Extensions of endomorphisms of C(X)

For a compact space X we consider extending endomorphisms of the algebra C(X) to be endomorphisms of Arens-Hoffman and Cole extensions of C(X). Given a non-linear, monic polynomial p is an element of C(X)[t], with C(X)[t]/pC(X)[t] semi-simple, we show that if an endomorphism of C(X) extends to the Arens-Hoffman extension with respect to p, then it also extends to the simple Cole extension with respect to p. We show that the converse to this is false. For a locally connected, metric X we characterize the algebraically closed C(X) in terms of the extendability of endomorphisms to Arens-Hoffman and to simple Cole extensions.