Comparison between algebraic and topological K-theory of locally convex algebras

Abstract

This paper is concerned with the algebraic K-theory of locally convex algebras stabilized by operator ideals, and its comparison with topological K-theory. We show that the obstruction for the comparison map between algebraic and topological K-theory to be an isomorphism is (absolute) algebraic cyclic homology and prove the existence of an 6-term exact sequence. We show that cyclic homology vanishes in the case when J is the ideal of compact operators and L is a Frechet algebra with bounded app. unit. This proves the generalized version of Karoubi's conjecture due to Mariusz Wodzicki and announced in his paper "Algebraic K-theory and functional analysis", First European Congress of Mathematics, Vol. II (Paris, 1992), 485--496, Progr. Math., 120, Birkh\"auser, Basel, 1994. We also consider stabilization with respect to a wider class of operator ideals, called sub-harmonic. We study the algebraic K-theory of the tensor product of a sub-harmonic ideal with an arbitrary complex algebra and prove that the obstruction for the periodicity of algebraic K-theory is again cyclic homology. The main technical tools we use are the diffeotopy invariance theorem of Cuntz and the second author (which we generalize), and the excision theorem for infinitesimal K-theory, due to the first author.