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

Twisted K-theory has its origins in the author's PhD thesis [K1]:
http://www.numdam.org/item?id=ASENS_1968_4_1_2_161_0
and in the paper with P. Donovan
http://www.numdam.org/item?id=PMIHES_1970__38__5_0

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
http://anziamj.austms.org.au/JAMSA/V47/Part3/Rosenberg.html,
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 <max.karoubi@gmail.com>