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 SDG denote the class of simple, unital, and infinite-dimensional AH algebras with slow dimension growth, and let A be an element of SDG. We apply our stability theorems to obtain the following: (i) A has strict comparison of positive elements; (ii) W (A) is recovered functorially from the Elliott invariant of A; (iii) The lower semicontinuous dimension functions on A are weak-∗ dense in the dimension functions on A; (iv) 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 SDG, conjectures of Blackadar and Handelman from the early 1980s.