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Commutativity of C*-algebras and associativity of JB*-algebras

Published online by Cambridge University Press:  28 June 2011

B. Iochum ,
G. Loupias  and
A. Rodriguez-Palacios
B. Iochum
Affiliation:
Centre de Physique Théorique, CNRS-Luminy, Case 907, F-13288, Marseille Cedex 09, France
G. Loupias
Affiliation:
Département de Physique Mathématique, Université des Sciences et Techniques du Languedoc, 34060 Montpellier Cedex, France
A. Rodriguez-Palacios
Affiliation:
Universidad de Granada, Facultad de Ciencias, Departamento de Analisis Matematico, 18071 Granada, Spain
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Abstract

A n.c. JB*-algebra is associative and commutative if and only if it has no non-zero nilpotent elements. This generalizes the analogous theorem of Kaplansky for C*-algebras. Different characterizations of associativity are given.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 106 , Issue 2 , September 1989 , pp. 281 - 291
Copyright
Copyright © Cambridge Philosophical Society 1989

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References

[1] Alfsen, E. M. and Effros, E. G.. Structure in real Banach spaces II. Ann. of Math. (2) 9 (1972), 129–173.CrossRefGoogle Scholar
[2] Alfsen, E. M., Shultz, F. W. and Størmee, E.. A Gelfand-Neumark theorem for Jordan algebras. Adv. in Math. 28 (1978), 11–56.Google Scholar
[3] Alvermann, K. and Janssen, G.. Real and complex non-commutative Jordan Banach algebras. Math. Z. 185 (1984), 105–113.CrossRefGoogle Scholar
[4] Aupetit, B.. Propriétés Spectrales des Algèbres de Banach. Lecture Notes in Math. vol. 735 (Springer-Verlag, 1979).CrossRefGoogle Scholar
[5] Benslimane, M., Fernandez, A., Garcia, E. and Kaidi, A. M.. Non-commutative Jordan algebras containing minimal inner ideals and without nilpotent elements. (Preprint, 1988.)Google Scholar
[6] Bonsall, F. F. and Duncan, J.. Complete Normed Algebras. Ergeb. Math. Grenzgeb. no. 80 (Springer-Verlag, 1973).Google Scholar
[7] Bouldin, R.. The numerical range of a product II. J. Math. Anal. Appl. 33 (1971), 212–219.Google Scholar
[8] Braun, R. B.. Structure and representations of non-commutative C*-Jordan algebras. Manuscripta Math. 41 (1983), 139–171.CrossRefGoogle Scholar
[9] Braun, R. B.. A Gelfand-Neumark theorem for C*-alternative algebras. Math. Z. 185 (1984), 225–242.Google Scholar
[10] Braun, R. B., Kaup, W. and Upmeier, H.. A holomorphic characterization of Jordan C*-algebras. Math. Z. 161 (1978), 277–290.Google Scholar
[11] Bunce, L. J.. On compact action in JB-algebras. Proc. Edinburgh Math. Soc. 26 (1983), 353–360.CrossRefGoogle Scholar
[12] Crabb, M. J., Duncan, J. and McGregor, C. M.. Characterizations of commutativity for C*-algebras. Glasgow Math. J. 15 (1974), 172–175.CrossRefGoogle Scholar
[13] Dineen, S. and Timoney, R. M.. The centroid of a JB*-triple system. Math. Scand. (to appear).Google Scholar
[14] Doran, R. S. and Belfi, V. A.. Characterizations of C*-algebras: The Gelfand-Naimark Theorem (Marcel Dekker, 1986).Google Scholar
[15] Duncan, J.. Review of [26]. Math. Rev. 87e:46067.Google Scholar
[16] Duncan, J. and Taylor, P. J.. Norm inequalities for C*-algebras. Proc. Roy. Soc. Edinburgh Sect. A 75 (1975/1976), 119–129.CrossRefGoogle Scholar
