2
$\begingroup$

A Banach space $X$ has the weak Phillips property if the canonical projection $X^{***}\to X^{*}$ is sequentially weak$^{*}$-weak continuous [FreedmanÜlger2000, Ülger2001].

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?

Improve this question
$\endgroup$
1
  • $\begingroup$ For the records: $K(E)$ is an M-ideal in $B(E)$ by Corollary VI.5.4 in [zbmath.org/?q=an:0789.46011] referenced by Dirk Werner in his answer below. $\endgroup$
    – Onur Oktay
    Commented Dec 12, 2021 at 12:29

1 Answer 1

Reset to default
4
$\begingroup$

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 This is essentially an unpublished result due to Hermann Pfitzner, see III.3.6 and III.3.7 in $M$-ideals in Banach spaces and Banach algebras by P. Harmand, W. Werner and myself (Zbl 0789.46011) along with the fact that $K(E)$ is an $M$-ideal in its bidual $L(E)$.

Improve this answer
$\endgroup$
1
  • $\begingroup$ Professor Werner, thank you very much. $\endgroup$
    – Onur Oktay
    Commented Dec 12, 2021 at 11:16

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .