Cyclic and separating vector
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The notion of a cyclic and separating vector is important in the theory of von Neumann algebras [1][2], 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 AΩ= {aΩ: a in A} is norm-dense in H. The element Ω is said to be separating if aΩ=0 with a in A implies a=0.
- Any element Ω of H defines a semi-norm p on A by p(a)=||aΩ||. Saying that Ω is separating is equivalent with saying that p is actually a norm.
- If Ω is cyclic for A then it is separating for the commutant A', which is the von Neumann algebra of all bounded operators in H which commute with all operators of A. Indeed, if a belongs to A' and satisfies aΩ=0 then one has for all b in A that 0=baΩ=abΩ. Because the set of bΩ with b in A is dense in H this implies that a vanishes on a dense subspace of H. By continuity this implies that a vanishes everywhere. Hence, Ω is separating for A'.
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 [2].
'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 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)=(aΩ,Ω), 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.