Download PDF by Mikael Rørdam, Erling Størmer (auth.): Classification of Nuclear C*-Algebras. Entropy in Operator

By Mikael Rørdam, Erling Størmer (auth.)

ISBN-10: 364207605X

ISBN-13: 9783642076053

ISBN-10: 3662048256

ISBN-13: 9783662048252

This EMS quantity involves components, written via top scientists within the box of operator algebras and non-commutative geometry. the 1st half, written by means of M.Rordam entitled "Classification of Nuclear, easy C*-Algebras" is on Elliotts category software. The emphasis is at the class by means of Kirchberg and Phillips of Kirchberg algebras: in simple terms endless, 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 special C*-algebras. The classificatin of finite uncomplicated C*-algebras beginning with AF-algebras, and carrying on with with AF- and AH-algberas) is roofed, yet quite often with no proofs. the second one half, written by means of E.Stormer entitled "A Survey of Noncommutative Dynamical Entropy" is a survey of the idea of noncommutative entropy of automorphisms of C*-algebras and von Neumann algebras from its initiation via Connes and Stormer in 1975 until 2001. the most definitions and resuls are mentioned and illustrated with the major examples within the conception. This publication should be priceless to graduate scholars and researchers within the box of operator algebras and similar areas.

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V, then an element x in K K (A, B) is invertible if and only ifyo(x): Ko(A) -+ Ko(B) and Yl(X): KI(A) -+ Kl(B) are invertible mappings. (iii) Two C* -algebras A and B in . V are K K -equivalent if and only if Ko(A) ~ Ko(B) and KI(A) ~ Kl(B). It is clear that (iii) follows from (i) and (ii). For every pair of countable abelian groups (Go, G j) there is an abelian C* -algebra B with Ko(B) ~ Go and K 1 (B) ~ G I. Hence a C* -algebra A satisfies the universal coefficient theorem if and only if it is K -abelian.

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. 5) becomes an approximate intertwining when 1/In is replaced by Ad Vn o1/ln and flin is replaced by Adun+1 0 flin. [tfollows moreover that A is isomorphic to Band there are isomorphisms fIi: A ---+ Band 1/1: B ---+ A with 1/1 = fIi- 1 such that f3oo,n+ I 0 flin ~u fIi 0 aoo,n and aoo,n 0 1/In ~u 1/1 0 f3oo,n for all n.

Hence A does not have real rank zero and the set of extreme points of trace simplex T (A) is homeomorphic to X (so that X can be recovered from A). By the assumption that X is contractible, C(X) and C have isomorphic K -theory, and hence (Ko(A), Ko(A)+, [I A ]o, Kl (A» :;::: (Q, Q n lR+, 1, (OD :;::: (Ko(Q), Ko(Q)+, [10]0, Kl (Q», where Q is the universal UHF-algebra with Ko(Q) :;::: Q; and the state space S(Ko(A» is a singleton, leaving only one choice for the mapping r A: T (A) ----+ S(KO(A».

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Classification of Nuclear C*-Algebras. Entropy in Operator Algebras by Mikael Rørdam, Erling Størmer (auth.)


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