Entire cyclic cohomology of Banach algebras and characters of \(\theta\)- summable Fredholm modules (Q1110781)

The author introduces in this article a new cyclic cohomology theory for Banach algebras. A cycle in this theory (an ``entire'' cyclic cocycle) is a cocycle in the total complex of the (B,b)-bicomplex (or of the bicomplex of Tsygan-Loday-Quillen) that does not necessarily have finite support but satisfies a certain growth condition with respect to the given Banach algebra norm. This allows to define the Chern character for a ``\(\theta\)-summable'' Fredholm module (H,D) over the Banach algebra A (i.e. H is a \({\mathbb{Z}}/2\)-graded Hilbert space, A is represented by operators of degree 0, D is of degree 1, [D,a] is bounded for \(a\in A\), and \(e^{-\theta D^ 2}\) is trace class for all \(\theta >0)\) in such a way that the index of D relative to a projection p in A is obtained by evaluating the cyclic cohomology class ch(D) representing the Chern character of D on the cyclic homology class ch(p) representing the Chern character of p. The notation of a \(\theta\)-summable module is a weakening of the finite summability condition fullfilled by elliptic operators on finite- dimensional compact manifolds. \(\theta\)-summable Fredholm modules arise in certain infinite-dimensional contexts. Entire cyclic cocycles correspond to traces on QA or EA which are continuous with respect to a certain topology. If T is such a trace and e a projection in A, then the author obtains the following interesting formula for the evaluation of the cocycle corresponding to T on the cycle corresponding to e (ch \(T| ch e)=T(Fe/\sqrt{1-(qe)^ 2})\) where F is the universal element of square 1 in EA.