  Let $X$ be a complex Banach space. The connection between algebra
homomorphisms defined on subalgebras of the Banach algebra
$\ell^{1}(\mathbb{N}_0)$ and the algebraic structure of Ces\`{a}ro sums of a
linear operator $T\in \mathcal{B}(X)$ is established. In particular, we show
that every $(C, \alpha)$-bounded operator $T$ induces - and is in fact
characterized - by such an algebra homomorphism. Our method is based on some
sequence kernels, Weyl fractional difference calculus and convolution Banach
algebras that are introduced and deeply examined. To illustrate our results,
improvements to bounds for Abel means, new insights on the $(C,\alpha)$
boundedness of the resolvent operator for temperated $\alpha$-times integrated
semigroups, and examples of bounded homomorphisms are given in the last
section.
