K for rings; spectral theory in Banach algebras; K for Banach algebras; Bott periodically and cyclic six-term exact sequence.
MATH 792 or MATH 792 and 960.
We will begin with the definition and basic properties of the functor K from rings to abelian groups, which associates to a ring R the abelian group whose elements are (roughly speaking) formal differences of isomorphism classes of finitely generated projective modules over R, with group operation modeled on direct summing of modules. Said another way, this group is the natural structure to consider of one wants to classify idempotent matrices over R in a reasonable way.
It turns out that K is best behaved and most readily calculated when the ring in question is a Banach algebra over the complex numbers. The reason for this is the existence is this setting of a companion functor K (roughly, invertible matrices modulo the connected component of the identity) which interacts usefully with K. After a tour of the relevant aspects of Banach algebra theory, we will develop K for Banach algebras and show how a short exact sequence of Banach algebras gives rise to a six-term cyclic exact sequence of K-groups. The course will conclude with some extended examples.
(All of the K's above, with the exception of the last, need subscripts. In order of appearance, starting with the summary in italics, these subscripts are: 0, 1, 0, 0, 1, 0, 1 .)
(Paschke 2007 )