Suppose I have a Banach space $E$ (which may be finite dimensional if you wish), a Hilbert space $H$ and a tensor $\tau \in H\otimes E$ in the algebraic tensor product. There are lots of ways to choose a measure $\mu$ and an isometry (not assumed surjective) $\theta:H\rightarrow L^2(\mu)$. Then $(\theta\otimes\iota)\tau \in L^2(\mu)\otimes E \subseteq L^2(\mu;E)$, and so I can compute the norm of $(\theta\otimes\iota)\tau$ in the vector-valued space $L^2(\mu;E)$.
Is there an intrinsic (or simple, etc.) characterisation of the infimum (over all choices of $\theta$ and $\mu$) of this norm?
(The infimum is non-zero, assuming $\tau\not=0$, as it's always larger than the injective tensor norm. But it's not obvious to me that you actually get a norm on $H\otimes E$ from this).
If $E$ is a Hilbert space, then the norm is independent of the choice of $\mu$ and $\theta$; you just get the Hilbert space tensor product norm. But what if, say, $E$ is a finite-dimensional $\ell^\infty$ space?