On the approximation property for Banach spaces predual to H∞-spaces

A major open problem asks about the (Grothendieck) approximation property for the space H∞:=H∞(D) of bounded holomorphic functions in the unit disk D⊂C. Motivated by this problem we establish approximation properties for Banach spaces predual to the spaces H∞(Ω) for Ω being finite direct products of starlike domains and unbranched coverings of strongly pseudoconvex domains in Stein manifolds.