K-THEORY FOR FRÉCHET ALGEBRAS

Abstract

We define K-theory for Fréchet algebras (assumed to be locally multiplicatively convex) so as to simultaneously generalize K-theory for σ-C*-algebras and K-theory for Banach algebras. The main results on K-theory of σ-C*-algebras, which are analogs of standard theorems on representable K-theory of spaces, carry over to the more general case. Our theory also gives the expected results in two other cases. If the invertible elements of a Fréchet algebra are an open set, as is the case for dense subalgebras of C*-algebras closed under holomorphic functional calculus, then our theory agrees with the result of applying the Banach algebra definition. For commutative unital Fréchet algebras, our K-theory is the same as the representable K-theory of the maximal ideal space with its compactly generated topology.