For any $$\hbox {C}^*$$-algebra $${\mathcal {A}}$$, we give a Banach $$*$$-algebra with approximate identity which $$C([-1, 1], {\mathcal {A}})$$, the $$\hbox {C}^*$$-algebra of all $${\mathcal {A}}$$-valued continuous functions on [0, 1], is its $$\hbox {C}^*$$-envelope. We show that $$C([-1, 1], {\mathcal {A}})$$ is $$*$$-isomorphic to a $$\hbox {C}^*$$-subalgebra of bounded continuous functions from self-adjoint elements of the closed unital ball of $${\mathcal {E}}$$ to $${\mathcal {A}}\otimes {\mathcal {E}}$$ for any unital $$\hbox {C}^*$$-algebra $${\mathcal {E}}$$. Furthermore, for any $$\hbox {C}^*$$-algebra $${\mathcal {A}}$$ and numerical semigroup S we give a pre-$$\hbox {C}^*$$-algebra with completion $$C([0, 1], {\mathcal {A}})$$ via Cauchy extensions of $$\hbox {C}^*$$-algebras. It is also shown that the Dirichlet extension of $${\mathcal {A}}$$ is $$*$$-isomorphic to $$C([0, 1], {\mathcal {A}})$$. Finally, we introduce the notion of M-Cauchy envelope of $$\hbox {C}^*$$-algebras, where M is an at most countable commutative monoid.