Good spectral triples, associated Lie groups of Campbell-Baker-Hausdorff type and unimodularity

Abstract

The notion of good spectral triple is initiated. We prove firstly that any regular spectral triple may be embedded in a good spectral triple, so that, in non-commutative geometry, we can restricts to deal only with good spectral triples. Given a good spectral triple K=(A,H,D), we prove that A is naturally endowed with a topology, called the K-topology, making it into an unital Frechet pre C*-algebra, and that the group Inv(A) of its invertible elements has a canonical structure of Frechet Lie group of Campbell-Baker-Hausdorff type open in its Lie algebra A; moreover, for any n>0 one has that K_n=(M_n(A), H\otimes C^n,D\otimes I_n) is still a good spectral triple. One deduces three important consequences.