Abstract

It is known that if m⩾3 and B is any ball in Cm with respect to some norm, say ∥⋅∥B, then there exists a linear map \mbox{L:(Cm,∥⋅∥∗B)→Mk} which is contractive but not completely contractive. The characterization of those balls in C2 for which contractive linear maps are always completely contractive, however, remains open. We answer this question for balls of the form ΩA in C2 and the balls in their norm dual, where ΩA={(z1,z2):∥z1A1+z2A2∥Op<1} for some pair of 2×2 matrices A1,A2.

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