Projective tensor product $\ell^2 \hat{\otimes}_\pi \ell^2$ In Ryan Introduction to Tensor Products of Banach spaces, Example 2.10, it is shown that the space $\ell^2 \hat{\otimes}_\pi \ell^2$, where $\hat{\otimes}_\pi$ denotes the projective tensor product and $\ell^2$ the usual space of square-summable sequences, contains $\ell^1$ as a subspace, and that $\ell^1$ is complemented in $\ell^2 \hat{\otimes}_\pi \ell^2$, that is, a subspace $M$ exists such that $\ell^2 \hat{\otimes}_\pi \ell^2 = \ell^1 \oplus M$. Is anything known about this space $M$? I'd be also intersted if there are other references that discuss the space $\ell^2 \hat{\otimes}_\pi \ell^2$ in more detail.
 A: $\ell^2$ has approximation property (AP) since it possess a Schauder basis. $\ell^2$ also has the Radon-Nikodym property (RNP) since every reflexive Banach space has RNP. You may deduce the following from the more general theorems in Ryan's book: If a Banach space $X$ is reflexive and has AP, then

*

*$X\hat{\otimes}_{\epsilon}X^* = K(X)$, the space of compact operators on $X$.

*$X\hat{\otimes}_{\pi}X^* = N(X)$, the space of nuclear operators on $X$

*$(X\hat{\otimes}_{\epsilon}X^*)^* = X\hat{\otimes}_{\pi}X^*$

*$(X\hat{\otimes}_{\pi}X^*)^* = B(X)$ the space of all bounded operators on $X$.

Thus, $\ell^2\hat{\otimes}_{\epsilon}\ell^2 = K(\ell^2)$, and $\ell^2\hat{\otimes}_{\pi}\ell^2 = N(\ell^2)$, which is the space of the trace class operators on $\ell^2$.
For the other question, given a complemented subspace $V$ of a Banach space $X$, $V$ may not have a unique complement. However, the complemented $\ell^1$ subspace given in Ryan's book is the closed linear span of the tensor diagonal $(e_k\otimes e_k)_{k\in\mathbb{N}}$, where $(e_k)_{k\in\mathbb{N}}$ can be chosen any orthonormal basis for $\ell^2$. Also, a particular projection (call it $P$) is also given in the book. One may simply define $M$ to be the range of the projection $I-P$, where $I$ is the identity map.
