Abstract: In joint work with David Blecher we extend Ueda's peak set theorem for subdiagonalsubalgebras 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. As Ueda did for finite von Neumann algebras, we show that this result 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. We also show that deep set theoretic issues dash hopes for extending the theorem to some other large general classes of von Neumann algebras. 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 cardinality of the underlying Hilbert space to not be `too large'.
