Karoubi defined K-theory classes given by Clifford-algebra module bundles, where a -module represents a class in and represents the trivial class if it extends to a -module.
We have a sequence of Clifford algebras which are generated by anticommuting square roots of . The sequence is periodic up to Morita equivalence; is , the algebra of real matrices, which is Morita equivalent to , and from then on it repeats every 8 with extra matrix dimensions thrown in.
Here we treat Clifford algebras as -graded algebras: while is Morita equivalent to as an algebra, it is not so as a graded algebra.
It turns out that can be represented geometrically by ‘bundles of Clifford modules’ over . Start with ; we know that elements of are ‘formal differences’ of vector bundles over . We can model the formal difference with an honest geometric object by using the -graded vector bundle , where is even and is odd. Such a thing should represent the zero class in K-theory just when and are isomorphic; this can be rephrased as saying that there exists an odd operator on (hence, taking to and vice versa) such that . But this just says that has an action of the first Clifford algebra .
More generally, Karoubi proved that for any , can be represented by -module bundles on modulo those such that the -action extends to a -action. When this is what we had above, since a -module is just a vector space. This allows us to deduce Bott periodicity for K-groups from the algebraic periodicity (up to Morita equivalence) of Clifford algebras.
Karoubi, Twisted K-theory old and new, (pdf)
Karoubi, Donovan, Graded Brauer groups and -theory with local coefficients (pdf)
Relation to twisted K-theory: