Ditkin Conditions

This paper is about the connection between certain Banach-algebraic properties of a commutative Banach algebra $E$ with unit and the associated commutative Banach algebra $C(X,E)$ of all continuous functions from a compact Hausdorff space $X$ into $E$. The properties concern Ditkin’s condition and bounded relative units. We show that these properties are shared by $E$ and $C(X,E)$. We also consider the relationship between these properties in the algebras $E$, $B$ and $\tilde{B}$ that appear in so-called admissible quadruples $(X,E,B,\tilde{B})$.