Trace-class operators on Hilbert modules and Haagerup tensor products

We show that the space of trace-class operators on a Hilbert module over a commutative C*-algebra, as defined and studied in earlier work of Stern and van Suijlekom (Journal of Functional Analysis, 2021), is completely isometrically isomorphic to a Haagerup tensor product of the module with its oper...

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Bibliographic Details
Published in:arXiv.org 2024-03
Main Authors: Crisp, Tyrone, Rosbotham, Michael
Format: Article
Language:English
Subjects:
Functional analysis
Hilbert space
Mathematical analysis
Modules
Operators (mathematics)
Tensors
Online Access:Get full text
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Summary:We show that the space of trace-class operators on a Hilbert module over a commutative C*-algebra, as defined and studied in earlier work of Stern and van Suijlekom (Journal of Functional Analysis, 2021), is completely isometrically isomorphic to a Haagerup tensor product of the module with its operator-theoretic adjoint. This generalises a well-known property of Hilbert spaces. In the course of proving this, we also obtain a new proof of a result of Stern-van Suijlekom concerning the equivalence between two definitions of trace-class operators on Hilbert modules.
ISSN:2331-8422