This work will be centered in commutative Banach subalgebras of the algebra of bounded linear operators defined on free Banach spaces of countable type. The main goal of this work will be to formulate a representation theorem for these operators through integrals defined by spectral measures type. In order to get this objective, we will show that, under special conditions, each one of these algebras is isometrically isomorphic to some space of continuous functions defined over a compact set. Then, we will identify such compact sets developing the Gelfand space theory in the non-Archimedean setting. This fact will allow us to define a measure which is known as spectral measure. As a second goal, we will formulate a matrix representation theorem for this class of operators in which the entries of the matrices will be integrals coming from scalar measures.

中文翻译：

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可数型自由巴拿赫空间算子的表示定理

这项工作将集中在可数类型的自由 Banach 空间上定义的有界线性算子的代数的交换 Banach 子代数上。这项工作的主要目标是通过谱测度类型定义的积分为这些算子制定表示定理。为了达到这个目标，我们将证明，在特殊条件下，这些代数中的每一个都与定义在紧集上的连续函数空间等距同构。然后，我们将确定在非阿基米德环境中发展 Gelfand 空间理论的这种紧集。这一事实将使我们能够定义一个称为频谱度量的度量。作为第二个目标，