Ueda’s peak set theorem for general von Neumann algebras

Abstract

Abstract. We extend Ueda's peak set theorem for subdiagonal subalgebras of tracial finite von Neumann algebras, to σ-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'.

Noncommutative topology and Jordan operator algebras

“…Finally in Section 9 we collect several miscellaneous results in the noncommutative topology of Jordan operator algebras, mostly Jordan versions of results from [, Section 6], [, Section 6], and [, Section 2]. This includes for example a strict Urysohn lemma, and a Urysohn lemma where the interpolating elements are “nearly positive” (that is, as close as we like to a contraction that is positive in the usual ‐algebraic sense).…”

On Vector-Valued Characters for Noncommutative Function Algebras

“…Note that by [5,Theorem 4.1] or [7,Lemma 5.8], the Gleason-Whitney type property that every weak* continuous state on A has at most one normal state extension to M , is equivalent to A + A * being weak* dense in M . In this case it is evident that any Φ : A → B(H) has at most one weak* continuous extension to M .…”

Quantum measurable cardinals

“…The von Neumann algebra version of Ueda's peak set theorem states that for every singular state φ on M there is a singular peak projection q with φ(q) = 1. It was shown in [7] that there are counterexamples to this statement if measurable cardinals exist. We will refine this conclusion.…”

“…In a recent paper [7] Blecher and Labuschagne investigated whether every von Neumann algebra verifies Ueda's peak set theorem [21]. It was discovered that the answer turns on the existence of singular states with a certain continuity property.…”

“…(In [7] this is stated for a countable decreasing sequence of projections, but the proof given there does not need this restriction.) It is singular if ψ(q) = 0 for every normal state ψ on M .…”