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An order-preserving representation theorem for complex Banach algebras and some examples

Published online by Cambridge University Press:  18 May 2009

A. C. Thompson  and
M. S. Vijayakumar
A. C. Thompson
Affiliation:
Dalhousie University, Halifax, N.S., Canada
M. S. Vijayakumar
Affiliation:
Dalhousie University, Halifax, N.S., Canada
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Let A be a complex Banach algebra with unit e of norm one. We show that A can be represented on a compact Hausdorff space ω which arises entirely out of the algebraic and norm structures of A. This space induces an order structure on A that is preserved by the representation. In the commutative case, ω is the spectrum of A, and we have a generalization of Gelfand's representation theorem for commutative complex Banach algebras with unit. Various aspects of this representation are illustrated by considering algebras of n × n complex matrices.

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 14 , Issue 2 , September 1973 , pp. 128 - 135
Copyright
Copyright © Glasgow Mathematical Journal Trust 1973

References

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