The Haar system in the preduals of hyperfinite factors

TL;DR

This paper constructs Schauder bases, including the Haar system, in the preduals of hyperfinite factors across various types, extending their applicability to non-commutative spaces and symmetric operator spaces.

Contribution

It introduces explicit Schauder bases in the preduals of hyperfinite factors of types II and III, broadening the understanding of their structure in non-commutative analysis.

Findings

Schauder bases are constructed in preduals of hyperfinite factors.

These bases form Schauder bases in symmetric spaces of measurable operators.

The results apply to non-commutative L^p-spaces across different types of factors.

Abstract

We shall present examples of Schauder bases in the preduals to the hyperfinite factors of types $\hbox{II}_1$, $\hbox{II}_\infty$, $\hbox{III}_\lambda$, $0 < \lambda \leq 1$. In the semifinite (respectively, purely infinite) setting, these systems form Schauder bases in any associated separable symmetric space of measurable operators (respectively, in any non-commutative $L^p$-space).