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On the commutativity of states in von Neumann algebras

Andrzej Łuczak

Abstract

The notion of commutativity of two normal states on a von Neumann algebra was defined some time ago by means of the Pedersen-Takesaki theorem. In this note we aim at generalizing this notion to an arbitrary number of states, and obtaining some results on so defined joint commutativity. Also relations between commutativity and broadcastability of states are investigated.

On the commutativity of states in von Neumann algebras

Abstract

The notion of commutativity of two normal states on a von Neumann algebra was defined some time ago by means of the Pedersen-Takesaki theorem. In this note we aim at generalizing this notion to an arbitrary number of states, and obtaining some results on so defined joint commutativity. Also relations between commutativity and broadcastability of states are investigated.

Paper Structure

This paper contains 3 sections, 5 theorems, 42 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Commutativity of states

Key Result

Theorem 1

Let $\omega$ be a faithful normal state on a von Neumann algebra $\mathscr{M}$, and let $\varphi$ be a normal state on $\mathscr{M}$. The following conditions are equivalent

Theorems & Definitions (9)

  • Theorem 1: Pedersen-Takesaki
  • Proposition 2
  • proof
  • Definition
  • Theorem 3
  • proof
  • Theorem 4
  • Theorem 5
  • proof