Frobenius reciprocity and the Haagerup tensor product

Crisp, Tyrone

doi:10.1090/tran/7203

Frobenius reciprocity and the Haagerup tensor product
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Trans. Amer. Math. Soc. 370 (2018), 6955-6972 Request permission

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 the Haagerup tensor product with the operator-theoretic adjoint of Rieffel’s induction bimodule.

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