Crossed products for interactions and graph algebras

Abstract

We consider Exel's interaction $(V,H)$ over a unital $C^{\ast }$-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 $C^{\ast }(E)$. By describing a partial homeomorphism of ${\hat{F}}_E$ dual to $(V,H)$ we find Cuntz-Krieger uniqueness theorem, criteria for gauge-invariance of all ideals and simplicity of $C^{\ast }(E)$ as results concerning reversible noncommutative dynamics. We also provide a new approach to calculation of $K$-theory of $C^{\ast }(E)$ using only an induced partial automorphism of $K_0(F_E)$ and the six-term exact sequence.

The term complete interaction changed to corner interaction. This version is accepted to Integral Equations and Operator Theory

References in corpus (6)

• When a C*-algebra is a coefficient algebra for a given endomorphism
• Exel's crossed product and crossed products by completely positive maps
• Topological freeness for Hilbert bimodules
• Dynamical systems of type (m,n) and their C*-algebras
• Ideals of the core of C*-algebras associated with self-similar maps
• Interactions and dynamical systems of type (n,m) - A case study

Cited by in corpus (5)

• Topological aperiodicity for product systems over semigroups of Ore type
• Crossed products by endomorphisms of $C_0(X)$-algebras
• Pure infiniteness and ideal structure of $C^{\ast }$-algebras associated to Fell bundles
• Exel's crossed product and crossed products by completely positive maps
• Ideal structure of crossed products by endomorphisms via reversible extensions of $C^{\ast }$-dynamical systems