C*-Correspondences, Hilbert Bimodules, and their L^p Versions
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Date
2024-01-10
Authors
Delfin Ares de Parga, Alonso
Journal Title
Journal ISSN
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Publisher
University of Oregon
Abstract
This dissertation initiates the study of $L^p$-modules, which are modules over $L^p$-operator algebras inspired by Hilbert modules over C*-algebras. The primary motivation for studying $L^p$-modules is to explore the possibility of defining $L^p$ analogues of Cuntz-Pimsner algebras.
The first part of this thesis consists of investigating representations of C*-correspondences on pairs of Hilbert spaces. This generalizes the concept of representations of Hilbert bimodules
introduced by R. Exel in \cite{Exel1993}. We present applications of representing a correspondence on a pair of Hilbert spaces $(\Hi_0, \Hi_1)$,
such as obtaining induced representations of both $\Li_A(\X)$ and $\mathcal{K}_A(\X)$ on $\Hi_1$,
and giving necessary and sufficient conditions on
an $(A,B)$ C*-correspondences to admit a Hilbert $A$-$B$-bimodule structure.
The second part is concerned with the theory of $L^p$-modules. Here we present a thorough treatment of $L^p$-modules, including morphisms between them and techniques for constructing new $L^p$-modules. We then useour results on representations for C*-correspondences to motivate and develop the theory of $L^p$-correspondences, their representations, the $L^p$-operator algebras they generate, and present evidence that well-known $L^p$-operator algebras can be constructed from $L^p$-correspondences via $L^p$-Fock representations.
Due to the technicality that comes with dealing with direct sums of $L^p$-correspondences and interior
tensor products, we only focus on two particular examples for which a Fock space construction can be carried out. The first example deals with the $L^p$-module $(\ell_d^p, \ell_d^q)$, for which we exhibit a covariant $L^p$-Fock representation that yields an $L^p$-operator algebra isometrically isomorphic to $\mathcal{O}_d^p$, the $L^p$-analogue of the Cuntz-algebra $\mathcal{O}_d$ introduced by N.C. Phillips in \cite{ncp2012AC}. The second example involves a nondegenerate $L^p$-operator algebra $A$ with a bicontractive approximate identity together with an isometric automorphism $\varphi_A \in \op{Aut}(A)$. In this case, we also present an algebra associated to a covariant $L^p$-Fock representation, but due to the current lack of knowledge of universality of the $L^p$-Fock representation, we only show that there is a contractive map from the crossed product $F^p(\Z, A, \varphi_A)$ to this algebra.
This dissertation includes unpublished material.
Description
Keywords
C*-correspondences, Hilbert Modules, L^p Cuntz-Pimsner algebras, L^p-Correspondences, L^p-Modules, L^p-Operator Algebras