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Register a new userThe non-selfadjoint approach to the Hao-Ng isomorphism
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In an earlier work, the authors proposed a non-selfadjoint approach to the Hao-Ng isomorphism problem for the full crossed product, depending on the validity of two conjectures stated in the broader context of crossed products for operator algebras. By work of Harris and Kim, we now know that these conjectures in the generality stated may not always be valid. In this paper we show that in the context of hyperrigid tensor algebras of C*-correspondences, each one of these conjectures is equivalent to the Hao-Ng problem. This is accomplished by studying the representation theory of non-selfadjoint crossed products of C*-correspondence dynamical systems; in particular we show that there is an appropriate dilation theory. A large class of tensor algebras of C*-correspondences, including all regular ones, are shown to be hyperrigid. Using Hamanas injective envelope theory, we extend earlier results from the discrete group case to arbitrary locally compact groups; this includes a resolution of the Hao-Ng isomorphism for the reduced crossed product and all hyperrigid C*-correspondences. A culmination of these results is the resolution of the Hao-Ng isomorphism problem for the full crossed product and all row-finite graph correspondences; this extends a recent result of Bedos, Kaliszewski, Quigg and Spielberg.
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Using non-selfadjoint techniques, we establish the Hao-Ng isomorphism for the reduced crossed product and all discrete groups. For the full crossed product an analogous result holds for all discrete groups but the C*-correspondences involved have to be hyperrigid. These results are obtained by calculating the C*-envelope of the reduced crossed product of an operator algebra by a discrete group.
This paper is an expanded version of the lectures I delivered at the Indian Statistical Institute, Bangalore, during the OTOA 2014 conference.
Let $mathcal{M}$ be a von Neumann algebra, and let $0<p,qleinfty$. Then the space $Hom_mathcal{M}(L^p(mathcal{M}),L^q(mathcal{M}))$ of all right $mathcal{M}$-module homomorphisms from $L^p(mathcal{M})$ to $L^q(mathcal{M})$ is a reflexive subspace of the space of all continuous linear maps from $L^p(mathcal{M})$ to $L^q(mathcal{M})$. Further, the space $Hom_mathcal{M}(L^p(mathcal{M}),L^q(mathcal{M}))$ is hyperreflexive in each of the following cases: (i) $1le q<pleinfty$; (ii) $1le p,qleinfty$ and $mathcal{M}$ is injective, in which case the hyperreflexivity constant is at most $8$.
We prove that the isomorphism relation for separable C$^*$-algebras, and also the relations of complete and $n$-isometry for operator spaces and systems, are Borel reducible to the orbit equivalence relation of a Polish group action on a standard Borel space.
In this paper we construct a Chern-Weil isomorphism for the equivariant Brauer group of R^n-actions on a principal torus bundle, where the target for this isomorphism is a dimensionally reduced Cech cohomology group. From this point of view, the usual forgetful functor takes the form of a connecting homomorphism in a long exact sequence in dimensionally reduced cohomology.
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