A NOTE ON COMMUTATIVE GELFAND THEORY FOR REAL BANACH ALGEBRAS

Abstract. Pfaffenberger and Phillips [2] consider a real and uni-tal case of the classical commutative Gelfand theorem and obtain two representation theorems. One is to represent a unital real com-mutative 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. 1. Preliminary and results Let B be a real commutative Banach algebra and ΦB the set of non-zero R-algebra homomorphism ϕ: B → C. Then we have ‖ϕ ‖ def= sup‖a‖≤1 |ϕ(a) | ≤ 1 for each ϕ ∈ ΦB. Actually, suppose that there exists an a ∈ B such that ‖a ‖ < 1 and |ϕ(a) | = 1. Set θ = − argϕ(a) and then eiθϕ(a) = 1. Also set b = n=1 an cosnθ and c = n=1 an sinnθ. Elementary trigonometric identities lead t