Commutants and bicommutants of algebras of unbounded operators

Abstract

The first purpose of this paper is to show that for each Op*-algebra (ℳ,𝒟) whose weak commutant ℳ′w is an algebra, there exists a closed Op*-algebra (ℳ̂,𝒟̂), which is the smallest extension of (ℳ,𝒟) satisfying ℳ̂w =ℳ′w and ℳ̂w 𝒟̂ =𝒟̂. The second purpose is to characterize an unbounded bicommutant ℳ″wσ of an Op*-algebra ℳ. The third purpose is to generalize the well-known Radon–Nikodym theorem for von Neumann algebras to Op*-algebras ℳ satisfying the von Neumann density type theorem ℳ̄ t*s =ℳ″wσ.