Sakai characterization theorem - Imported Knowlpedia Content

Statement

The Sakai characterization theorem states that a $C^*$ -algebra $M$ is a $W^*$ -algebra if and only if there is a Banach space $E$ such that $M$ is isometrically isomorphic, as a Banach space, to $E^*$ . In that case $E$ is canonically isometrically isomorphic to the predual $M_*$ , and $M$ admits a faithful normal representation whose image is a concrete von Neumann algebra. Thus being a Banach dual forces the operator-algebraic weak-star structure; it is not an additional hypothesis.

Content of the theorem

The difficult direction begins with only the $C^*$ -algebra operations, norm, and an isometric identification $M=E^*$ . Sakai's theorem shows that the resulting weak-star topology is compatible with multiplication and involution, identifies the normal functionals, and yields a concrete weak-operator-closed realization Sakai, Theorem 1.16.7. Conversely, the canonical predual of any concrete von Neumann algebra makes it a Banach dual.

Uniqueness and terminology

The predual obtained in the theorem is unique up to the canonical isometric isomorphism that preserves evaluation on $M$ . Accordingly, abstract $W^*$ -algebras and concrete von Neumann algebras are equivalent viewpoints, although a concrete realization still includes a choice of Hilbert space and representation. The phrase “dual $C^*$ -algebra” is potentially ambiguous in older literature and should not replace the precise Banach-dual formulation.

References

Shôichirô Sakai, $C^*$ -Algebras and $W^*$ -Algebras, Springer, 1971; Classics in Mathematics reprint, 1998. DOI record. Relevant: Theorem 1.16.7 and the abstract characterization of $W^*$ -algebras.
Masamichi Takesaki, Theory of Operator Algebras I, Springer, 1979. DOI record. Relevant: Chapter III on preduals, normal representations, and concrete von Neumann algebras.