On the structure of tensor products of l_p spaces

We examine some structural properties of (injective and projective) tensor products of $\ell_p$-spaces (projections, complemented subspaces, reflexivity, isomorphisms, etc.). We combine these results with combinatorial arguments to address the question of primarity for these spaces and their duals. Our main results are: \medbreak \item{(1)} If $1<p<\infty$, then $B(\ell_p)\approx B(L_p)$ ($B(X)$ consists of the bounded linear operators on $X$). \medbreak \item{(2)} If ${1\over p_i}+{1\over p_j}\leq1$ for every $i\neq j$, or if all of the $p_i$'s are equal, then $\ell_{p_1}\hat{\otimes}\cdots \hat{\otimes}\ell_{p_N}$ is primary. \medbreak \item{(3)} $\ell_p$ embeds into $\ell_{p_1}\hat{\otimes}\cdots \hat{\otimes}\ell_{p_N}$ if and only if there exists $A\subset \{1,2,\cdots,n\}$ such that ${1\over p}=\min\{\sum_{i\in A}{1\over p_i},1\}$. \medbreak \item{(4)} If $1\leq p<\infty$ and $m\geq1$, then the space of homogeneous analytic polynomials ${\cal P}_m(\ell_p)$ and the symmetric tensor product of $m$ copies of $\ell_p$ are primary.