Regularity conditions for vector-valued function algebras

Z. Barqi () and M. Abtahi ()
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Z. Barqi: Damghan University
M. Abtahi: Damghan University

Indian Journal of Pure and Applied Mathematics, 2024, vol. 55, issue 2, 439-450

Abstract: 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.

Keywords: Vector-valued function algebra; Regularity condition; Ditkin’s condition; Bounded relative unit; Tensor product; 46J10; 46J20 (search for similar items in EconPapers)
Date: 2024
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DOI: 10.1007/s13226-023-00376-4

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