Wold decomposition and C*-envelopes of self-similar semigroup actions on graphs

Abstract

We study the Wold decomposition for representations of a self-similar semigroup P action on a directed graph E. We then apply this decomposition to the case where P=N to study the C*-envelope of the associated universal non-selfadjoint operator algebra AN,E by carefully constructing explicit non-trivial dilations for non-boundary representations. In particular, it is shown that the C*-envelope of AN,E coincides with the self-similar graph C*-algebra OZ,E.