An application of sympolic dynamics: Diagonal-preserving Gauge-invariant isomorphisms of graph C*-algebras

Specialeforsvar ved Michael Staal-Olsen

Titlel An application of symbolic dynamics: Diagonal-preserving Gauge-invariant isomorphisms of graph C*-algebras

Abstract: We provide an overview of the basic theory concerning symbolic dynamics and graph C*-algebras. We introduce étale groupoids and study the corresponding groupoid C*-algebra which turns out to be the natural groupoid analogue of the group-C*-algebra of a discrete group. Our studies will provide a reconstruction Theorem for groupoids similar to the one introduced by Renault, and it will allow us to follow the ideas of a characteri-zation of strong shift equivalence and one-sided eventual conjugacy which are due to Carlsen and Rout. The characterization is based on a study of graph C*-algebras equipped with generalised gauge-actions. As it turns out, two Cuntz-Krieger algebras are *-isomorphic by a diagonal-preserving and gauge-intertwining *-isomorphism if and only if the corresponding one-sided shifts are eventually conjugate, and the stabilization of two Cuntz-Krieger algebras are *-isomorphic by a diagonal-preserving gauge-intertwining *-isomorphism if and only if the corresponding two-sided shifts are conjugate. We further-more apply Elliott’s Classification Theorem for AF-algebras and provide a reformulation of shift equivalence by means of theory related to graph C*-algebras. The reformulation asserts that two integral matrices are shift equivalent if and only if the Fixed-Point algebras of the corresponding Cuntz-Krieger algebras are *-isomorphic in a certain sense that involves approximative unitary equivalence

Vejleder:  Søren Eilers
Censor:    Wojcieck Szymanski, SDU