Abstract :
We present methods for the computation of the Hochschild and cyclic-type continuous homology and cohomology of some locally convex strict inductive limits image of Fréchet algebras Am. In the pure algebraic case it is known that, for the cyclic homology of A, image for all ngreater-or-equal, slanted0 [Cyclic Homology, Springer, Berlin, 1992, E.2.1.1]. We show that, for a locally convex strict inductive system of Fréchet algebras image such that0→Am→Am+1→Am+1/Am→0is topologically pure for each m and for continuous Hochschild and cyclic homology, similar formulas hold. For such strict inductive systems of Fréchet algebras we also establish relations between the continuous cohomology of A and Am, mset membership, variantN. For example, for the continuous cyclic cohomology image and image, mset membership, variantN, we show the exactness of the following short sequence, for all ngreater-or-equal, slanted0,imagewhere image is the first derived functor of the projective limit. We give explicit descriptions of continuous periodic and cyclic homology and cohomology of a LF-algebra image which is a locally convex strict inductive limit of amenable Banach algebras Am, where for each m, Am is a closed ideal of Am+1.
