Recall that a Banach space $X$ has the property (K) if every $w^{*}$-convergent sequence in $X^{*}$ admits a convex block subsequence which converges with respect to the Mackey topology. The property (K) was invented by S. Kwapien to provide an alternative approach to some results of N. Kalton and A. Pelczynski on subspaces of $L_{1}[0,1]$. A. Pelczynski noted that $L_{1}(\mu)$ ($\mu$ is a finite measure) has the property (K). Further, Schur spaces, Grothendieck spaces and strongly weakly compactly generated spaces enjoy the property (K). It is known that $c_{0}$ and $C[0,1]$ fail the property (K). But it seems not easy to check that $C[0,1]$ fails the property (K).

Question. How to check that $C[0,1]$ fails the property (K)?

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    $\begingroup$ It would help if you would give references when you mention relatively obscure concepts. Where did Kwapien introduce property (K). I thought it was first defined in my paper with Figiel and Pelczynski.$$ $$ Figiel, Tadeusz; Johnson, William B.; Pełczyński, Aleksander Some approximation properties of Banach spaces and Banach lattices. Israel J. Math. 183 (2011), 199--231. $\endgroup$ – Bill Johnson Sep 20 '20 at 16:48
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    $\begingroup$ Oh, we defined a property called property (k) that is (probably) different from property (K). Kalton and Pelczynski defined property (K) somewhat differently; namely, where the convex block subsequence $y_n^*$ satisfies $y_n^*(z_n) \to 0$ for every weakly null sequence $(z_n)$ in $X$. Is that equivalent to your definition of property (K)? $\endgroup$ – Bill Johnson Sep 20 '20 at 17:21
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    $\begingroup$ Kalton, N. J.(1-MO); Pełczyński, A.(PL-PAN) Kernels of surjections from ℒ1-spaces with an application to Sidon sets. Math. Ann. 309 (1997), no. 1, 135--158. $\endgroup$ – Bill Johnson Sep 20 '20 at 17:22
  • $\begingroup$ A. Aviles and J. Rodriguez (Convex combinations of weak*-convergent sequences and the Mackey topology, Mediterr J. Math. 2016) mentioned that property (K) was invented by Kwapien, but did not give the reference. $\endgroup$ – Dongyang Chen Sep 21 '20 at 0:30
  • $\begingroup$ In your paper with Figiel and Pelczynski, a weakening of property (K), called property (k), was introduced and proved that a Banach space $X$ would fail property (k) if $X$ contains a complemented copy of $c_{0}$. This answers my question. $\endgroup$ – Dongyang Chen Sep 21 '20 at 0:35

Let $(x_{n}^{*})_{n}$ be a weak*-null sequence in $X^{*}$. The following are equivalent:

(1)$\sup\limits_{x\in K}|\langle x^{*}_{n},x\rangle|\rightarrow 0$ for each weakly compact subset $K$ in $X$;

(2)$|\langle x^{*}_{n},x_{n}\rangle|\rightarrow 0$ for each weakly null sequence $(x_{n})_{n}$ in $X$.

Indeed, if (1) is false, there exist a subsequence $(x^{*}_{k_{n}})_{n}$ of $(x^{*}_{n})_{n}$, a sequence $(x_{n})_{n}$ in $K$ and $\epsilon_{0}>0$ so that $|\langle x^{*}_{k_{n}},x_{n}\rangle|>\epsilon_{0}$ for all $n$. Since $K$ is weakly compact, there is a subsequence $(x_{n_{m}})_{m}$ of $(x_{n})_{n}$ that converges weakly to $x\in K$. Let us define a weakly null sequence $(z_{n})_{n}$ in $X$ by $z_{k_{n_{m}}}=x_{n_{m}}-x$ and $z_{n}=0$ otherwise. By (2), $\langle x^{*}_{n},z_{n}\rangle\rightarrow 0$. Note that $\langle x^{*}_{n},x\rangle\rightarrow 0$. This implies that $\langle x^{*}_{k_{n_{m}}},x_{n_{m}}\rangle\rightarrow 0$, a contradiction.

It follows from the above fact that my property (K) is equivalent to the Kalton-Pelczynski version of property (K).

  • $\begingroup$ Thanks, Dongyang. $\endgroup$ – Bill Johnson Sep 21 '20 at 19:41

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