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Weak exactness and amalgamated free product of von Neumann algebras

Kai Toyosawa

Abstract

We show that the amalgamated free product of weakly exact von Neumann algebras is weakly exact. This is done by using a universal property of Toeplitz-Pimsner algebras and a locally convex topology on bimodules of von Neumann algebras, which is used to characterize weakly exact von Neumann algebras.

Weak exactness and amalgamated free product of von Neumann algebras

Abstract

We show that the amalgamated free product of weakly exact von Neumann algebras is weakly exact. This is done by using a universal property of Toeplitz-Pimsner algebras and a locally convex topology on bimodules of von Neumann algebras, which is used to characterize weakly exact von Neumann algebras.
Paper Structure (6 sections, 7 theorems, 53 equations)

This paper contains 6 sections, 7 theorems, 53 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Amalgamated free product
  4. A relative topology on C$^{*}$-bimodules and weak exactness
  5. Toeplitz-Pimsner algebra
  6. Weak exactness of amalgamated free product

Key Result

Theorem 1.1

Let $M_{i}$, $i\in I$, be weakly exact $\sigma$-finite von Neumann algebras such that each $M_{i}$ contains a copy of a fixed von Neumann subalgebra $(B,\varphi)$ with a faithful normal state $\varphi$. Assume each $M_{i}$ admits a faithful normal conditional expectation $E_{i}\colon M_{i}\to B$. Th

Theorems & Definitions (15)

  • Theorem 1.1
  • Corollary 1.2
  • Lemma 2.1
  • Example 2.2
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • Lemma 3.1
  • proof
  • ...and 5 more