A vanishing theorem for operations on entire cyclic cohomology
Derivations of an algebra \(A\) act on the Hochschild and cyclic homology of \(A\), and the induced action on periodic cyclic homology is trivial. \textit{E. Getzler} in ``Cartan homotopy formula and the Gauss-Manin connection'' (to appear) extended this action of derivations to a Lie operation of the Hochschild cohomology \(H^* (A,A)\) on the \((b,B)\)- bicomplex of \(A\) and showed that the induced operation on the periodic cyclic homology is trivial. This work is devoted to establish a similar vanishing theorem for entire cyclic cohomology: \[ -\qquad \text{The pairing } H^* (A,A) \otimes HE^* (A) \to HE^* (A) \text{ is trivial.} \] The proof uses a representation theorem of \textit{A. Connes} ``Entire cyclic cohomology of Banach algebras and characters of \(\theta\)- summable Fredholm modules'' [ibid. 1, No. 6, 519-548 (1988; Zbl 0657.46049)].
scientific article; zbMATH DE number 682607