Cyclic cohomology associated with difference operators and anticommutators
type: Item
instance of: scholarly article
title: Cyclic cohomology associated with difference operators and anticommutators
zbMATH Open document ID: 0643.47038
DOI: 10.1007/BF01195800
published in: Integral Equations and Operator Theory
publication date: 1987-01-01T00:00:00Z
review text: The author has studied the cyclic cohomology associated with difference operators and obtained four theorems. Theorem 1. Let \({\mathcal A}\) be an algebra in \(L(H)\), \(\delta\) be a difference operator on \({\mathcal A}\), satisfying \(\delta\) \(f\in L^ p(H)\), \(f\in\mathcal A\). Let \[ \psi_ n(f_ 0,...,f_ n)=\text{Tr}(\delta (f_ 0)...(f_ n)),\quad n-1\geq p \] for \(f_ 0,...,f_ n\in\mathcal A\). Then \(\psi_ n\) is in the cyclic cohomology class containing \(bR_{n-2}\psi_{n-2}\), where \(n\geq p+2\) and \(n\) is even. The corollary of Theorem 1 shows that under the condition of Theorem 1, if \(p=1\) and \(n\) is even, then the \(n\)-th cyclic cohomology class is determined by \(\psi_ 0(f_ 0f_ 1...f_ n).\) The author next considers the difference operators on the algebras of polynomials of Laurent polynomials \(p(A,B)\) of operators \(A,B\in L(H)\) satisfying \(\{A,B\}=AB+BA\in L^ 1(H)\). Under some conditions, the trace formula for \(\{p(A,B),q(A,B)\}\) and the concrete form of cyclic cocycles are given by the theorems 2 and 3. By Theorem 4, the cyclic cohomology associated with the difference operator defined by invertible weak contraction is determined.
Mathematics Subject Classification ID: 47B39
Mathematics Subject Classification ID: 47B47
Mathematics Subject Classification ID: 46M20
zbMATH DE Number: 4048450
zbMATH Keywords: anticommutation
zbMATH Keywords: cyclic cohomology associated with difference operators
zbMATH Keywords: difference operators on the algebras of polynomials of Laurent polynomials
zbMATH Keywords: trace formula
zbMATH Keywords: cyclic cocycles
zbMATH Keywords: invertible weak contraction
author: Daoxing Xiao
MaRDI profile type: Publication
cites work: Toeplitz type operators, determining functions, principal functions, and trace formulas
cites work: Anti-commutators and the related trace formulas
description: scientific article; zbMATH DE number 4048450