Note that: The following, stronger result holds if A is a *-algebra (an algebra that is closed under adjoints) and unital (i.e., contains the identity operator 1).
For a proof, see Proposition 5 of Part I, Chapter 1 of von Neumann algebras.
[2] Proposition If A is a *-algebra of bounded linear operators on H and 1 belongs to A, then Ω is cyclic for A if and only if it is separating for the commutant A′.
A special case occurs when A is a von Neumann algebra, in which case a vector Ω that is cyclic and separating for A is also cyclic and separating for the commutant A′.
Every element Ω of the Hilbert spaceH defines a positive linear functional ωΩ on a *-algebra A of bounded linear operators on H via the inner product ωΩ(a) = (aΩ,Ω), for all a in A.