is a trace if and only if $G$ is unimodular). It is convenient to work with this weight. A natural weight on the commutant $\rho(G)''$ of ${\mathcal{L}}G$ in $B(L^2 G)$ is given by $\psi(x^*x) = \|f\|_{L^2G}^2$ if $x = \rho(f)$ with $f
\in L^2G$ and $\psi(x^*x)=\infty$ otherwise.

The non-commutative $L^p$-space of ${\mathcal{L}}G$. {#subsect=Connes-Hilsum}
----------------------------------------------------

We recall the Connes-Hilsum construction [@Con], [@Hilsum] of $L^p({\mathcal{L}}G)$ in the particular case when ${\mathcal{L}}G$ is the von Neumann algebra of $G$. The space $D(L^2 G,\psi)$ of $\psi$-bounded elements of $L^2G$ is the set of functions $\xi \in L^2G$ such that $f \mapsto \xi \ast f$ is bounded on $L^2G$. We will also denote this operator by $\lambda(\xi)$. Given a weight $\omega$ on ${\mathcal{L}}G$, the spatial derivative $d\omega/d\psi$ is the unique positive self-adjoint operator such that $\omega(\lambda(\xi) \lambda(\xi)^\ast) = \| (d\omega/d\psi)^{1/2} \xi\|_{L^2G}^2$ for all $\psi$-bounded $\xi$. For example, gives that $d\varphi/d\psi = \Delta$. The spatial derivative of an element $\omega$ of the predual is defined by $d \omega/d\psi = u d |\omega|/d\psi$ where $\omega = u |\omega|$ is the polar decomposition of $\omega$. The set $L^1({\mathcal{L}}G)$ defined as $\{ d\omega/d\psi,\omega \in {\mathcal{L}}G_*\}$ is then a linear space (the sum being the closure of the sum), and 