[For part I see the author in Mathematika 34, No.2, 160-171 (1987; Zbl 0619.30040).]
Let K be a compact subset of the complex plane, C(K) be the set of all continuous functions on K and R(K) be the algebra of functions in C(K) which are uniform limits of rational functions with poles off K. For \(\phi\) smooth and f locally integrable, the localization operator of Vitushkin, \(T_{\phi}\), solves an inhomogeneous \({\bar \partial}\)- equation, namely (\(\partial /\partial \bar z)(T_{\phi}f)=\phi [(\partial /\partial \bar z)(f)]\). A (planar) T-invariant algebra A is a closed subalgebra of C(K) such that R(K)\(\subseteq A\) and \(T_{\phi}A\subseteq A\) for all smooth \(\phi\) with compact support.
A Banach space B is said to have the Banach approximation property (BAP) if there exists a sequence \(\{P_ n\}^{\infty}_{n=1}\) of finite dimensional linear operators on B such that \(P_ nf\) converges to f for all \(f\in B\). A Banach space B has the Dunford-Pettis property (DPP) if, whenever \(\{f_ n\}^{\infty}_{n=1}\) is a sequence in B tending weakly to 0 and \(\{F_ n\}^{\infty}_{n=1}\) is a sequence in \(B^*\) tending weakly to 0, then \(\lim_{n\to \infty}F_ n(f_ n)=0.\)
Planar T-invariant algebras are known to have the BAP [T. W. Gamelin, UCLA Course Notes 1975)] and the DPP [J. A. Cima and R. M. Timoney, Mich. Math. J. 34, 99-104 (1987; Zbl 0617.46058)]. In part I of this work, a definition of T-invariance was proposed for subalgebras of functions defined on a compact subset of a non-compact Riemann surface and this was applied to the theory of qualitative uniform approximation. In this paper, we establish that the T-invariant algebras on Riemann surfaces also always have the BAP and DPP. It should be noted that in the paper, a line is missing in the definition of the Dunford-Pettis property.