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


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.

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

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Correspondence to Fereshteh Sady.

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Communicated by Vesko Valov.

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Pourghobadi, Z., Sady, F. & Tavani, M.N. Zero product preserving functionals on \(C(\varOmega )\)-valued spaces of functions. Ann. Funct. Anal. 11, 459–472 (2020).

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

Mathematics Subject Classification

  • 47B38
  • 47B48
  • 46J10