[Paper Review] Metrics on states from actions of compact groups

Marc A. Rieffel|ArXiv.org|Jul 16, 1998

Advanced Operator Algebra ResearchMathematics20 references157 citations

TL;DR

This paper establishes that metrics on the state space of a unital C*-algebra, derived from ergodic actions of compact Lie groups via Dirac operators or associated norms on the Lie algebra, induce the same topology as the weak-* topology. The key contribution is proving that the metric topology induced by a Dirac operator arising from such an action agrees with the standard weak-* topology on the state space, generalizing Connes' noncommutative metric framework to group-theoretic settings.

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.

Motivation & Objective

• To establish conditions under which metrics on the state space of a C*-algebra, derived from group actions, induce the weak-* topology.
• To extend Connes' noncommutative metric framework to C*-algebras with ergodic actions of compact Lie groups.
• To show that metrics defined via norms on the Lie algebra or length functions on the group yield topologies equivalent to the weak-* topology.
• To verify that the metric topology from a Dirac operator associated with such an action coincides with the weak-* topology.

Proposed method

• Define a Lipschitz semi-norm using the operator norm of the commutator [D, a] for a in the smooth subalgebra A^∞.
• Construct a Dirac operator D on a Hilbert space completion of A^∞ ⊗ S, where S is a spinor module, using the Lie algebra action and a representation on the Clifford algebra.
• Use an orthonormal basis of the dual Lie algebra to express D as a sum of operators involving the group action and Clifford multiplication.
• Establish a comparison between the Dirac-induced semi-norm L(a) = ||[D, λ_a]|| and the norm ||da|| from the Lie algebra action, proving both upper and lower bounds.
• Apply a comparison lemma and topological arguments to show that the metric topology induced by L agrees with the weak-* topology.
• Verify that the resulting metric is finite and well-defined on the state space, and that the topology it generates matches the standard weak-* topology.

Experimental results

Research questions

• RQ1 Does the metric topology induced by a Dirac operator on a C*-algebra with an ergodic compact group action agree with the weak-* topology?
• RQ2 Can metrics on the state space be constructed from norms on the Lie algebra of a compact group acting ergodically on a C*-algebra, and do they yield the weak-* topology?
• RQ3 How do length functions on a compact group relate to metrics on the state space of a C*-algebra under ergodic group action?
• RQ4 What is the relationship between the Lipschitz semi-norm defined via a Dirac operator and the intrinsic geometry of the group action?

Key findings

• The metric topology induced by the Dirac operator on the state space of a unital C*-algebra under an ergodic compact Lie group action agrees with the weak-* topology.
• A lower bound is established for the semi-norm L(a) = ||[D, a]|| in terms of the norm ||da|| from the Lie algebra, ensuring topological equivalence.
• The comparison lemma allows transferring topological control from the Lie algebra norm to the Dirac-induced metric.
• The result holds for any inner product on the dual of the Lie algebra, showing robustness of the construction.
• The construction applies to non-commutative tori, which carry ergodic torus actions, linking to string theory contexts.