Adjoint functors and Hilbert modules

Operator Algebra Seminar by Tyrone Crisp, UCPH.

Blecher has shown that every suitably continuous functor between the categories of Hilbert modules over C*-algebras A and B is given by tensor product with an A-B bimodule. In the algebraic setting, every tensor-product functor has a right adjoint, given by Hom. In the C*-setting, the extra symmetry imposed by the *-operation means that a right adjoint is automatically a left adjoint as well, and this makes adjointability a far more restrictive property.

Results of Kajiwara, Pinzari and Watatani provide necessary and sufficient conditions for a bimodule to admit an adjoint functor. These conditions, which have to do with the finiteness of certain Jones-type indices, immediately rule out many naturally occurring examples. We will introduce a weaker notion of adjunction, specially tailored to Hilbert modules, which applies to more examples while retaining many of the desirable properties of adjoint functors. As an application, we obtain an adjoint (in the usual sense) to parabolic induction for tempered unitary representations of reductive Lie groups.

This is joint work with Pierre Clare and Nigel Higson.