Smooth *-Algebras

Looking for the universal covering of the smooth non-commutative torus leads to a curve of associative multiplications on the space O M (R) ∼= OC(R ) of Laurent Schwartz which is smooth in the deformation parameter ~. The Taylor expansion in ~ leads to the formal Moyal star product. The non-commutative 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. Table of contents 0. Introduction 1. Smooth ∗-algebras 2. The non-commutative torus 3. The smooth Heisenberg algebra 4. Appendix: Calculus in infinite dimensions and convenient vector spaces 0. Introduction The noncommutative torus in its topological version (C-completion) as well as in its smooth version [6] is one of the most important examples in noncommutative geometry. Beside the fact that the classical tools of differential geometry have unambiguous generalizations to it, it provides a very nontrivial example of noncommutative geometry satisfying the axioms of [7] (see also in [8], [9]). We looked at its smooth version and asked for its universal covering. We found the Heisenberg plane as it is presented in this paper: a twisted convolution on a carefully chosen space of distributions, namely the topological dual space O M of the Schwartz space OM 1991 Mathematics Subject Classification. 46L87, 46L60.