On normed Jordan algebras which are Banach dual spaces |
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Authors: | Frederic W Shultz |
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Affiliation: | Mathematics Department, Wellesley College, Wellesley, Massachusetts 02181 U.S.A. |
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Abstract: | Alfsen, Shultz, and Størmer have defined a class of normed Jordan algebras called JB-algebras, which are closely related to Jordan algebras of self-adjoint operators. We show that the enveloping algebra of a JB-algebra can be identified with its bidual. This is used to show that a JB-algebra is a dual space iff it is monotone complete and admits a separating set of normal states; in this case the predual is unique and consists of all normal linear functionals. Such JB-algebras (“JBW-algebras”) admit a unique decomposition into special and purely exceptional summands. The special part is isomorphic to a weakly closed Jordan algebra of self-adjoint operators. The purely exceptional part is isomorphic to C(X, M38) (the continuous functions from X into M38). |
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