Separating vector - Imported Knowlpedia Content

Definition

Let $M\subseteq B(H)$ be a von Neumann algebra on a Hilbert space. A vector $\xi\in H$ is separating for $M$ if

x\xi=0\text{ for }x\in M\quad\Longrightarrow\quad x=0.

Equivalently, the linear map $M\to H$ , $x\mapsto x\xi$ , is injective. For a representation $\pi:A\to B(H)$ , “separating for $\pi(A)$ ” has the same meaning; it is stronger than faithfulness of $\pi$ , since faithfulness only requires each nonzero algebra element to act nontrivially on at least one vector.

Duality with cyclicity

A vector $\xi$ is separating for $M$ exactly when it is cyclic for the commutant $M'$ . Dually, $\xi$ is cyclic for $M$ exactly when it is separating for $M'$ . The proof uses the orthogonal projection onto $\overline{M'\xi}$ : this projection belongs to $M''$ , and for a von Neumann algebra the identity $M''=M$ turns separation into density Kadison–Ringrose, §5.5.

Examples and failure

For $M=B(H)$ with $\dim H>1$ , no vector is separating: a nonzero projection onto $\xi^\perp$ annihilates $\xi$ . In the standard representation of $L^\infty(X,\mu)$ on $L^2(X,\mu)$ , a vector is separating precisely when it is nonzero almost everywhere. A faithful normal state represented by the GNS construction has a cyclic vector, but that vector becomes separating for the represented von Neumann algebra only under the appropriate faithfulness hypothesis.

Role in modular theory

A von Neumann algebra with a vector that is both cyclic and separating is in standard position for Tomita–Takesaki theory. On the dense domain $M\xi$ , one defines the antilinear operator $S_0(x\xi)=x^*\xi$ . Separation makes this definition unambiguous, while cyclicity makes its domain dense. The closure and polar decomposition of $S_0$ then produce the modular operator and modular conjugation.

References

Richard V. Kadison and John R. Ringrose, Fundamentals of the Theory of Operator Algebras, Volume I: Elementary Theory, American Mathematical Society, 1997. AMS DOI record. Relevant: §5.5 on cyclic and separating vectors.
Masamichi Takesaki, Theory of Operator Algebras I, Springer, 1979. DOI record. Relevant: Chapter VI on cyclic and separating vectors and modular theory.