BMO spaces of $\sigma$-finite von Neumann algebras and Fourier-Schur multipliers on $S{U}_q(2)$

Abstract

We consider semi-group BMO spaces associated with an arbitrary $\sigma$-finite von Neumann algebra $(\mathcal{M},\phi )$. 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 $S{U}_q(2)$.

This version corrects a mistake in the appendix; in particular, we no longer construct a predual for BMO but only for the associated row and column spaces. For the published version the corrections are included in an erratum

References in corpus (1)

• The Haagerup approximation property for von Neumann algebras via quantum Markov semigroups and Dirichlet forms