Entire cyclic cohomology of \({\mathbb{Z}}/2\)-graded Banach algebras
Since \textit{A. Connes} introduced the concept of cyclic cohomology groups in his fundamental paper [Publ. Math., Inst. Hautes Etud. Sci. 62, 257--360 (1985; Zbl 0592.46056); see also C. R. Acad. Sci., Paris, Sér. I 296, 953--958 (1983; Zbl 0534.18009)], several variants and applications of the concept were proposed within the framework of K-theory (Kasparov KK-group), theory of p-summable Fredholm modules and so on. A \({\mathbb{Z}}_ 2\)-graded variant was developed by the author by means of the graded differential envelope \(\Omega(A)\) for a \({\mathbb{Z}}_ 2\)-graded k-algebra \(A\) [Operator algebras and mathematical physics, Proc. Summer. Conf., Iowa City/Iowa 1985, Contemp. Math. 62, 221--235 (1987; Zbl 0612.46048)]. The paper under review is a \({\mathbb{Z}}_ 2\)-analog of the theory of entire cyclic cohomology of B-algebras [\textit{A. Connes}, \(K\)-Theory 1, No. 6, 519--548 (1988; Zbl 0657.46049)]. The technique of the paper is based upon the notions of t-paratrace and t-parabracket of a \({\mathbb{Z}}_ 2\)-graded differential k-algebra \(\Omega =(\Omega^+\oplus \Omega^-,d)\), \(k={\mathbb{R}}\) or \({\mathbb{C}}\); a t-parabracket is defined as \[ \{w,W\}_ t:=[w,W]-t(-1)^{\partial w} dw dW\quad (t\in k) \] with \(w,W\in \Omega^+\cup \Omega^-\) and a grade \(\partial w\) for \(w\), and a t-paratrace \(\phi\) of \(\Omega\) is \(\phi \in \Omega^*\) such that \(\phi\) (\(\{\) \(\}_ t)=0\) for all t-parabrackets \(\{\) \(\}_ t\) (the t-paratraces may be treated as continuous functionals for an appropriate \(\pi\)-norm). The paper contains the routine consequences of the definitions in Appendix C (and includes for completeness three more ones) with an extension to a bigraded differential algebra such as just \(\Omega(A)\). The main results of the paper are the following: for unital \({\mathbb{Z}}_ 2\)-graded Banach algebra \(A\) there is a bijection between even (odd) part of ``normalized'' entire cocycles, which represent the entire cyclic cohomology of \(A\), and the corresponding 1/2-paratraces of \(\Omega(A)\). Also, it turns out that there is a connection of these paratraces with the ordinary traces of special algebras of J. Cuntz and R. Zekri.
scientific article; zbMATH DE number 4105888
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