So today we discussed the multiplicative axiom, the last (and most important) of the three “B” axioms that characterize the analytic index.   This axiom says (in its simplest version) that if \(M=X\times Y\) is a product of manifolds and \(a\in K(TX)\), \(b\in K(TY)\), then

\[ \newcommand{\Index}{{\rm Index}} \Index(ab) = \Index(a)\Index(b). \]

In class we discussed the case where the symbols \(a,b\) are represented by differential operators and, following the original paper, we argued using the “double complex” definition of the product on K-theory.  (Jinpeng’s notes, which I’ll post in a moment, give an account of what we said in class.)  Here I’d like to explain how the formalism of \(\newcommand{\Z}{{\mathbb Z}} \Z_2\)-graded operators and vector spaces allows one to make the algebra more concise and conceptual.   (This then becomes part of the formalism that is taken over for Kasparov’s KK-theory.) Continue reading →