  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).
