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Let $E,F$ be Banach spaces and consider the projective tensor product $E \widehat\otimes F$. If $\tau \in E \widehat\otimes F$ with $\|\tau\|<1$ then by definition we can find $(x_n)\subseteq E$ and $(y_n)\subseteq F_n$ with $$ \tau = \sum_n x_n \otimes y_n \qquad \sum_n \|x_n\| \|y_n\| < 1. $$

Now suppose $T,S$ are (bounded) operators on $E, F$ respectively, so that $T\otimes S$ acts on $E \widehat\otimes F$. Suppose $\|(T\otimes S)\tau\| < 1$ as well as $\|\tau\|<1$. Can we find a representation $\tau=\sum_n x_n\otimes y_n$ with both $$\sum_n \|x_n\| \|y_n\| < 1 \qquad \sum_n \|T(x_n)\| \|S(y_n)\| < 1? $$

Motivation: I am mostly interested in the case when $E=F$ is a Hilbert space, and $T=S^{-1}$ is a positive (perhaps unbounded) operator. This is motivated by looking at automorphism groups of C$^*$-algebras.

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    $\begingroup$ Do we at least know this for $E=F=\ell_1(J)$, where in some sense we know the "best" representation for elements of $\ell_1(J)\widehat\otimes\ell_1(J) \cong \ell_1(J\times J)$ $\endgroup$
    – Yemon Choi
    Commented Apr 22, 2017 at 2:48

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