The commuting B.A.P. for Banach spaces

Summary

Introduction. In light of Enflo's famous counterexample to the approximation problem [3], the study of “weaker structures” has gained added importance. The most fruitful of these has been the bounded approximation property (B.A.P.) (see section 2 for the definitions), the π_γ-property, and the finite dimensional decomposition property (F.D.D.P.). Johnson, Rosenthal and Zippin [11] examined certain relationships between these weaker structures. Since 1970, this paper has been the standard reference for people working in the area. Essentially no further positive progress has been made on the important problem of finding general conditions which imply the F.D.D.P. for a Banach space X.

Enflo's example [3] was the first in a long series of important counterexamples in the area. Figiel and Johnson [4] then showed the existence of a Banach space which has the approximation property (A.P.) but fails the B.A.P.. Lindenstrauss (see [14]) found a Banach space X with a basis so that X* is separable and fails the A.P. Recently, S. J. Szarek [18] constructed a Banach space with the F.D.D.P. which fails to have a basis.

In the sequel, we will see that the much ignored concept of commuting B.A.P. (C.B.A.P.) plays a central role in passing from “weaker structures” to a F.D.D. for a separable Banach space. In section 2 we give the definitions and review the work to date on these problems. Section 3 is a study of C.B.A.P. and what conditions imply its existence.