Strong Morita equivalence of C*-algebras - Imported Knowlpedia Content

Definition

Two $C^*$ -algebras $A$ and $B$ are strongly Morita equivalent when there exists an $A$ - $B$ imprimitivity bimodule $X$ . Concretely, $X$ is a full right Hilbert $B$ -module with a nondegenerate $*$ -homomorphism

A\longrightarrow\mathcal L_B(X)

whose image is exactly $\mathcal K_B(X)$ , the $C^*$ -algebra of compact adjointable operators on $X$ . Equivalently, $X$ carries compatible full $A$ - and $B$ -valued inner products satisfying

{}_A\langle x,y\rangle z=x\langle y,z\rangle_B.

The existence of such an $X$ , not a chosen algebra isomorphism, is the defining equivalence.

Equivalence relation and representations

Strong Morita equivalence is reflexive, symmetric, and transitive. The conjugate bimodule reverses an equivalence, while the interior tensor product composes two of them. Rieffel induction through $X$ gives an equivalence between the nondegenerate representation categories of $A$ and $B$ Rieffel, induced representations.

Ideals, primitive spectra, and many structural properties correspond under this equivalence. In particular, type I and continuous-trace behavior are Morita invariant.

Examples and non-examples

For any Hilbert space $H$ , the module $H$ is a $\mathcal K(H)$ - $\mathbb C$ imprimitivity bimodule, so $\mathcal K(H)$ is strongly Morita equivalent to $\mathbb C$ . More generally, $M_n(A)$ is strongly Morita equivalent to $A$ .

Morita equivalence is weaker than $*$ -isomorphism: $\mathcal K(H)$ and $\mathbb C$ are not isomorphic when $H$ is infinite-dimensional. A general $C^*$ -correspondence is a near miss; without fullness and the identification of the left action with compact operators, it need not implement an equivalence.

References

M. A. Rieffel, “Induced Representations of $C^*$ -Algebras,” Advances in Mathematics 13 (1974), 176–257. DOI record. Relevant: imprimitivity bimodules, induction, and equivalence of representation theories.
I. Raeburn and D. 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 and strong Morita equivalence.