Nonself-adjoint Semicrossed Products by Abelian Semigroups

Fuller, Adam Hanley

Canadian journal of mathematics, Tome 65 (2013) no. 4, pp. 768-782

Résumé

Let $S$ be the semigroup $S=\sum\nolimits_{i=1}^{\oplus k}{{{S}_{i}}}$ , where for each $i\in I,{{S}_{i}}$ is a countable subsemigroup of the additive semigroup ${{\mathbb{R}}_{+}}$ containing 0. We consider representations of $S$ as contractions ${{\left\{ {{T}_{s}} \right\}}_{s\in S}}$ on a Hilbert space with the Nica-covariance property: $T_{s}^{*}{{T}_{t}}={{T}_{t}}T_{s}^{*}$ 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 $S$ 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).

Détail

DOI : 10.4153/CJM-2012-051-8

Mots-clés : 47L55, 47A20, 47L65, semicrossed product, crossed product, C*-envelope, dilations