Efimov spaces and the separable quotient problem for spaces Cp(K)

The classic Rosenthal-Lacey theorem asserts that the Banach space C(K) of continuous real-valued maps on an infinite compact space K has a quotient isomorphic to c or ℓ2. More recently, Ka̧kol and Saxon [20] proved that the space Cp(K) endowed with the pointwise topology has an infinite-dimensional separable quotient algebra iff K has an infinite countable closed subset. Hence Cp(βN) lacks infinite-dimensional separable quotient algebras. This motivates the following question: (⁎) Does Cp(K) admit an infinite-dimensional separable quotient (shortly SQ) for any infinite compact space K? Particularly, does Cp(βN) admit SQ? Our main theorem implies that Cp(K) has SQ for any compact space K containing a copy of βN. Consequently, this result reduces problem (⁎) to the case when K is an Efimov space (i.e. K is an infinite compact space that contains neither a non-trivial convergent sequence nor a copy of βN). Although, it is unknown if Efimov spaces exist in ZFC, we show, making use of some result of R. de la Vega (2008) (under ◊), that for some Efimov space K the space Cp(K) has SQ. Some applications of the main result are provided.