## Frobenius reciprocity and the Haagerup tensor product

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

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## Additional Information

**Tyrone Crisp**- Affiliation: Max Planck Institute for Mathematics, Vivatsgasse 7, 53111 Bonn, Germany
- Address at time of publication: Department of Mathematics, Radboud University Nijmegen, P.O. Box 9010, 6500GL Nijmegen, The Netherlands
- MR Author ID: 782294
- Email: t.crisp@math.ru.nl
- Received by editor(s): October 18, 2016
- Received by editor(s) in revised form: February 3, 2017
- Published electronically: May 30, 2018
- © Copyright 2018 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**370**(2018), 6955-6972 - MSC (2010): Primary 46M15; Secondary 22D30, 46L07
- DOI: https://doi.org/10.1090/tran/7203
- MathSciNet review: 3841838