An approximate universal coefficient theorem

Lin, Huaxin

doi:10.1090/S0002-9947-05-03696-2

An approximate universal coefficient theorem
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Trans. Amer. Math. Soc. 357 (2005), 3375-3405 Request permission

Abstract:

An approximate Universal Coefficient Theorem (AUCT) for certain $C^*$-algebras is established. We present a proof that Kirchberg-Phillips’s classification theorem for separable nuclear purely infinite simple $C^*$-algebras is valid for $C^*$-algebras satisfying the AUCT instead of the UCT. It is proved that two versions of AUCT are in fact the same. We also show that $C^*$-algebras that are locally approximated by $C^*$-algebras satisfying the AUCT satisfy the AUCT. As an application, we prove that certain simple $C^*$-algebras which are locally type I are in fact isomorphic to simple AH-algebras. As another application, we show that a sequence of residually finite-dimensional $C^*$-algebras which are asymptotically nuclear and which asymptotically satisfies the AUCT can be embedded into the same simple AF-algebra.

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