Normal functional - Imported Knowlpedia Content

Definition

Let $M$ be a von Neumann algebra. A normal functional on $M$ is a bounded linear functional $\varphi:M\to\mathbb C$ that is continuous for the ultraweak topology. Equivalently, $\varphi$ belongs to the canonical predual $M_*$ , under the isometric identification $M=(M_*)^*$ . Thus a normal functional is precisely a scalar-valued normal linear map. Normality is an additional continuity property specific to von Neumann algebras; an arbitrary bounded functional on the underlying $C^*$ -algebra need not be normal.

Order characterization

If $\varphi$ is positive, normality is equivalent to preservation of suprema of bounded increasing nets:

\varphi\!\left(\sup_i x_i\right)=\sup_i\varphi(x_i)

for every bounded increasing net $(x_i)$ in $M_+$ . It is also enough to test this condition on increasing nets of projections. Replacing nets by sequences is generally insufficient unless suitable countability hypotheses are imposed. This order-continuity criterion is developed in Takesaki, Chapter III, §2.

Positive decomposition and states

Every normal functional is a linear combination of four positive normal functionals. Its adjoint $\varphi^*(x)=\overline{\varphi(x^*)}$ , real and imaginary parts, and polar decomposition all remain within $M_*$ . A positive normal functional of norm one is a normal state. Positivity and normalization are therefore extra properties; they are not part of the definition of a normal functional.

Concrete model

For $M=B(H)$ , each normal functional has the form

\varphi(x)=\operatorname{Tr}(Tx)

for a unique trace-class operator $T$ , and $\|\varphi\|=\|T\|_1$ . It is positive exactly when $T$ is positive. Singular bounded functionals on $B(H)$ lie outside the trace-class predual and show why norm continuity alone does not imply normality.

References

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