Equivariant cyclic cocycles on the Boutet de Monvel symbol algebra

TL;DR

This paper constructs a cyclic cocycle on the symbol algebra of Boutet de Monvel operators, linking index formulas for boundary value problems to K-theory pairings, and extends this to equivariant cases involving group actions.

Contribution

It introduces a new cyclic cocycle on the Boutet de Monvel symbol algebra and its equivariant extension, providing a novel interpretation of index formulas in boundary value problems.

Findings

Established a cyclic cocycle on Boutet de Monvel's symbol algebra.

Connected index formulas to K-theory pairings via the cocycle.

Extended the construction to equivariant settings with group actions.

Abstract

We construct a periodic cyclic cocycle on the symbol algebra of Boutet de Monvel operators and use it to interpret the index formula for elliptic pseudodifferential boundary value problems due to Fedosov as the Chern--Connes pairing of the classes in $K$-theory of elliptic symbols with this cyclic cocycle. We also consider the equivariant case. Namely, we construct a periodic cyclic cocycle on the crossed product of the algebra of symbols with a group acting on this algebra by automorphisms. Such crossed products arize in index theory of nonlocal boundary value problems with shift operators.