We prove that separable
-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
-algebras
provided that one algebra has stable rank one; close
-algebras
must have affinely homeomorphic spaces of lower-semicontinuous
quasitraces; strict comparison is preserved by sufficient closeness of
-algebras. We also
examine
-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
-algebras
have isomorphic Cuntz semigroups when one algebra absorbs the Jiang–Su algebra
tensorially.