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On non-unital Jordan–Banach algebras

Published online by Cambridge University Press:  24 October 2008

R. R. Smith
R. R. Smith
Affiliation:
Texas A & M University, College Station, Texas 77843
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Extract

A unital JB-algebra is a Jordan algebra A with identity together with a complete norm satisfying, for all a, b ∈ A,

(i) (a2b) a = a2(ba),

(ii) ∥a2∥ = ∥a∥2,

(iii) ∥ab∥ ≤ ∥a∥ ∥b∥,

(iv) ∥a2 + b2∥ ≥ ∥a2∥, ∥b2∥.

(It should be noted that axiom (iii) is a consequence of (ii) and (iv).) Such spaces have been studied by several authors (3, 6, 11), and as a consequence their structure is now quite well understood. Many of the results of these papers, while relying on the existence of an identity for their proofs, can be formulated for algebras which lack this property. C*-algebra theory and operator theory abound in examples of spaces which fail to be unital JB-algebras only in this one respect, and this motivates the study of the general case undertaken in this note.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 82 , Issue 3 , November 1977 , pp. 375 - 380
Copyright
Copyright © Cambridge Philosophical Society 1977

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References

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