A non-reflexive Grothendieck space that does not containl∞
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A compact spaceS is constructed such that, in the dual Banach spaceC(S)*, every weak* convergent sequence is weakly convergent, whileC(S) does not have a subspace isomorphic tol∞. 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 spaceT is constructed such thatC(T) does not containl∞ 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.
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