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Search: MSC category 47L65 ( Crossed product algebras (analytic crossed products) )

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1. CJM Online first

Bickerton, Robert T.; Kakariadis, Evgenios T.A.
 Free Multivariate w*-Semicrossed Products: Reflexivity and the Bicommutant Property We study w*-semicrossed products over actions of the free semigroup and the free abelian semigroup on (possibly non-selfadjoint) w*-closed algebras. We show that they are reflexive when the dynamics are implemented by uniformly bounded families of invertible row operators. Combining with results of Helmer we derive that w*-semicrossed products of factors (on a separable Hilbert space) are reflexive. Furthermore we show that w*-semicrossed products of automorphic actions on maximal abelian selfadjoint algebras are reflexive. In all cases we prove that the w*-semicrossed products have the bicommutant property if and only if the ambient algebra of the dynamics does also. Keywords:reflexivity, semicrossed productCategories:47A15, 47L65, 47L75, 47L80

2. CJM 2012 (vol 65 pp. 768)

 Nonself-adjoint Semicrossed Products by Abelian Semigroups 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, dilationsCategories:47L55, 47A20, 47L65