# nLab Hilbert W*-module

## Definition

A Hilbert W*-module is a Hilbert C*-module $M$ over a von Neumann algebra $A$ such that $M$ admits a predual as a Banach space.

## Properties

For any von Neumann algebra $A$, the category of Hilbert W*-modules over $A$ is equivalent to the category of W*-representations of $A$.

The equivalence is implemented by the following functors.

Given a Hilbert W*-module $M$, we send it to the completion of $M\otimes_A L^2(A)$, where $L^2(A)$ is the Haagerup standard form of $A$.

Given a W*-representation $R$, we send it to the internal hom $Hom_A(L^2(A), R)$, which is a Hilbert W*-module over $A$.

Last revised on October 18, 2021 at 13:44:17. See the history of this page for a list of all contributions to it.