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On the topology of pointwise convergence on the boundaries of \(L_1\)-preduals. (English) Zbl 1170.46019

A real Banach space \(X\) is said to be an \(L_1\)-predual if its dual \(X^\ast\) is isometric to \(L_1(\mu)\) for some suitable measure \(\mu\). The space of continuous functions on a compact set and the space of affine continuous functions on a Choquet simplex are some examples of such spaces. A subset \(B\) of the closed unit ball of \(X^\ast\) is said to be a boundary if for each \(x\in X\), there exists a \(b^\ast \in B\) such that \(b^\ast(x)=\|x\|\). There is a natural topology, say \(\tau\), on \(X\) induced by pointwise convergence on \(B\).
Suppose that \(X\) is such that Ext\((B_{X^\ast})\), the set of extreme points of the dual unit ball (which is a boundary), is weak\(^\ast\)-Lindelöf. The contents of this interesting paper are best described by a weaker version quoted in the abstract, namely, for any sequence \(\{x_n\}_{n \geq 1}\), the closure w.r.t \(\tau\) is separable w.r.t the topology generated by the norm.

MSC:

46B25 Classical Banach spaces in the general theory
41A50 Best approximation, Chebyshev systems
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