Nonself-adjoint semicrossed products by abelian semigroups

TL;DR

This paper studies nonself-adjoint semicrossed product algebras arising from Nica-covariant representations of abelian semigroups acting on operator algebras, establishing dilation results and computing their C*-envelopes.

Contribution

It proves the existence of unique minimal isometric Nica-covariant dilations for these representations and calculates the C*-envelope of the associated semicrossed product algebra.

Findings

Unique minimal isometric Nica-covariant dilations exist for all such representations.

The C*-envelope of the isometric nonself-adjoint semicrossed product algebra is explicitly determined.

The results extend the understanding of nonself-adjoint operator algebras generated by semigroup actions.

Abstract

Let $\mathcal{S}$ be the semigroup $\mathcal{S}=\sum^{\oplus k}_{i=1}\Sc{S}_i$, where for each $i\in I$, $\mathcal{S}_i$ is a countable subsemigroup of the additive semigroup $\B{R}_+$ containing 0. We consider representations of $\mathcal{S}$ as contractions $\{T_s\}_{s\in\mathcal{S}}$ on a Hilbert space with the Nica-covariance property: $T_s^*T_t=T_tT_s^*$ whenever $t\wedge s=0$. We show that all such representations have a unique minimal isometric Nica-covariant dilation. This result is used to help analyse the nonself-adjoint semicrossed product algebras formed from Nica-covariant representations of the action of $\mathcal{S}$ on an operator algebra $\mathcal{A}$ by completely contractive endomorphisms. We conclude by calculating the $C^*$-envelope of the isometric nonself-adjoint semicrossed product algebra (in the sense of Kakariadis and Katsoulis).