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The Cuntz semigroup and stability of close C*-algebras

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.