nLab Kac algebra

Idea

Kac algebras are one of the standard setups for Tannaka duality involving functional analysis. In a spirit of generalizing Pontrjagin duality to nonabelian locally compact groups they generalize locally compact groups and their Tannaka duals in a framework of von Neumann algebras with additional structure where a multiplicative unitary plays a central role. They are named after George Kac.

Literature

The standard monograph on Kac algebras is

• M. Enock, J. M. Schwartz, Kac algebras and duality of locally compact groups, Springer-Verlag, 1992, , x+257 pp. gBooks, MR94e:46001

See also

• Takehiko Yamanouchi, An example of Kac algebra actions on Von Neumann algebra, Proc. Amer. Math. Soc.. 119, No. 2 (1993) 503-511 jstor
• M. Enock, L. Vaînerman, Deformation of a Kac algebra by an abelian subgroup, Commun.Math. Phys. 178 (1996) 571–595 doi
• Leonid Vaĭnerman, 2-cocycles and twisting of Kac algebras, Commun. Math. Phys. 191 (1998) 697-721 doi
• Teodor Banica, Compact Kac algebras and commuting squares, J. Funct. Anal. 176 (2000) 80-99 arXiv:math.QA/9909156

A related, but different setup of Hopf-von Neumann algebras is in

• L. I. Vaĭnerman, G. I. Kac, Nonunimodular ring groups and Hopf-von Neumann algebras,doi Math. USSR Sbornik 23:2 (1974) 185
• Shahn Majid, Hopf-von Neumann algebra bicrossproducts, Kac algebra bicrossproducts and

  classical Yang-Baxter equations_, J. Funct. Anal. 95 (1991) 291-319

category: algebra, functional analysis