Invariants of algebraic derivations and automorphisms in Banach algebras

We prove that if a semisimple real or complex Banach algebra A possesses an algebraic derivation whose invariants are algebraic, then A is finite-dimensional. This result is a full generalization of a recent result by Haily, Kaidi and Palacios (2011) [15] for the case of inner derivations in complex semisimple Banach algebras. The analogous result for automorphism case is also obtained.