Description of projective and injective tensor products $\ell^2 \otimes \ell^2$? The following question is probably too elementary and/or well-known for MathOverflow, so I'll try here:
Let $\ell^2 \mathbin{\hat\otimes_\pi} \ell^2$ and $\ell^2 \mathbin{\hat\otimes_\varepsilon} \ell^2$ refer to the (completed) projective and injective tensor products (as defined, say, in Wikipedia), as Banach spaces, of the Hilbert space $\ell^2 = \{u\colon\mathbb{N}\to\mathbb{R} : \sum_{k=0}^{+\infty} u_k^2 < +\infty\}$ of square-summable sequences with itself.
I understand that it is not easy to describe these spaces, but I wonder if it is still possible to give a reasonably concrete condition for a “sequence of sequences” (i.e., a function $\mathbb{N}^2 \to \mathbb{R}$) to belong to one or the other?
More precisely, if we consider the continuous linear form $e_k^*\colon\ell^2\to\mathbb{R}$ which maps $u \in \ell^2$ to its $k$-th term $\langle u, e_k\rangle$, then the tensor product $e_m^* \otimes e_n^*$ defines a continuous linear form of norm $1$ on either $\ell^2 \mathbin{\hat\otimes_\pi} \ell^2$ or $\ell^2 \mathbin{\hat\otimes_\varepsilon} \ell^2$, so an element $v$ in one of these spaces defines an array $J(v)\colon (m,n) \mapsto (e_m^* \otimes e_n^*)(v)$, which belongs to $\ell^\infty(\mathbb{N}^2)$ (the space of bounded functions $\mathbb{N}^2\to\mathbb{R}$).  This, in turn, defines a continuous linear map $J_\alpha \colon \ell^2 \mathbin{\hat\otimes_\alpha} \ell^2 \to \ell^\infty(\mathbb{N}^2)$ (of norm $1$) for $\alpha \in \{\pi,\varepsilon\}$.  I guess I have four questions:


*

*Is $J_\pi$ injective?  (Can $\ell^2 \mathbin{\hat\otimes_\pi} \ell^2$ be seen as a space of functions $\mathbb{N}^2\to\mathbb{R}$?)

*Is $J_\varepsilon$ injective?  (Can $\ell^2 \mathbin{\hat\otimes_\varepsilon} \ell^2$ be seen as a space of functions $\mathbb{N}^2\to\mathbb{R}$?)

*What is the image of $J_\pi$?  (When does a function $\mathbb{N}^2\to\mathbb{R}$ belong to $\ell^2 \mathbin{\hat\otimes_\pi} \ell^2$?)

*What is the image of $J_\varepsilon$?  (When does a function $\mathbb{N}^2\to\mathbb{R}$ belong to $\ell^2 \mathbin{\hat\otimes_\varepsilon} \ell^2$?)
 A: Here's a positive answer to the first two questions, and a very partial answer to the other two.  Let the reference [R] stand for Ryan's book Introduction to Tensor Products of Banach Spaces (2002).
Let $\mathcal{B}$ be the vector space of continous bilinear forms $\ell^2 \times \ell^2 \to \mathbb{R}$ with the usual norm $\|B\| := \sup\{|B(x,y)| : \|x\|\leq 1, \|y\|\leq 1\}$ for $B \in \mathcal{B}$.  There is an obvious map $I \colon \mathcal{B} \to \ell^\infty(\mathbb{N}^2)$ taking $B$ to $(B(e_m,e_n))$; this map $I$ is injective and continuous of norm $1$ (injectivity of $I$ follows from the fact that the $e_n$ span a dense subset of $\ell^2$; the fact that $\|I\|\leq 1$ is clear from the definition of the norm on $\mathcal{B}$; and that $\|I\|=1$ follows from the fact that $\|I(e_m^*\otimes e_n^*)\| = 1$).
Now the maps $J_\pi$ and $J_\varepsilon$ defined in the question factor through this $I\colon \mathcal{B} \to \ell^\infty(\mathbb{N}^2)$, the factors being the natural maps, which we also denote $J_\pi$ and $J_\varepsilon$, from $\ell^2 \mathbin{\hat\otimes_\alpha} \ell^2$ (for $\alpha$ in $\{\pi,\varepsilon\}$) to $\mathcal{B}$.  So to answer the first two questions positively, it is enough to show the injectivity of these $J_\alpha \colon \ell^2 \mathbin{\hat\otimes_\alpha} \ell^2 \to \mathcal{B}$.
In the case of $J_\varepsilon \colon \ell^2 \mathbin{\hat\otimes_\varepsilon} \ell^2 \to \mathcal{B}$, injectivity is almost by definition: as explained in [R, §3.1], $J_\varepsilon$ is norm preserving and we can see $\ell^2 \mathbin{\hat\otimes_\varepsilon} \ell^2$ as the closure of $\ell^2 \otimes \ell^2$ inside $\mathcal{B}$.  Its image is the set of "approximable" bilinear forms.
In the case of $J_\pi \colon \ell^2 \mathbin{\hat\otimes_\pi} \ell^2 \to \mathcal{B}$, injectivity is not automatic (see [R, §2.6]).  Its image is the set of "nuclear" bilinear forms.  Nevertheless, since $\ell^2$ is a Hilbert space, as a Banach space it has the approximation property ([R, ex. 4.4]), so by [R, cor. 4.8], the map $J_\pi$ is indeed injective.
To summarize, $J_\varepsilon$ is an isometric embedding which identifies $\ell^2 \mathbin{\hat\otimes_\varepsilon} \ell^2$ with the closure of $\ell^2 \otimes \ell^2$ inside $\mathcal{B}$ or "approximable" bilinear forms; as for $J_\pi$, it is injective of norm $1$, and its image consists of "nuclear" bilinear forms.
It is also worth noting that $\ell^2 \mathbin{\hat\otimes_\varepsilon} \ell^2$ can also be identified with compact operators $\ell^2 \to \ell^2$ ([R, prop. 4.12]), and that $\ell^2 \mathbin{\hat\otimes_\pi} \ell^2$ isometrically embeds in the dual of $\ell^2 \mathbin{\hat\otimes_\varepsilon} \ell^2$ ([R, thm. 4.14]).
I still don't have a clear picture of how one can tell whether an element of $\mathcal{B}$, let alone $\ell^\infty(\mathbb{N}^2)$, is approximable resp. nuclear, nor do I know if a concrete description can be given.  I can give one example, though: the bilinear form corresponding to the Hilbert space dot product on $\ell^2$, whose image by $I\colon \mathcal{B} \to \ell^\infty(\mathbb{N}^2)$ takes the value $1$ on the diagonal and $0$ elsewhere, is in the image of $J_\varepsilon$ but not of $J_\pi$.
