Predual of a von Neumann algebra

For a von Neumann algebra $M$ , its predual $M_*$ is the Banach space of ultraweakly continuous linear functionals on $M$ , with the norm inherited from the Banach dual $M^*$ . Evaluation gives an isometric identification

M=(M_*)^*.

Under this identification, the weak-star topology $\sigma(M,M_*)$ is precisely the ultraweak topology on $M$ . Consequently, elements of $M_*$ are also called normal functionals. A fundamental uniqueness theorem says that $M_*$ is the unique Banach predual of $M$ up to the canonical isometric isomorphism compatible with the duality.

Concrete realization

If $M\subseteq B(H)$ is represented concretely, every element of $M_*$ has the form

x\longmapsto \operatorname{Tr}(ax),

after restricting a trace-class operator $a$ on $H$ to $M$ . Different trace-class operators can induce the same functional. More precisely, $M_*$ is isometrically the quotient of the trace-class operators by the annihilator of $M$ , rather than generally a distinguished subspace of the trace class.

Positive and normal functionals

Positive elements of $M_*$ are exactly the normal positive functionals on $M$ . Every element of $M_*$ is a linear combination of normal positive functionals, and normal states are the positive elements of norm one. This order structure makes monotone convergence inside $M$ visible to its predual.

Why the distinction matters

The full Banach dual $M^*$ usually contains singular functionals that are not ultraweakly continuous. Thus $M_*$ must not be confused with $M^*$ . Weak-star compactness, normal representations, and integration theory for von Neumann algebras use the dual pair $(M,M_*)$ , not the larger pairing $(M,M^*)$ .

References