Vector functional on a von Neumann algebra - Imported Knowlpedia Content

Definition

Let $M\subseteq B(H)$ be a concrete von Neumann algebra on a Hilbert space $H$ . For $\xi,\eta\in H$ , the vector functional $\omega_{\xi,\eta}:M\to\mathbb C$ is

\omega_{\xi,\eta}(x)=\langle x\xi,\eta\rangle .

Here the inner product is taken to be linear in its first variable. The functional is bounded and normal, and $\lVert\omega_{\xi,\eta}\rVert\leq\lVert\xi\rVert\lVert\eta\rVert$ . When $\eta=\xi$ , it is positive; if additionally $\lVert\xi\rVert=1$ and the representation is unital, it is a vector state.

Relation to the predual

Every vector functional belongs to the predual $M_*$ . Conversely, every $\omega\in M_*$ can be written as an absolutely convergent series

\omega=\sum_{n=1}^{\infty}\omega_{\xi_n,\eta_n}, \qquad \sum_n\lVert\xi_n\rVert^2<\infty,\quad \sum_n\lVert\eta_n\rVert^2<\infty.

Thus vector functionals linearly generate the predual, although a given normal functional need not be a single vector functional in the chosen representation Kadison–Ringrose, §5.2.

Concrete examples

For $M=B(H)$ ,

\omega_{\xi,\eta}(x)=\operatorname{Tr}(\theta_{\xi,\eta}x),

where $\theta_{\xi,\eta}$ is the rank-one trace-class operator determined by the same inner-product convention. Hence vector functionals are exactly the predual functionals represented by rank-one trace-class operators. A normal state with a density operator of rank greater than one is not a vector state in the identity representation of $B(H)$ , but it is a sum of positive vector functionals.

Dependence on representation and conventions

Vector-functional status depends on the concrete representation $M\subseteq B(H)$ , whereas normality is intrinsic to $M$ . In a standard form, every positive normal functional is represented by a unique vector in the natural positive cone. With the alternative convention that the Hilbert-space inner product is linear in the second variable, authors reverse the vector order in the notation so that $x\mapsto\omega_{\xi,\eta}(x)$ remains linear in $x$ .

References

Richard V. Kadison and John R. Ringrose, Fundamentals of the Theory of Operator Algebras, Volume I: Elementary Theory, American Mathematical Society, 1997. DOI record. Relevant: §5.2 on vector functionals, ultraweak continuity, and the predual of a concrete von Neumann algebra.
Masamichi Takesaki, Theory of Operator Algebras I, Springer, 1979. DOI record. Relevant: Chapter III, §2 on normal functionals and concrete predual representations.