A noncommutative Brooks–Jewett Theorem

In classical measure theory the Brooks–Jewett Theorem provides a finitely-additive-analogue to the Vitali–Hahn–Saks Theorem. In this paper, it is studied whether the Brooks–Jewett Theorem allows for a noncommutative extension. It will be seen that, in general, a bona-fide extension is not valid. Indeed, it will be shown that a C * -algebra A satisfies the Brooks–Jewett property if, and only if, it is Grothendieck, and every irreducible representation of A is finite-dimensional; and a von Neumann algebra satisfies the Brooks–Jewett property if, and only if, it is topologically equivalent to an abelian algebra.