A Banach space $X$ has the *weak Phillips property* if the canonical projection $X^{\*\*\*}\to X^{\*}$ is sequentially weak$^{\*}$-weak continuous [[FreedmanÜlger2000](https://doi.org/10.1090/S0002-9939-00-05703-8), [Ülger2001](https://doi.org/10.4064/cm87-2-1)].


Let $1<p<\infty$ and $E = (\oplus\_{n=1}^{\infty}\ell^1\_n)\_{\ell^p}$ be the $\ell^p$ direct sum of $(\ell^1\_n)\_{n=1}^{\infty}$.


**Question:** Does the space of compact operators $K(E)$ have the weak Phillips property?



Does $K( (\oplus_{n=1}^{\infty}\ell^1_n)_{\ell^p})$ have the weak Phillips property?
Yes, it does. This is essentially an unpublished result due to Hermann Pfitzner, see III.3.6 and III.3.7 in *$M$-ideals in Banach spaces and Banach algebras* by P. Harmand, W. Werner and myself ([Zbl 0789.46011](https://zbmath.org/?q=an%3A0789.46011)) along with the fact that $K(E)$ is an $M$-ideal in its bidual $L(E)$.


