[Paper Review] C*-algebraic quantum Gromov-Hausdorff distance

Hanfeng Li|ArXiv.org|Nov 28, 2003

Advanced Operator Algebra Research40 references19 citations

TL;DR

This paper introduces a $C^*$-algebraic quantum Gromov-Hausdorff distance that distinguishes the multiplicative structures of $C^*$-algebras—addressing a key limitation of Rieffel’s original quantum distance. By incorporating the algebraic structure via a generalized Leibniz rule, the new distance extends the classical Gromov-Hausdorff distance, ensures quantum completeness and compactness theorems, and provides criteria for continuity of parameterized families of compact quantum metric spaces.

ABSTRACT

We introduce a new quantum Gromov-Hausdorff distance between C*-algebraic compact quantum metric spaces. Because it is able to distinguish algebraic structures, this new distance fixes a weakness of Rieffel's quantum distance. We show that this new quantum distance has properties analogous to the basic properties of the classical Gromov-Hausdorff distance, and we give criteria for when a parameterized family of C*-algebraic compact quantum metric spaces is continuous with respect to this new distance.

Motivation & Objective

• To address the limitation of Rieffel’s quantum Gromov-Hausdorff distance, which fails to distinguish non-isomorphic $C^*$-algebras due to its reliance solely on state spaces.
• To develop a quantum distance that incorporates the multiplicative structure of $C^*$-algebras, ensuring algebraic distinctions are preserved.
• To establish a quantum version of Gromov’s completeness and compactness theorems within the $C^*$-algebraic framework.
• To provide criteria for the continuity of parameterized families of $C^*$-algebraic compact quantum metric spaces with respect to the new distance.
• To demonstrate that known convergence results—such as noncommutative tori and matrix algebras converging to coadjoint orbits—hold under this new distance.

Proposed method

• Define the $C^*$-algebraic quantum Gromov-Hausdorff distance as a modified Gromov-Hausdorff distance directly on the order-unit spaces (or $C^*$-algebras), rather than on their state spaces.
• Utilize a generalized Leibniz rule to ensure compatibility between the Lipschitz seminorm and the $C^*$-algebraic structure, which is essential for distinguishing algebraic isomorphism types.
• Construct the distance using balls in the order-unit space equipped with a Lipschitz seminorm derived from a group action and a length function on a compact group $G$.
• Apply the theory of continuous fields of $C^*$-algebras and strongly continuous group actions to define continuous families of quantum metric spaces.
• Establish continuity criteria based on the convergence of multiplicity functions of irreducible representations in the dual group $\hat{G}$ of a compact group $G$.
• Prove that convergence of multiplicity functions $\mathrm{mul}({\mathcal{A}}_t, \gamma) \to \mathrm{mul}({\mathcal{A}}_{t_0}, \gamma)$ for all $\gamma \in \hat{G}$ implies convergence in the $C^*$-algebraic quantum distance.

Experimental results

Research questions

• RQ1 Can a quantum Gromov-Hausdorff distance be defined directly on $C^*$-algebras that distinguishes non-isomorphic algebras based on their multiplicative structure?
• RQ2 Does the new distance extend the classical Gromov-Hausdorff distance in the same way as Rieffel’s original construction?
• RQ3 What conditions ensure that limits of $C^*$-algebraic compact quantum metric spaces under this new distance remain $C^*$-algebras?
• RQ4 Under what conditions is a parameterized family of $C^*$-algebraic compact quantum metric spaces continuous with respect to the new distance?
• RQ5 Do known convergence results in noncommutative geometry—such as noncommutative tori and matrix algebras converging to coadjoint orbits—hold under this new distance?

Key findings

• The $C^*$-algebraic quantum Gromov-Hausdorff distance successfully distinguishes non-isomorphic $C^*$-algebras by incorporating their multiplicative structure, unlike Rieffel’s original distance.
• The new distance satisfies a quantum version of Gromov’s completeness and compactness theorems, provided the Lipschitz seminorms satisfy a generalized Leibniz rule.
• A parameterized family of $C^*$-algebraic compact quantum metric spaces is continuous with respect to the new distance if and only if the multiplicity functions of irreducible representations converge at each point.
• The continuity of noncommutative tori and matrix algebras converging to integral coadjoint orbits of compact semisimple Lie groups is confirmed under this new distance.
• The set of points in a compact metric space $T$ where the family $({\mathcal{A}}_t, L_t)$ fails to be continuous is a nowhere dense $F_\sigma$ set, implying continuity holds at residual points.
• The $C^*$-algebraic quantum distance is equivalent to the order-unit quantum distance introduced in prior work, but with the added advantage of algebraic structure sensitivity.