Extensions of endomorphisms of 𝐶(𝑋)

Feinstein, J.; Oliver, T.

doi:10.1090/S0002-9939-06-08441-3

Extensions of endomorphisms of $C(X)$
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by J. F. Feinstein and T. J. Oliver PDF

Proc. Amer. Math. Soc. 135 (2007), 109-117 Request permission

Abstract:

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 $p\in C(X)[t]$, 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.

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