Separable (and metrizable) infinite dimensional quotients of \(C_p(X)\) and \(C_c(X)\) spaces. In honour of Manuel López-Pellicer (Q2337496)

This survey paper presents problems and results concerning the properties of a Tychonoff space \(X\) which ensure the existence of an infinite-dimensional separable quotient of \(C_p(X)\) (``\(C_p(X)\) has SQ''). The problem is motivated by the Rosenthal-Lacey theorem on \(C(K)\)'s with infinite compact spaces \(K\) and by the result of \textit{J. Kąkol} and \textit{S. A. Saxon} [Proc. Am. Math. Soc. 145, No. 9, 3829--3841 (2017; Zbl 1383.46003)] characterizing those \(X\) for which \(C_p(X)\) admits an infinite-dimensional quotient algebra. Further discussed results are in particular the following ones. A theorem of \textit{J. Kąkol} and \textit{W. Śliwa} [J. Math. Anal. Appl. 457, No. 1, 104--113 (2018; Zbl 1383.46002)] says that if \(K\) is a compact space containing \(\beta\mathbb N\), then \(C_p(K)\) admits SQ. This helps to reduce the problem to the case of Efimov spaces \(K\). An example of Kakol and Śliwa of Efimov \(K\) with \(C_p(K)\) admitting SQ is presented under \(\diamond\). Two theorems of \textit{T. Banakh} et al. [Rev. R. Acad. Cienc. Exactas Fís. Nat., Ser. A Mat., RACSAM 113, No. 4, 3015--3030 (2019; Zbl 1437.46029)] giving conditions on a pseudocompact \(X\) under which \(C_p(X)\) admits even a metrizable SQ are presented. This is related to a Josefson-Nissenzweig type property. The discussion includes additional observations and open problems. Some proofs of known results are also indicated or even fully presented. For the entire collection see [Zbl 1419.46002].