Rieffel induction - Imported Knowlpedia Content

Definition

Let $E$ be a $C^*$ -correspondence from $A$ to $B$ , with left action $\varphi:A\to\mathcal L_B(E)$ , and let $\pi:B\to\mathcal B(H_\pi)$ be a nondegenerate representation. Rieffel induction through $E$ is the representation of $A$ on the internal tensor product $E\otimes_\pi H_\pi$ defined by

\operatorname{Ind}_E(\pi)(a)(\xi\otimes h) =(\varphi(a)\xi)\otimes h.

Here $E\otimes_\pi H_\pi$ is obtained by balancing $\xi b\otimes h=\xi\otimes\pi(b)h$ , quotienting the null space of its induced inner product, and completing. Adjointability of $\varphi(a)$ makes the formula well defined and bounded.

Action on intertwiners

If $T:H_\pi\to H_\rho$ intertwines two representations of $B$ , then

1_E\otimes T:E\otimes_\pi H_\pi\longrightarrow E\otimes_\rho H_\rho

intertwines their induced representations. Induction therefore acts functorially on nondegenerate representation categories and preserves unitary equivalence. Tensor-product associativity identifies induction through a composite correspondence with successive induction.

Imprimitivity theorem

When $E$ is an $A$ - $B$ imprimitivity bimodule, Rieffel induction is an equivalence between the categories of nondegenerate representations of $B$ and $A$ . Induction through the conjugate bimodule $\widetilde E$ is a quasi-inverse. Consequently, strong Morita equivalence preserves the representation-theoretic structure encoded by ideals and irreducible representations Rieffel, §§5–6.

Examples and scope

For the identity $B$ - $B$ correspondence $E=B$ , the map $b\otimes h\mapsto\pi(b)h$ identifies $\operatorname{Ind}_B(\pi)$ with $\pi$ . For a full Hilbert $B$ -module $E$ , induction implements the equivalence between $B$ and $\mathcal K_B(E)$ .

References

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