The K-theory of graded rings

The topological K-theory of Z/2-graded C*-algebras plays an important role in operator theory. Historically, finding the definition was rather subtle, for several reasons. For example, simply taking the K-theory of a category of graded modules does not give the 'correct' definition, as I will explain. The definition of Z/2-graded C*-algebras by Karoubi is mostly outmoded because of Kasparov's KK-theory, whose technical definitions are more amenable to computations in index theory. The aim of this talk is to revive Karoubi's version by explaining how it is a natural higher-algebraic generalization of the classic Atiyah-Bott-Shapito paper on Clifford algebras. This approach also suggests a definition of K-theory for arbitrary Z/2-graded rings.