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Zero product preserving maps of operator-valued functions

Authors: Wen-Fong Ke, Bing-Ren Li and Ngai-Ching Wong
Translated by:
Journal: Proc. Amer. Math. Soc. 132 (2004), 1979-1985
MSC (2000): Primary 46E40, 47B33
Published electronically: December 15, 2003
MathSciNet review: 2053969
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Abstract: Let $X,Y$ be locally compact Hausdorff spaces and ${\mathcal M}$, ${\mathcal N}$ be Banach algebras. Let $\theta: C_0(X,{\mathcal M}) \to C_0(Y, {\mathcal N})$ be a zero product preserving bounded linear map with dense range. We show that $\theta$ is given by a continuous field of algebra homomorphisms from ${\mathcal M}$ into ${\mathcal N}$if ${\mathcal N}$ is irreducible. As corollaries, such a surjective $\theta$ arises from an algebra homomorphism, provided that ${\mathcal M}$ is a $W^*$-algebra and ${\mathcal N}$ is a semi-simple Banach algebra, or both ${\mathcal M}$ and ${\mathcal N}$ are $C^*$-algebras.

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Additional Information

Wen-Fong Ke
Affiliation: Department of Mathematics, National Cheng Kung University, Tainan 701, Taiwan

Bing-Ren Li
Affiliation: Institute of Mathematics, Chinese Academy of Sciences, Beijing 100080, China

Ngai-Ching Wong
Affiliation: Department of Applied Mathematics, National Sun Yat-sen University, Kaohsiung 804, Taiwan

Keywords: Zero product preserving maps, Banach algebra homomorphisms
Received by editor(s): July 25, 2002
Received by editor(s) in revised form: March 7, 2003
Published electronically: December 15, 2003
Communicated by: David R. Larson
Article copyright: © Copyright 2003 American Mathematical Society