  It is shown that if a Markov map $T$ on a noncommutative probability space
$\mathcal{M}$ has a spectral gap on $L_2(\mathcal{M})$, then it also has one on
$L_p(\mathcal{M})$ for $1<p<\infty$. For fixed $p$, the converse also holds if
$T$ is factorizable. These results are also new for classical probability
spaces.
