By Mikael Rørdam, Erling Størmer (auth.)
This EMS quantity comprises 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 class application. The emphasis is at the type by way of Kirchberg and Phillips of Kirchberg algebras: simply countless, basic, nuclear separable C*-algebras. This class result's defined virtually with complete proofs ranging from Kirchbergs tensor product theorems and Kirchbergs embedding theorem for special C*-algebras. The classificatin of finite basic C*-algebras beginning with AF-algebras, and carrying on with with AF- and AH-algberas) is roofed, yet usually 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 concept. This e-book may be valuable to graduate scholars and researchers within the box of operator algebras and comparable areas.
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Additional info for Classification of Nuclear C*-Algebras. Entropy in Operator Algebras
Assume that there are isomorphisms ao: Ko(A) --+ Ko(B), al : KI (A) --+ KI (B), and an affine homeomorphism y: T(B) --+ T(A), such that ao(Ko(A)+) = Ko(B)+, ao([lA]O) = [lBla, and the diagram y ~ T(A) T(B)---""* rB 1 lrA S(Ko(B» --::---.... 3> S(Ko(A» aD commutes. Then A is isomorphic to B and there is an isomorphism cp: A --+ B satisfying Ko(cp) = ao, KI(cp) = ai, and T(cp) = y. 15) for a partial confirmation of the Elliott conjecture in the finite, simple, unital case. 3). 11) and of stable rank one; or it may be that one mus\ refine the invariant (for example as suggested below).
An AH-algebra can be the inductive limit of two different sequences, one that has slow dimension growth and another that does not. 6]. 2. 2). The following three conditions are equivalent: (i) A is simple; (ii) for every natural number nand for every a in An such that Cfoo,n (a) is non-zero, there is m :::: n such that Cfm,n (a) is full in Am; (iii) for every natural number n andfor every a in An such that Cfoo,n (a) is non-zero, there is m :::: n such that Cfm,n(a)(x) =F Ofor every x in X m. 3 (Dadarlat-Nagy-Nemethi-Pasnicu).
This conjecture is verified for purely infinite C* -algebras in the so-called VCT-class . I . l proved by Kirchberg and Phillips. As mentioned above, there are examples of simple, separable C* -algebras of type (Inf) that are not purely infinite (). 3 having the same invariant as a purely infinite C*-algebra (cf. 3). 3 should therefore most likely be reformulated either by requiring that A and B are purely infinite (not just of type (Inf) or by refining the invariant for example by including the semi group of Murray-von Neumann equivalence classes of projections.
Classification of Nuclear C*-Algebras. Entropy in Operator Algebras by Mikael Rørdam, Erling Størmer (auth.)