Jerzy Kąkol (UAM, Poznań), About Banach-Mazur’s problem from the 1930s

Oddział Łódzki Polskiego Towarzystwa Matematycznego

W dniu 6 grudnia 2023 roku (ŚRODA) o godz. 11:00, w sali D103 Wydziału Matematyki i Informatyki Uniwersytetu  Łódzkiego  przy ul. Banacha 22 w Łodzi

Prof. dr hab. Jerzy Kąkol (Wydział Matematyki i Informatyki Uniwersytetu im. Adama Mickiewicza w Poznaniu)

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ABOUT BANACH-MAZUR’S PROBLEM FROM THE 1930s

Streszczenie:

One of the famous unsolved problems in Functional Analysis asks (Banach-Mazur's problem (1932)) if every infinite-dimensional Banach space can be mapped by a continuous linear operator onto an infinite-dimensional separable Banach space. This problem is known under the name Separable quotient problem. For many concrete Banach spaces the answer is positive, for example, reflexive Banach spaces, or even weakly compactly generated Banach spaces. Argyros, Dodos and Kanellopoulos, proved that every dual Banach space has a separable quotient. On the other hand, Rosenthal showed that all Banach spaces C(X) of continuous (real-valued) functions on X have a quotient isomorphic to c₀ or l₂. We provide several useful methods to examine which Banach spaces admit a separable quotient, and the same problem will be studied for spaces C_p(X) with the pointwise topology. The talk gathers also quite new results. A connection with Efimov compact spaces X will be also discussed.

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