K-théorie multiplicative et homologie cyclique. (Multiplicative K- theory and cyclic homology). (French) Zbl 0601.18007

Let A be a locally convex topological algebra over a commutative ring containing the rational numbers. The author defines multiplicative K- theory groups \({\mathcal K}_ r(A)\). Under suitable circumstances there is an exact sequence involving \({\mathcal K}_ r(A)\), the topological K-theory \(K_ r^{top}(A)\), and the cyclic homology of A. (The author gives a definition of cyclic homology using non-commutative differential forms.) Let \(K_ r(A)\) be the Quillen algebraic K-theory of A. Then the homomorphism \(K_ r(A)\to K_ r^{top}(A)\) factors through \({\mathcal K}_ r(A)\).
Reviewer: R.J.Steiner


18F25 Algebraic \(K\)-theory and \(L\)-theory (category-theoretic aspects)
55N15 Topological \(K\)-theory
46L80 \(K\)-theory and operator algebras (including cyclic theory)