2004/10/01 by J. F. Feinstein, Feinstein, J. F., T. J. Oliver +1
Mathematics · #46J10 #47B48 #Advanced Operator Algebra Research #Advanced Topics in Algebra #FOS: Mathematics #Functional Analysis (math.FA) #Holomorphic and Operator Theory #math.FA #msc:46J10 #msc:47B48
paper · pdf · doi:10.48550/arxiv.math/0410025
12 pages latex, 2 figures
arxiv created 2004/10/01 · openalex publication_date 2004/10/01 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 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.