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

作者：Toms Andrew S^*

来源：Proceedings of the London Mathematical Society, 2008, 96: 1-25.

DOI：10.1112/plms/pdm023

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.

• 出版日期2008-1