Characterization of amenability by a factorization property of the group von Neumann algebra

TL;DR

This paper establishes that a locally compact group's amenability is characterized by a specific factorization property of its group von Neumann algebra, providing a new perspective on group properties through operator algebra structures.

Contribution

It proves that amenability of a group is equivalent to a particular factorization property of its von Neumann algebra, addressing open problems in the field.

Findings

Amenability characterized by von Neumann algebra factorization

Provides a new operator algebraic criterion for group amenability

Partially resolves open questions by Hu and Neufang

Abstract

We show that the amenability of a locally compact group $G$ is equivalent to a factorization property of $VN(G)$ which is given by $ VN(G) = <VN(G)^*VN(G)>$. This answer partially two problems proposed by Z. Hu and M. Neufang in their article \textit{Distinguishing properties of Arens irregularity}, Proc. Amer. Math. Soc. 137 (2009), no. 5, 1753-1761.