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Abstract
It was shown by M. Pimsner that given a separable, commutative C*-algebra A and an automorphism α ∈ Aut(A), the conditions AF-embeddable, quasidiagonal, and stably finite, are equivalent for A ×α Z. Moreover, Pimsner characterized these finiteness conditions in terms of ''pseudoperiodic points'' in the underlying dynamical system. An analogous result was obtained by N. Brown in the case where A is an AF–algebra. In this case, these finiteness conditions are characterized by a K-theoretic condition on the automorphism [α] of K0(A).
We generalize these results to certain graph algebras and Cuntz–Pimnser algebras. In particular, if E is a second countable topological graph that is either discrete, totally disconnected, or compact with no sinks, then these finiteness conditions are equivalent and can be characterized by combinatorial conditions on the graph E. Moreover, if A is an AF–algebra and H is a separable C*-correspondence over A, then these finiteness conditions are equivalent for the Cuntz–Pimsner algebra O A(H). In addition, if H is ''regular'', the finiteness conditions are characterized by a K-theoretic condition on the endomorphism [H] of K 0(A).
In the process, we prove a few other results which may be of independent interest. In particular, we calculate the K-theory of the crossed product OA(H) ×γ T where (A, H) is a ''regular'' C*-correspondence and γ is the usual gauge action. If A is an AF–algebra and H is a separable C*-correspondence, we show OA(H) ×γ T is AF. Finally, if E is a compact graph with no sinks and C*(E) is finite, we show C*(E) ≅ C( E∞) ×σ Z where E∞ is the infinite path space of E and σ is the ''backward shift'' on E ∞.
