By Mikael Rørdam, Erling Størmer (auth.)

This EMS quantity contains elements, written by means of top scientists within the box of operator algebras and non-commutative geometry. the 1st half, written through M.Rordam entitled "Classification of Nuclear, uncomplicated C*-Algebras" is on Elliotts type application. The emphasis is at the class by way of Kirchberg and Phillips of Kirchberg algebras: merely endless, uncomplicated, nuclear separable C*-algebras. This category result's defined virtually with complete proofs ranging from Kirchbergs tensor product theorems and Kirchbergs embedding theorem for specified C*-algebras. The classificatin of finite easy C*-algebras beginning with AF-algebras, and carrying on with with AF- and AH-algberas) is roofed, yet typically with out proofs. the second one half, written by way of E.Stormer entitled "A Survey of Noncommutative Dynamical Entropy" is a survey of the speculation of noncommutative entropy of automorphisms of C*-algebras and von Neumann algebras from its initiation by way of Connes and Stormer in 1975 until eventually 2001. the most definitions and resuls are mentioned and illustrated with the major examples within the thought. This e-book can be beneficial to graduate scholars and researchers within the box of operator algebras and similar areas.

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**Example text**

1 one finds that This implies that IICBoo,m+2 0 flim+1 0 am+I,n)(a) - (f3oo,m+1 0 flim am,n)(a) II 0 = IIf3oo,m+2((flim+1 oam)(am,n(a)) - (f3m+1 o flim)(am,n(a))) II < 8m +8m+ 1 for all m 2: mo. 3) is Cauchy and therefore convergent. 3. 5) be a (possibly non-commutative) diagram of separable C* -algebras An and Bn and * -homomorphisms flin and 1/In such that A and B are the inductive limits of the upper, respectively, the lower row in the diagram. Suppose that 1/In+1 0 flin ~u an and flin 0 1/In ~u f3n for all n.

8). 4]). A simple, purely infinite, separable C* -algebra A is said to be classifiable if it belongs to the UCT class ~V and the map H(A, B) -+ KL(A, B) is bijective for every simple, purely infinite C* -algebra B. 1 below and in the fact that there are many classifiable, purely infinite, simple C*-algebras. It was shown in [119] and [57] that for every pair of countable abelian groups Go, GI there is a classifiable, purely infinite, simple C*algebra A with Ko(A) ~ Go and K I (A) ~ G I. 2). 1. Let A and B be stable, simple, purely infinite, separable, classifiable C* -algebras.

F 0 An)(t))Z(t)*, for all fin C(1l') and for all t in [0,1]. 4). In general, when k i= 0, cp is not approximately unitarily equivalent to a standard *-homomorphism given by eigenvalue functions. 7 (The Goodearl Algebras). Goodearl described in [68] a particularly nice class of simple, unital AH-algebras, some having real rank zero and some not. Take a compact Hausdorff space X, take sequences {k n }~ \ and {In} ~ \ of positive integers such that k n divides kn+ \ for each n and In < kn+ \ / kn, and take points Xn,i in X for i = 1, 2, ...