Hostname: page-component-78c5997874-4rdpn Total loading time: 0 Render date: 2024-11-16T15:06:07.193Z Has data issue: false hasContentIssue false
  • English
  • Français

On the Structure and Tensor Products of JC-Algebras

Published online by Cambridge University Press:  20 November 2018

Harald Hanche-Olsen
Harald Hanche-Olsen*
Affiliation:
Insititute for Energy Technology, Kjeller, Norway
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Norm closed (or weakly closed) Jordan algebras of self-adjoint operators on a Hilbert space were initially studied by Topping, Effros, and Stormer [15], [4], [12], [13]. These works are very “spatial”, in that the algebras are considered in one given representation. The introduction of their abstract counterparts, the JB- and JBW-algebras, has led to an increased interest in this subject. The author hopes this paper will support the view that a more “space-free” approach is fruitful, even if only the “concrete” algebras are under study. In accordance with this view, a “JC-algebra” in this paper will mean a normed Jordan algebra over the reals, which is isometrically isomorphic to a norm closed Jordan algebra of self-adjoint operators.

Some of the results in this paper are closely related to, or rewordings of, results in the above-mentioned papers. However, I feel that the present approach is sufficiently different to be of interest in itself. In particular, many of the technical difficulties associated with earlier approaches are avoided.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 35 , Issue 6 , 01 December 1983 , pp. 1059 - 1074
Copyright
Copyright © Canadian Mathematical Society 1983

References

1. Alfsen, E. M., Shultz, F. W. and Stormer, E.. A Gelfand-Neumark theorem for Jordan algebras, Adv. Math. 28 (1978), 11–56.Google Scholar
2. Alfsen, E. M., Hanche-Olsen, H. and Shultz, F. W., State spaces of C*-algebras, Acta Math. 744 (1980), 267–305.Google Scholar
3. Edwards, C. M., Ideal theory in JB-algebras. J. London Math. Soc. (2) 16 (1977), 507–513.Google Scholar
4. Effros, E. and Stormer, E., Jordan algebras of self adjoint operators, Trans. Amer. Math. Soc. 727 (1967), 313–316.Google Scholar
5. Giordano, T., Antiautomorphisms involutifs des facteurs de von Neumann injectifs, thesis (1981).Google Scholar
6. Hanche-Olsen, H., A note on the bidual of a JB-algebra, Math. Zeit. 175 (1980), 29–31.Google Scholar
7. Hanche-Olsen, H., >Split faces and ideal structure of operator algebras, Math. Scand. 48 (1981), 137–144.Google Scholar
8. Kirillov, A. A., Elements of the theory of representations (Springer-Verlag, Berlin, Heidelberg, New York, 1976).CrossRefGoogle Scholar
9. Robertson, A. G., Automorphisms of spin factors and decomposition of positive maps, Quart. J. Math. Oxford (2) 34 (1983), 87–96.Google Scholar
10. Shultz, F. W., On normed Jordan algebras which are Banach dual spaces, J. Funct. Anal. 31 (1979), 360–376.Google Scholar
11. Storrner, E., On the Jordan structure of C*-algebras, Trans. Amer. Math. Soc. 120 (1965), 438–447.Google Scholar
12. Storrner, E., Jordan algebras of type I, Acta Math. 115 (1966), 165–184.Google Scholar
13. Storrner, E., Irreducible Jordan algebras of self adjoint operators, Trans. Amer. Math. Soc. 130 (1968), 153–166.Google Scholar
14. Storrner, E., Real structure in the hyperfinite factor, Duke Math. J. 47 (1980), 145–154.Google Scholar
15. Topping, D., Jordan algebras of self adjoint operators, Mem. Amer. Math. Soc. 53 (1965).Google Scholar
16. Storrner, E., An isomorphism invariant for spin factors, J. Math. Mech. 75 (1966), 1055–1064.Google Scholar