Null projections and noncommutative function theory in operator algebras

Abstract

We study projections in the bidual of a $C^*$-algebra $B$ that are null with respect to a subalgebra $A$, that is projections $p\in B^{**}$ satisfying $|\phi|(p)=0$ for every $\phi\in B^*$ annihilating $A$. In the separable case, $A$-null projections are precisely the peak projections in the bidual of $A$ at which the subalgebra $A$ interpolates the entire $C^*$-algebra $B$. These are analogues of null sets in classical function theory, on which several profound results rely. This motivates the development of a noncommutative variant, which we use to find appropriate `quantized' versions of some of these classical facts. Through a delicate generalization of a theorem of Varopoulos, we show that, roughly speaking, sufficiently regular interpolation projections are null precisely when their atomic parts are. As an application, we give alternative proofs and sharpenings of some recent peak-interpolation results of Davidson and Hartz for algebras on Hilbert function spaces, also illuminating thereby how earlier noncommutative peak-interpolation theory may be applied. In another direction, given a convex subset of the state space of $B$, we characterize when the associated Riesz projection is null. This is then applied to various important topics in noncommutative function theory, such as the F.& M. Riesz property, the existence of Lebesgue decompositions, the description of Henkin functionals, and Arveson's noncommutative Hardy spaces (maximal subdiagonal algebras).