Stability in the Cuntz semigroup of a commutative C*‐algebra

Let A be a C*‐algebra. The Cuntz semigroup W ( A ) is an analogue for positive elements of the semigroup V ( A ) of Murray‐von Neumann equivalence classes of projections in matrices over A . We prove stability theorems for the Cuntz semigroup of a commutative C*‐algebra which are analogues of classical stability theorems for topological vector bundles over compact Hausdorff spaces. Let S D G denote the class of simple, unital, and infinite‐dimensional AH algebras with slow dimension growth, and let A be an element of S D G . We apply our stability theorems to obtain the following: A has strict comparison of positive elements;W ( A ) is recovered functorially from the Elliott invariant of A ; the lower semicontinuous dimension functions on A are weak‐* dense in the dimension functions on A ; the dimension functions on A form a Choquet simplex. Statement (ii) confirms a conjecture of Perera and the author, while statements (iii) and (iv) confirm, for S D G , conjectures of Blackadar and Handelman from the early 1980s.