  We study the Wiener-Hopf factorization for L\'evy processes $X_t$ with
completely monotone jumps. Extending previous results of L.C.G. Rogers, we
prove that the space-time Wiener-Hopf factors are complete Bernstein functions
of both the spatial and the temporal variable. As a corollary, we prove
complete monotonicity of: (a) the tail of the distribution function of the
supremum of $X_t$ up to an independent exponential time; (b) the Laplace
transform of the supremum of $X_t$ up to a fixed time $T$, as a function of
$T$. The proof involves a detailed analysis of the holomorphic extension of the
characteristic exponent $f(\xi)$ of $X_t$, including a peculiar structure of
the curve along which $f(\xi)$ takes real values.
