Metrics on states from actions of compact groups

TL;DR

This paper explores various methods to define metrics on the state space of a $C^*$-algebra using compact Lie group actions, ensuring these metrics align with the weak-$*$ topology.

Contribution

It introduces multiple approaches to metric construction on state spaces via group actions and verifies their topological equivalence with the weak-$*$ topology.

Findings

Metrics from length functions, norms, and Dirac operators induce the same topology as weak-$*$.

The methods unify different metric constructions under a common topological framework.

Results facilitate analysis of state spaces in noncommutative geometry.

Abstract

Let a compact Lie group act ergodically on a unital $C^*$-algebra $A$. We consider several ways of using this structure to define metrics on the state space of $A$. These ways involve length functions, norms on the Lie algebra, and Dirac operators. The main thrust is to verify that the corresponding metric topologies on the state space agree with the weak-$*$ topology.