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

### Authors:

- Jerzy Kąkol,
- Wiesław Śliwa

### Abstract

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 ℓ_{2}. More recently, Ka̧kol and Saxon [20] proved that the space C_{p}(K) endowed with the pointwise topology has an infinite-dimensional separable quotient algebra iff K has an infinite countable closed subset. Hence C_{p}(βN) lacks infinite-dimensional separable quotient algebras. This motivates the following question: (⁎) Does C_{p}(K) admit an infinite-dimensional separable quotient (shortly SQ) for any infinite compact space K? Particularly, does C_{p}(βN) admit SQ? Our main theorem implies that C_{p}(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 C_{p}(K) has SQ. Some applications of the main result are provided.

- Record ID
- UAM229062e205b341c9852e4025838e8577
- Author
- Journal series
- Journal of Mathematical Analysis and Applications, ISSN 0022-247X
- Issue year
- 2018
- Vol
- 457
- Pages
- 104-113
- ASJC Classification
- ;
- DOI
- DOI:10.1016/j.jmaa.2017.08.010 opening in a new tab
- Language
- en English
- Score (nominal)
- 40
- Score source
- journalList
- Score
- = 40.0, 19-03-2020, ArticleFromJournal
- Publication indicators
- = 6; : 2018 = 1.187; : 2018 = 1.188 (2) - 2018=1.219 (5)

- Uniform Resource Identifier
- https://researchportal.amu.edu.pl/info/article/UAM229062e205b341c9852e4025838e8577/

* presented citation count is obtained through Internet information analysis and it is close to the number calculated by the Publish or Perishopening in a new tab system.