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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.

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