Total positivity in stable semigroups

We characterize the total positivity in space-time of real strictly stable semigroups. In the positive case, this solves a problem which had been raised by Karlin. In the drifted Cauchy case, this concludes a study which we had initiated in a previous paper. The case of isotropic stable semigroups is also investigated. We apply these results to the bell-shape and monotone likelihood properties of certain real stable densities.