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

The main result is the following theorem. Let be a commutative Banach algebra with radical R, where the factor algebra is isomorphic to the algebra of all continuous functions on a totally disconnected compact space. If rⁿ¹ /n 0 as n uniformly for r R, rl, then the algebra is strongly decomposable, i.e., there exists a closed subalgebra B isomorphic to such that=BR.This is a strengthening of the theorem of A. Ya. Khelemskii, who assumed. There are 4 references.Translated from Matematicheskie Zametki, Vol. 2, No. 6, pp. 589–592, December, 1967.