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arXiv · 1605.06023

Frobenius reciprocity and the Haagerup tensor product

Tyrone Crisp
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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.

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Tyrone Crisp. 2017-02-07. Frobenius reciprocity and the Haagerup tensor product. https://arxiv.org/abs/1605.06023

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