We prove that separable C*-algebras which are completely close in a natural
uniform sense have isomorphic Cuntz semigroups, continuing a line of research
developed by Kadison - Kastler, Christensen, and Khoshkam. This result has
several applications: we are able to prove that the property of stability is
preserved by close C*-algebras provided that one algebra has stable rank one;
close C*-algebras must have affinely homeomorphic spaces of
lower-semicontinuous quasitraces; strict comparison is preserved by sufficient
closeness of C*-algebras. We also examine C*-algebras which have a positive
answer to Kadison's Similarity Problem, as these algebras are completely close
whenever they are close. A sample consequence is that sufficiently close
C*-algebras have isomorphic Cuntz semigroups when one algebra absorbs the
Jiang-Su algebra tensorially.Comment: 26 pages; typos fixe
