# Is the canonical map $\mathfrak L(X,E)\:\hat\otimes_\pi\:\mathfrak L(Y,F)\to\mathfrak L(X\:\hat\otimes_\pi\:Y,E\:\hat\otimes_\pi\:F)$ injective?

If $$A,B$$ are $$\mathbb R$$-Banach spaces, let $$A\:\hat\otimes_\pi\:B$$ denote the completion of the algebraic tensor product of $$A$$ and $$B$$ with respect to the projective norm. Let $$X,Y,E,F$$ be $$\mathbb R$$-Banach spaces. If $$S\in\mathfrak L(X,E)$$ and $$T\in\mathfrak L(Y,F)$$, let $$S\otimes_\pi T$$ denote the unique bounded linear operator from $$X\:\hat\otimes_\pi\:Y$$ to $$E\:\hat\otimes_\pi\:F$$ with $$(S\otimes_\pi T)(x\otimes y)=Sx\otimes Ty\;\;\;\text{for all }(x,y)\in X\times Y\tag1.$$ Note that $$\mathfrak L(X,E)\times\mathfrak L(Y,F)\to\mathfrak L(X\:\hat\otimes_\pi\:Y,E\:\hat\otimes_\pi\:F)\;,\;\;\;(S,T)\mapsto S\otimes_\pi T\tag2$$ is a bounded bilinear operator and hence there is a unique bounded linear operator $$\iota$$ from $$\mathfrak L(X,E)\:\hat\otimes_\pi\:\mathfrak L(Y,F)$$ to $$\mathfrak L(X\:\hat\otimes_\pi\:Y,E\:\hat\otimes_\pi\:F)$$ with $$\iota(S\otimes T)=S\otimes_\pi T\;\;\;\text{for all }(S,T)\in\mathfrak L(X,E)\times\mathfrak L(Y,F)\tag3.$$

Is $$\iota$$ injective?

I've seen that the notation $$S\otimes T$$ is often used for $$S\otimes_\pi T$$. With that notation it's not clear if it refers to the element of $$\mathfrak L(X,E)\otimes\mathfrak L(Y,F)$$ or the element in $$\mathfrak L(X\:\hat\otimes_\pi\:Y,E\:\hat\otimes_\pi\:F)$$. So, I guess they can be identified.

EDIT: If the claim is not true in this generality, feel free to add additional assumptions (like the approximation property) on the involved spaces.

## 1 Answer

Not in general. Consider the particular case when $E=F=\mathbb{R}$, then we are talking about the injectivity of the canonical map $$X' \hat{\otimes}_\pi Y' \to \big( X \hat{\otimes}_\pi Y \big)'$$ where the $'$ denotes the dual space. However, this map factors as $$X' \hat{\otimes}_\pi Y' \to X' \hat{\otimes}_\varepsilon Y' \to \big( X \hat{\otimes}_\pi Y \big)'$$ where the first arrow is simply the fact that $\varepsilon \le \pi$, and the second one can be found e.g. in Section 6.1 of Defant and Floret's Tensor norms and operator ideals. If the composition is injective then the first map $$X' \hat{\otimes}_\pi Y' \to X' \hat{\otimes}_\varepsilon Y'$$ has to be injective, but this is well-known not to be the case in general (and this is closely related to approximation properties of the spaces involved; see Section 5 in the aforementioned book).

• But, at least for your choice $E=F=\mathbb R$, it's correct if $Y$ has the approximation property, right? If so, the claim remain correct for more general $E,F$? May 4, 2018 at 23:10
• It's been a while, but I'd still be interested in the question from my previous comment. May 16, 2020 at 19:11