Cyclic Cohomology of Projective Limits of Topological Algebras

Abstract

We present methods for the computation of the Hochschild and cyclic continuous cohomology and homology of some locally convex topological algebras. Let (Aα, T α,β ) (Λ, ) be a reduced projective system of complete Hausdorff locally convex algebras with jointly continuous multiplications, and let A be the projective limit algebra A = lim← α Aα. We prove that, for the continuous cyclic cohomology HC * and continuous periodic cohomology HP * of A and Aα, α ∈ Λ, for all n 0, HC n (A) = lim→ α HC n (Aα), the inductive limit of HC n (Aα), and, for k = 0, 1, HP k (A) = lim→ α HP k (Aα). For a projective limit algebra A = lim← m Am of a countable reduced projective system (Am, T m, ) N of Fréchet algebras, we also establish relations between the cyclic-type continuous homology of A and Am, m ∈ N. For example, we show the exactness of the following short sequence for all n 0:We present a class of Fréchet algebras A for which the continuous periodic cohomology HP k (A), k = 0, 1, is isomorphic to the continuous cyclic cohomology HC 2 +k (A) starting from some integer . We apply the above results to calculate the continuous cyclic-type homology and cohomology of some Fréchet locally m-convex algebras.

Twisted K-theory, K-homology, and bivariant Chern–Connes type character of some infinite dimensional spaces

“…Let {A n , f n } be a countable inverse system of topological algebras and continuous homomorphisms. The inverse system is called reduced if the canonical maps lim ← −n A n → A m have dense range for all m ∈ N. The periodic cyclic homology of the inverse limit A = lim ← −n A n of a reduced countable inverse system of Fréchet algebras can be computed from the following short exact sequence (see Theorem 5.4 of [43])…”

Hochschild cohomology of tensor products of topological algebras

“…Consider the Fréchet locally C * -algebra L(H) of continuous linear operators T on H that leave each H i invariant and satisfy T j P ij = P ij T j for all i < j, where T j = T | Hj : T j (η) = T (η) for η ∈ H j and P ij is the projection from H j onto H i . By [15,Example 6.6], for all n 0, we obtain H n (L(H), L(H)) = {0}.…”

Cyclic cohomology of certain nuclear Fréchet algebras and DF algebras

“…* , H c * and H p * are Hochschild homology, cyclic homology and periodic cyclic homology of the mixed complex (ΩA + ,b,B) in LCS, see [17].…”

“…For biflat Banach algebras A, Helemskii proved A = A 2 = Im π A [7, Proposition 7.2.6] and gave the description of the cyclic homology HC * and cohomology HC * groups of A in [8]. Later the author generalized Helemskii's result to inverse limits of biflat Banach algebras [17,Theorem 6.2] and to locally convex strict inductive limits of amenable Banach algebras [18,Corollary 4.9].…”