# On the structure of tensor products of l_p spaces

Research paper by **Alvaro Arias, Jeff Farmer**

Indexed on: **07 Feb '94**Published on: **07 Feb '94**Published in: **Mathematics - Functional Analysis**

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#### 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 ${1\over p_i}+{1\over p_j}\leq1$ 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.