Separable (and metrizable) infinite dimensional quotients of Cp(X) and Cc(X) spaces

0505923 - MÚ 2020 RIV CH eng C - Konferenční příspěvek (zahraniční konf.)
Kąkol, Jerzy
Separable (and metrizable) infinite dimensional quotients of Cp(X) and Cc(X) spaces.
Descriptive Topology and Functional Analysis II. Cham: Springer, 2019 - (Ferrando, J.), s. 175-189. Springer Proceedings in Mathematics & Statistics, 286. ISBN 978-3-030-17375-3. ISSN 2194-1009.
[2nd Meeting in Topology and Functional Analysis, In Honour of Manuel López-Pellicer Mathematical Work. Elche (ES), 07.06.2018-08.06.2018]
Grant CEP: GA ČR GF16-34860L
Institucionální podpora: RVO:67985840
Klíčová slova: the separable quotient problem * spaces of continuous functions * quotient spaces * the Josefson-Nissenzweig theorem * Efimov space
Obor OECD: Pure mathematics
https://link.springer.com/chapter/10.1007%2F978-3-030-17376-0_10

The famous Rosenthal-Lacey theorem states that for each infinite compact set K the Banach space C(K) of continuous real-valued functions on a compact space K admits a quotient which is either an isomorphic copy of c or ℓ2. Whether C(K) admits an infinite dimensional separable (or even metrizable) Hausdorff quotient when the uniform topology of C(K) is replaced by the pointwise topology remains as an open question. The present survey paper gathers several results concerning this question for the space Cp(K) of continuous real-valued functions endowed with the pointwise topology. Among others, that Cp(K) has an infinite dimensional separable quotient for any compact space K containing a opy of βN. Consequently, this result reduces the above question to the case when K is a Efimov space (i.e. K is an infinite compact space that contains neither a non-trivial convergent sequence nor a copy of βN). On the other hand, although it is unknown if Efimov spaces exist in ZFC, we note under (applying some result due to R. de la Vega), that for some Efimov space K the space Cp(K) has an infinite dimensional (even metrizable) separable quotient. The last part discusses the so-called Josefson–Nissenzweig property for spaces Cp(K), introduced recently in [3], and its relation with the separable quotient problem for spaces Cp(K).
Trvalý link: http://hdl.handle.net/11104/0297257