Ergodic Actions and Spectral Triples

TL;DR

This paper constructs spectral triples from ergodic actions of compact Lie groups on C*-algebras, linking algebraic ergodicity with analytic properties like finite summability, and provides numerous examples including noncommutative tori.

Contribution

It introduces a general method to build spectral triples from ergodic group actions, ensuring properties like real structure, finite summability, and satisfying Connes' axioms.

Findings

Constructed spectral triples from ergodic actions of compact Lie groups.

Established essential self-adjointness of the unbounded operators involved.

Provided examples including noncommutative tori and quantum Heisenberg manifolds.

Abstract

In this article, we give a general construction of spectral triples from certain Lie group actions on unital C*-algebras. If the group G is compact and the action is ergodic, we actually obtain a real and finitely summable spectral triple which satisfies the first order condition of Connes' axioms. This provides a link between the "algebraic" existence of ergodic action and the "analytic" finite summability property of the unbounded selfadjoint operator. More generally, for compact G we carefully establish that our (symmetric) unbounded operator is essentially selfadjoint. Our results are illustrated by a host of examples - including noncommutative tori and quantum Heisenberg manifolds.