Cyclic-separating duality

Statement

Let $M\subseteq B(H)$ be a von Neumann algebra on a Hilbert space, and let $M'$ be its commutant. The cyclic-separating duality says that, for every $\xi\in H$ ,

\xi\text{ is cyclic for }M \quad\Longleftrightarrow\quad \xi\text{ is separating for }M',

and

\xi\text{ is separating for }M \quad\Longleftrightarrow\quad \xi\text{ is cyclic for }M'.

Here cyclic means $\overline{M\xi}=H$ , while separating means $x\xi=0\Rightarrow x=0$ . The von Neumann hypothesis supplies $M''=M$ , which is essential to the converse implications.

Proof mechanism

If $\xi$ is cyclic for $M$ and $y\in M'$ satisfies $y\xi=0$ , then $yx\xi=xy\xi=0$ for every $x\in M$ . Density of $M\xi$ forces $y=0$ , so $\xi$ is separating for $M'$ .

Conversely, let $p$ be the orthogonal projection onto $\overline{M\xi}$ . This subspace and its orthogonal complement are invariant under every unitary in $M$ , hence $p\in M'$ . If $\xi$ is separating for $M'$ , then $(1-p)\xi=0$ implies $p=1$ , so $\overline{M\xi}=H$ . Interchanging $M$ and $M'$ , and using $M''=M$ , proves the second equivalence.

Examples and scope

For the multiplication algebra $M=L^\infty(X,\mu)$ on $L^2(X,\mu)$ , where $\mu$ is a finite measure, the constant function $1$ is both cyclic and separating; here $M'=M$ . By contrast, if $\dim H>1$ , every nonzero vector is cyclic for $B(H)$ , but no vector is separating for $B(H)$ . The commutant $B(H)'=\mathbb C1$ reverses those two properties exactly as the duality predicts.

For an arbitrary concrete $C^*$ -algebra $A\subseteq B(H)$ , cyclicity for $A$ still implies separation for $A'$ . The converse naturally concerns the bicommutant $A''$ , not necessarily $A$ , because the projection argument only places the relevant projection in $A'$ .

References

Richard V. Kadison and John R. Ringrose, Fundamentals of the Theory of Operator Algebras, Volume I: Elementary Theory, American Mathematical Society, 1997 reprint. AMS DOI record. Relevant: Proposition 5.5.11 on cyclic and separating vectors for commutants.