Positive isometric Fourier multipliers on non-commutative $L^p$-spaces

For a locally compact group $G$, let $\mathcal{L}G$ denote its left group von Neumann algebra and let $L^p(\mathcal{L}G)$, $1\le p\le \mathrm{\infty }$, be the corresponding non-commutative $L^p$-space. Given $\phi \in L^{\mathrm{\infty }}(G)$, we study the Fourier multiplier $M_{\phi ,p}$ acting on $L^p(\mathcal{L}G)$. We prove that for any $p\mathrm{\ne }2$, the operator $M_{\phi ,p}$ is a positive surjective isometry if and only if $\phi$ coincides locally almost everywhere with a continuous character of $G$. This characterization extends results obtained recently (jointly with C.~Arhancet) in the unimodular setting.