  We show that the spectral radius for the action of a self map $f$ of a smooth
projective variety (over an arbitrary base field) on its $\ell$-adic cohomology
is achieved on the $f^*$-stable sub-algebra generated by any ample class. This
generalizes a result of Esnault-Srinivas who had obtained an analogous result
for automorphisms of surfaces. Over $\mathbb{C}$ we also show that this
sub-algebra is naturally an irreducible representation of a
Looijenga-Lunts-Verbitsky type Lie algebra acting on the cohomology of a smooth
projective variety.
