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Nonself-adjoint Semicrossed Products by Abelian Semigroups

Open Access article
 Printed: Aug 2013
  • Adam Hanley Fuller,
    Pure Mathematics Deptartment, University of Waterloo, Waterloo, ON N2L 3G1
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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 semicrossed product, crossed product, C*-envelope, dilations
MSC Classifications: 47L55, 47A20, 47L65 show english descriptions Representations of (nonselfadjoint) operator algebras
Dilations, extensions, compressions
Crossed product algebras (analytic crossed products)
47L55 - Representations of (nonselfadjoint) operator algebras
47A20 - Dilations, extensions, compressions
47L65 - Crossed product algebras (analytic crossed products)

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