Cyclic cohomology associated with difference operators and anticommutators
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.
cyclic cohomology associated with difference operators
difference operators on the algebras of polynomials of Laurent polynomials
scientific article; zbMATH DE number 4048450