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Nonself-adjoint Semicrossed Products by Abelian Semigroups

Published online by Cambridge University Press:  20 November 2018

Adam Hanley Fuller
Adam Hanley Fuller*
Affiliation:
Department of Pure Mathematics, University of Waterloo, WaterlooON N2L 3G1
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Abstract

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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).

Keywords

47L5547A2047L65semicrossed productcrossed productC*-envelopedilations
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 65 , Issue 4 , 01 August 2013 , pp. 768 - 782
Copyright
Copyright © Canadian Mathematical Society 2013

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