^{1} Mathematics, Department of Mathematics and Computer Science (IMADA), Faculty of Science, SDU^{2} Department of Mathematics and Computer Science (IMADA), Faculty of Science, SDU^{3} Department of Mathematics and Computer Science (IMADA), Faculty of Science, SDU

DOI:

10.1007/s00020-014-2166-5

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 complemented 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 dual to (V,H) we find the fundamental structure theorems for C*(E), such as Cuntz–Krieger uniqueness theorem, as results concerning reversible noncommutative dynamics on 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(F_E) and the six-term exact sequence

Type:

Journal article

Language:

English

Published in:

Integral Equations and Operator Theory, 2014, Vol 80, Issue 3, p. 415-451