## Metrics on states from actions of compact groups (1998)

Venue: | Doc. Math |

Citations: | 44 - 5 self |

### BibTeX

@ARTICLE{Rieffel98metricson,

author = {Marc A. Rieffel},

title = {Metrics on states from actions of compact groups},

journal = {Doc. Math},

year = {1998},

pages = {215--229}

}

### Years of Citing Articles

### OpenURL

### Abstract

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. Connes [Co1, Co2, Co3] has shown us that Riemannian metrics on non-commutative spaces (C ∗-algebras) can be specified by generalized Dirac operators. Although in this setting there is no underlying manifold on which one then obtains an ordinary metric, Connes has shown that one does obtain in a simple way an ordinary metric on the state space of the C ∗-algebra, generalizing the Monge-Kantorovich metric on probability measures [Ra] (called the “Hutchinson metric ” in the theory of fractals [Ba]). But an aspect of this matter which has not received much attention so far [P] is the question of when the metric topology (that is, the topology from the metric coming from a Dirac operator) agrees with the underlying weak- ∗ topology on the state space. Note that for locally compact spaces their topology agrees with the weak- ∗ topology coming from viewing points as linear functionals (by evaluation) on the algebra of continuous functions vanishing at infinity.