4 problems

Let $X$ be a valid norm, meaning a norm on $\mathcal{S}_1^{\tilde k'\!, n}\otimes\mathcal{S}_1^{\tilde k'\!, n}\otimes\ell_2^{k\tilde k'}$ satisfying … for every $x$, together with…

Let $X$ be a norm on the algebraic tensor product $\mathcal{S}_1^{\tilde k'\!, n}\otimes\mathcal{S}_1^{\tilde k'\!, n}\otimes\ell_2^{k\tilde k'}$. Call $X$ valid if … for every $x$…

Weaker-form conjecture. There exists a dimension-independent constant $\beta<1$ such that

Let $n$, $k$, and $\tilde k'$ be the parameters in the construction, let $\mathcal{S}_1^{\tilde k'\!, n}$ denote the corresponding trace-class space, let…