Unique predual theorem - Imported Knowlpedia Content

Statement

Let $M$ be a von Neumann algebra. The unique predual theorem states that its predual $M_*$ is the unique Banach space predual of $M$ , up to the canonical isometric isomorphism compatible with the dual pairing. More explicitly, if a Banach space $E$ and a linear isometry $\Phi:M\to E^*$ exhibit $M$ as a dual Banach space, then $E$ identifies isometrically with $M_*$ through the functionals it induces on $M$ . Consequently the weak-star topology $\sigma(M,M_*)$ is intrinsic to the von Neumann algebra's normed $*$ -algebra structure.

Meaning of uniqueness

The assertion is stronger than the existence of some abstract isometry between two predual spaces. Each element of a proposed predual determines a bounded functional on $M$ , and the theorem says that the resulting subspace of $M^*$ is exactly $M_*$ . Thus the dual pairing, and not merely the Banach-space isomorphism class, is fixed.

Consequences for normality

Because the weak-star topology is unique, weak-star continuous functionals on $M$ are exactly the normal functionals, independently of which dual presentation is initially chosen. Weak-star continuous $*$ -homomorphisms, normal states, and weak-star compactness are therefore intrinsic notions rather than artifacts of a concrete representation on a Hilbert space.

Context and caution

Many dual Banach spaces have inequivalent preduals, so uniqueness is a special rigidity property of von Neumann algebras. The theorem concerns isometric dual realizations. It should not be weakened to a claim that every Banach space merely isomorphic to $M$ as a Banach space carries the same predual or weak-star topology.

References

Shôichirô Sakai, C-Algebras and W-Algebras, Springer, 1998. Publisher record. Relevant: Theorem 1.13.2 on uniqueness of the predual.
Masamichi Takesaki, Theory of Operator Algebras I, Springer, 2002. Publisher record. Relevant: Chapter III, §2 on the predual and normal functionals.