Let $A$ be a p-Banach algebra and $\alpha$ a two-sided ideal of $A$ with a complete p-norm stronger than the p-norm inherited from $A$. By integral methods, we give here a holomorphic functional calculus relatively to $\alpha$ which coïncides with the holomorphic functional calculus defined in $A\vert\alpha$\, considered as a quotient quasi-Banach algebra. As application, we get a version of Šilov's decomposition theorem. We also give the spectral mapping theorem and an integral formula for the image of the k-th derivative of a holomorphic function.
"Calcul fonctionnel holomorphe dans les algèbres p-Banach quotients." Bull. Belg. Math. Soc. Simon Stevin 11 (4) 625 - 633, November 2004. https://doi.org/10.36045/bbms/1102689126