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

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{B}\mathrm{M}\mathrm{O}$($\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 $S{U}_q(2)$.