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

TL;DR

This paper introduces the concept of good spectral triples in non-commutative geometry, showing their embedding properties, topological structure, and associated Lie groups of Campbell-Baker-Hausdorff type, with implications for spectral triple stability.

Contribution

It defines good spectral triples, proves their universal embedding property, and establishes the Lie group structure of invertible elements in the associated algebra.

Findings

Any regular spectral triple can be embedded into a good spectral triple.

The algebra in a good spectral triple has a natural Frechet topology making it a pre C*-algebra.

The invertible elements form a Frechet Lie group of Campbell-Baker-Hausdorff type.

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.