Differentiability Properties of an Abstract Autonomous Composition Operator

We define an `abstract' composition operator of elements of a suitable Banach space, with the elements of a commutative Banach algebra with unity, and produce theorems of $r$-th order differentiability for such operator. We also show that, under general conditions, if the Banach space and the Banach algebra are function spaces, then the `abstract' composition coincides with the ordinary composition of functions, and present some applications to the composition operator in function spaces function spaces.