By Mikael Rørdam, Erling Størmer (auth.)

This EMS quantity involves components, written by means of major 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: only countless, uncomplicated, nuclear separable C*-algebras. This class result's defined nearly with complete proofs ranging from Kirchbergs tensor product theorems and Kirchbergs embedding theorem for distinctive 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 normally with out proofs. the second one half, written through 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 through Connes and Stormer in 1975 until 2001. the most definitions and resuls are mentioned and illustrated with the most important examples within the thought. This ebook can be helpful to graduate scholars and researchers within the box of operator algebras and similar areas.

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**Example text**

8 in that an ordered group is simple if and only if it has no non-trivial ideals. Let A be a C*-algebra for which (Ko(A), Ko(A)+) is an ordered abelian group. If 1 is an ideal in A and if l: 1 -+ A is the inclusion map, then KO(l)(Ko(l)) is an ideal in (KoCA), KoCA)+). This gives a canonical map from the ideal lattice of A to the ideal lattice of Ko(A). 2 (Property (IP». ~(A). Equivalently, a C* -algebra has the ideal property if and only if each ideal in A is generated (as an ideal) by its projections.

There is a canonical functor from the category C*-alg of separable C*-algebras (with usual *-homomorphisms) to KK that maps a C* -algebra to itself, and a *-homomorphism q;: A ---+ B to K K (q;) in K K (A, B). 1 (Kasparov). >- K K (A, B) "'. ~ A. >- 0. These properties actually characterize K K -theory as shown by Higson in [74] (based on work of Cuntz). To formulate Higson's result we need the following definition: A natural transformation between two functors F] and F2 from the category of separable C* -algebras to the category of abelian groups is a collection of maps aA: F] (A) ---+ F2(A), one map for each separable C*-algebra A, such that F](A) aA FJ (

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, ... , In. Put Fn = {x n , \ , Xn,2, ... , xn'/n}' Associate to this set of data the sequence where CPn is the unital *-homomorphism given by CPn(f)(X) = diag(J(xn,d, f(X n ,2), ... , f(xn,/n).