Josefson–Nissenzweig property for Cp-spaces

0508352 - MÚ 2020 RIV ES eng J - Článek v odborném periodiku
Banakh, T. - Kąkol, Jerzy - Śliwa, W.
Josefson–Nissenzweig property for Cp-spaces.
Revista de la Real Academia de Ciencias Exactas, Fisicas y Naturales. Roč. 113, č. 4 (2019), s. 3015-3030. ISSN 1578-7303. E-ISSN 1579-1505
Grant CEP: GA ČR GF16-34860L
Institucionální podpora: RVO:67985840
Klíčová slova: Efimov space * quotient spaces * spaces of continuous functions * The Josefson–Nissenzweig theorem
Obor OECD: Pure mathematics
Impakt faktor: 1.406, rok: 2019
Způsob publikování: Open access
http://dx.doi.org/10.1007/s13398-019-00667-8

The famous Rosenthal–Lacey theorem asserts that for each infinite compact space K the Banach space C(K) admits a quotient isomorphic to Banach spaces c or ℓ 2 . The aim of the paper is to study a natural variant of this result for the space C p (X) of continuous real-valued maps on a Tychonoff space X with the pointwise topology. Following Josefson–Nissenzweig theorem for infinite-dimensional Banach spaces we introduce a corresponding property (called Josefson–Nissenzweig property, briefly, the JNP) for C p (X) -spaces. We prove: for a Tychonoff space X the space C p (X) satisfies the JNP if and only if C p (X) has a quotient isomorphic to c0:={(xn)n∈N∈RN:xn→0} (with the product topology of R N ) if and only if C p (X) contains a complemented subspace isomorphic to c. The last statement provides a C p -version of the Cembranos theorem stating that the Banach space C(K) is not a Grothendieck space if and only if C(K) contains a complemented copy of the Banach space c with the sup-norm topology. For a pseudocompact space X the space C p (X) has the JNP if and only if C p (X) has a complemented metrizable infinite-dimensional subspace. An example of a compact space K without infinite convergent sequences with C p (K) containing a complemented subspace isomorphic to c is given.
Trvalý link: http://hdl.handle.net/11104/0299282