Freeness of Linear and Quadratic Forms in von Neumann Algebras

TL;DR

This paper characterizes semicircular distributions through the freeness of linear and quadratic forms in noncommutative random variables within a tracial W*-probability space, under relaxed moment conditions.

Contribution

It provides a new characterization of semicircular distributions based on freeness properties in noncommutative probability, extending previous results.

Findings

Semicircular distribution characterized by freeness of forms

Relaxed moment conditions in the characterization

Advances understanding of noncommutative probability structures

Abstract

We characterize semicircular distribution by the freeness of linear and quadratic forms in noncommutative random variables from a tracial $W^*$-probability space with relaxed moment conditions.