Diagonal comparison of ample C*-diagonals

Abstract: We introduce diagonal comparison, a regularity property of diagonal pairs where the sub-C*-algebra has totally disconnected spectrum, and establish its equivalence with the concurrence of strict comparison of the ambient C*-algebra and dynamical comparison of the underlying dynamics induced by the partial action of the normalisers. As an application, we show that for diagonal pairs arising from principal minimal transformation groupoids with totally disconnected unit space, diagonal comparison is equivalent to tracial Z-stability of the pair and that it is implied by finite diagonal dimension. In-between, we show that any projection of the diagonal sub-C*-algebra can be uniformly tracially divided, and explore a property of conditional expectations onto abelian sub-C*-algebras, namely containment of every positive element in the hereditary subalgebra generated by its conditional expectation. We show that the expectation associated to a C*-pair with finite diagonal dimension is always hereditary in that sense, and we give an example where this property does not occur.