Crossed products for interactions and graph algebras

Abstract

We consider Exel's interaction $(V,H)$ over a unital $C^*$-algebra $A$, such that $V(A)$ and $H(A)$ are hereditary subalgebras of $A$. For the associated crossed product, we obtain a uniqueness theorem, ideal lattice description, simplicity criterion and a version of Pimsner-Voiculescu exact sequence. These results cover the case of crossed products by endomorphisms with hereditary ranges and complementary kernels. As model examples of interactions not coming from endomorphisms we introduce and study in detail interactions arising from finite graphs. The interaction $(V,H)$ associated to a graph $E$ acts on the core $F_E$ of the graph algebra $C^*(E)$. By describing a partial homeomorphism of $\widehat{F}_E$ dual to $(V,H)$ we find Cuntz-Krieger uniqueness theorem, criteria for gauge-invariance of all ideals and simplicity of $C^*(E)$ as results concerning reversible noncommutative dynamics. We also provide a new approach to calculation of $K$-theory of $C^*(E)$ using only an induced partial automorphism of $K_0(F_E)$ and the six-term exact sequence.