Representation theorems for operators on free Banach spaces of type countable

Abstract: This work will be centered in commutative Banach subalgebras of the algebra of bounded linear operators defined on a Free Banach spaces of countable type. The main goal of this work wil 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 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 whose entries of these matrices will be integrals coming from scalar measures.