On Banach spaces with the commuting bounded approximation property (Q803877)

The paper is concerned with separable Banach spaces X which admit a sequence of bounded linear operators \(R_ n: X\to X\) of finite rank satisfying \(R_ nR_ m=R_{\min (n,m)}\) if \(n\neq m\) and \(\lim_{n}R_ nx=x\) for all \(x\in X\). Such a sequence is called commuting approximating sequence. \(R_ n-R_{n-1}\) is defined to factor through \(\ell_ p\) for some \(1\leq p\leq \infty\), if there are linear oprators \(T_ n: X\to \ell_ p\) and \(S_ n: \ell_ p\to X\) with \(S_ nT_ n=R_ n-R_{n-1}\) and \(\sup_{n}\| S_ n\|,\sup_{n}\| T_ n\| <\infty.\) Then the main result is the following: Let \((R_ n)\) be a commuting approximating sequence such that \(R_ n- R_{n-1}\) factors through \(\ell_ p\) for some p. Then \(X\oplus \ell_ p\), if \(1\leq p<\infty\), and \(X\oplus c_ 0\), if \(p=\infty\), has a basis. As an example the disc algebra A is considered. Here the preceding assumptions are easily verified which provides an elementary (if indirect) proof for the fact that A has a basis.