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Domains of quantum metrics on AF algebras

Konrad Aguilar, Katrine von Bornemann Hjelmborg, Frederic Latremoliere

Abstract

Given a compact quantum metric space (A, L), we prove that the domain of L coincides with A if and only if A is finite dimensional. We then show how one can explicitly build many quantum metrics with distinct domains on infinite-dimensional AF algebras. In the last section, we provide a strategy for calculating the distance between certain states in these quantum metrics, which allow us to calculate the distance between pure states in these quantum metrics on the quantized interval and on the Cantor space.

Domains of quantum metrics on AF algebras

Abstract

Given a compact quantum metric space (A, L), we prove that the domain of L coincides with A if and only if A is finite dimensional. We then show how one can explicitly build many quantum metrics with distinct domains on infinite-dimensional AF algebras. In the last section, we provide a strategy for calculating the distance between certain states in these quantum metrics, which allow us to calculate the distance between pure states in these quantum metrics on the quantized interval and on the Cantor space.
Paper Structure (3 sections, 14 theorems, 50 equations)

This paper contains 3 sections, 14 theorems, 50 equations.

Table of Contents

  1. Introduction
  2. Domain and dimension
  3. Explicit calculations of some quantum metrics

Key Result

Theorem 1.4

Aguilar-Latremoliere15 Let $A=\overline{\cup_{n \in {\mathds{N}}}A_n}^{\|\cdot\|_A}$ be a unital AF-algebra equipped with a faithful tracial state $\tau$ such that $A_0={\mathds{C}}1_A$ and $1_A\in A_n$ for every $n \in {\mathds{N}}$. For each $n \in {\mathds{N}}$, let denote the unique $\tau$-preserving conditional expectation onto $A_n$ (that is, $\tau \circ E_n=\tau$ for all $n \in {\mathds{N}

Theorems & Definitions (30)

  • Definition 1.1
  • Definition 1.2: Rieffel98a
  • Definition 1.3: Brown-Ozawa
  • Theorem 1.4
  • Theorem 2.1
  • proof
  • Corollary 2.2
  • proof
  • Lemma 2.3
  • proof
  • ...and 20 more