We are concerned with the development of the more general real case of the classical theorem of Gelfand on representation of a complex commutative unital Banach algebra as an algebra of continuous functions defined on a compact Hausdorff space. To that end, we use only intrinsic methods which do not depend on the complexification of the algebra, and obtain two representation theorems for commutative unital real Banach algebras as algebras of continuous real (respectively, complex) functions on the compact space of real-valued (respectively, complex-valued) $$\mathbb R$$ -algebra homomorphisms.

中文翻译：

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实巴拿赫代数的格尔凡德理论

我们关注 Gelfand 经典定理的更一般实例的发展，该定理将复杂的交换单位 Banach 代数表示为在紧致 Hausdorff 空间上定义的连续函数的代数。为此，我们仅使用不依赖于代数复化的内在方法，并获得可交换单位实巴拿赫代数的两个表示定理作为实值紧空间上连续实（分别为复）函数的代数（分别为复值） $$\mathbb R$$ -代数同态。