A negative answer follows from this paper of Graham. In fact, my naive intuition on Varopoulos and tilde algebras turned out to be completely incorrect. Hope to find more on that in Graham & McGehee "Essays in commutative harmonic analysis".
The link in the answer now that, but it seems quite likely that the intended paper is Colin C. Graham: On a Banach algebra of Varopoulos, Journal of Functional Analysis, Volume 4, Issue 3, 1969, DOI: 10.1016/0022-1236(69)90001-9. — Martin Sleziak1 hour ago
Throughout, let $(\mathcal{M},\tau)$ be a von Neumann algebra $\mathcal{M}$, acting on a Hilbert space $H$, with normal semifinite faithful trace $\tau$.
In the survey article by Pisier and Xu, the non-commutative $L^p$ space $1\leq p<\infty$ is defined as follows. Let $S_+$ be the set of all p...
The link in the question no longer works, but it seems quite likely that the intended paper is Edward Nelson: Notes on non-commutative integration, Journal of Functional Analysis Volume 15, Issue 2, February 1974, Pages 103-116, DOI: 10.1016/0022-1236(74)90014-7 — Martin Sleziak5 mins ago
Throughout, let $(\mathcal{M},\tau)$ be a von Neumann algebra $\mathcal{M}$, acting on a Hilbert space $H$, with normal semifinite faithful trace $\tau$.
In the survey article by Pisier and Xu, the non-commutative $L^p$ space $1\leq p<\infty$ is defined as follows. Let $S_+$ be the set of all p...
