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Regularity conditions for vector-valued function algebras

Author

Listed:
  • Z. Barqi

    (Damghan University)

  • M. Abtahi

    (Damghan University)

Abstract

We consider several notions of regularity, including strong regularity, bounded relative units, and Ditkin’s condition, in the setting of vector-valued function algebras. Given a commutative Banach algebra A and a compact space X, let $${\varvec{A}}$$ A be a Banach A-valued function algebra on X and let $${\mathfrak {A}}$$ A be the subalgebra of $${\varvec{A}}$$ A consisting of scalar-valued functions. This paper is about the connection between regularity conditions of the algebra $${\varvec{A}}$$ A and the associated algebras $${\mathfrak {A}}$$ A and A. That $${\varvec{A}}$$ A inherits a certain regularity condition $${\mathscr {P}}$$ P to $${\mathfrak {A}}$$ A and A is the easy part of the problem. We investigate the converse and show that, under certain conditions, $${\varvec{A}}$$ A receives $${\mathscr {P}}$$ P form $${\mathfrak {A}}$$ A and A. The results apply to tensor products of commutative Banach algebras as they are included in the class of vector-valued function algebras.

Suggested Citation

  • Z. Barqi & M. Abtahi, 2024. "Regularity conditions for vector-valued function algebras," Indian Journal of Pure and Applied Mathematics, Springer, vol. 55(2), pages 439-450, June.
    • Handle: RePEc:spr:indpam:v:55:y:2024:i:2:d:10.1007_s13226-023-00376-4
    DOI: 10.1007/s13226-023-00376-4
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