[Settfa] Tuesday 17/05, Jerzy Kakol

[kubis at math.cas.cz]

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 Tuesday 17th May, 10:00am 
 
  Place: IM in konírna 

 
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 Speaker:Jerzy Kakol, Adam Mickiewicz University / IM CAS
 Title: On Grothendieck spaces $C(X)$ and complemented copies of $(c_0)_p$ in spaces $C_p(X\times Y)$ and $C_p(X,E)$

 Abstract  

 It is known that the Banach space $C(X\times Y)$ always contains a complemented copy of the Banach space $c_{0}$ for infinite compact spaces $X$ and $Y$ (Cembranos-Freniche). The aim of the talk is to summarize several (older and very recent) results, conceptsand ideas concerning the corresponding results for the spaces $C_{p}(X\times Y)$ ofcontinuous functions on $X\times Y$ endowed with the pointwisetopology. For example, one shows a theorem (implying also Cembranos-Freniche result) stating that for all infiniteTychonoff spaces $X$ and $Y$ the space $C_{p}(X\times Y)$ contains either a complemented copy of$\mathbb{R}^{\omega}$ or a complemented copy of the space$(c_{0})_{p}=\{(x_n)_{n\in\omega}\in \mathbb{R}^\omega\colon x_n\to0\}$, both endowed with the product topology. On theother hand, assuming the ContinuumHypothesis, there are examples of pseudocompact spaces $X$ suchthat $C_{p}(X\times X)$ doesnot contain a complemented copy of $(c_{0})_{p}$. This approach provides 
 new concepts related with the Grothendieck spaces $C(K)$ and $C(K,E)$. Several applications will be provided.

 For more information see the seminar web page at 
 https://calendar.math.cas.cz/set-theory-and-analysis-actual .