Given a von Neumann algebra M, then the weak$^*$ (or extended) Haagerup tensor product of M with itself is the collection of $\tau\in M\overline\otimes M$ with $$\tau=\sum_i x_i\otimes y_i$$ the sum converging sigma-weakly, where $$\Big\|\sum_i x_ix_i^*\Big\|<\infty, \quad \Big\|\sum_i y_i^*y_i\Big\|<\infty$$ again, these sums of positives being in the sigma-weak sense. Let $\sigma:M\overline\otimes M\rightarrow M\overline\otimes M$ be the swap map. Notice that the extended Haagerup tensor product is not symmetric under $\sigma$.
However, suppose that I happen to know that both $\tau$ and $\sigma(\tau)$ are in the extended Haagerup tensor product. Can I find a "symmetric" expression for $\tau$, similar to that above (surely it is too much to hope that, say, also $\sum_i x_i^*x_i$ and $\sum_i y_iy_i^*$ converge, but is there something a little weaker?)
Pisier and Oikhberg studied something similar(ish) in a Proc EMS paper, but I don't know of any other sources in the literature.
Edit: I should say that I'm also interested in the case when actually $\tau=\sigma(\tau)$.
