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 and Their Modules—an operator space approach, Vol. 30, London Mathematical Society Mongraphs