Algebraic entropy for smooth projective varieties

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 l- 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 sur-faces. Over 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.