Harmonic analysis and BMO-spaces of free Araki-Woods factors

TL;DR

This paper extends the theory of BMO-spaces and multipliers to free Araki-Woods factors, providing new interpolation results, examples, and bounds for operators in this non-tracial von Neumann algebra setting.

Contribution

It generalizes BMO-space interpolation theorems to arbitrary von Neumann algebras and constructs specific multipliers for free Araki-Woods algebras.

Findings

Interpolation theorems for BMO-spaces in non-tracial von Neumann algebras

Examples of multipliers on free Araki-Woods algebras including $L_ ext{infty} ightarrow ext{BMO}$ multipliers

$L_p$-bounds for a generalized Hilbert transform

Abstract

We consider semi-group BMO-spaces associated with arbitrary von Neumann algebras and prove interpolation theorems. This extends results by Junge-Mei for the tracial case. We give examples of multipliers on free Araki-Woods algebras and in particular we find $L_\infty \rightarrow {\rm BMO}$ multipliers. We also provide $L_p$-bounds for a natural generalization of the Hilbert transform.