Ueda’s peak set theorem for general von Neumann algebras

Abstract:

We extend Ueda’s peak set theorem for subdiagonal subalgebras of tracial finite von Neumann algebras to $\sigma$-finite von Neumann algebras (that is, von Neumann algebras with a faithful state, which includes those on a separable Hilbert space or with separable predual). To achieve this extension, completely new strategies had to be invented at certain key points, ultimately resulting in a more operator algebraic proof of the result. Ueda showed in the case of finite von Neumann algebras that his peak set theorem is the fountainhead of many other very elegant results, like the uniqueness of the predual of such subalgebras, a highly refined F & M Riesz type theorem, and a Gleason-Whitney theorem. The same is true in our more general setting, and indeed we obtain a quite strong variant of the last mentioned theorem. We also show that set theoretic issues dash hopes for extending the theorem to some other large general classes of von Neumann algebras, for example finite or semi-finite ones. Indeed certain cases of Ueda’s peak set theorem for a von Neumann algebra $M$ may be seen as ‘set theoretic statements’ about $M$ that require the sets to not be ‘too large’.