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Let

  • $U,H$ be separable $\mathbb R$-Hilbert spaces
  • $U\:\hat\otimes_2\:H$ denote the Hilbert-Schmidt tensor product of $U$ and $H$
  • $\mathfrak L_1(E)$ denote the space of nuclear linear operators on $E$, for any $\mathbb R$-Banach space $E$
  • $T\in\mathfrak L_1(U\:\hat\otimes_2\:H)$

We can show that there is a unique element $\operatorname{tr}_U(T)$ of $\mathfrak L_1(U)$ with $$\operatorname{tr}\left(\operatorname{tr}_U(T)L\right)=\operatorname{tr}\left(T\left(L\otimes_2\operatorname{id}_H\right)\right)\;\;\;\text{for all }L\in\mathfrak L(U),\tag1$$ where $\operatorname{tr}$ denotes the trace functional and $L\otimes_2\operatorname{id}_H$ denotes the Hilbert-Schmidt tensor product of bounded linear operators.

Are we able to prove a similar result for the projective tensor product? To be precise: Let

  • $X,Y$ be $\mathbb R$-Banach spaces with the approximation property
  • $X\:\hat\otimes_\pi\:Y$ denote the projective tensor product of $X$ and $Y$
  • $T\in\mathfrak L_1(X\:\hat\otimes_\pi\:Y)$

Is there a unique element $\operatorname{tr}_X$ of $\mathfrak L_1(X)$ with $$\operatorname{tr}\left(\operatorname{tr}_X(T)L\right)=\operatorname{tr}\left(T\left(L\otimes_\pi\operatorname{id}_Y\right)\right)\;\;\;\text{for all }L\in\mathfrak L(X),\tag2$$ where $L\otimes_\pi\operatorname{id}_Y$ denotes the projective tensor product of bounded linear operators?

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