[Paper Review] Group C*-algebras as compact quantum metric spaces

Marc A. Rieffel|ArXiv.org|May 17, 2002

Advanced Operator Algebra ResearchMathematics31 references18 citations

TL;DR

This paper establishes that group C*-algebras of ℤ^d equipped with a length function—either word-length or the restriction of a norm on ℝ^d—define compact quantum metric spaces via Connes' Dirac operator construction. Using the cosphere algebra and a novel metric compactification of metric spaces, the authors prove that the Lipschitz seminorm induces the weak-* topology on the state space, confirming the Lip-norm condition for these groups and their twisted versions, including non-commutative tori.

ABSTRACT

Let $\ell$ be a length function on a group $G$, and let $M_{\ell}$ denote the operator of pointwise multiplication by $\ell$ on $\bell^2(G)$. Following Connes, $M_{\ell}$ can be used as a ``Dirac'' operator for $C_r^*(G)$. It defines a Lipschitz seminorm on $C_r^*(G)$, which defines a metric on the state space of $C_r^*(G)$. We investigate whether the topology from this metric coincides with the weak-* topology (our definition of a ``compact quantum metric space''). We give an affirmative answer for $G = {\mathbb Z}^d$ when $\ell$ is a word-length, or the restriction to ${\mathbb Z}^d$ of a norm on ${\mathbb R}^d$. This works for $C_r^*(G)$ twisted by a 2-cocycle, and thus for non-commutative tori. Our approach involves Connes' cosphere algebra, and an interesting compactification of metric spaces which is closely related to geodesic rays.

Motivation & Objective

• To determine whether the metric on the state space of a group C*-algebra, induced by Connes' Dirac operator from a length function, coincides with the weak-* topology.
• To extend this analysis to twisted group C*-algebras, including non-commutative tori, via 2-cocycle deformations.
• To develop and apply a new compactification of metric spaces, termed the 'metric compactification', related to geodesic rays and Busemann points.
• To establish the Lip-norm property for ℤ^d with word-length or norm-induced length functions, ensuring the metric topology matches the weak-* topology.
• To explore the structure of the cosphere algebra and its role in verifying the compact quantum metric space condition.

Proposed method

• Define a Lipschitz seminorm Lℓ on the dense *-subalgebra Cc(G) of Cr*(G) using the commutator [Mℓ, πf], where Mℓ is multiplication by a length function ℓ on ℓ²(G).
• Construct a metric ρL on the state space S(Cr*(G)) via ρL(μ,ν) = sup{|μ(a)−ν(a)| : a ∈ Cc(G), Lℓ(a) ≤ 1}.
• Introduce the 'metric compactification' of a locally compact metric space, generalizing Gromov's construction, to analyze boundary behavior and geodesic rays.
• Use the action of G on the metric compactification and its boundary to study finite orbits and amenability, linking to the cosphere algebra.
• Apply the Stone–Weierstrass theorem to show that the algebra of continuous functions on the metric compactification equals the closure of the group algebra, implying the compactification is well-behaved.
• Verify that the cosphere algebra Sℓ* A is isomorphic to C*(G, C(∂hG), α) for F2, and use this to analyze the structure of the spectral triple.

Experimental results

Research questions

• RQ1 Does the metric on the state space of Cr*(ℤ^d) induced by the Dirac operator from a length function coincide with the weak-* topology?
• RQ2 Can the Lip-norm condition be verified for twisted group C*-algebras of ℤ^d, including non-commutative tori?
• RQ3 How does the metric compactification of a metric space relate to geodesic rays and Busemann points?
• RQ4 What is the role of the cosphere algebra in characterizing the metric structure of group C*-algebras?
• RQ5 Under what conditions does the action of G on the boundary of the metric compactification yield finite orbits, enabling the verification of the Lip-norm property?

Key findings

• For ℤ^d with a word-length function or the restriction of a norm on ℝ^d, the seminorm Lℓ defined by [Mℓ, πf] is a Lip-norm, so the metric topology on the state space coincides with the weak-* topology.
• The metric compactification of ℤ^d with such length functions coincides with the hyperbolic compactification, and the boundary consists of Busemann points corresponding to geodesic rays.
• The cosphere algebra for the spectral triple (Cr*(ℤ^d), ℓ²(ℤ^d), Mℓ) is isomorphic to C*(ℤ^d, C(∂hℤ^d), α), and the action on the boundary is amenable.
• The construction extends to twisted group C*-algebras Cr*(ℤ^d, c) for any 2-cocycle c, including non-commutative tori, and the Lip-norm property holds in these cases as well.
• The proof relies on showing that the algebra C(Ḡ^ℓ) of continuous functions on the metric compactification equals C(Ḡ^h), implying the compactification is topologically well-behaved and the boundary is rich enough to separate points.
• The method fails for F2 due to lack of finite orbits in the boundary action, leaving the Lip-norm property open for free groups.