Simplicial Normalisation in the Entire Cyclic Cohomology of Banach Algebras

摘要

Entire cyclic cohomology was introduced by Connes as the de Rham type cohomology of a Banach algebra. It is calculated by means of a certain Z(2)-graded complex of cochains defined on A. The authors have then only even or odd cochains, even cochains being sequences of possibly infinite length (f(0), f(2), ..., f(2n), ...) where for each n, f(2n) is a 2n + 1-linear functional on A. Odd cochains are defined in a similar way. Inside this cochain complex there is a subcomplex of entire cochains that is of cochains satisfying certain growth conditions. A priori there are two ways to define this complex. The authors say that a cochain is simplicially normalized if for any n, f(2n)(a(0), a(1), ..., a(2n)) is zero whenever A(i) = 1 for some i = or > 1. The entire cyclic theory as introduced by Connes uses cochains which are not simplicially normalized in this sense, whereas there are other formulations of the same theory which are simplificially normalized. It is only natural to ask if the two formulations are equivalent. In the paper the authors show that the entire cyclic cohomology has the simplicial normalization property and therefore one can use either approach to define the theory.