Periodic Cyclic Cocycles on the Boutet de Monvel Symbol Algebra

Abstract

Boutet de Monvel constructed an algebra of boundary value problems for pseudodifferential operators on a manifold with boundary. We define periodic cyclic cocycles on the algebra of symbols of Boutet de Monvel operators. Cocycles of this form enable one to interpret the index formula for elliptic pseudodifferential boundary value problems in the Boutet de Monvel calculus due to Fedosov as the Chern–Connes pairing with the classes in $$K$$ -theory defined by elliptic symbols. We also consider the equivariant case. Namely, we construct periodic cyclic cocycles on the crossed product of the symbol algebra by a group acting on this algebra by automorphisms. Such crossed products arise in index theory of nonlocal boundary value problems with shift operators. DOI 10.1134/S1061920822040021