The positive contractive part of a Noncommutative $L^p$-space is a complete Jordan invariant

TL;DR

This paper demonstrates that the positive part of the closed unit ball in non-commutative L^p-spaces uniquely characterizes the underlying von Neumann algebra as a complete Jordan invariant.

Contribution

It establishes that the metric structure of the positive part of the unit ball in non-commutative L^p-spaces serves as a complete invariant for von Neumann algebras.

Findings

The positive part of the unit ball is a complete Jordan invariant.

This invariant characterizes the von Neumann algebra uniquely.

The result applies for all 1 ≤ p ≤ ∞.

Abstract

Let $1\leq p \leq +\infty$. We show that the positive part of the closed unit ball of a non-commmutative $L^p$-space, as a metric space, is a complete Jordan $^*$-invariant for the underlying von Neumann algebra.