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Free Multivariate w*-Semicrossed Products: Reflexivity and the Bicommutant Property

  • Robert T. Bickerton,
    Newcastle University , Newcastle , NE1 7 RU , United Kingdom
  • Evgenios T.A. Kakariadis,
    Newcastle University , Newcastle , NE1 7 RU , United Kingdom
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Abstract

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 product reflexivity, semicrossed product
MSC Classifications: 47A15, 47L65, 47L75, 47L80 show english descriptions Invariant subspaces [See also 47A46]
Crossed product algebras (analytic crossed products)
Other nonselfadjoint operator algebras
Algebras of specific types of operators (Toeplitz, integral, pseudodifferential, etc.)
47A15 - Invariant subspaces [See also 47A46]
47L65 - Crossed product algebras (analytic crossed products)
47L75 - Other nonselfadjoint operator algebras
47L80 - Algebras of specific types of operators (Toeplitz, integral, pseudodifferential, etc.)
 

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