Hostname: page-component-5c6d5d7d68-qks25 Total loading time: 0 Render date: 2024-08-30T20:28:42.970Z Has data issue: false hasContentIssue false
  • English
  • Français

Equivalent norms on Banach Jordan algebras

Published online by Cambridge University Press:  24 October 2008

M. A. Youngson
M. A. Youngson
Affiliation:
Heriot-Watt University
Get access Rights & Permissions [Opens in a new window]

Extract

1. Introduction. Recently Kaplansky suggested the definition of a suitable Jordan analogue of B*-algebras, which we call J B*-algebras (see (10) and (11)). In this article, we give a characterization of those complex unital Banach Jordan algebras which are J B*-algebras in an equivalent norm. This is done by generalizing results of Bonsall ((3) and (4)) to give necessary and sufficient conditions on a real unital Banach Jordan algebra under which it is the self-adjoint part of a J B*-algebra in an equivalent norm. As a corollary we also obtain a characterization of the cones in a Banach Jordan algebra which are the set of positive elements of a J B*-algebra.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 86 , Issue 2 , September 1979 , pp. 261 - 270
Copyright
Copyright © Cambridge Philosophical Society 1979

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

(1)Albert, A. A.On a certain algebra of quantum mechanics. Ann. of Math. (2), 35 (1934), 65–73.CrossRefGoogle Scholar
(2)Alfsen, E. M., Shultz, F. W. and Størmer, E. A.Gelfand-Neumark Theorem for Jordan algebras. Adv. in Math. 28 (1978), 11–56.CrossRefGoogle Scholar
(3)Bonsall, F. F.Locally multiplicative wedges in Banach algebras. Proc. London Math. Soc. (3) 30 (1975), 239–256.CrossRefGoogle Scholar
(4)Bonsall, F. F.Jordan subalgebras of Banach algebras. Proc. Edin. Math. Soc. 21 (1978), 103–110.CrossRefGoogle Scholar
(5)Bonsall, F. F. and Duncan, J.Complete normed algebras (Berlin, Heidelberg, New York, Springer–Verlag, 1973).CrossRefGoogle Scholar
(6)Cuntz, J.Locally C*-Equivalent Algebras. J. Funct. Anal. 23 (1976), 95–106.CrossRefGoogle Scholar
(7)Devapakkiam, C. V.Jordan algebras with continuous inverse. Math. Jap. 16 (1971), 115–125.Google Scholar
(8)Jacobson, N.Structure and representations of Jordan algebras (Amer. Math. Soc. Collo-quium Publications 39, Providence, 1968).CrossRefGoogle Scholar
(9)Jordan, P., von Neumann, J. and Wigner, E.On an algebraic generalisation of the quantum mechanical formulation. Ann. of Math. (2) 35 (1934), 29–64.CrossRefGoogle Scholar
(10)Wright, J. D. M.Jordan C*-algebras. Mich. Math. J. 24 (1977), 291–302.CrossRefGoogle Scholar
(11)Youngson, M. A.A Vidav theorem for Banach Jordan algebras. Math. Proc. Cambridge Philos. Soc. 84 (1978), 263–272.CrossRefGoogle Scholar