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A note on commutation relations and finite dimensional approximations

Fernando Lledó, Diego Martínez

Abstract

In this article we show that the main C*-algebras describing the canonical commutation relations of quantum physics, i.e., the Weyl and resolvent algebras, are in the class of Følner C*-algebras, a class of C*-algebras admitting a kind of finite approximations of Følner type. In particular, we show that the tracial states of the resolvent algebra are uniform locally finite dimensional.

A note on commutation relations and finite dimensional approximations

Abstract

In this article we show that the main C*-algebras describing the canonical commutation relations of quantum physics, i.e., the Weyl and resolvent algebras, are in the class of Følner C*-algebras, a class of C*-algebras admitting a kind of finite approximations of Følner type. In particular, we show that the tracial states of the resolvent algebra are uniform locally finite dimensional.
Paper Structure (5 sections, 8 theorems, 26 equations)

This paper contains 5 sections, 8 theorems, 26 equations.

Table of Contents

  1. Introduction
  2. Acknowledgements:
  3. Fø lner C*-algebras
  4. The Weyl algebra
  5. The resolvent algebra and finite dimensional approximations

Key Result

Lemma 2.3

Let ${\mathcal{A}}$ be a unital Fø lner C*-algebra and denote by $\varphi_n\colon{\mathcal{A}}\to M_{k(n)}(\mathbb{C})$ a net of c.c.p. maps which are asymptotically multiplicative in the $\|\cdot\|_{2,\mathrm{tr}}$-norm and asymptotically isometric in the operator norm. Define $e_n:=\varphi_n(\math

Theorems & Definitions (21)

  • Definition 2.1
  • Lemma 2.3
  • proof
  • Proposition 2.4
  • proof
  • Remark 2.5
  • Proposition 2.6
  • proof
  • Definition 2.7
  • Theorem 2.8
  • ...and 11 more