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The structure of type I JBW-algebras

Published online by Cambridge University Press:  24 October 2008

P. J. Stagey
P. J. Stagey
Affiliation:
La Trobe University
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Extract

A complete normed real Jordan algebra A with 1 is said to be a JB-algebra if and only if, for all a, b ∈ A,

where o denotes the Jordan product on A. If A is a Banach dual space then A is known as a JBW-algebra. In this case, by Theorem 3·9 of (7),

where Aex is an exceptional Jordan algebra and Asp is Jordan isomorphic to a JW-algebra (i.e. a weakly closed Jordan algebra of self-adjoint operators on a complex Hilbert space). Theorem 3–9 of (7) gives a description of Aex and so in this paper attention will be restricted to JW-algebras. Such an algebra A is said to be of type I if there exists an idempotent e in A with central cover equal to 1 such that e is abelian i.e. such that eAe is commutative as a set of operators. (See § 7 of (11) for further details.) A type I algebra A is said to be of type In, where n is some cardinal number, if there exists an orthogonal family {eα: α ∈ Λ} of n abelian projections in A such that Σαeα = 1 and such that, for each α ≠ β, there exists a symmetry sαβ in A with eβ = sαβeαsαβ. (A symmetry is an element sαβ of A such that ). By theorems 15 and 16 of (11) any type I JW-algebra A can be uniquely decomposed into a direct sum of type In(α) JW-algebras for suitable cardinal numbers n(α). Thus in order to investigate the structure of type I JW-algebras it is sufficient to consider type In JW-algebras.

Information

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 90 , Issue 3 , November 1981 , pp. 477 - 482
Copyright
Copyright © Cambridge Philosophical Society 1981

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References

REFERENCES

(1)Alfsen, E. M., Hanche-Olsen, H. and Shultz, F. W.State spaces of C*-algebras. Acta Math. 144 (1980), 267–305.CrossRefGoogle Scholar
(2)Alfsen, E. M., Shuiltz, F. W. and Størmer, E.A Gelfand-Neumark theorem for Jordan algebras. Advances in Math. 23 (1973), 11–56.Google Scholar
(3)Dixmier, J.Les algèbres d'opérateurs dans l'espace Hubertien (Gauthier-Villars, Paris, 1969).Google Scholar
(4)Ionescu, Tulcea A. and Ionescu, Tulcea C.Topics in the theory of lifting (Ergebnisse der Mathematik und ihrer Grenzgebiete 43, Springer, Berlin-Heidelberg-New York, 1969).Google Scholar
(5)Kaplansky, I.Rings of operators (Benjamin, New York, 1963).Google Scholar
(6)Sakai, S.C*-algebras and W*-algebras (Ergebnisse der Mathematik und ihrer Grenzgebiete 60, Springer, Berlin-Heidelberg-New York, 1971).Google Scholar
(7)Shultz, F. W.On normed Jordan algebras which are Banach dual spaces. J. Funct. Anal. 31 (1979), 360–376.CrossRefGoogle Scholar
(8)Stacey, P. J.Type I2 JBW-algebras. Quart. J. Math. Oxford (2), (1981, to appear).Google Scholar
(9)Stacey, P. J.Local and global splittings in the state space of a JB-algebra. Math. Ann., (1981, to appear).CrossRefGoogle Scholar
(10)Størmer, E.Jordan algebras of type I. Acta Math. 115 (1966), 165–184.CrossRefGoogle Scholar
(11)Topping, D.Jordan algebras of self-adjoint operators (Mem. Amer. Math. Soc. 53, Amer. Math. Soc., Providence, R.I., 1965).CrossRefGoogle Scholar