Singular state on a von Neumann algebra - Imported Knowlpedia Content

Definition

Let $M$ be a von Neumann algebra. A state $\varphi:M\to\mathbb C$ is singular if every normal positive functional $\psi$ satisfying $0\leq\psi\leq\varphi$ is zero. Thus a singular state has no nonzero normal positive part hidden beneath it in the order on functionals. This is stronger than merely being non-normal: a convex combination of a normal state and a singular state is non-normal but not singular. The convention is the noncommutative analogue of a purely finitely additive measure in the Yosida–Hewitt decomposition.

Equivalent projection criterion

Takesaki's projection criterion says that a positive functional $\varphi$ on $M$ is singular exactly when every nonzero projection $e\in M$ dominates a nonzero projection $f\leq e$ with $\varphi(f)=0$ Takesaki, Theorem 1. This formulation shows that singularity is distributed throughout the projection lattice; it is not simply failure of ultraweak continuity at one particular increasing net.

Normal-singular decomposition

Every positive functional $\omega\in M^*$ decomposes uniquely as

\omega=\omega_{\mathrm n}+\omega_{\mathrm s},

where $\omega_{\mathrm n}$ is normal and $\omega_{\mathrm s}$ is singular. For a state, the two summands are positive but need not themselves be states: their values at $1$ add to $1$ . The state is normal precisely when $\omega_{\mathrm s}=0$ , and singular precisely when $\omega_{\mathrm n}=0$ . This is the operator-algebraic normal-singular decomposition Takesaki, pp. 365–366.

Examples and non-examples

On $\ell^\infty(\mathbb N)$ , evaluation along a free ultrafilter is a singular state. It vanishes on every finitely supported sequence even though the corresponding finite-coordinate projections increase strongly to $1$ . On $B(H)$ for infinite-dimensional $H$ , any state that factors through the quotient by $K(H)$ is singular.

Finite-dimensional von Neumann algebras have no singular states because every linear functional on them is normal. A nonnormal state with a nonzero normal summand is the decisive near-miss: nonnormality alone does not imply singularity.

References

Masamichi Takesaki, “On the Singularity of a Positive Linear Functional on Operator Algebra,” Proceedings of the Japan Academy 35 (1959), 365–366. DOI record. Relevant: the opening definition, normal-singular decomposition, and Theorem 1.
Richard V. Kadison and John R. Ringrose, Fundamentals of the Theory of Operator Algebras, Volume II: Advanced Theory, AMS, 1997. Publisher record. Relevant: §7.1 on normal and singular functionals.