Brown–Green–Rieffel stabilization theorem

Statement

Let $A$ and $B$ be $\sigma$ -unital $C^*$ -algebras, equivalently $C^*$ -algebras admitting countable approximate identities, and let $\mathcal K$ be the $C^*$ -algebra of compact operators on a separable infinite-dimensional Hilbert space. The Brown–Green–Rieffel stabilization theorem states that $A$ and $B$ are strongly Morita equivalent if and only if

A\otimes_{\min}\mathcal K\cong B\otimes_{\min}\mathcal K

as $C^*$ -algebras. Thus, under the countability hypothesis, Morita equivalence becomes ordinary $*$ -isomorphism after stabilization by compact operators.

Mechanism of the theorem

An imprimitivity bimodule places $A$ and $B$ as complementary full corners of its linking algebra. After tensoring with $\mathcal K$ , the $\sigma$ -unitality hypothesis supplies enough countable matrix units to identify these full corners with the stabilized linking algebra. Conversely, isomorphic stabilizations are Morita equivalent, and each algebra is Morita equivalent to its stabilization. This is the main result of Brown–Green–Rieffel.

Consequences and examples

The theorem turns many Morita-invariant questions into isomorphism questions for stable algebras. For example, $M_n(A)$ and $A$ have isomorphic stabilizations and are strongly Morita equivalent. Likewise, $\mathcal K(H)$ is strongly Morita equivalent to $\mathbb C$ for separable infinite-dimensional $H$ , since $\mathcal K(H)\otimes\mathcal K\cong\mathcal K$ .

Hypotheses and limitations

References

Lawrence G. Brown, Philip Green, and Marc A. Rieffel, “Stable Isomorphism and Strong Morita Equivalence of $C^*$ -Algebras,” Pacific Journal of Mathematics 71 (1977), 349–363. DOI record. Relevant: the main stabilization theorem and the counterexamples without countable approximate identities.
Iain Raeburn and Dana P. Williams, Morita Equivalence and Continuous-Trace $C^*$ -Algebras, Mathematical Surveys and Monographs 60, American Mathematical Society, 1998. DOI record. Relevant: Chapter 3 on imprimitivity bimodules, linking algebras, and stabilization.