Double duals and Hilbert modules

Lin, Huaxin

doi:10.2140/apde.2025.18.1531

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Double duals and Hilbert modules

Huaxin Lin

Vol. 18 (2025), No. 6, 1531–1566

DOI: 10.2140/apde.2025.18.1531

Abstract

Let $A$ be a $C^{*}$ -algebra, $H$ be a Hilbert $A$ -module and $K (H)$ be the closure of the set of finite-rank module maps. We show that the $W^{*}$ -algebra of all bounded $A^{* *}$ -module maps on the smallest self-dual Hilbert $A^{* *}$ -module containing $H$ is isomorphic to $K {(H)}^{* *}$ as $W^{*}$ -algebras. We also show that the unit ball of $H$ is closed in $H^{♯}$ , the dual of $H$ in an $A$ -weak topology of $H^{♯}$ , and the unit ball of $H$ is also dense in the unit ball of $H^{♯}$ in a weak* topology. Some versions of the Kaplansky density theorem for Hilbert $C^{*}$ -modules are also presented.

Keywords

Hilbert $C^*$-modules, double duals

Mathematical Subject Classification

Primary: 46L05, 46L08

Secondary: 46L35

Milestones

Received: 17 December 2023

Revised: 5 April 2024

Accepted: 8 June 2024

Published: 29 May 2025

Authors

Huaxin Lin
Shanghai Institute for Mathematics and Interdisciplinary Sciences Shanghai China

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Vol. 18, No. 6, 2025