nLab Hopf C-star-algebra

under construction

1. Idea

A Hopf $C^\ast$-algebra [Vaes & Van Dale 2001] is a C-star algebra equipped with structure and property analogous to that of a Hopf algebra structure on the underlying associative algebra.

A weak $C^\ast$-Hopf algebra according to Böhm & Szlachanyi is a star-weak Hopf algebra such that has a faithful star-representation on a Hilbert space.

With suitable definitions, the central Tannaka duality-property of Hopf algebras (that their representation category is a rigid monoidal category with fiber functor) is lifted to the operator algebra context: the $C^\ast$-representation category of a (weak) $C^\ast$-Hopf algebra is a rigid monoidal C-star-category with fiber functor. (Böhm-Szlachanyi).

2. References

On $C^\ast$ algebras equipped with a suitable coproduct, but without an antipode (hence just $C^\ast$-bialgebras):

• S. Baaj, Georges Skandalis, Unitaires multiplicatifs et dualité pour les produits croisés de $C^\ast$-algèbres. Ann. scient. Ec. Norm. Sup., 4e série, 26 (1993), 425–488.

• J.-M. Vallin, $C^\ast$-algèbres de Hopf et $C^\ast$-algèbres de Kac. Proc. London Math. Soc. (3)50 (1985), 131–174.

On the issue of how to add the definition of the antipode in the $C^\ast$-context:

• Stefaan Vaes, Alfons Van Daele: Hopf $C^\ast$-algebras, Proc. London Math. Soc. 82 3 (2001) 337-384 [arXiv:math/9907030, doi:10.1112/S002461150101276X]

Weak $C^\ast$-Hopf algebras and their C-star categories of representations are discussed in

• Gabi Böhm, Kornel Szlachanyi: A Coassociative $C^\ast$-Quantum Group with Non-Integral Dimensions [arXiv:q-alg/9509008]

  Weak $C^\ast$-Hopf algebras [pdf]