MR3257881 Reviewed Hadwin, Don Approximate double commutants in von Neumann algebras and C∗-algebras. Oper. Matrices 8 (2014), no. 3, 623–633. (Reviewer: Francesco Tschinke) 46L10 (46L05

Abstract

In this paper, the author proves an asymptotic version of the double commutant theorem, in a particular set-up of commutative C∗ -algebras. More precisely, he considers the relative approximate double commutant of a C ∗-algebra with unit, and, using a theorem of characterization for a commutative C∗-subalgebra with unit (inspired by a well-known result due to Kadison for a von Neumann sub-algebra of type I), and from a theorem based on a Machado result, he proves that if A is a commutative C∗-subalgebra of a C∗-algebra B centrally prime with unit, then A is equal to its relative approximate double commutant. In the case where B is a von Neumann algebra, a distance formula is found.