BMO spaces of $\sigma$-finite von Neumann algebras and Fourier-Schur multipliers on $SU_q(2)$

TL;DR

This paper extends the theory of BMO spaces to $\sigma$-finite von Neumann algebras, proving they are Banach spaces with interpolation properties, and introduces Fourier-Schur multipliers on the quantum group $SU_q(2)$.

Contribution

It generalizes BMO space properties to $\sigma$-finite von Neumann algebras and introduces Fourier-Schur multipliers on $SU_q(2)$.

Findings

BMO spaces admit a predual in the $\sigma$-finite case

BMO spaces are Banach spaces and interpolate with $L_p$ spaces

Fourier-Schur multipliers exist on $SU_q(2)"

Abstract

We consider semi-group BMO spaces associated with an arbitrary $\sigma$-finite von Neumann algebra $(\mathcal{M}, \varphi)$. We prove that the associated row and column BMO spaces always admit a predual, extending results from the finite case. Consequently, we can prove that the semi-group BMO spaces considered are Banach spaces and they interpolate with $L_p$ as in the commutative situation, namely $[\mathrm{BMO}(\mathcal{M}), L_p^\circ(\mathcal{M})]_{1/q} \approx L_{pq}^\circ(\mathcal{M})$. We then study a new class of examples. We introduce the notion of Fourier-Schur multiplier on a compact quantum group and show that such multipliers naturally exist for $SU_q(2)$.