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Pfaffenberger and Phillips  consider a real and unital case of the classical commutative Gelfand theorem and obtain two representation theorems. One is to represent a unital real commutative Banach algebra A as an algebra of continuous functions on the unital homomorphism space ΦA. The other is to represent A as an algebra of continuous sections on the maximal ideal space MA. In this note, we point out that similar theorems for non-unital case hold and show that two representation theorems are essentially identical.
real commutative Banach algebras
real algebra homomorphisms
commutative Gelfand theory.
Mathematical Journal of Okayama University
Department of Mathematics, Faculty of Science, Okayama University