$M_d$-multipliers of a locally compact group

We show that the space $M_d(G)$ of $M_d$-multipliers of a locally compact group $G$ is isometrically isomorphic to the Banach space of bounded functionals on the $d$-fold Haagerup tensor product of $L^1(G)$ vanishing on the kernel of the convolution map. Consequently, we see that $M_d(G)$ is isometrically isomorphic to the dual space of $X_d(G)$, the completion of $L^1(G)$ in the dual of $M_d(G)$. We also show that $M_d$-type-approximation-properties are inherited to lattices.