Noncommutative differential geometry, quantization, and smooth symmetries of the C*-algebras associated to topological dynamics

Abstract

Noncommutative differential geometric structures are considered for a class of simple C*-algebras. This structure is defined in terms of smooth Lie group actions on the C*-algebra in question together with a certain quantization mapping motivated directly by the known cohomological obstructions for the quantum mechanical quantization correspondence. We show that such a quantization mapping may be constructed for the C*-algebras associated to antisymmetric bi-characters and for the Cuntz/Cuntz-Krieger C*-algebras associated to topological dynamics. A certain curvature obstruction is defined in terms of the quantization mapping. It is shown that existence of smooth Lie group actions is determined by the curvature obstruction.