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We study the Wiener-Hopf factorization for L\'evy processes $X_t$ withcompletely monotone jumps. Extending previous results of L.C.G. Rogers, weprove that the space-time Wiener-Hopf factors are complete Bernstein functionsof both the spatial and the temporal variable. As a corollary, we provecomplete monotonicity of: (a) the tail of the distribution function of thesupremum of $X_t$ up to an independent exponential time; (b) the Laplacetransform 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 thecharacteristic exponent $f(\xi)$ of $X_t$, including a peculiar structure ofthe curve along which $f(\xi)$ takes real values.

Fluctuation theory for L\'evy processes with completely monotone jumps