Geodesic
An order-preserving representation theorem for complex Banach algebras and some examples
Glasgow mathematical journal, Tome 14 (1973) no. 2, pp. 128-135

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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. 
Thompson, A. C.; Vijayakumar, M. S. An order-preserving representation theorem for complex Banach algebras and some examples. Glasgow mathematical journal, Tome 14 (1973) no. 2, pp. 128-135. doi: 10.1017/S0017089500001877
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