# Journal of Operator Theory

Volume 75, Issue 2, Spring 2016 pp. 299-317.

Self-similar graph $C^*$-algebras and partial crossed products

**Authors**:
Ruy Exel (1) and Charles Starling (2)

**Author institution:**(1) Depart. de Matem\'atica, Campus Universit\'ario
Trindade CEP 88.040-900 Florian\'opolis SC, Brasil

(2) Department of Mathematics and Statistics,
University of Ottawa, K1N 6N5 Canada

**Summary: **In a recent paper, Pardo and the first named author introduced a class of
$C^*$-algebras which are constructed from an action of a group on a graph.
This class was shown to include many $C^*$-algebras of interest,
including all Kirchberg algebras in the UCT class.
In this paper, we study the conditions under which these algebras can be realized as partial
crossed products of commutative $C^*$-algebras by groups. In addition, for any
$n\geqslant 2$ we present a large class of groups such that for any group $H$ in this class,
the Cuntz algebra $\mathcal{O}_n$ is isomorphic to a partial crossed product of a
commutative $C^*$-algebra by $H$.

**DOI: **http://dx.doi.org/10.7900/jot.2015mar04.2072

**Keywords: **$C^*$-algebra, partial crossed product, inverse semigroup, self-similar group

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