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 e 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.