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

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