Selfadjointness criteria for operator algebras
Let \(A\) be a closed subalgebra of the algebra \(B(H)\) of all bounded linear operators on a Hilbert space \(H\). Assume that for every completely isometric representation \(\pi:A\to B(H)\), the commutant \((\pi(A))'\) is a selfadjoint. Then the author proves that \(A\) itself must be a selfadjoint algebra. If \(A\) is closed in the weak* topology of \(B(H)\), then one need only consider weak*-continuous completely isometric representations \(\pi\). The proofs use methods developed in the study of operator spaces.
scientific article; zbMATH DE number 1449202