The classical theorem of Gelfand provides a representation of a commutative complex unital Banach algebra as a subalgebra of ๐’žโ„‚(๐’ฆ) of continuous complex-valued functions defined on a compact Hausdorff space ๐’ž. 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 ๐’œ does admit a Gelfand representation a โ†’ รข as in the complex case, where each รข: ๐’ฆ โ†’ โ„‚ is a continuous function. However, if we attempt to represent a commutative real Banach algebra as a subalgebra of ๐’ž(๐’ฆ) 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 ๐’œ that will imply that the representation of ๐’œ 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.

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