Hostname: page-component-cd9895bd7-dzt6s Total loading time: 0 Render date: 2024-12-26T00:54:51.012Z Has data issue: false hasContentIssue false
  • English
  • Français

Traces On Jordan Algebras

Published online by Cambridge University Press:  20 November 2018

Gert Kjærgård Pedersen  and
Erling Størmer
Gert Kjærgård Pedersen
Affiliation:
University of Copenhagen, Copenhagen, Denmark
Erling Størmer
Affiliation:
University of Oslo, Blindern, 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.

In the theory of Jordan algebras one encounters several definitions of the trace, and it is sometimes unclear whether the different notions are equivalent or not. If we restrict attention to the so–called JB–algebras studied in [2] and their weakly closed analogues JBW–algebras [8], we shall in the present note show that the different concepts are all equivalent for JBW–algebras, and that the conditions not involving projections are equivalent for JB–algebras. Among the seven equivalent conditions we shall consider, the second (ii) was used by Alfsen and Shultz [1] to show that if the JBW–algebra, is the self–adjoint part of a von Neumann algebra, then the condition characterizes traces on the enveloping von Neumann algebra. Condition (iii) appears in Robertson's paper [7] together with the implication (ii) ⇒ (iii). The inequality (iv) is a Jordan analogue of Gardner's inequality | ϕ(x) ≦ ϕ(|x|)|, [3], characterizing traces on Cast;–algebras.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 34 , Issue 2 , 01 April 1982 , pp. 370 - 373
Copyright
Copyright © Canadian Mathematical Society 1981

References

1. Alfsen, E. M. and Shultz, F. W., Non-commutative spectral theory for affine functions on convex sets, Mem. Amer. Math. Soc. 172 (1976).Google Scholar
2. Alfsen, E. M., Shultz, F. W. and Stormer, E., A Gelfand-Neumark theorem for Jordan algebras, Adv. in Math. 28 (1978), 11–56.Google Scholar
3. Gardner, L. T., An inequality characterizes the trace, Can. J. Math. 31 (1979), 1322– 1328.Google Scholar
4. Janssen, G., Réelle Jordanalgebren mit endlicher Spur, Manuscripta Math. 13 (1974), 237–273.Google Scholar
5. Jacobson, N., Structure and representations of Jordan algebras Amer. Math. Soc. Colloq. Publ. 39 (Amer. Math. Soc, Providence, R.I., 1968).Google Scholar
6. Pedersen, G. K., C*-algebras and their automorphism groups London Math. Soc. Monographs 14 (Academic Press, London/New York, 1979).Google Scholar
7. Robertson, A. G., Projections and proximity maps on Jordan operator algebras, to appear.Google Scholar
8. Shultz, F. W., On normed Jordan algebras which are Banach dual spaces, J. Functional Anal. 31 (1979), 360–376.Google Scholar
9. Topping, D., Jordan algebras of self-adjoint operators, Mem. Amer. Math. Soc. 53 (1965).Google Scholar