Cyclic and separating vector

In mathematics, the notion of a cyclic and separating vector is important in the theory of von Neumann algebras, and in particular in Tomita–Takesaki theory. A related notion is that of a vector which is cyclic for a given operator. The existence of cyclic vectors is guaranteed by the Gelfand–Naimark–Segal (GNS) construction.
Definitions.
Given a Hilbert space "H" and a linear space "A" of bounded linear operators in "H", an element Ω of "H" is said to be "cyclic" for "A" if the linear space "A"Ω = {"a"Ω: "a" ∈ "A"} is norm-dense in "H". The element Ω is said to be "separating" if "a"Ω = 0 with "a" in "A" implies "a" = 0.
The following stronger result holds if "A" is a *-algebra (an algebra which is closed under taking adjoints) and contains the identity operator 1. For a proof, see Proposition 5 of Part I, Chapter 1 of.
Proposition If "A" is a *-algebra of bounded linear operators in "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. Then a vector Ω which is cyclic and separating for "A" is also cyclic and separating for the commutant "A′"
Positive linear functionals.
A positive linear functional "ω" on a *-algebra "A" is said to be "faithful" if "ω"("a") = 0, where "a" is a positive element of "A," implies "a" = 0.
Every element Ω of "H" defines a positive linear functional "ω" on a *-algebra "A" of bounded linear operators in "H" by the relation "ω"("a") = ("a"Ω,Ω) for all "a" in "A". If "ω" is defined in this way and "A" is a C*-algebra then "ω" is faithful if and only if the vector Ω is separating for "A". Note that a von Neumann algebra is a special case of a C*-algebra.
Proposition Let "φ" and "ψ" be elements of "H" which are cyclic for "A". Assume that "ω" = "ω". Then there exists an isometry "U" in the commutant "A′" such that "φ" = "Uψ".