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Projectivity of Hilbert modules over uniform algebras

In this paper we study the projectivity of Hilbert modules over uniform algebras, first in the category of Banach modules and then in the smaller category of Hilbert modules. We prove that such modules are never projective as Banach modules in the case where the uniform algebras have a metrizable character space without any isolated points. On the other hand, we give an example, whenever the character space of the uniform algebra is metrizable, of a one-dimensional module, which is projective as a Hilbert module.