On the structure of tensor products of l_p spaces

Abstract

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 $\frac{1}{p_i}+\frac{1}{p_j}\le 1$ for every $i\ne j$, 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 $\frac{1}{p}=\min \{{\sum }_{i\in A}\frac{1}{p_i},1\}$. \medbreak \item{(4)} If $1\le p<\infty$ and $m\ge 1$, then the space of homogeneous analytic polynomials ${\mathcal{P}}_m({\ell }_p)$ and the symmetric tensor product of $m$ copies of ${\ell }_p$ are primary.