Equivariant cyclic cocycles on the Boutet de Monvel symbol algebra

Boutet de Monvel constructed an algebra of boundary value problems for pseudodifferential operators on manifolds with boundary. This algebra was studied a lot and it plays an important role in index theory. We construct periodic cyclic cocycles on the symbol algebra of Boutet de Monvel operators and use them 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 arise in index theory of nonlocal boundary value problems with shift operators. This is joint work with Andrei Boltachev. https://doi.org/10.48550/arXiv.2201.09987