[17] Edwards, C. M.. On Jordan W*-algebras. Bull. Sci. Math. (2) 104 (1980), 393–403.Google Scholar
[18] Hanche-Olsen, H. and Størmer, E.. Jordan operators algebras. Monographs Stud. Math. no. 21 (Pitman, 1984).Google Scholar
[19] Holbrook, J. A. R.. On the power-bounded operators of Sz-Nagy and Foias. Acta Sci. Math. (Szeged) 29 (1968), 299–310.Google Scholar
[20] Huruya, T.. The normed space numerical index of C*-algebras. Proc. Amer. Math. Soc. 63 (1977), 289–290.Google Scholar
[21] Iochum, B.. Cônes Autopolaires et Algèbres de Jordan. Lecture Notes in Math. vol. 1049 (Springer-Verlag, 1984).Google Scholar
[22] Kaidi, A. M., Martinez, J. and Rodriguez, A.. On a non-associative Vidav-Palmer theorem. Quart. J. Math. Oxford Ser. 2 32 (1981), 435–442.Google Scholar
[23] Kaplansky, I.. Rings of Operators. Math. Lecture Note Ser. (Benjamin, 1968).Google Scholar
[24] Lin, C. S.. On commutativity of C*-algebras. Glasgow Math. J. 29 (1987), 93–97.Google Scholar
[25] Martinez, J.. JV-algebras. Math. Proc. Cambridge Philos. Soc. 87 (1980), 47–50.Google Scholar
[26] Martinez, J., Mena, J. F., Paya, R. and Rodriguez, A.. An approach to numerical ranges without Banach algebra theory. Illinois J. Math. 29 (1985), 609–625.Google Scholar
[27] Sz-Nagy, B. and Foias, C.. Harmonic Analysis of Operators in Hilbert Space (Akadémiai Kiadó, 1970).Google Scholar
[28] Ogasawara, T.. A theorem on operator algebras. J. Sci. Hiroshima Univ. Ser. A 18 (1955), 307–309.Google Scholar
[29] Paya, R., Perez, J. and Rodriguez, A.. Non-commutative Jordan C*-algebras. Manuscripta Math. 37 (1982), 87–120.Google Scholar
[30] Paya, R., Perez, J. and Rodriguez, A.. Type I factor representations of non-commutative JB*-algebras. Proc. London Math. Soc. (3) 48 (1984), 428–444.CrossRefGoogle Scholar
[31] Rodriguez, A.. A Vidav-Palmer theorem for Jordan C*-algebras and related topics. J. London Math. Soc. (2) 22 (1980), 318–332.Google Scholar
[32] Rodriguez, A.. Non-associative normed algebras spanned by Hermitian elements. Proc. London Math. Soc. (3) 47 (1983), 258–274.Google Scholar
[33] Schafer, R. D., An Introduction to Non-associative Algebras (Academic Press, 1966).Google Scholar
[34] Sherman, S.. Order in operator algebras. Amer. J. Math. 73 (1981), 227–232.Google Scholar
[35] Shultz, F. W.. On normed Jordan algebras which are Banach dual spaces. J. Fund. Anal. 31 (1979), 360–376.CrossRefGoogle Scholar
[36] Taylor, D. C.. The strict topology for double centralizer algebras. Trans. Amer. Math. Soc. 150 (1970), 633–643.Google Scholar
[37] Upmeier, H.. Symmetric Banach Manifolds and Jordan C*-algebras. North-Holland Math. Stud. no. 104 (North-Holland, 1985).Google Scholar
[38] Wright, J. D. M.. Jordan C*-algebras. Michigan Math. J. 24 (1977), 291–302.CrossRefGoogle Scholar
[39] Youngson, M. A.. A Vidav theorem for Banach Jordan algebras. Math. Proc. Cambridge Philos. Soc. 84 (1978), 263–272.CrossRefGoogle Scholar
[40] Youngson, M. A.. Hermitian operators on Banach Jordan algebras. Proc. Edinburgh Math. Soc. 22 (1979), 93–104.CrossRefGoogle Scholar
[41] Youngson, M. A.. Non unital Banach Jordan algebras and C*-triple systems. Proc. Edinburgh Math. Soc. 24 (1981), 19–31.Google Scholar