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arXiv · 2307.10108

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

Boyu Li·Dilian Yang
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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 $\mathcal{O}_{\mathbb{Z}, E}$.

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BibTeXRIS

Boyu Li, Dilian Yang. 2023-07-19. Wold Decomposition and C*-envelopes of Self-Similar Semigroup Actions on Graphs. https://doi.org/10.1016/j.jmaa.2024.128395

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