Hostname: page-component-cd9895bd7-mkpzs Total loading time: 0 Render date: 2024-12-25T02:24:39.461Z Has data issue: false hasContentIssue false

A Vidav theorem for Banach Jordan algebras

Published online by Cambridge University Press:  24 October 2008

M. A. Youngson
Affiliation:
University of Edinburgh

Abstract

In the present article we prove a result characterizing the Jordan analogues of B*-algebras among the complex Banach Jordan algebras in terms of the algebra and norm structures alone.

Type
Research Article
Copyright
Copyright © Cambridge Philosophical Society 1978

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. 35 (1934), 6573.CrossRefGoogle Scholar
(2)Alfsen, E. M., Shultz, F. W. and Størmer, E.A Gelfand–Neumark theorem for Jordan algebras. Preprint, University of Oslo, 1975.Google Scholar
(3)Arveson, W. B.An invitation to C*-algebras (Berlin, Heidelberg, New York, Springer-Verlag, 1976).CrossRefGoogle Scholar
(4)Bonsall, F. F.Jordan algebras spanned by Hermitian elements of a Banach algebra. Math. Proc. Cambridge Philos. Soc. 81 (1977), 313.CrossRefGoogle Scholar
(5)Bonsall, F. F. and Duncan, J.Complete normed algebras (Berlin, Heidelberg, New York, Springer–Verlag, 1973).CrossRefGoogle Scholar
(6)Bonsall, F. F. and Duncan, J.Numerical ranges of operators on normed spaces and of elements of normed algebras (London Math. Soc. Lecture Note Series 2, Cambridge University Press, 1971).CrossRefGoogle Scholar
(7)Civin, P. and Yood, B.Lie and Jordan structures in Banach algebras. Pacific J. Math. 15 (1965), 775797.CrossRefGoogle Scholar
(8)Devapakkiam, C. V.Jordan algebras with continuous inverse. Math. Japan 16 (1971). 115125.Google Scholar
(9)Glennie, C. M.Some identities valid in special Jordan algebras but not valid in all Jordan algebras. Pacific J. Math. 16 (1966), 4759.CrossRefGoogle Scholar
(10)Glimm, J. G. and Kadison, R. V.Unitary operators in C*-algebras. Pacific J. Math. 10 (1960), 347356.CrossRefGoogle Scholar
(11)Harris, L. A. Bounded symmetric homogeneous domains in infinite dimensional space. Lecture Notes in Mathematics 364 (Berlin, Heidelberg, New York, Springer-Verlag, 1974).Google Scholar
(12)Jacobson, N.Struture and representations of Jordan algebras (Amer. Math. Soc. Colloquium Publications 39, Providence, 1968).CrossRefGoogle Scholar
(13)Jordan, P., von Neumann, J. and Wigner, E.On an algebraic generalisation of the quantum mechanical formulation. Ann. of Math. 35 (1934), 2964.CrossRefGoogle Scholar
(14)Kadison, R. V.A generalised Schwarz inequality and algebraic invariants for operator algebras. Ann. of Math. 56 (1952), 494503.CrossRefGoogle Scholar
(15)Kaplansky, I.Normed algebras. Duke Math. J. 16 (1949), 399418.CrossRefGoogle Scholar
(16)Palmer, T. W.Characterisations of C*-algebras. Bull. Amer. Math. Soc. 74 (1968), 538540.CrossRefGoogle Scholar
(17)Palmer, T. W.Characterisations of C*-algebras. II. Trans. Amer. Math. Soc. 148 (1970), 577588.Google Scholar
(18)Segal, I. E.Irreducible representations of operator algebras. Bull. Amer. Math. Soc. 53 (1947), 7388.CrossRefGoogle Scholar
(19)Sherman, S.On Segal's postulates for general quantum mechanics. Ann. of Math. 64 (1956), 593601.CrossRefGoogle Scholar
(20)Størmer, E.On the Jordan structure of C*-algebras. Trans. Amer. Math. Soc. 120 (1965), 438447.Google Scholar
(21)Vidav, I.Eine metrische Kennzeichnung der selbstradjungierten Operatoren. Math. Z. 66 (1956), 121128.CrossRefGoogle Scholar
(22)Wright, J. D. M. Jordan C*-algebras. Preprint.Google Scholar
(23)Wright, J. D. M. and Youngson, M. A.A Russo Dye theorem for Jordan C*-algebras. Functional Analysis: Surveys and Recent Results (North Holland Press, 1977).Google Scholar