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 In the theory of von Neumann algebras, a part of the mathematical field of functional analysis, Tomita–Takesaki theory is a method for constructing modular automorphisms of von Neumann algebras from the polar decomposition of a certain involution ...

. A related notion is that of a vector which is

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for a given operator. The existence of cyclic vectors is guaranteed by the Gelfand–Naimark–Segal (GNS) construction.

Definitions

Given a

Hilbert space In mathematics, Hilbert spaces (named after David Hilbert) allow generalizing the methods of linear algebra and calculus from (finite-dimensional) Euclidean vector spaces to spaces that may be infinite-dimensional. Hilbert spaces arise natural ...

''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 = is norm-dense in ''H''. The element Ω is said to be ''separating'' if ''a''Ω = 0 with ''a'' in ''A'' implies ''a'' = 0. * Any element Ω of ''H'' defines a

semi-norm In mathematics, particularly in functional analysis, a seminorm is a vector space norm that need not be positive definite. Seminorms are intimately connected with convex sets: every seminorm is the Minkowski functional of some absorbing disk and ...

''p'' on A by ''p''(''a'') = , , ''a''Ω, , . Saying that Ω is separating is equivalent with saying that ''p'' is actually a

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. * 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 ''a''Ω = 0 then one has for all ''b'' in ''A'' that 0 = ''ba''Ω = ''ab''Ω. Because the set of ''b''Ω 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. ''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 In mathematics (specifically linear algebra, operator theory, and functional analysis) as well as physics, a linear operator A acting on an inner product space is called positive-semidefinite (or ''non-negative'') if, for every x \in \mathop(A), \la ...

of ''A,'' implies ''a'' = 0. Every element Ω of ''H'' defines a positive linear functional ''ω''_Ω on a *-algebra ''A'' of bounded linear operators in ''H'' by the relation ''ω''_Ω(''a'') = (''a''Ω,Ω) for all ''a'' in ''A''. If ''ω''_Ω is defined in this way and ''A'' is a

C*-algebra In mathematics, specifically in functional analysis, a C∗-algebra (pronounced "C-star") is a Banach algebra together with an involution satisfying the properties of the adjoint. A particular case is that of a complex algebra ''A'' of continuous ...

then ''ω''_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

. ''Proposition'' Let ''φ'' and ''ψ'' be elements of ''H'' which are cyclic for ''A''. Assume that ''ω''_''φ'' = ''ω''_''ψ''. Then there exists an

isometry In mathematics, an isometry (or congruence, or congruent transformation) is a distance-preserving transformation between metric spaces, usually assumed to be bijective. The word isometry is derived from the Ancient Greek: ἴσος ''isos'' mea ...

''U'' in the commutant ''A′'' such that ''φ'' = ''Uψ''.

References

{{reflist Linear operators Operator theory