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Guillemin transform and Toeplitz representations for operators on singular manifolds. (English) Zbl 1071.46039
Booß-Bavnbek, Bernhelm (ed.) et al., Spectral geometry of manifolds with boundary and decomposition of manifolds. Proceedings of the workshop, Roskilde, Denmark, August 6–9, 2003. Providence, RI: American Mathematical Society (AMS) (ISBN 0-8218-3536-X/pbk). Contemporary Mathematics 366, 281-306 (2005).
Summary: A new approach to the construction of index formulas for elliptic operators on singular manifolds is suggested on the basis of the $$K$$-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.
For the entire collection see [Zbl 1057.58001].

##### MSC:
 46L87 Noncommutative differential geometry 46L80 $$K$$-theory and operator algebras (including cyclic theory) 58J40 Pseudodifferential and Fourier integral operators on manifolds