Regularity of certain commutative Banach rings

• Published in: Journal of mathematical analysis and applications Vol. 517; no. 1; p. 126589
• Main Authors: Lee, Cheuk-Yin, Leung, Chi-Wai
• Format: Journal Article
• Language: English
• Published: Elsevier Inc 01.01.2023
• Subjects: Berkovich spectrum; Commutative Banach ring; Gelfand transform; Commutative Banach ring; Berkovich spectrum; Gelfand transform
• ISSN: 0022-247X; 1096-0813
• DOI: 10.1016/j.jmaa.2022.126589

Let (A,‖⋅‖) be a commutative unital Banach ring. Let M(A) be the Berkovich spectrum of A, i.e., the set of all bounded multiplicative non-zero semi-norms defined on A which is endowed with the pointwise convergence topology. As in the case of complex Banach algebras, we introduce the notions of the hull-kernel topology on M(A) and the regularity of a commutative unital Banach ring. We show that the Banach algebra of all bounded continuous k-valued functions defined on a zero-dimensional topological space is regular, where k is a complete valuation field. Hence, we see that the Berkovich spectrum of this algebra is a Hausdorff space in the hull-kernel topology. Next, we use the tool of Gelfand transform to characterize a Banach ring of finite Berkovich spectrum. From this, we see that if A is uniform, i.e., ‖f2‖=‖f‖2 for all f in A, and has finite Berkovich spectrum, then it is regular.