New multiplicativity results for qubit maps

Let $\Phi$ be a trace-preserving, positivity-preserving (but not necessarily completely positive) linear map on the algebra of complex $2 \times 2$ matrices, and let $\Omega$ be any finite-dimensional completely positive map. For $p=2$ and $p \geq 4$, we prove that the maximal $p$-norm of the product map \Phi \ot \Omega is the product of the maximal $p$-norms of $\Phi$ and $\Omega$. Restricting $\Phi$ to the class of completely positive maps, this settles the multiplicativity question for all qubit channels in the range of values $p \geq 4$.Comment: 14 pages; original proof simplified by using Gorini and Sudarshan's classification of extreme affine maps on R^