Tokarev's theorem on Banach lattices which are Grothendieck spaces

When browsing the literature, I have found the following theorem of E. Tokarev:

Let $X$ be a Banach lattice with weakly sequentially complete dual space. Then for any Banach space $Y$, every unconditionally converging operator $T\colon X\to Y$ is weakly compact. (In other words, Banach lattices with weakly sequentially duals have Pełczyński's property (V).)

(This is Theorem 1.1 here.)

This theorem, if true, would have a number of fantastic consequences. For instance, a very nice theorem of W. B. Johnson would follow easily:

Suppose that $X$ is a Banach space with local unconditional structure. Then either $X$ is super-reflexive or $X$ contains $\ell_\infty^n$'s uniformly or $X$ contains $\ell_1^n$'s uniformly complemented.

I can produce a rather lengthy list of further applications (some of them, I believe, new and which would be useful to me for other purposes). However, I must admit I don't understand the proof. (For instance, I don't understand why the measures $\tilde{\mu}_n$ are well defined, and why we can pass to a convergent subsequence of $y_n^*$'s.)

After a while, I decided to apply the so-called psychological argument: if this theorem is so strong, let us look on other results quoting it. Unfortunately, neither google nor mathscinet can find anything. I also tried to contact the author but it seems he doesn't check his mailbox. Hence my question:

Does this theorem have a chance to be true? Or maybe there is an easy counter-example?

1 Answer

There is a counterexample in

Figiel, T.; Ghoussoub, N.; Johnson, W. B. On the structure of nonweakly compact operators on Banach lattices. Math. Ann. 257 (1981), no. 3, 317–334.

• Thank you! I wish there were a possibility of marking such papers on mathscinet (like the one I mentioned in my question) in order to warn potential readers. – Tomasz Kania Jun 13 '14 at 19:40
• @TomekKania: +1000 (Not to mention MRs that claim a paper says X when it actually says Y.) – Yemon Choi Jun 15 '14 at 19:00