Standard Hilbert C*-module - Imported Knowlpedia Content

Definition

Let $A$ be a $C^*$ -algebra. The standard Hilbert $A$ -module, denoted $H_A$ or $\ell^2(A)$ , is the space of sequences $a=(a_n)_{n\geq1}$ in $A$ for which the partial sums $\sum_{n=1}^N a_n^*a_n$ converge in norm. It is a right $A$ -module by componentwise multiplication, and

\langle a,b\rangle_A=\sum_{n=1}^{\infty}a_n^*b_n.

The series defining this inner product converges in norm. Equipped with the induced norm, $H_A$ is a Hilbert $C^*$ -module. It is also the completion of the finitely supported sequences in $A$ .

Coordinates and terminology

If $A$ is unital, the sequences having $1_A$ in one coordinate and zero elsewhere form the canonical coordinate vectors. For nonunital $A$ , those vectors need not belong to $H_A$ , although finitely supported $A$ -valued sequences remain dense. Accordingly, “standard” or “free” describes the module's universal role and should not be read as asserting the existence of a Hilbert-space orthonormal basis in every case.

Finite and countable standard modules

The finite column module $A^n$ embeds as the first $n$ coordinates of $H_A$ . Splitting the coordinates into two infinite subsets gives a unitary Hilbert-module isomorphism

H_A\oplus H_A\cong H_A.

This absorption of countably many coordinates is the elementary model for stabilization phenomena.

Role in stabilization

Every countably generated Hilbert $A$ -module is isomorphic to an orthogonally complemented submodule of $H_A$ . Equivalently, adjoining one copy of $H_A$ absorbs such a module. This is the content of the Kasparov stabilization theorem and makes $H_A$ the standard ambient module in $KK$ -theory.

References

E. Christopher Lance, Hilbert C-Modules: A Toolkit for Operator Algebraists*, Cambridge University Press, 1995. Publisher record. Relevant: Chapter 2 on the standard module and countably generated modules.