Guillemin Transform and Toeplitz Representations for Operators on Singular Manifolds
V. Nazaikinskii, G. Rozenblum, A. Savin, B. Sternin
TL;DR
The paper develops a K-theory and cyclic cohomology framework for index formulas of elliptic operators on manifolds with conical singularities by treating the symbol algebra as the primary object. It extends the Toeplitz quantization paradigm to singular spaces via a generalized Guillemin transform, proving equivalence with the standard pseudodifferential quantization and producing an index formula expressed through the Chern–Connes character of the symbol algebra. It also introduces a resolution-based mechanism to generate new equivalent Toeplitz quantizations, both bounded and unbounded, enabling local index representations through positive spectral projections of self-adjoint operators and offering a path toward a Dirac-type operator framework in the singular setting.
Abstract
An approach to the construction of index formulas for elliptic operators on singular manifolds is suggested on the basis of K-theory of algebras and cyclic cohomology. The equivalence of Toeplitz and pseudodifferential quantizations, well known in the case of smooth closed manifolds, is extended to the case of manifolds with conical singularities. We describe a general construction that permits one, for a given Toeplitz quantization of a C^*-algebra, to obtain a new equivalent Toeplitz quantization provided that a resolution of the projection determining the original quantization is given.