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Lectures on Noncommutative Geometry

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Lectures on Noncommutative Geometry Lectures on Noncommutative Geometry Masoud Khalkhali Contents 1 Introduction 2 2 From C∗-algebras to noncommutative spaces 3 2.1 Gelfand-Naimark theorems . 3 2.2 GNS,KMS,andtheflowoftime . 7 2.3 From groups to noncommutative spaces . 13 2.4 Continuous fields of C∗-algebras ................ 20 2.5 Noncommutative tori . 22 3 Beyond C∗-algebras 26 3.1 Algebras stable under holomorphic functional calculus . 26 3.2 Almost commutative and Poisson algebras . 30 3.3 Deformationtheory. 36 4 Sources of noncommutative spaces 40 4.1 Noncommutative quotients . 41 4.2 Hopf algebras and quantum groups . 46 5 Topological K-theory 51 5.1 The K0 functor ......................... 52 5.2 The higher K-functors ..................... 60 arXiv:math/0702140v2 [math.QA] 12 Apr 2007 5.3 Bott periodicity theorem . 64 5.4 Furtherresults .......................... 65 5.5 Twisted K-theory ........................ 67 5.6 K-homology............................ 68 6 Cyclic Cohomology 70 6.1 Cycliccocycles .......................... 71 6.2 Connes’spectralsequence . 75 6.3 Topological algebras . 77 1 6.4 Thedeformationcomplex . 78 6.5 Cyclichomology ......................... 80 6.6 Connes-Cherncharacter. 85 6.7 Cyclicmodules .......................... 90 6.8 Hopf cyclic cohomology . 93 Abstract This text is an introduction to a few selected areas of Alain Connes’ noncom- mutative geometry written for the volume of the school/conference “Non- commutative Geometry 2005” held at IPM Tehran. It is an expanded version of my lectures which was directed at graduate students and novices in the subject. 1 Introduction David Hilbert once said “No one will drive us from the paradise which Can- tor created for us” [70]. He was of course referring to set theory and the vast new areas of mathematics which were made possible through it. This was particularly relevant for geometry where our ultimate geometric intuition of space so far has been a set endowed with some extra structure (e.g. a topol- ogy, a measure, a smooth structure, a sheaf, etc.). With Alain Connes’ non- commutative geometry [18, 22, 24], however, we are now gradually moving into a new ‘paradise’, a ‘paradise’ which contains the ‘Hilbertian paradise’ as one of its old small neighborhoods. Interestingly enough though, and I hasten to say this, methods of functional analysis, operator algebras, and spectral theory, pioneered by Hilbert and his disciples, play a big role in Connes’ noncommutative geometry. The following text is a greatly expanded version of talks I gave during the conference on noncommutative geometry at IPM, Tehran. The talks were directed at graduate students, mathematicians, and physicists with no background in noncommutative geometry. It inevitably covers only certain selected parts of the subject and many important topics such as: metric and spectral aspects of noncommutative geometry, the local index formula, connections with number theory, and interactions with physics, are left out. For an insightful and comprehensive introduction to the current state of the art I refer to Connes and Marcolli’s article ‘A Walk in the Noncommutative Garden’ [34] in this volume. For a deeper plunge into the subject one can’t do better than directly going to Connes’ book [24] and original articles. 2 I would like to thank Professors Alain Connes and Matilde Marcolli for their support and encouragement over a long period of time. Without their kind support and advice this project would have taken much longer to come to its conclusion, if ever. My sincere thanks go also to Arthur Greenspoon who took a keen interest in the text and kindly and carefully edited the entire manuscript. Arthur’s superb skills resulted in substantial improvments in the original text. 2 From C∗-algebras to noncommutative spaces Our working definition of a noncommutative space is a noncommutative al- gebra, possibly endowed with some extra structure. Operator algebras, i.e. algebras of bounded linear operators on a Hilbert space, provided the first really deep insights into this noncommutative realm. It is generally agreed that the classic series of papers of Murray and von Neumann starting with [103], and Gelfand and Naimark [61] are the foundations upon which the theory of operator algebras is built. The first is the birthplace of von Neu- mann algebras as the noncommutative counterpart of measure theory, while in the second C∗-algebras were shown to be the noncommutative analogues of locally compact spaces. For lack of space we shall say nothing about von Neumann algebras and their place in noncommutative geometry (cf. [24] for the general theory as well as links with noncommutative geometry). We start with the definition of C∗-algebras and results of Gelfand and Naimark. References include [7, 24, 49, 57]. 