Lifting some approximation properties from a dual space X′ to the Banach space X

For a fixed Banach operator ideal A, we characterize A-compact sets (in the sense of Carl and Stephani) that are determined by c(0) via the Banach composition ideal A omicron R-infinity with R-infinity the Banach ideal of Fourie and Swart. This characterization allows us to relate kappa(A)-approximation properties on a Banach space and kappa(B)-approximation properties on its dual space, where A and B are ideals linked by some classical procedures. These approximation properties have been widely studied in several papers in the last years.