  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.
