Excision of equivariant cyclic cohomology of topological algebras
The author defines an equivariant cyclic (co)homology theory for Fréchet algebras with continuous action of a compact group. It is defined by a twisted version of the Loday-Quillen bicomplex and is supposed to be the target of a Chern character from equivariant \(K\)- theory. (For example, the equivariant cyclic homology of \(\mathbb{C}\) in degree zero equals the space of class functions on \(G\).) In this context an equivariant Hochschild theory, an equivariant periodic cyclic theory and a Connes-Gysin exact sequence relating equivariant cyclic- and equivariant Hochschild-(co)homology are developed. It is shown that an equivariant \(H\)-unitality condition for a \(G\)-algebra \(J\) (acyclicity of a twisted bar-construction) implies excision, i.e. the existence of a six-term exact sequence for the equivariant periodic cyclic (co)homology groups of the algebras in an equivariant extension with equivariant linear splitting and kernel equal to \(J\). Examples of equivariantly \(H\)- unital algebras are \(G\)-algebras with isometric \(G\)-action (in a strong sense) possessing a uniformly bounded (left) approximate unit as is explained at the end of the paper.
scientific article; zbMATH DE number 222031