Function algebras with a strongly precompact unit ball (Q2436864)

Let \(\mu\) be a finite positive Borel measure with compact support \(K \subset \mathbb{C}\). Since \(L^{\infty}(\mu)\) can be regarded as an algebra of multiplication operators on the Hilbert space \(L ^{2}(\mu)\), \textit{J. Froelich} and \textit{M. Marsalli} [ibid. 115, No. 2, 454--479 (1993; Zbl 0796.47031)] gave a characterization of the subalgebras \(\mathcal{R}\subset L^{\infty} (\mu)\) with the property that the unit ball of \(\mathcal{R}\) is precompact in the strong operator topology. They put a particular interest on the following subalgebras of \(L^{\infty}(\mu)\): \(P(K)\) the uniform closure of the analytic polynomials, \(R(K)\) the uniform closure of the rational functions with poles outside \(K\), \(A(K)\) the algebra of all continuous functions on \(K\) that are analytic on the interior of \(K\) denoted \(\operatorname{int}(K)\), \(C(K)\) the algebra of all continuous functions on \(K\). They proved that, if the restriction of the measure \(\mu\) 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 \(\mu\) to the boundary of each component of \(\mathbb{C}\setminus K\) is discrete. Froelich and Marsalli also gave necessary and sufficient conditions for the algebras \(L^{\infty}(\mu)\), \(C(K)\) and \(P(K)\) to have a strongly precompact unit ball. In this paper, the authors provide three examples that go to clarify the results of Froelich and Marsalli; in particular, it is shown that the sufficient conditions for the unit ball of \(A(K)\) and the one of \(R(K)\) to be strongly precompact are not necessary. Among the main results, the authors prove that, if \(L,K\) are two compact sets such that \(L\subset K\) and if \(\mathcal{R}\) is a closed subspace of \(C(K)\), then the natural mapping \(\mathcal{R}\hookrightarrow L^{2}(\mu)\) is compact for every measure \(\mu\) supported on \(L\) if and only if the restriction mapping \(R:\mathcal{R}\rightarrow C(L)\) does not fix a copy of \(\ell_{1}\).