Recently, it was proved by Tikuisis, White and Winter that any faithful trace on a separable, nuclear C * -algebras in the UCT class is quasidiagonal. Building on their work, we generalise the result, and show that any faithful, amenable trace on a separable, exact C * -algebras in the UCT class is quasidiagonal. We also prove that any amenable trace on a separable, exact, quasidiagonal C * -algebra in the UCT class is quasidiagonal.2000 Mathematics Subject Classification. 46L05, 46L35, 46L80.
Abstract. Elliott and Kucerovsky stated that a non-unital extension of separable C * -algebras with a stable ideal, is nuclearly absorbing if and only if the extension is purely large. However, their proof was flawed. We give a counter example to their theorem as stated, but establish an equivalent formulation of nuclear absorption under a very mild additional assumption to being purely large. In particular, if the quotient algebra is nonunital, then we show that the original theorem applies. We also examine how this effects results in classification theory.
I present a new proof of Kirchberg's O2-stable classification theorem: two separable, nuclear, stable/unital, O2-stable C * -algebras are isomorphic if and only if their ideal lattices are order isomorphic, or equivalently, their primitive ideal spaces are homeomorphic. Many intermediate results do not depend on pure infiniteness of any sort. 2010 Mathematics Subject Classification. 46L05, 46L35.
We show that every nuclear O∞-stable * -homomorphism with a separable exact domain has nuclear dimension at most 1. In particular separable, nuclear, O∞-stable C * -algebras have nuclear dimension 1. We also characterise when O∞-stable C * -algebras have finite decomposition rank in terms of quasidiagonality and primitive-ideal structure, and determine when full O2-stable * -homomorphisms have nuclear dimension 0.
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