nLab Hilbert W*-module


A Hilbert W*-module is a Hilbert C*-module MM over a von Neumann algebra AA such that:

  • MM admits a predual M *M_{\ast} as a Banach space;

  • for all mMm \in M, the map m, A:MA\langle m, - \rangle_{A}: M \to A is normal (i.e. the dual of some map A *M *A_{*} \to M_{*}),

where , A:M¯×MA\langle -,- \rangle_{A}: \overline{M} \times M \rightarrow A is the AA-valued inner product on MM, and A *A_{*} is the predual of AA.

The above definition is equivalent to the definitions using a notion of self-duality (See Blecher-Merdy, Lemma 8.5.4).


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.


William Paschke?, Inner Product Modules over B *B^\ast-algebras, Trans. Amer. Math. Soc., 182, 1973 (link)

David Blecher?, On Selfdual Hilbert Modules, Fields Institute Communications, 13, 1997

David Blecher?, Christian Le Merdy?, Operator and Their Modules—an operator space approach, Vol. 30, London Mathematical Society Mongraphs

Last revised on May 9, 2022 at 16:16:43. See the history of this page for a list of all contributions to it.