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Zero product preserving functionals on \(C(\varOmega )\)-valued spaces of functions

  • Ziba Pourghobadi
  • Fereshteh SadyEmail author
  • Masoumeh Najafi Tavani
Original Paper

Abstract

Let X be a compact Hausdorff space and \(\varOmega \) be a locally compact \(\sigma \)-compact space. In this paper we study (real-linear) continuous zero product preserving functionals \(\varphi : A \longrightarrow {\mathbb {C}}\) on certain subalgebras A of the Fréchet algebra \(C(X,C(\varOmega ))\). The case that \(\varphi \) is continuous with respect to a specified complete metric on A will also be discussed. In particular, for a compact Hausdorff space K we characterize \(\Vert \cdot \Vert \)-continuous linear zero product preserving functionals on the Banach algebra \(C^1([0,1],C(K))\) equipped with the norm \(\Vert f\Vert =\Vert f\Vert _{[0,1]}+\Vert f'\Vert _{[0,1]}\), where \(\Vert \cdot \Vert _{[0,1]}\) denotes the supremum norm. An application of the results is given for continuous ring homomorphisms on such subalgebras.

Keywords

Zero product preserving functionals Vector-valued spaces of functions Ring homomorphisms 

Mathematics Subject Classification

47B38 47B48 46J10 

Notes

Acknowledgements

The authors would like to thank the referee for his/her invaluable comments and suggestions.

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.

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Copyright information

© Tusi Mathematical Research Group (TMRG) 2019

Authors and Affiliations

  1. 1.Department of Mathematics, Science and Research BranchIslamic Azad UniversityTehranIran
  2. 2.Department of Pure Mathematics, Faculty of Mathematical SciencesTarbiat Modares UniversityTehranIran
  3. 3.Department of Mathematics, Faculty of Basic SciencesIslamic Azad University, Islamshahr BranchTehranIran

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