Optimal log-Sobolev inequality and hypercontractivity for positive semigroups on M2(C)

We study positivity and contractivity properties for semigroups on M-2(C), compute the optimal log-Sobolev constant and prove hypercontractivity for the class of positive semigroups leaving invariant both subspaces generated by the Pauli matrices sigma(0), sigma(3) and sigma(1), sigma(2). The optimal log-Sobolev constant turns out to be bigger than the usual one arising in several commutative and noncommutative contexts when the semigroup acts on the off-diagonal matrices faster than on diagonal matrices. These results are applied to the semigroup of the Wigner-Weisskopf atom.