Sequences of operator algebras converging to odd spheres in the quantum Gromov-Hausdorff distance

TL;DR

This paper constructs a sequence of Toeplitz algebras on generalized Bergman spaces and proves their convergence to the algebra of continuous functions on odd spheres using quantum Gromov-Hausdorff distance, extending previous work on quantum metric spaces.

Contribution

It introduces a new quantum metric space structure on Toeplitz algebras and demonstrates their convergence to odd spheres, expanding the class of quantum spaces approximated by matrix algebras.

Findings

Sequence of Toeplitz algebras converges to odd spheres in quantum Gromov-Hausdorff distance.

Provides a new example of quantum metric space convergence.

Extends Rieffel's approximation results to odd spheres.

Abstract

Marc Rieffel had introduced the notion of the quantum Gromov-Hausdorff distance on compact quantum metric spaces and found a sequence of matrix algebras that converges to the space of continuous functions on $2$-sphere in this distance. One finds applications of similar approximations in many places in the theoretical physics literature. In this paper, we have defined a compact quantum metric space structure on the sequence of Toeplitz algebras on generalized Bergman spaces and have proved that the sequence converges to the space of continuous function on odd spheres in the quantum Gromov-Hausdorff distance.