process

Twisted K-theory – old and new 127identified with the bundle of endomorphisms End.S C ˚ S /, where S C and S arethe even and odd “spinors” associated to the Spin c -structure.In order to define graded twisted K-theory in the infinite-dimensional case, wefollow the same pattern as in §2. For instance, let us take a graded bundle of algebras Amodelled on M 2 .K/: it has 2 invariants, one in H 1 .XI Z=2/, the other in H 3 .XI Z/(and not just in the torsion part of this group). We then define K A .X/ as the gradedK-theory of the graded algebra .X;A/, according to §1. The same definition holdsfor bundles of graded algebras modelled on K K D KŒx=.x 2 1/ D K y˝ Kx.If C 0;1 is the Clifford algebra C C with its usual graded structure, the generalresults of §1 show that the group K A .X/ fits into an exact sequence:K .A y˝C 0;1 /1.X/ K.A/1.X/ K A .X/ K.A y˝C 0;1/ .X/ K .A/ .X/where K .A/i.X/ denotes in general the K i -group of the Banach algebra .X;A/.Let us now assume that A is oriented modelled on M 2 .K/ (which means that thestructural group of A may be reduced to PU 0 .H ˚ H/; see below or 3.2 in the finitedimensionalsituation). We are going to show that A may be written as M 2 .A 0 /, withthe obvious grading, A 0 being an ungraded bundle of algebras modelled on K. For thispurpose, we write the commutative diagram S1S 1 U.H/U.H/ U.H/PU.H / PU 0 .H ˚ H/where the first horizontal map is the identity and the others are induced by the diagonal.This shows that H 1 .XI PU.H // H 1 .XI PU 0 .H ˚ H//, which is equivalent tosaying that A may be written as M 2 .A 0 / for a certain bundle of algebras A 0 .Therefore, A y˝C 0;1 is Morita equivalent to A 0 A 0 and K A .X/ is the K-theory ofthe ring homomorphism (more precisely the functor defined by the associated extensionof the scalars, as we shall consider in other situations)A 0 A 0 M 2 .A 0 /defined by .a; b/ 7!0b a0 (no grading). We should also note that A 0 is Morita equivalentto A as an ungraded bundle of algebras. Since the usual K-theory (resp. gradedK-theory) is invariant under Morita equivalence (resp. graded Morita equivalence), theprevious considerations lead to the following theorem:Theorem 3.1. Let A be an oriented bundle of graded algebras modelled on M 2 .K/.Then K A .X/ is isomorphic to K .A/ .X/ via the identification above.