Abstract
Takesaki [5] poses the question of how much information about aC *-algebraA is contained in its representation theory. He gives it a precise meaning in the following setting: One can furnish the set Rep (A:H) of all representations ofA in a suitable Hilbert spaceH with a topology, with an action of the unitary groupG ofB(H) on it, and with an addition. The setA F of operator fields Rep (A:H)→B(H) commuting with the action ofG and addition, called the admissible operator fields, turn out to form aW *-algebra isomorphic to the bidual ofA with Arens multiplication or with the universal enveloping von Neumann algebra ofA. Takesaki shows in the separable case thatA can be identified inA F as the set of continuous admissible operator fields, and leaves the same question open for arbitraryC *-algebras. Changing the structures on Rep(A:H) slightly, it is shown here that this result obtains in the general case as well. The proof proceeds along the lines set up in [5] but makes no use of the representation theory of NGCR algebras.
Similar content being viewed by others
References
Bauer, H.: Konvexität in topologischen Vektorräumen. Lecture notes, Universität Hamburg, Hamburg 1964/65.
Dixmier, J.: LesC *-algèbres et leurs représentations. Paris: Gauthiers-Villars 1964.
Ernest, J.: A new group algebra for locally compact groups. Amer. J. Math.86, 467–492 (1964); — Can. J. Math.17 (1965); — Hopf-von Neumann Algebras. Preprint University of California (1965), and literature cited therein.
Takeda, Z.: Conjugate spaces of operator algebras. Proc. Japan. Acad.30, 90–95 (1954).
Takesaki, M.: A duality in the representation theory ofC *-algebras. Ann. Math.85, 370–382 (1967).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Bichteler, K. A generalization to the non-separable case of Takesaki's duality theorem forC *-algebras. Invent Math 9, 89–98 (1969). https://doi.org/10.1007/BF01389891
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01389891