  Let $G$ be a locally compact unimodular group, and let $\phi$ be some
function of $n$ variables on $G$. To such a $\phi$, one can associate a
multilinear Fourier multiplier, which acts on some $n$-fold product of the
non-commutative $L_p$-spaces of the group von Neumann algebra. One may also
define an associated Schur multiplier, which acts on an $n$-fold product of
Schatten classes $S_p(L_2(G))$. We generalize well-known transference results
from the linear case to the multilinear case. In particular, we show that the
so-called `multiplicatively bounded $(p_1,\ldots,p_n)$-norm' of a multilinear
Schur multiplier is bounded above by the corresponding multiplicatively bounded
norm of the Fourier multiplier, with equality whenever the group is amenable.
Further, we prove that the bilinear Hilbert transform is not bounded as a
vector valued map $L_{p_1}(\mathbb{R}, S_{p_1}) \times L_{p_2}(\mathbb{R},
S_{p_2}) \rightarrow L_{1}(\mathbb{R}, S_{1})$, whenever $p_1$ and $p_2$ are
such that $\frac{1}{p_1} + \frac{1}{p_2} = 1$. A similar result holds for
certain Calder\'on-Zygmund type operators. This is in contrast to the
non-vector valued Euclidean case.
