Classification of Cuntz-Krieger algebras up to stable isomorphism

Strong classification of purely infinite Cuntz-Krieger algebras

“…The proofs of Theorems 3.1 and 3.2 are given at the end of Section 7. We immediately get the following corollary, where the first half of course is already proved earlier in [Res06].…”

“…In Section 6, we show how to describe the isomorphisms these matrix operations induce on the (stabilized) graph C * -algebras and their reduced filtered K-theory. In [ERRS16c], it was proved that Cuntz splicing a graph gives a graph whose C * -algebra is stably isomorphic to the C * -algebra of the original graph (generalizing a result in [Res06]). As we are proving a strong classification result, we will need to know more about what this stable isomorphism induces on the reduced filtered K-theory.…”

“…The outline of the proof of Theorem 3.1 is as follows. We emulate the previous proofs that go from filtered Ktheory data to stable isomorphism or flow equivalence, as in [BH03,Boy02,Res06]. A key component of those proofs is manipulation of the matrix B E , in particular that we can perform certain basic row and column operations without changing the stable isomorphism class or the flow equivalence class, depending on the context (cf.…”

“…A nondegenerate matrix A with entries from {0, 1} satisfies Condition (II) if and only if the Cuntz-Krieger algebra O A is purely infinite if and only if the Cuntz-Krieger algebra O A has finitely many ideals if and only if the Cuntz-Krieger algebra O A has real rank zero. See [Res06] and the references therein.…”

“…We denote these by FK red (Ψ) and FK s red (Ψ), respectively. There is a nice description of the ideal lattice and the K-theory of all subquotients as well as of the occurring cyclic six-term exact sequences in terms of the underlying matrix (see [Res06] and the references therein). …”