Journal of Operator Theory

Volume 95, Issue 2, Spring 2026 pp. 519-535.

Trace-class operators on Hilbert modules and Haagerup tensor products

Authors: Tyrone Crisp (1), Michael Rosbotham (2)
Author institution: (1) Department of Mathematics and Statistics, University of Maine, Orono ME, 04469, U.S.A.
(2) Department of Mathematics and Statistics, University of Maine, Orono ME, 04469, U.S.A.

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 (\textit{J. Funct. Anal.}, 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.

DOI: http://dx.doi.org/10.7900/jot.
Keywords: Hilbert module, Haagerup tensor product, trace-class operators

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