BMO spaces of $\sigma $-finite von Neumann algebras and Fourier–Schur multipliers on ${\rm SU}_q(2)$
Volume 262 / 2022
Studia Mathematica 262 (2022), 45-91
MSC: Primary 43A15, 46L67, 46L51.
DOI: 10.4064/sm201202-18-6
Published online: 7 October 2021
Abstract
We consider semigroup BMO spaces associated with an arbitrary $\sigma $-finite von Neumann algebra $(\mathcal {M}, \varphi )$. We prove that BMO always admits a predual, extending results from the finite case. Consequently, we can prove—in the current setting of BMO—that they are Banach spaces and they interpolate with $L_p$ as in the commutative situation, namely $[{\rm BMO} (\mathcal {M}), L_p^\circ (\mathcal {M})]_{1/q} \approx L_{pq}^\circ (\mathcal {M})$. 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 ${\rm SU} _q(2)$.
