Perturbations of C∗-algebraic Invariants

Kadison and Kastler introduced a metric on the set of all C∗-algebras on a fixed Hilbert space. In this paper structural properties of C∗-algebras which are close in this metric are examined. Our main result is that the property of having a positive answer to Kadison’s similarity problem transfers to close C∗-algebras. In establishing this result we answer questions about closeness of commutants and tensor products when one algebra satisfies the similarity property. We also examine K-theory and traces of close C∗-algebras, showing that sufficiently close algebras have isomorphic Elliott invariants when one algebra has the similarity property.