Contractivity and complete contractivity for finite dimensional Banach spaces

It is known that if m >= 3 and B is any ball in C-m with respect to some norm, say parallel to.parallel to(B), then there exists a linear map L: (C-m,parallel to.parallel to(B)*) -> M-k which is contractive but not completely contractive. The characterization of those balls in C-2 for which contractive linear maps are always completely contractive, however, remains open. We answer this question for balls of the form Omega(A) in C-2 and the balls in their norm dual, where Omega(A) = { (z(1), z(2)) : parallel to z(1) A(1) + z(2)A(2)parallel to Op < 1} for some pair of 2 x 2 matrices A(1), A(2).