fa.functional analysis - Symmetric (extended) Haagerup tensor product

Given a von Neumann algebra M, then the weak$^*$ (or extended) Haagerup tensor product of M with itself is the collection of $tauin Moverlineotimes M$ with

$$tau=sum_i x_iotimes y_i$$ the sum converging sigma-weakly, where $$Big|sum_i x_ix_i^*Big|<infty, quad Big|sum_i y_i^*y_iBig|<infty$$ again, these sums of positives being in the sigma-weak sense. Let $sigma:Moverlineotimes Mrightarrow Moverlineotimes M$ 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 $tau=sigma(tau)$.

• Next differential topology - Stokes' theorem etc., for non-Hausdorff manifolds
• Previous real analysis - Integrability of derivatives