Countably generated Hilbert C*-module - Imported Knowlpedia Content

Definition

Let $A$ be a $C^*$ -algebra and let $E$ be a right Hilbert $A$ -module. The module $E$ is countably generated if there is a sequence $(x_n)_{n\geq1}\subseteq E$ such that

E=\overline{\operatorname{span}}\{x_na:n\geq1,\ a\in A\},

where the closure is taken in the Hilbert-module norm. “Generated” therefore means topologically generated as a right $A$ -module, not algebraically generated by finitely supported combinations. A finite generating family is allowed by repeating entries or appending zeros.

Equivalent compact-operator criterion

The module $E$ is countably generated precisely when its algebra $\mathcal K(E)$ of compact Hilbert-module operators has a countable approximate identity. This converts a module generation condition into the $\sigma$ -unitality of a $C^*$ -algebra Lance, Chapter 2.

Stabilization

The Kasparov stabilization theorem says that every countably generated $E$ satisfies

E\oplus H_A\cong H_A,

where $H_A$ is the standard Hilbert $A$ -module. Thus countable generation is the exact smallness hypothesis that permits $E$ to occur as an orthogonally complemented submodule of one standard module.

Examples and scope

Every $A^n$ and $H_A=\ell^2(A)$ is countably generated. A closed submodule of a countably generated Hilbert module need not be countably generated without hypotheses on $A$ . Countable generation also does not mean that $E$ possesses a countable orthonormal basis: Hilbert $C^*$ -modules generally need not admit such bases.

References

E. Christopher Lance, Hilbert $C^*$ -Modules: A Toolkit for Operator Algebraists, Cambridge University Press, 1995. DOI record. Relevant: Chapter 2 on countably generated modules, compact operators, and stabilization.