nLab
Hilbert W*-module

Definition

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

Properties

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

The equivalence is implemented by the following functors.

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

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

Last revised on July 3, 2021 at 13:36:07. See the history of this page for a list of all contributions to it.