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 $M_{\ast}$ as a Banach space;

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

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

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

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 $A$.

Related concepts

• measurable field of Hilbert spaces

• Serre-Swan theorem

• W*-representation

• Hilbert C*-module

• von Neumann algebra

• duality between geometry and algebra

References

William Paschke?, Inner Product Modules over $B^\ast$-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 Algebras and Their Modules: An operator space approach, Vol. 30, London Mathematical Society Monographs, 2004