REAL BANACH ALGEBRAS AS 𝒞 (𝒦) ALGEBRAS

The classical theorem of Gelfand provides a representation of a commutative complex unital Banach algebra as a subalgebra of 𝒞_ℂ(𝒦) of continuous complex-valued functions defined on a compact Hausdorff space 𝒞. Since the complex algebras can be regarded as a subclass of the real algebras, it is natural to ask what can be said about this larger class. As it happens, a real commutative Banach algebra 𝒜 does admit a Gelfand representation a → â as in the complex case, where each â: 𝒦 → ℂ is a continuous function. However, if we attempt to represent a commutative real Banach algebra as a...

The classical theorem of Gelfand provides a representation of a commutative complex unital Banach algebra as a subalgebra of 𝒞_ℂ(𝒦) of continuous complex-valued functions defined on a compact Hausdorff space 𝒞. Since the complex algebras can be regarded as a subclass of the real algebras, it is natural to ask what can be said about this larger class. As it happens, a real commutative Banach algebra 𝒜 does admit a Gelfand representation a → â as in the complex case, where each â: 𝒦 → ℂ is a continuous function. However, if we attempt to represent a commutative real Banach algebra as a subalgebra of 𝒞(𝒦) of continuous real-valued functions in the same fashion, complications arise and in the general case it need not even be true. In this article, we will look at two conditions on 𝒜 that will imply that the representation of 𝒜 as a space of continuous functions consists only of real-valued functions. The methods we use are intrinsic, that is to say, they do not rely on the complexification of the algebra.