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Let us recall Phelp's property-$U$: A subspace $Y\subset X$ is said to have property-$U$ if every $y^*\in Y^*$ has unique norm preserving extension over $X$. $Y$ is weak Hahn-Banach smooth if $y^*$ has unique norm preserving extension if it is norm attaining on $S_Y$.

Let $K(\ell_1), B(\ell_1)$ represent the spaces of compact and bounded linear operators on $\ell_1$ respectively. It is known that $K(\ell_1)$ does not have property-$U$ in $B(\ell_1)$. One way to see this is $K(\ell_1)$ has $1\frac{1}{2}$-ball property but not 2-ball property (see MR0557239 (80m:46019)). A subspace has 2-ball property if and only if it has $1\frac{1}{2}$-ball property and property-$U$. My question is the following.

Is $K(\ell_1)$ in $B(\ell_1)$ a weak Hahn-Banach smooth subspace?

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  • $\begingroup$ Which paper are you referring to? $\endgroup$
    – Dirk Werner
    Commented Aug 15, 2021 at 9:36
  • $\begingroup$ This is 'Best approximation and intersection properties of balls' by David Yost, appeared in Bull Aust Math Soc (1979) $\endgroup$
    – Tanmoy Paul
    Commented Aug 15, 2021 at 13:50
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    $\begingroup$ The equivalence that you mention is Theorem 4 in Yost's paper in Math. Scand. 50, 100-110 (1982); Zbl 0479.46008. It seems to me that the proof only uses weak Hahn-Banach smoothness. So the answer to your question is no. $\endgroup$
    – Dirk Werner
    Commented Aug 15, 2021 at 20:16

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