An asymptotic double commutant theorem for 𝐶*-algebras

Abstract

An asymptotic version of von Neumann’s double commutant theorem is proved in which C ∗ {C^{\ast }} -algebras play the role of von Neumann algebras. This theorem is used to investigate asymptotic versions of similarity, reflexivity, and reductivity. It is shown that every nonseparable, norm closed, commutative, strongly reductive algebra is selfadjoint. Applications are made to the study of operators that are similar to normal (subnormal) operators. In particular, if T is similar to a normal (subnormal) operator and π \pi is a representation of the C ∗ {C^{\ast }} -algebra generated by t, then π ( T ) \pi (T) is similar to a normal (subnormal) operator.