Cyclic-type cohomology of strict inductive limits of Fréchet algebras

Abstract

We present methods for the computation of the Hochschild and cyclic-type continuous homology and cohomology of some locally convex strict inductive limits A = lim m → A m of Fréchet algebras A m . In the pure algebraic case it is known that, for the cyclic homology of A, HC n ( A ) = lim m → HC n ( A m ) for all n ⩾ 0 [Cyclic Homology, Springer, Berlin, 1992, E.2.1.1]. We show that, for a locally convex strict inductive system of Fréchet algebras ( A m ) m = 1 ∞ such that 0 → A m → A m + 1 → A m + 1 / A m → 0 is 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 A m , m ∈ N . For example, for the continuous cyclic cohomology HC n ( A ) and HC n ( A m ) , m ∈ N , we show the exactness of the following short sequence, for all n ⩾ 0 , 0 → lim m ← ( 1 ) HC n - 1 ( A m ) → HC n ( A ) → lim i ← HC n ( A m ) → 0 , where lim m ← ( 1 ) 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 A = lim m → A m which is a locally convex strict inductive limit of amenable Banach algebras A m , where for each m, A m is a closed ideal of A m + 1 .