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

TL;DR

This paper explores the Wold decomposition for self-similar semigroup actions on graphs and determines the C*-envelope of associated operator algebras, revealing it coincides with a self-similar C*-algebra.

Contribution

It introduces a Wold decomposition for self-similar semigroup actions on graphs and explicitly identifies the C*-envelope of related non-selfadjoint algebras.

Findings

Wold decomposition for self-similar semigroup actions established

C*-envelope of the algebra matches the self-similar C*-algebra

Explicit dilations constructed for non-boundary representations

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 $P=\mathbb{N}$ 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 $\mathcal{O}_{\mathbb{Z}, E}$.