Abstract
We consider Exel’s interaction \({(\mathcal{V, H})}\) over a unital C*-algebra A, such that \({\mathcal{V}(A)}\) and \({\mathcal{H}(A)}\) 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 complemented kernels. As model examples of interactions not coming from endomorphisms we introduce and study in detail interactions arising from finite graphs.
The interaction \({(\mathcal{V, H})}\) associated to a graph E acts on the core \({\mathcal{F}_E}\) of the graph algebra C*(E). By describing a partial homeomorphism of \({\widehat{\mathcal{F}_E}}\) dual to \({(\mathcal{V, H})}\) we find the fundamental structure theorems for C*(E), such as Cuntz–Krieger uniqueness theorem, as results concerning reversible noncommutative dynamics on \({\mathcal{F}_E}\) . We also provide a new approach to calculation of K-theory of C*(E) using only an induced partial automorphism of \({K_0(\mathcal{F}_E)}\) and the six-term exact sequence.
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This work was in part supported by Polish National Science Centre Grant Number DEC-2011/01/D/ST1/04112.
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Kwaśniewski, B.K. Crossed Products for Interactions and Graph Algebras. Integr. Equ. Oper. Theory 80, 415–451 (2014). https://doi.org/10.1007/s00020-014-2166-5
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DOI: https://doi.org/10.1007/s00020-014-2166-5