Frobenius reciprocity and the Haagerup tensor product

Abstract

In the context of operator-space modules over C ∗ C^* -algebras, we give a complete characterisation of those C ∗ 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 H of a locally compact group G G admits a left adjoint in this setting if and only if H H is cocompact in G G . The adjoint functor is given by the Haagerup tensor product with the operator-theoretic adjoint of Rieffel’s induction bimodule.