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Operator algebras in Solovay's model


We develop the theory of operator algebras in the Solovay model obtained by applying Solovay's construction to a countable transitive model of the axiom of constructibility. We exposit an approach to verifying familiar theorems in the Solovay model by appealing to its absoluteness properties. In this way, we verify a substantial portion of the elementary theory of operator algebras. In this Solovay model, we define a continuum analog of the ultraweak topology, which we call the continuum-weak topology. We define a V*-algebra to be a concrete algebra of operators that is closed in the continuum-weak topology. We show that every separable C*-algebra has an enveloping V*-algebra. If the separable C*-algebra is commutative, then its enveloping V*-algebra is isomorphic to the V*-algebra of bounded complex-valued functions on its spectrum. More generally, if the separable C*-algebra is type I, then its enveloping V*-algebra is isomorphic to a direct sum of type I factors, one for each irreducible representation.

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