Frobenius reciprocity and the Haagerup tensor product

Tyrone Crisp
2018 Transactions of the American Mathematical Society  
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
more » ... 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.
doi:10.1090/tran/7203 fatcat:uaud5ylh3zawrghdh326tjkg44