AF algebras in the quantum Gromov-Hausdorff propinquity space

Abstract

For unital AF algebras with faithful tracial states, we provide criteria for convergence in the quantum Gromov-Hausdorff propinquity. For any unital AF algebra, we introduce new Leibniz Lip-norms using quotient norms. Next, we show that any unital AF algebra is the limit of its defining inductive sequence of finite dimensional C*-algebras in quantum propinquity given any Lip-norm defined on the inductive sequence. We also establish sufficient conditions for when a *-isomorphism between two AF algebras produces a quantum isometry in the context of various Lip-norms.

32 pages. Some clarifications made - especially in section 4. This article, in part, generalizes results from arXiv:1511.07114 and was originally part of arXiv:1608.07016v1, which was split due to length

References in corpus (2)

• Spectral triples for AF C*-algebras and metrics on the Cantor set
• Convergence of Quotients of AF Algebras in Quantum Propinquity by Convergence of Ideals

Cited by in corpus (1)

• Convergence of Quotients of AF Algebras in Quantum Propinquity by Convergence of Ideals