2.1 Gelfand-Naimark theorems By an algebra in these notes we shall mean an associative algebra over the field of complex numbers C. Algebras are not assumed to be commutative or unital, unless explicitly specified so. An involution on an algebra A is a conjugate linear map a 7→ a∗ satisfying (ab)∗ = b∗a∗ and (a∗)∗ = a for all a and b in A. By a normed algebra we mean an algebra A such that A is a normed vector space and kabk ≤ kakkbk, for all a, b in A. If A is unital, we assume that k1k = 1. A Banach algebra is a normed algebra which is complete as a metric space. One of the main 3 consequences of completeness is that norm convergent series are convergent; ∞ n in particular if kak < 1 then the geometric series n=1 a is convergent. From this it easily follows that the group of invertibleP elements of a unital Banach algebra is open in the norm topology. Definition 2.1. A C∗-algebra is an involutive Banach algebra A such that for all a ∈ A the C∗-identity kaa∗k = kak2 (1) holds. A morphism of C∗-algebras is an algebra homomorphism f : A −→ B which preserves the ∗ structure, i.e. f(a∗)= f(a)∗, for all a ∈ A. The C∗-identity (1) puts C∗-algebras in a unique place among all Ba- nach algebras, comparable to the unique position enjoyed by Hilbert spaces among all Banach spaces. Many facts which are true for C∗-algebras are not necessarily true for an arbitrary Banach algebra. For example, one can n 1 show, using the spectral radius formula ρ(a) = Lim ka k n coupled with the C∗-identity, that the norm of a C∗-algebra is unique. In fact it can be shown that a morphism of C∗-algebras is automatically contractive in the sense that for all a ∈ A, kf(a)k ≤ kak. In particular they are always continuous. It ∗ follows that if (A, k k1) and (A, k k2) are both C -algebras then kak1 = kak2, for all a ∈ A. Note also that a morphism of C∗-algebras is an isomorphism if and only if it is one to one and onto. Isomorphisms of C∗-algebras are necessarily isometric. Example 2.1. let X be a locally compact Hausdorff space and let A = C0(X) denote the algebra of continuous complex valued functions on X vanishing at infinity. Equipped with the sup norm and the involution defined by complex conjugation, A is easily seen to be a commutative C∗-algebra. It is unital if and only if X is compact, in which case A will be denoted by C(X). By a fundamental theorem of Gelfand and Naimark, to be recalled ∗ below, any commutative C -algebra is of the form C0(X) for a canonically defined locally compact Hausdorff space X. 4 Example 2.2. The algebra A = L(H) of all bounded linear operators on a complex Hilbert space H endowed with the operator norm and the usual adjoint operation is a C∗-algebra. The crucial C∗-identity kaa∗k = kak2 is easily checked. When H is finite dimensional of dimension n we obtain the matrix algebra A = Mn(C). A direct sum of matrix algebras A = Mn1 (C) ⊕ Mn2 (C) ⊕···⊕ Mnk (C) is a C∗-algebra as well. It can be shown that any finite dimensional C∗- algebra is unital and is a direct sum of matrix algebras [7, 49]. In other words, finite dimensional C∗-algebras are semisimple. Any norm closed subalgebra of L(H) which is also closed under the adjoint map is clearly a C∗-algebra. A nice example is the algebra K(H) of compact operators on H. By definition, a bounded operator T : H → H is called compact if it is the norm limit of a sequence of finite rank operators. By the second fundamental theorem of Gelfand and Naimark, to be discussed below, any C∗-algebra is realized as a subalgebra of L(H) for some Hilbert space H. Let A be an algebra. A character of A is a nonzero multiplicative linear map χ : A → C. If A is unital then necessarily χ(1) = 1. Let A denote the set of characters of A. It is also known as the maximal spectrum of A. If A is a Banach algebra b it can be shown that any character of A is automatically continuous and has norm one. We can thus endow A with the weak* topology inherited from A∗, the continuous dual of A. The unit ball of A∗ is compact in the weak* b topology and one can deduce from this fact that A is a locally compact Hausdorff space. It is compact if and only if A is unital. b When A is a unital Banach algebra there is a one to one correspondence between characters of A and the set of maximal ideals of A: to a character χ we associate its kernel, which is a maximal ideal, and to a maximal ideal I one associates the character χ : A → A/I ≃ C.
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