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Trace-class operators on Hilbert modules and Haagerup tensor products

Tyrone Crisp, Michael Rosbotham

TL;DR

The paper establishes that the space of trace-class operators on a Hilbert $A$-module $F$ over a commutative $C^*$-algebra $A$ is completely isometrically isomorphic to the Haagerup tensor product $F^* \otimes_A^{\mathrm{h}} F$, extending the Hilbert-space result $H^* \otimes^{\mathrm{h}} H \cong \operatorname{L}^1(H)$. By employing frames of multipliers, it generalises the SvS definition of trace-class operators to settings where frames may not exist and proves the trace is independent of the chosen frame, with localisation linking $A$-valued traces to pointwise traces on fibres. The core argument combines localisation, Haagerup tensor techniques, and an operator-space framework: first for free modules $F=C_0(\widehat{A},H)$ and then for general $F$ via an adjointable isometry $\theta$, yielding a complete isometry $F^* \otimes_A^{\mathrm{h}} F \cong \operatorname{L}^1_A(F)$. An explicit harmonic-analysis example shows how these tensor products produce concrete descriptions of Haagerup-type bimodules, connecting representation induction with fibrewise trace-class structures. Overall, the work provides a practical realisation of Haagerup tensor products in the commutative setting and enhances descent theory applications through concrete operator-space identifications.

Abstract

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.

Trace-class operators on Hilbert modules and Haagerup tensor products

TL;DR

The paper establishes that the space of trace-class operators on a Hilbert -module over a commutative -algebra is completely isometrically isomorphic to the Haagerup tensor product , extending the Hilbert-space result . By employing frames of multipliers, it generalises the SvS definition of trace-class operators to settings where frames may not exist and proves the trace is independent of the chosen frame, with localisation linking -valued traces to pointwise traces on fibres. The core argument combines localisation, Haagerup tensor techniques, and an operator-space framework: first for free modules and then for general via an adjointable isometry , yielding a complete isometry . An explicit harmonic-analysis example shows how these tensor products produce concrete descriptions of Haagerup-type bimodules, connecting representation induction with fibrewise trace-class structures. Overall, the work provides a practical realisation of Haagerup tensor products in the commutative setting and enhances descent theory applications through concrete operator-space identifications.

Abstract

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.
Paper Structure (5 sections, 11 theorems, 36 equations)

This paper contains 5 sections, 11 theorems, 36 equations.

Table of Contents

  1. Introduction
  2. Frames of multipliers and trace-class operators
  3. Trace-class operators on $C_0(\widehat{A},H)$
  4. Operator-space structure on $\operatorname{L}^1_A(F)$
  5. An example from harmonic analysis

Key Result

Theorem 2.4

The following are equivalent for a Hilbert module $F$ over a commutative $C^*$-algebra $A$: Moreover, if $(\beta_i)$ is a frame of multipliers for $F$ then for each $\eta\in F$ the sum $\sum_{i=1}^\infty {\vert {\beta_i}\rangle}\langle \beta_i\, |\, \eta\rangle$ converges in norm to $\eta$. ∎

Theorems & Definitions (28)

  • Example 2.1
  • Definition 2.2: RT
  • Example 2.3
  • Theorem 2.4: RT
  • Definition 2.5
  • Theorem 2.7: SvS
  • proof
  • Definition 2.8
  • Theorem 3.1: cf. SvS
  • Definition 3.2
  • ...and 18 more