Clifford Modules and Twisted K-theory

The setting is the following: V is a real vector bundle on a compact space X, provided with a non degenerate quadratic form to which we associate a bundle of (real or complex) Clifford algebras denoted by C(V ); the quadratic form is implicit in this notation. We denote by M(V ) the Grothendieck group associated to the category of (real or complex) vector bundles provided with a structure of (twisted) Z/2-graded C(V )-module. Another way to describe M(V ) is to consider the bundle V ⊕1, where the symbol “1” denotes the trivial vector bundle of rank one with a positive quadratic form. Then M(V ) is just the Grothendieck group K(Λ1) of the category P(Λ1) whicho objects are finitely generated projective modules over Λ1. The notation Λn means in general Λ⊗̂C, where Λ is the ring of continuous sections of the Z/2-graded bundle C(V ) and C is the Clifford algebra of R with a positive quadratic form.