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

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 $L_{\infty }\to \mathrm{BMO}$ multipliers. We also provide $L_p$-bounds for a natural generalization of the Hilbert transform.