach algebras. Chapter 3 is mainly about the Gelfand transform and its consequences. So, the Gelfand transform on commutative Banach algebras and C∗-algebras is discussed in Section 3.1, the continuous functional calculus is presented in Section 3.2, and finally the Gelfand duality between commutative C∗- algebras and locally compact and Hausdorff topological spaces is studied in Section 3.3. We begin our study of abstract C∗-algebras in Chapter 4. Positivity in C∗- algebras, approximate units, ideals of C∗-algebras, hereditary C∗-subalgebras and multiplier algebras are the main topics covered in this chapter. Finally, these notes end in Chapter 5, where we present various topics concerning the C∗-algebra B(H) of bounded operators on a Hilbert space H. We begin this chapter with presenting necessary notions and materials about Hilbert spaces. Elementary topics about bounded operators on Hilbert spaces are discussed in Section 5.2. We discuss three important examples of concrete C∗-algebras in Section 5.3 including the reduced group C∗-algebra of a locally compact group G. Three locally convex topologies on the C∗-algebra B(H), specifically the strong, weak and strong-∗ operator topologies are discussed in Section 5.4. The Borel functional calculus in B(H) is presented in Section 5.5. Projections in B(H) and the polar decomposition of elements of B(H) are studied in Section 5.6. In Section 5.7, C∗-algebras of compact operators are studied briefly. Finally, the von Neumann bicommutant theorem is presented in Section 5.8. Although we have tried to present every topic as easy and self contained as possible, we have left many little details to readers in the form of exercises amongst the main part of the text. We also added some exercises at the end of each chapter. In order to distinguish between these two groups of the exercises, we named the latter group “problems”. We only used a limited number of references to prepare these notes, but we give a long list of books related to the subject. We hope this list helps student and beginners to find complementary topics related to C∗-algebras. We welcome any suggestions and comments related to these notes, especially regarding possible mistakes, typos or suggesting new examples, exercises and/or top- ics. Please, send your comments to [email protected] or [email protected]. Chapter 2 Banach algebras and spectral theory C∗-algebras are a special type of Banach algebras. Therefore many fundamental facts about Banach algebras are usually applicable in the theory of C∗-algebras too. Besides, some C∗-algebras are obtained from some Banach algebras, for instance, the reduced group C∗-algebra, see Example 5.3.4. Therefore we devote this chapter to the study of several topics in Banach algebras which are relevant to the theory of C∗-algebras. In Section 2.1, we gather basic definitions and facts concerning Banach alge- bras and give some examples of Banach algebras and C∗-algebras. A detailed study of the Banach algebra L1(G) associated to a locally compact group G is given in Section 2.2. Although the materials presented in this section are not necessary for the basic theory of C∗-algebras, we include this section for several reasons: First, L1(G) appears naturally in applications of the theory of C∗-algebras in harmonic analysis. Secondly, L1(G) motivates some constructions in C∗-algebras. And finally, it provides us with many examples of Banach algebras which are neither commuta- tive nor the algebras of bounded operators on some Banach spaces. The spectrum of an element of a Banach algebra is introduced and studied in Section 2.3. The algebra of compact operators on a Banach space is another general example of Ba- nach algebras. The spectral theory of compact operators is much richer than the spectral theory of general elements of Banach algebras and it is used in the study of C∗-algebras of compact operators on Hilbert spaces. Therefore we devote Section 2.4 to a detailed study of this topic. Finally, in Section 2.5, we discuss the holomor- phic functional calculus in Banach algebras. It is a useful theory which enables us to construct new elements in a Banach algebra by applying certain holomorphic func- tions defined over the spectrum of an element of Banach algebra. We include this section , because we also discuss the continuous functional calculus in C∗-algebras 3 4 CHAPTER 2. BANACH ALGEBRAS AND SPECTRAL THEORY in Section 3.2 and the Borel functional calculus in the C∗-algebra B(H) of bounded operators on a Hilbert space H in Section 5.5. Thus all the three major functional calculi related to C∗-algebras are covered. Before starting our study of Banach algebras, we recall some well known the- orems from functional analysis. Their proofs can be found in standard texts on functional analysis such as [19, 34, 41]. Theorem 2.0.1. [Uniform boundedness theorem] Let E and F be two Banach spaces. Given a subset Σ B(E, F ), if the set T x ; T Σ is bounded for ev- ery x X, then Σ is bounded,⊆ that is the set T{k; T k Σ ∈is bounded.} ∈ {k k ∈ } Theorem 2.0.2. [Open mapping theorem] Every onto bounded linear map T : E F → between two Banach spaces is open. Theorem 2.0.3. [Closed graph theorem] A linear map T : E F between two Ba- nach spaces is bounded if and only if its graph is a closed subs→et of E F . × Theorem 2.0.4. [The Banach-Alaoglu theorem] Assume O is a neighborhood of 0 in a topological vector space V . The subset ρ V ∗; ρ(x) 1 for all x O V ∗ is weak-∗ compact. In particular the closed unit{ ∈ ball of| V ∗ is| ≤ weak-∗ compact.∈ }⊆ For the proof of the following two theorems see Theorem 3.6 and 3.7 of [41]. Theorem 2.0.5. [The Hahn-Banach theorem] Assume X is a locally convex topo- logical vector space and M is a subspace of X.
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