The Banach space $E=(\bigoplus_{n=1}^{\infty}\ell^1_n)_{\ell^p}$ for $1<p<\infty$ shows up in various places in the literature to construct counterexamples. The purpose of this post is to accumulate the properties of this space in a single page.
I have made a short list for a start. (My question is) Please write down in the answers or comments below the other properties that are known to you, so I can add them to this list for everyone's (esp. students', beginners', and amateurs') benefit.
Thanks very much in advance.
$E$ has a basis, so have the approximation property (AP).
$E$ is reflexive but not superreflexive, since $\ell^1$ is finitely representable on $E$.
$K(E)^* = E \mathbin{\hat{\otimes}_{\pi}} E^*$ since $E$ has AP and RNP. Also,$\hspace{4mm}B(E) = \big( E \mathbin{\hat{\otimes}_{\pi}} E^* \big)^* = K(E)^{**}$.
$K(E)^*$ has RNP [Ruess1984]. As a result, $K(E)$ contains no copy of $\ell^1$.
$K(E)$ is an M-ideal in $B(E)$ [HWW1993, Corollary 6.5.4]. Thus, every closed subspace of $K(E)$ has property (V) [HWW1993, Corollary 3.3.7].
$ap(K(E)) = \{0\}$ since $E$ has AP [DuncanUlger1992, Proposition 3.3].
$wap(K(E)) = K(E)^*$ since $E$ is reflexive, see e.g. [DalesLau2005, p.54]. Equivalently, $K(E)$ is Arens regular. This could be derived from the fact that $K(E)$ has property (V), since every Banach algebra with property (V) is Arens regular.
$B(E)$ is not Arens regular [Daws2004, Corollary 2].
$B(E)$ does not have property (V), since every Banach algebra with property (V) is Arens regular.
$B(E)$ contains a complemented copy of $\ell^1$ [Kania2013].
$B(E)$ is not a Grothendieck space [Kania2013].
