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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
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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

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