On a condition on the radical of a Banach algebra ensuring strong decomposability

Abstract

The main result is the following theorem. Let$\mathfrak{A}$ be a commutative Banach algebra with radical R, where the factor algebra$\mathfrak{A}/R$ is isomorphic to the algebra of all continuous functions on a totally disconnected compact space. If ∥rⁿ∥¹ /n → 0 as n →∞ uniformly for r ε R, ∥r∥≤l, then the algebra$\mathfrak{A}$ is strongly decomposable, i.e., there exists a closed subalgebra B⊂$\mathfrak{A}$ isomorphic to$\mathfrak{A}/R$ such that$\mathfrak{A}$=B⊕R.This is a strengthening of the theorem of A. Ya. Khelemskii, who assumed$\left\| {r^n } \right\|^{1/n^2 } \to 0$. There are 4 references.