Module homomorphisms from Frechet algebras

We first study some propertiesý ýof A-module homomorphisms θ:X→Yý, ýwhere Xý ýand Y are Fréchet A-modules and A is a unitalý ýFréchet algebraý. ýThen we show that if there exists aý ýcontinued bisection of the identity for Aý, ýthen θ isý ýautomatically continuous under certain condition on Xý. ýIný ýparticularý, ýevery homomorphism from A into certainý ýFréchet algebras (including Banach algebra) is automaticallyý ýcontinuousý. ýFinallyý, ýwe show that every unital Fréchetý ýalgebra with a continued bisection of the identityý, ýisý ýfunctionally continuousý.