An introduction to the classification of amenable \(C^*\)-algebras (Q2778492)

In noncommutative topology and noncommutative geometry, noncommutative spaces are \(C^*\)-algebras. It is interesting to classify these objects in some sense. Two major tools in noncommutative geometry and topology are the \(K\)-theory (or \(KK\)-theory) and cyclic theories like \(HP_*\) (resp., bivariant cyclic theories). About a decade ago, George A. Elliot initiated the program of classification of \(C^*\)-algebras using \(K\)- (or \(KK\)-)theories. The book under review is devoted to expose recent developments in this area.NEWLINENEWLINENEWLINEThe author starts in Chapters 1-3 with the background of the theory of \(C^*\)-algebras. In Chapter 4, he succeeds to describe the first interesting class of the so-called \(AT\)-algebras which are inductive limits of finite products of algebras of finite dimensional matrices over the algebras of functions on finite subsets in the torus \(\mathbf T = \mathbf S^1\). Elliot's Theorem (Theorem 4.6.4) characterizes simple \(AT\)-algebras with real rank zero through a quadruple of \(K\)-theory data \((K_0(A),K_0(A)_+,[1_A],K_1(A))\).NEWLINENEWLINENEWLINEIn Chapter 5 and 6, the author exposes the theory of classification of \(AH\)-algebras. In particular, he proves the theorem of Elliot and Gong that simple \(AH\)-algebras with slow dimensional growth and with real rank zero are also characterized by the quadruple of \(K\)-theory data \((K_0(A),K_0(A)_+,[1_A],K_1(A))\).NEWLINENEWLINENEWLINEBeing an introduction to the subject, the book is a well-readable and comprehensive guide for those who want to study and work with this theory. From the publisher's description: ``The book contains many new proofs and some original results related to the classification of amenable \(C^*\)-algebras''.