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Jordan algebras spanned by Hermitian elements of a Banach algebra

Published online by Cambridge University Press:  24 October 2008

F. F. Bonsall
F. F. Bonsall
Affiliation:
University of Edinburgh
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Extract

The Vidav–Palmer theorem [(11), (5), (2) (p. 65)] characterizes C*-algebras among Banach algebras in terms of the algebra and norm structure alone, without reference to an involution, in the following way. Let B denote a complex unital Banach algebra, and let Her (B) denote the set of Hermitian elements of B, that is the elements of B with real numerical ranges. In this notation, the Vidav–Palmer theorem tells us that if

then B is isometrically isomorphic to a C*-algebra of operators on a Hilbert space, with the Hermitian elements corresponding to the self-adjoint operators in the C*-algebra.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 81 , Issue 1 , January 1977 , pp. 3 - 13
Copyright
Copyright © Cambridge Philosophical Society 1977

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References

REFERENCES

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