Locally Noncommutative Spacetimes

Abstract

A rather new physically well-motivated notion of locally noncommutative spacetimes–a refinement of the notion noncommutative spacetimes–is reviewed and discussed. It is shown that the latter model can be realized in the framework of formal deformation quantization using star products as well as in the framework of Rieffel's strict deformation quantization. For the convergent setting Rieffel's former results for $C^*$-algebras are generalized to pro-$C^*$-algebras and applied to actions well-suited to the idea of locally noncommutative spacetimes.