Abstract

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.

You do not currently have access to this article.