  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
$SU_q(2)$.
