On the topology of pointwise convergence on the ... - CARMA
On the topology of pointwise convergence on the ... - CARMA
On the topology of pointwise convergence on the ... - CARMA
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Introducti<strong>on</strong>We shall say that a Banach space (X, ‖·‖) is an L 1 -predual ifX ∗ is isometric to L 1 (µ) for some suitable measure µ. Someexamples <str<strong>on</strong>g>of</str<strong>on</strong>g> L 1 -preduals include (C(K), ‖ · ‖ ∞ ), and moregenerally, <str<strong>on</strong>g>the</str<strong>on</strong>g> space <str<strong>on</strong>g>of</str<strong>on</strong>g> c<strong>on</strong>tinuous affine functi<strong>on</strong>s <strong>on</strong> a Choquetsimplex endowed with <str<strong>on</strong>g>the</str<strong>on</strong>g> supremum norm. The o<str<strong>on</strong>g>the</str<strong>on</strong>g>rnoti<strong>on</strong> we shall c<strong>on</strong>sider in this talk is that <str<strong>on</strong>g>of</str<strong>on</strong>g> a boundary.Specifically, for a n<strong>on</strong>-trivial Banach space X over R we saythat a subset B <str<strong>on</strong>g>of</str<strong>on</strong>g> B X ∗, <str<strong>on</strong>g>the</str<strong>on</strong>g> closed unit ball <str<strong>on</strong>g>of</str<strong>on</strong>g> X ∗ , is aboundary, if for each x ∈ X <str<strong>on</strong>g>the</str<strong>on</strong>g>re exists a b ∗∈ B suchthat b ∗ (x) = ‖x‖. The prototypical example <str<strong>on</strong>g>of</str<strong>on</strong>g> a boundaryis Ext(B X ∗) - <str<strong>on</strong>g>the</str<strong>on</strong>g> set <str<strong>on</strong>g>of</str<strong>on</strong>g> all extreme points <str<strong>on</strong>g>of</str<strong>on</strong>g> B X ∗, but <str<strong>on</strong>g>the</str<strong>on</strong>g>reare many o<str<strong>on</strong>g>the</str<strong>on</strong>g>r interesting examples. In a recent paper byMoors and Reznichenko <str<strong>on</strong>g>the</str<strong>on</strong>g> authors investigated <str<strong>on</strong>g>the</str<strong>on</strong>g> <str<strong>on</strong>g>topology</str<strong>on</strong>g><strong>on</strong> a Banach space X that is generated by Ext(B X ∗) and,more generally, <str<strong>on</strong>g>the</str<strong>on</strong>g> <str<strong>on</strong>g>topology</str<strong>on</strong>g> <strong>on</strong> X generated by an arbitrary