$G$ is unimodular. Although ${\mathcal{L}}G$ is often semi-finite, it is more natural to view the $L^p$-spaces of ${\mathcal{L}}G$ as contructed from this weight. As is well-known, the non-commutative $L^p$-spaces then have the property that $L^p$ and $L^q$ do not naturally intersect for $p \neq q$. It turns out that the most convenient way to define the $L^p$-Fourier multiplier is by realizing $L^p({\mathcal{L}}G)$ in the Connes-Hilsum construction (Subsection \[subsect=Connes-Hilsum\]) associated to the Plancherel weight on ${\mathcal{L}}G'  = {\mathcal{R}}G$ (the von Neumann algebra of $L^2G$ associated to the right translations). Indeed, this realizes $L^p({\mathcal{L}}G)$ as a space of unbounded operators on $B(L^2 G)$, and the naive definition of $T_\phi^p$ would just be the restriction of $T_\phi$ to $L^p({\mathcal{L}}G)$. Since in general (*i.e.* unless $G$ is unimodular) $L^p({\mathcal{L}}G) \cap B(L^2(G)) = \{0\}$ this is a bit to naive, and the precise definition along these lines is given in the Definition-Proposition \[dfprop=def\_\_Tphip\].

An important observation we make is that although $L^p({\mathcal{L}}G)$ might not contain nonzero bounded operators, its corners in fact lie in the Schatten class ${\mathcal{S}}_p$. More precisely, for $F \subset G$ relatively compact we can (carefully) define the corner $P_F x P_F$ of any element $x \in L^p({\mathcal{L}}G)$, and we prove in 