Hostname: page-component-7479d7b7d-wxhwt Total loading time: 0 Render date: 2024-07-10T23:23:21.375Z Has data issue: false hasContentIssue false
  • English
  • Français

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
Get access Rights & Permissions [Opens in a new window]

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

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

[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