Twisted K-theory, old and new, by Max Karoubi

Twisted K-theory has its origins in the author's PhD thesis [K1]: and in the paper with P. Donovan

The objective of this paper is to revisit the subject in the light of new developments inspired by Mathematical Physics. See for instance E. Witten (hep-th/9810188), J. Rosenberg, C. Laurent-Gentoux, J.-L. Tu, P. Xu (ArXiv math/0306138) and M.F. Atiyah, G. Segal (ArXiv math/0407054), among many authors.

The unifiyng theme in our presentation is the notion of K-theory of graded Banach algebras,implicit in [K1], from which most of the classical theorems in twisted K-theory are derived.

We also prove some new results in the subject: a Thom isomorphism in this setting, explicit computations in the equivariant case and new cohomology operations. Incidentally, the version of K-theory developed by M.F. Atiyah and M. Hopkins in ArXiv math/0302128 is already present in [K1,section 3.3]. See section 6.16 of the paper or K-theory preprint Nr 424 for more details.

Max Karoubi <>