Imprimitivity bimodule

Let $A$ and $B$ be $C^*$ -algebras. An $A$ - $B$ imprimitivity bimodule is a $C^*$ -correspondence ${}_AE_B$ , hence an $(A,B)$ -bimodule, such that the closed span of $\langle E,E\rangle_B$ is $B$ and the left action is a $*$ -isomorphism

A\longrightarrow \mathcal K_B(E).

The first condition is right fullness; the second makes the left action faithful, nondegenerate, and exactly the generalized compact operators. Existence of such an $E$ means that $A$ and $B$ are strongly Morita equivalent.

Two-inner-product formulation

For $\xi,\eta\in E$ , define the left $A$ -valued inner product by requiring that ${}_A\langle\xi,\eta\rangle$ act as the rank-one operator $\theta_{\xi,\eta}$ . Then

{}_A\langle\xi,\eta\rangle\zeta =\xi\langle\eta,\zeta\rangle_B.

The two inner products are full and induce the same norm. Conversely, a full left and right Hilbert bimodule satisfying this compatibility identity gives the compact-operator formulation above.

Standard examples

Every $C^*$ -algebra $A$ is an $A$ - $A$ imprimitivity bimodule with ${}_A\langle x,y\rangle=xy^*$ and $\langle x,y\rangle_A=x^*y$ . More generally, any full right Hilbert $B$ -module $E$ is a $\mathcal K_B(E)$ - $B$ imprimitivity bimodule. Hilbert spaces give the special case $\mathcal K(H)\sim_M\mathbb C$ .

Distinction from a correspondence

An arbitrary correspondence may have a nonfaithful left action, may fail to be full on the right, or may act by adjointable operators outside $\mathcal K_B(E)$ . Any of these failures prevents it from being an imprimitivity bimodule. Thus “Hilbert $C^*$ -bimodule” is potentially ambiguous unless fullness and the compatibility condition are stated.

References

Marc A. Rieffel, “Induced representations of C-algebras,” Advances in Mathematics* 13 (1974), 176–257. DOI record. Relevant: §§2–6 on rigged modules, imprimitivity, and induced representations.
Iain Raeburn and Dana P. Williams, Morita Equivalence and Continuous-Trace C-Algebras*, American Mathematical Society, 1998. AMS DOI record. Relevant: Chapter 3 on imprimitivity bimodules.