Twisted K-theory, old and new

Twisted K-theory has its origins in the author's PhD thesis [27] : 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 generalizations and 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 (math/0306138) and M.F. Atiyah, G. Segal (math/0407054), among many authors. The unifiyng theme in our presentation is the notion of K-theory of graded Banach algebras,implicit in [27], 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 (in the graded and ungraded cases).