Zero product preserving maps on Banach algebras of Lipschitz functions

Let ( K , d ) be a non-empty, compact metric space and α ∈ ] 0 , 1 [ . Let A be either lip α ( K ) or Lip α ( K ) and let B be a commutative unital Banach algebra. We show that every continuous linear map T : A → B with the property that T ( f ) T ( g ) = 0 whenever f , g ∈ A are such that f g = 0 i...

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Bibliographic Details
Published in:Journal of mathematical analysis and applications 2010-09, Vol.369 (1), p.94-100
Main Authors: Alaminos, J., Extremera, J., Villena, A.R.
Format: Article
Language:English
Subjects:
Algebra
Algebra of Lipschitz functions
Commutative rings and algebras
Disjointness preserving map
Exact sciences and technology
Fourier algebra
Functional analysis
General topology
Map preserving zero product
Mathematical analysis
Mathematics
Sciences and techniques of general use
Topology. Manifolds and cell complexes. Global analysis and analysis on manifolds
Weighted Fourier algebra
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Summary:Let ( K , d ) be a non-empty, compact metric space and α ∈ ] 0 , 1 [ . Let A be either lip α ( K ) or Lip α ( K ) and let B be a commutative unital Banach algebra. We show that every continuous linear map T : A → B with the property that T ( f ) T ( g ) = 0 whenever f , g ∈ A are such that f g = 0 is of the form T = w Φ for some invertible element w in B and some continuous epimorphism Φ : A → B .
ISSN:0022-247X
1096-0813
DOI:10.1016/j.jmaa.2010.02.041