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A non-reflexive Grothendieck space that does not contain \(l_{\infty }\). (English) Zbl 1358.46007
Summary: A compact space \(S\) is constructed such that, in the dual Banach space \(C(S)^*\), every weak\(^*\) convergent sequence is weakly convergent, while \(C(S)\) does not have a subspace isomorphic to \(l_{\infty }\). The construction introduces a weak version of completeness for Boolean algebras, here called the Subsequential Completeness Property. A related construction leads to a counterexample to a conjecture about holomorphic functions on Banach spaces. A compact space \(T\) is constructed such that \(C(T)\) does not contain \(l_{\infty }\) but does have a “bounding” subset that is not relatively compact. The first of the examples was presented at the International Conference on Banach spaces, Kent, Ohio, 1979.

46B03 Isomorphic theory (including renorming) of Banach spaces
46G20 Infinite-dimensional holomorphy
32A10 Holomorphic functions of several complex variables
Full Text: DOI
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