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

Kód výsledku v IS VaVaI

<a href="https://www.isvavai.cz/riv?ss=detail&h=RIV%2F67985840%3A_____%2F19%3A00505923" target="_blank" >RIV/67985840:_____/19:00505923 - isvavai.cz</a>

Výsledek na webu

<a href="http://dx.doi.org/10.1007/978-3-030-17376-0_10" target="_blank" >http://dx.doi.org/10.1007/978-3-030-17376-0_10</a>

DOI - Digital Object Identifier

<a href="http://dx.doi.org/10.1007/978-3-030-17376-0_10" target="_blank" >10.1007/978-3-030-17376-0_10</a>

Jazyk výsledku

angličtina

Název v původním jazyce

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

Popis výsledku v původním jazyce

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).

Název v anglickém jazyce

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

Popis výsledku anglicky

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).

Druh

D - Stať ve sborníku

CEP obor

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OECD FORD obor

10101 - Pure mathematics

Projekt

<a href="/cs/project/GF16-34860L" target="_blank" >GF16-34860L: Logika a topologie v Banachových prostorech</a><br>

Návaznosti

I - Institucionalni podpora na dlouhodoby koncepcni rozvoj vyzkumne organizace

Rok uplatnění

2019

Kód důvěrnosti údajů

S - Úplné a pravdivé údaje o projektu nepodléhají ochraně podle zvláštních právních předpisů

Název statě ve sborníku

Descriptive Topology and Functional Analysis II

ISBN

978-3-030-17375-3

ISSN

2194-1009

e-ISSN

—

Počet stran výsledku

15

Strana od-do

175-189

Název nakladatele

Springer

Místo vydání

Cham

Místo konání akce

Elche

Datum konání akce

7. 6. 2018

Typ akce podle státní příslušnosti

WRD - Celosvětová akce

Kód UT WoS článku

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