Normal functional - Knowlpedia

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.

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.