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

Abstract

This paper is concerned with the algebraic K-theory of locally convex C -algebras stabilized by operator ideals, and its comparison with topological K-theory. We show that if L is locally convex and J a Fréchet operator ideal, then all the different variants of topological K-theory agree on the completed projective tensor product L ⊗ ˆ J , and that the obstruction for the comparison map K ( L ⊗ ˆ J ) → K top ( L ⊗ ˆ J ) to be an isomorphism is (absolute) algebraic cyclic homology. We prove the existence of an exact sequence (Theorem 6.2.1) ▪ We show that cyclic homology vanishes in the case when J is the ideal of compact operators and L is a Fréchet algebra whose topology is generated by a countable family of sub-multiplicative seminorms and admits an approximate right or left unit which is totally bounded with respect to that family (Theorem 8.3.3). This proves the generalized version of Karoubi's conjecture due to Mariusz Wodzicki and announced in his paper [M. Wodzicki, Algebraic K-theory and functional analysis, in: First European Congress of Mathematics, Vol. II, Paris, 1992, in: Progr. Math., vol. 120, Birkhäuser, Basel, 1994, pp. 485–496]. We also consider stabilization with respect to a wider class of operator ideals, called sub-harmonic. Every Fréchet ideal is sub-harmonic, but not conversely; for example the Schatten ideal L p is sub-harmonic for all p > 0 but is Fréchet only if p ⩾ 1 . We prove a variant of the exact sequence above which essentially says that if A is a C -algebra and J is sub-harmonic, then the obstruction for the periodicity of K ∗ ( A ⊗ C J ) is again cyclic homology (Theorem 7.1.1). This generalizes to all algebras a result of Wodzicki for H-unital algebras announced in [M. Wodzicki, Algebraic K-theory and functional analysis, in: First European Congress of Mathematics, Vol. II, Paris, 1992, in: Progr. Math., vol. 120, Birkhäuser, Basel, 1994, pp. 485–496]. The main technical tools we use are the diffeotopy invariance theorem of Cuntz and the second author (which we generalize in Theorem 6.1.6), and the excision theorem for infinitesimal K-theory, due to the first author.