Function algebras with a strongly precompact unit ball

• Published in: Journal of functional analysis Vol. 265; no. 7; pp. 1357 - 1366
• Main Authors: Lacruz, Miguel, Rodríguez-Piazza, Luis
• Format: Journal Article
• Language: English
• Published: Elsevier Inc 01.10.2013
• Subjects: Discrete measure; Function algebra; Strong operator topology; Discrete measure; Function algebra; Strong operator topology
• ISSN: 0022-1236; 1096-0783
• DOI: 10.1016/j.jfa.2013.05.035

Let μ be a finite positive Borel measure with compact support K⊆C, and regard L∞(μ) as an algebra of multiplication operators on the Hilbert space L2(μ). Then consider the subalgebra A(K) of all continuous functions on K that are analytic on the interior of K, and the subalgebra R(K) defined as the uniform closure of the rational functions with poles outside K. Froelich and Marsalli showed that if the restriction of the measure μ to the boundary of K is discrete then the unit ball of A(K) is strongly precompact, and that if the unit ball of R(K) is strongly precompact then the restriction of the measure μ to the boundary of each component of C\K is discrete. The aim of this paper is to provide three examples that go to clarify the results of Froelich and Marsalli; in particular, it is shown that the converses to both statements are false.