Journal of Operator Theory

Volume 47, Issue 1, Winter 2002 pp. 169-186.

Partial dynamical systems and $C^*$-algebras generated by partial isometries

Authors: Ruy Exel (1), Marcelo Laca (2), and John Quigg (3)
Author institution: (1) Departamento de Matematica, Universidade Federal de Santa Catarina, 88010--970 Florianopolis SC, Brasil
(2) Department of Mathematics, University of Newcastle, NSW 2308, Australia
(3) Department of Mathematics, Arizona State University, Tempe, Arizona 85287, USA

Summary: A collection of partial isometries whose range and initial projections satisfy a specified set of conditions often gives rise to a partial representation of a group. The corresponding $C^*$-algebra is thus a quotient of the universal $C^*$-algebra for partial representations of the group, from which it inherits a crossed product structure, of an abelian $C^*$-algebra by a partial action of the group. This allows us to characterize faithful representations and simplicity, and to study the ideal structure of these $C^*$-algebras in terms of amenability and topological freeness of the associated partial action. We also consider three specific applications: to partial representations of groups, to Toeplitz algebras of quasi-lattice ordered groups, and to Cuntz-Krieger algebras.

Keywords: Partial representation, partial action, partial group algebra, Cuntz-Krieger algebra, crossed product, faithful representation

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