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One-sided James’ compactness theorem. (English) Zbl 1362.46005

Let \(E\) be a real Banach space, \(E^*\) its dual, \(A,B\) convex subsets of \(E\). It is known that in James’ classical theorem (a bounded closed convex subset of \(E\) is weakly compact if all functionals in \(E^*\) attain their supremum on it) it is not always necessary to test the sup-attainment of all functionals. This point of view becomes apparent in the main result of the paper under review, a “one-sided” James-like theorem: Suppose that the dual unit ball of \(E\) is weak\(^*\) convex block compact (meaning that each bounded sequence in the dual admits weak\(^*\)-convergent convex blocks), that \(A\) and \(B\) are bounded closed and convex with strictly positive distance. If every \(x^*\in E^*\) such that \[ \sup\langle x^*(B)\rangle<\inf\langle x^*(A)\rangle \] attains its infimum on \(A\) and its supremum on \(B\), then both \(A\) and \(B\) are weakly compact. (The theorem answers a question of Delbaen concerning the case \(E=L^1\) on a probability space and \(B=\{0\}\). The term “one-sided” refers to the fact that in the theorem only those \(x^*\)’s intervene that separate \(A\) and \(B\), so \(A\) lies on one side of \(x^*\).)
This interesting approach turns out to be quite natural, as on the way to the proof of the main result, the authors develop ‘one-sided’ analogues of well-known ingredients like the \((I)\)-generation or like conditions ensuring that the norm and the weak\(^*\)-closures of the convex hull of a (not necessarily bounded) subset of \(E^*\) coincide. Also, a one-sided version of the classical Bishop-Phelps theorem is proved. Final remarks show that some results are sharp and mention the (intriguing) conjecture that the main result (and a second, related one) should hold without convex block weak\(^*\)-compactness.

MSC:

46A50 Compactness in topological linear spaces; angelic spaces, etc.
46B50 Compactness in Banach (or normed) spaces
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