Smooth *-algebras

Abstract

Looking for the universal covering of the smooth noncommutative torus leads to a curve of associative multiplications on the space O′M (ℝ2n) ≅ OC(ℝ2n) of L. Schwartz which is smooth in the deformation parameter (latin small letter h with stroke). The Taylor expansion in (latin small letter h with stroke) leads to the formal Moyal star product. The noncommutative torus and this version of the Heisenberg plane are examples of smooth *-algebras: smooth in the sense of having many derivations. A tentative definition of this concept is given.