  We study the problem of extending a state on an abelian $C^*$- subalgebra to
a tracial state on the ambient $C^*$-algebra. We propose an approach that is
well-suited to the case of regular inclusions, in which there is a large supply
of normalizers of the subalgebra. Conditional expectations onto the subalgebra
give natural extensions of a state to the ambient $C^*$-algebra; we prove that
these extensions are tracial states if and only if certain invariance
properties of both the state and conditional expectations are satisfied. In the
example of a groupoid $C^*$-algebra, these invariance properties correspond to
invariance of associated measures on the unit space under the action of
bisections. Using our framework, we are able to completely describe the tracial
state space of a Cuntz-Krieger graph algebra. Along the way we introduce
certain operations called graph tightenings, which both streamline our
description and provides connections to related finiteness questions in graph
$C^*$-algebras. Our investigation has close connections with the so-called
unique state extension property and its variants.

Operator Algebras