JournalsjncgVol. 14, No. 2pp. 487–529

Descent of Hilbert CC*-modules

  • Tyrone Crisp

    Radboud University Nijmegen, The Netherlands, and University of Maine, Orono, USA
Descent of Hilbert $C$*-modules cover

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Abstract

Let FF be a right Hilbert CC*-module over a CC*-algebra BB, and suppose that FF is equipped with a left action, by compact operators, of a second CC*-algebra AA. Tensor product with FF gives a functor from Hilbert CC*-modules over AA to Hilbert CC*-modules over BB. We prove that under certain conditions (which are always satisfied if, for instance, AA is nuclear), the image of this functor can be described in terms of coactions of a certain coalgebra canonically associated to FF. We then discuss several examples that fit into this framework: parabolic induction of tempered group representations; Hermitian connections on Hilbert CC*-modules; Fourier (co)algebras of compact groups; and the maximal CC*-dilation of operator modules over non-self-adjoint operator algebras.

Cite this article

Tyrone Crisp, Descent of Hilbert CC*-modules. J. Noncommut. Geom. 14 (2020), no. 2, pp. 487–529

DOI 10.4171/JNCG/371