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# Nonself-adjoint Semicrossed Products by Abelian Semigroups

Published:2012-11-17
Printed: Aug 2013
Pure Mathematics Deptartment, University of Waterloo, Waterloo, ON N2L 3G1
 Format: LaTeX MathJax PDF

## Abstract

Let $\mathcal{S}$ be the semigroup $\mathcal{S}=\sum^{\oplus k}_{i=1}\mathcal{S}_i$, where for each $i\in I$, $\mathcal{S}_i$ is a countable subsemigroup of the additive semigroup $\mathbb{R}_+$ 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).
 Keywords: semicrossed product, crossed product, C*-envelope, dilations
 MSC Classifications: 47L55 - Representations of (nonselfadjoint) operator algebras 47A20 - Dilations, extensions, compressions 47L65 - Crossed product algebras (analytic crossed products)

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