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Modulus support functionals, Rajchman measures and peak functions. (English) Zbl 1467.46008

The complex Bishop-Phelps problem asks if for every closed bounded convex subset \(C\) of a complex Banach space \(X\), the linear functionals for which \(\sup_{x\in C} |\varphi(x)|\) is attained are dense in the dual space; such a \(\varphi\) is called a modulus support functional. This problem was solved in the negative by V. Lomonosov who constructed a counterexample, \(S\), in the predual \(L^1/H_0^1\) of \(H^\infty\) [V. Lomonosov, Isr. J. Math. 115, 25–28 (2000; Zbl 0954.46009)]. A candidate for a counterexample, \(S_0\), in \(c_0\) was suggested in [V. Kadets et al., in: The mathematical legacy of Victor Lomonosov. Berlin: De Gruyter. 157–188 (2020; Zbl 1482.47029)].
Solving a problem from the latter paper, the authors prove that there are in fact “many” modulus support functionals for \(S_0\); however, it remains open whether they are dense. For the proof the authors make use of a delicate connection to Rajchman measures.

MSC:

46B04 Isometric theory of Banach spaces
46A55 Convex sets in topological linear spaces; Choquet theory
52A07 Convex sets in topological vector spaces (aspects of convex geometry)
43A05 Measures on groups and semigroups, etc.
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