Frobenius reciprocity and the Haagerup tensor product

TL;DR

This paper characterizes when certain tensor product functors in operator space modules over C*-algebras have left adjoints, linking operator theory with group representation induction and establishing a Frobenius reciprocity theorem.

Contribution

It provides a complete characterization of C*-correspondences with adjoint functors and applies this to prove a Frobenius reciprocity theorem for group representations on operator spaces.

Findings

The functor of unitary induction admits a left adjoint if and only if the subgroup is cocompact.

The left adjoint is given by Haagerup tensor product with the operator-theoretic adjoint.

Established a connection between adjoint operators and adjoint functors in this setting.

Abstract

In the context of operator-space modules over C*-algebras, we give a complete characterisation of those C*-correspondences whose associated Haagerup tensor product functors admit left adjoints. The characterisation, which builds on previous joint work with N. Higson, exhibits a close connection between the notions of adjoint operators and adjoint functors. As an application, we prove a Frobenius reciprocity theorem for representations of locally compact groups on operator spaces: the functor of unitary induction for a closed subgroup H of a locally compact group G admits a left adjoint in this setting if and only if H is cocompact in G. The adjoint functor is given by Haagerup tensor product with the operator-theoretic adjoint of Rieffel's induction bimodule.