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

Result code in 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>

Result on the web

<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>

Result language

angličtina

Original language name

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

Original language description

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).

Czech name

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Czech description

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Type

D - Article in proceedings

CEP classification

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OECD FORD branch

10101 - Pure mathematics

Project

<a href="/en/project/GF16-34860L" target="_blank" >GF16-34860L: Logic and Topology in Banach spaces</a><br>

Continuities

I - Institucionalni podpora na dlouhodoby koncepcni rozvoj vyzkumne organizace

Publication year

2019

Confidentiality

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

Article name in the collection

Descriptive Topology and Functional Analysis II

ISBN

978-3-030-17375-3

ISSN

2194-1009

e-ISSN

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Number of pages

15

Pages from-to

175-189

Publisher name

Springer

Place of publication

Cham

Event location

Elche

Event date

Jun 7, 2018

Type of event by nationality

WRD - Celosvětová akce

UT code for WoS article

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