Wold Decomposition and C*-envelopes of Self-Similar Semigroup Actions on Graphs

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 $P=\mathbb{N}$ to study the C*-envelope of the associated universal non-selfadjoint operator algebra ${\mathcal{A}}_{\mathbb{N},E}$ by carefully constructing explicit non-trivial dilations for non-boundary representations. In particular, it is shown that the C*-envelope of ${\mathcal{A}}_{\mathbb{N},E}$ coincides with the self-similar C*-algebra ${\mathcal{O}}_{\mathbb{Z},E}$.