Smooth \(\star\)-algebras. (English) Zbl 1026.46062

Dubois-Violette, Michel; Kriegl, Andreas; Maeda, Yoshiaki; Michor, Peter W.

Maeda, Yoshiaki (ed.) et al., Noncommutative geometry and string theory. Proceedings of the international workshop, Keio Univ., Yokohama, Japan, March 16-22, 2001. Kyoto: Progress of Theoretical Physics, Prog. Theor. Phys., Suppl. 144, 54-78 (2001).

Summary: Looking for the universal covering of the smooth noncommutative torus leads to a curve of associative multiplications on the space \({\mathcal O}_M'(\mathbb{R}^{2n}) \cong {\mathcal O}_C(\mathbb{R}^{2n})\) of L. Schwartz which is smooth in the deformation parameter \(\hbar\). The Taylor expansion in \(\hbar\) 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.
For the entire collection see [Zbl 0989.00062].

MSC:

46L65	Quantizations, deformations for selfadjoint operator algebras
46L05	General theory of \(C^*\)-algebras
46L85	Noncommutative topology

Keywords:

noncommutative torus; Moyal star product; Heisenberg plane