A Katznelson-Tzafriri type theorem for Ces\`aro bounded operators

We extend the well-known Katznelson-Tzafriri theorem, originally posed for power-bounded operators, to the case of Ces`aro bounded operators of any order $\alpha >0.$ For this purpose, we use a functional calculus between a new class of fractional Wiener algebras and the algebra of bounded linear operators, whose existence is characterized by the Ces`aro boundedness. Finally, we apply the main theorem to get ergodicity results for the Ces`aro means of bounded operators.