Preface

C*-algebras are algebras of operators on Hilbert spaces. Operator algebras were introduced by John von Neumann in connection with the study of quantum mechanics in 1929 followed by the work of F. Murray and J. von Neumann which developed the basic theory of von Neumann algebras. Later, Gelfand and Naimark introduced the notion of C*-algebras. The Gelfand transformation shows that every unital commutative C*-algebra is isomorphic to the algebra of continuous functions on a compact Hausdorff space. Therefore it is also natural to view C*-algebras as non-commutative topological spaces. While original motivational examples are limited to the spectral theorem for a normal operator and unitary representations of locally compact groups, today, with the development of K-theory and KK-theory, its study grows to a diversified field which includes or is closely related to the index theory for subfactors, abstract harmonic analysis, group representation theory and non-commutative geometry, as well as the classification of amenable C*-algebras. The classification of simple amenable C*-algebras is often known as the Elliott program. Classification of amenable C*-algebras was initiated by G. A. Elliott through his paper [33] and was much publicized through his ICM lecture [34], though the Elliott program is predated by works of Glimm ([47]), Dixmier ([27]) and Bratteli ([10]), as well as Elliott’s classification of AF-algebras ([31]). The main goal is to use the Elliott invariant (a set of K-theoretic invariants) to completely determine amenable C*-algebras up to isomorphism. These notes will discuss some of the current developments. In the study of topology, one studies continuous maps between spaces. In C*algebra theory, which is often regarded as non-commutative topology, one studies homomorphisms from one C*-algebra to another. Given two homomorphisms φ1 and φ2 from a C*-algebra A into another C*-algebra B, one asks when these two homomorphisms are approximately or asymptotically unitarily equivalent. The latter means that there is a continuous path of unitaries {u(t) : t ∈ [1,∞)} in B such that φ1 is conjugate to φ2 eventually via the path. One of the features of these notes is that it presents a study of the asymptotic unitary equivalence of monomorphisms from a given unital separable C*-algebra to a unital separable simple C*-algebra. The so-called Basic Homotopy Lemma first appeared in a paper of O. Bratteli, G. A. Elliott, D. Evans and A. Kishimoto [9]. It studied a pair of unitaries in a unital simple C*-algebra. It was later developed to the following type of problem: Let A be a unital simple C*-algebra, let C be another unital C*-algebra and let φ : C → A be a unital homomorphism. Suppose that u is a unitary in A such that u almost commutes with the image of φ and there exists a continuous path of unitaries {u(t) : t ∈ [0, 1]} with u(0) = u and u(1) = 1A. The question is whether there exists a continuous path of unitaries {v(t) : t ∈ [0, 1]} in A such that v(0) = u