Function algebras with a strongly precompact unit ball

Let μ   be a finite positive Borel measure with compact support K⊆CK⊆C, and regard L∞(μ)L∞(μ) as an algebra of multiplication operators on the Hilbert space L2(μ)L2(μ). Then consider the subalgebra A(K)A(K) of all continuous functions on K that are analytic on the interior of K  , and the subalgebra R(K)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)A(K) is strongly precompact, and that if the unit ball of R(K)R(K) is strongly precompact then the restriction of the measure μ   to the boundary of each component of C\KC\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.