Kasparov stabilization theorem

Statement

Let $A$ be a $C^*$ -algebra, let $E$ be a countably generated Hilbert $A$ -module, and let $H_A=\ell^2(A)$ be the standard Hilbert $A$ -module. The Kasparov stabilization theorem states that there is a unitary Hilbert-module isomorphism

E\oplus H_A\cong H_A.

Here unitary means an adjointable $A$ -linear map whose adjoint is its inverse, in the sense of an adjointable operator. In particular, $E$ is isomorphic to an orthogonally complemented submodule of $H_A$ . No unitality assumption on $A$ is required; the countable-generation hypothesis on $E$ is essential.

Equivalent complemented form

From a unitary $U:E\oplus H_A\to H_A$ , the image $U(E\oplus0)$ is the range of an adjointable projection $P$ on $H_A$ , while $(1-P)H_A\cong H_A$ . Conversely, any adjointable projection with $PH_A\cong E$ and $(1-P)H_A\cong H_A$ yields the displayed absorption isomorphism. In particular, the theorem implies the frequently used weaker form that every countably generated Hilbert $A$ -module embeds as an orthogonally complemented submodule of $H_A$ .

Proof mechanism

One chooses a countable generating sequence for $E$ , rescales it so that the associated coordinate operators are bounded, and constructs an adjointable operator from $H_A$ with dense range in $E$ . A sequence of operator rotations then turns this approximate spanning data into a unitary between $E\oplus H_A$ and $H_A$ . The construction replaces the orthonormal-basis argument unavailable for general Hilbert $C^*$ -modules.

Consequences

Stabilization lets one represent countably generated Hilbert modules by adjointable projections on a single standard module. It is therefore a basic tool for compact module operators, Morita equivalence, and Kasparov's $KK$ -theory. The theorem does not extend to arbitrary, possibly uncountably generated Hilbert modules without additional hypotheses.

References

E. Christopher Lance, Hilbert C-Modules: A Toolkit for Operator Algebraists*, Cambridge University Press, 1995. Publisher record. Relevant: Theorem 6.2 and its proof.
G. G. Kasparov, “Hilbert $C^*$ -modules: Theorems of Stinespring and Voiculescu,” Journal of Operator Theory 4 (1980), 133–150. Journal record. Relevant: the stabilization theorem for countably generated modules.