In type III1 factors the ultrapower of the spectral subspace M(σ^φ, F) is a proper subset of the spectral subspace of the ultrapowered algebra.
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The bicentralizer problem for III1 factors has a positive solution if and only if the bicentralizer functor is a zeroset in the theory of III1 factors, and the class of such factors with trivial bicentralizer is ∀∃-axiomatizable.
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Ultrapowers of spectral subspaces
In type III1 factors the ultrapower of the spectral subspace M(σ^φ, F) is a proper subset of the spectral subspace of the ultrapowered algebra.
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Model theory and Connes' bicentralizer problem
The bicentralizer problem for III1 factors has a positive solution if and only if the bicentralizer functor is a zeroset in the theory of III1 factors, and the class of such factors with trivial bicentralizer is ∀∃-axiomatizable.