On a class of closed cocycles for algebras of non-formal, possibly unbounded, pseudodifferential operators

TL;DR

This paper constructs closed cocycles for algebras of non-formal pseudodifferential operators on S^1, extending known cocycles like the Schwinger cocycle to broader classes of operators.

Contribution

It introduces a method to generate 2k-closed cocycles for these algebras using Chern-Weil forms, generalizing existing cocycles such as the Schwinger cocycle.

Findings

Constructed 2k-closed cocycles for pseudodifferential operator algebras.

Extended the cohomology class of the Schwinger cocycle to non-classical operators.

Provided a new framework for analyzing algebraic structures of pseudodifferential operators.

Abstract

In this article, we consider algebras $\mathcal{A}$ of non-formal pseudodifferential operators over $S^1$ which contain $C^\infty(S^1),$ understood as multiplication operators. We apply a construction of Chern-Weil type forms in order to get $2k-$closed cocycles. For $k=1,$ we obtain a cocycle on the algebra of (maybe non classical) pseudodifferential operators with the same cohomology class as the Schwinger cocycle on the algebra of Classical pseudodifferential operators, previously extended and studied by the author on algebras of the same type.