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Analytic K-homology. (English) Zbl 0968.46058

Oxford Mathematical Monographs. Oxford: Oxford University Press. xviii, 405 p. £65.00/hbk (2000).
The book is devoted to expose the main steps of the construction the K-homology as some generalized homology theory and various applications, in particular of index map from elliptic operators to K-homology. The main idea was started with the BDF theory for classifying the unitary equivalent classes of extensions of \(C^*\)-algebras \(C(X)\) of continuous complex-valued functions on metrizable compacts \(X\), with help of the elementary \(C^*\)-algebra \(\mathcal K\) of compact operators in separable Hilbert space, i. e. unitary equivalent classes of extensions of the form \[ 0 \to \mathcal K \to E \to C(X) \to 0. \] Following the extension theory a “reasonable equivalence relation” between extensions should be what are “identically” on the third terms \(C(X)\) and “different up to automorphisms” on the first terms \(\mathcal K\). It reduces therefore to the unitary equivalence classes of extensions. It is also related to the inclusion of \(C(X)\) into the multiplier algebras of \(\mathcal K\), which is just the algebra \(L(H)\) of all bounded operators in a separable Hilbert space \(H\), modulo the compact operators. Dilation properties let to define the homomorphism which is reasonable to consider as the sum of two extensions and the trivial extension as the unity in the monoid of all unitary equivalence classes of extensions. It was then easy to define the inverse of an extension. One has therefore a group, the so called \(\mathcal Ext_{-1}(.)\) group. It is a functor from the category of metrizable compacts to the category of Abelian groups. Using suspension construction Brown-Douglas-Fillmore defined the higher functors \(\mathcal Ext_n()\). It was proven that really it is a generalized homology theory with a natural pairing with topological Atiyah K-theory. It were found at least three interesting applications of the theory to the classical problems:
1. classifying normal operators up to unitary equivalence and compact perturbations,
2. indices of pseudodifferential operators with on some specific domain, and
3. indices of some group \(C^*\)-algebras [namely, of the group of affine transformations of the real line and some related groups, Do Ngoc Diep, “Methods of Noncommutative Geometry for Groups \(C^*\)-algebras”, CRC Res. Notes Math. Ser. 416 (1999)].
The main question is whether we can classify by the same way the more general extensions of \(C^*\)-algebras of type \[ 0 \to \mathcal K \to E \to A \to 0 \] where A is an arbitrary \(C^*\)-algebras. The first 7 chapters of the book under review are devoted to answer this question. As a detailed discussion of these chapters let us quote from authors’ introduction:
“The first part (chapters 1-7) leads towards a proof of the Brown-Douglas-Fillmore theorem. Chapter 1 introduces the basic concepts of \(C^*\)-algebra theory and operator theory. In chapter 2 we develop the theory of Fredholm index, which, as we have indicated, is fundamental to analytic K-homology. The chapter also provides the basic definitions of \(C^*\)-extensions theory. Chapter 3 is devoted to a number of technical results in \(C^*\)-algebra theory: Stinespring’s Theorem, nuclearity and Voiculescu’s Theorem are some of the topics covered here. The fourth chapter is devoted to K-theory for \(C^*\)-algebras. The pace here is brisk, as many readers will already be familiar with K-theory, but the chapter is itself-contained and could also serve as a rapid introduction to the subject.
In chapter 5 we begin the study of K-homology proper. We define the K-homology of a \(C^*\)-algebra in terms of the K-theory of suitable dual algebra \(\mathcal D(A)\). We use the technical results of Chapter 3 to develop the analysis of these dual algebras, and we obtain the long exact sequence of K-homology associated to \(C^*\)-algebra extensions. Chapter 6 makes the connection between K-homology and coarse geometry – the study of geometric spaces from the point of view of their large-scale structure. The connection was originally explored from the perspective on index theory on non-compact manifolds – we shall return to this subject in the second half of the book – but our immediate business is to use coarse geometry to prove the homotopy invariance of K-homology, and thereby show that K-homology is a generalized homology theory in the sense of algebraic topology. By this stage we shall have assembled nearly all the tools we require to prove the Brown-Douglas-Fillmore Theorem. In Chapter 7 we complete the proof, assuming a certain naturality property of the index pairing between K-theory and K-homology. This naturality property is given a direct proof in Chapter 8, but for the reader who wants to use the BDF Theorem proved without getting involved in the second half of the book, we indicate a more circuitous way around the problem in the exercises to Chapter 7. We also prove the Universal Coefficient Theorem for K-homology, which can be thought of as a generalization of the BDF Theorem to higher-dimensional spaces \(X\).”
Kasparov considered the problem of classifying extensions of the more general type \[ 0 \to B \to E \to A \to 0 \] and constructed KK-theory which is a bi-variant K-functor, or operator K-theory. In Chapters 8 and 9 the authors discuss the relations between this analytic K-homology and the KK-theory. Like in the BDF theory, the analytic and KK-theory have applications to the index problem for elliptic differential operators in manifolds (last 3 chapters of this book) and to group \(C^*\)-algebras as indicated above. Let us quote again the authors’ description of the chapters:
“The second half of the book is centered around index theory. In chapter 8 we introduce Kasparov’s definition of K-homology in terms of Fredholm modules, and we show that his definition is equivalent to the duality-based one of chapter 5. Then we carry out some key computations involving boundary map and the index pairing. In chapter 9 we describe the product structure on K-homology. This complicated but powerful construction is Kasparov’s major contribution to the theory: it allows very simple proofs of the main properties of K-homology, and as we shall see it also connects in a very beautiful way with the theory of elliptic operators. That theory is the subject of Chapter 10; after reviewing the basic results of elliptic operator theory on manifolds, we show how an elliptic operator gives rise to a Fredholm module and therefore to a K-homology class, and how the Kasparov product on K-homology corresponds to the external product defined by Atiyah on the class of elliptic operators. In Chapter 11 we apply K-homology to prove the Atiyah-Singer Index Theorem, at least in an illustrative special case. The proof could be put onto one line – an indication of the power of K-homology theory. We also use K-homology to prove some related index theorems: the Toeplitz Index Theorem for strongly pseudoconvex domain in \(\mathbb C^n\), and (in exercise) the Callias Index Theorem for operators of ‘Dirac-Schrödinger’ type. All these results are rather simple consequences of basic calculations of K-homology theory. Finally, in Chapter 12, we introduce the topic of higher index theory. Here one contemplates an ‘index’ which is no longer an integer but an element of \(C^*\)-algebra K-theory group. Higher index theory turns out to be critically important to a number of geometric problems, of which we have selected the positive scalar curvature problem – which manifolds carry positive scalar curvature metrics? – as an illustrative and important example. Central to higher index theory are the Baum-Connes conjectures. Using our work on coarse geometry we shall be able to state the conjectures, prove them in some special cases, and describe their relationship to the positive scalar curvature problem.”
The book is the first systematic exposition of the analytic K-homology in book form. Each chapter supplies exercises and therefore, the book is an excellent source for advanced courses in the subject.

MSC:

46L80 \(K\)-theory and operator algebras (including cyclic theory)
46M18 Homological methods in functional analysis (exact sequences, right inverses, lifting, etc.)
58Jxx Partial differential equations on manifolds; differential operators
19Kxx \(K\)-theory and operator algebras
46-02 Research exposition (monographs, survey articles) pertaining to functional analysis
46Lxx Selfadjoint operator algebras (\(C^*\)-algebras, von Neumann (\(W^*\)-) algebras, etc.)

Citations:

Zbl 0944.70002
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