Let $G$ be a Haussdorf locally compact second countable topological group. Denote $\mu$ a Haar mesure on $G$. We can embed the algebraic tensor product $L^\infty(G) \otimes L^\infty(G)$ into $L^\infty (G \times G)$ (with mesure $\mu \otimes \mu$) via $f \otimes g(x,y) = f(x)g(y)$. This embedding is of course injective. My question is weither its image is dense in $L^\infty(G \times G)$ for either norm topology or $\sigma$-weak topology induced by the injection $f \mapsto [m_f : g \mapsto fg]$ of $L^\infty(G \times G)$ into $\mathcal{B}(L^2(G \times G))$.
Thanks in advance for help.