On automorphisms of the Banach space $\ell _\infty /c_0$

Piotr Koszmider¹ ; Cristóbal Rodríguez-Porras²

¹ Institute of Mathematics Polish Academy of Sciences Śniadeckich 8 00-656 Warszawa, Poland
² Departamento de Matemáticas Facultad de Ciencias Universidad de Los Andes 5101 Mérida, Venezuela and Equipe de Logique UFR de Mathématiques Université Denis Diderot Paris 7 75013 Paris, France

Fundamenta Mathematicae, Tome 235 (2016) no. 1, pp. 49-99.

Résumé

We investigate Banach space automorphisms $T:\ell _\infty /c_0\rightarrow \ell _\infty /c_0 $ focusing on the possibility of representing their fragments of the form $$T_{B,A}:\ell _\infty (A)/c_0(A)\rightarrow \ell _\infty (B)/c_0(B)$$ for $A, B\subseteq \mathbb {N}$ infinite by means of linear operators from $\ell _\infty (A)$ into $\ell _\infty (B)$, infinite $A\times B$-matrices, continuous maps from $B^*=\beta B\setminus B$ into $A^*$, or bijections from $B$ to $A$. This leads to the analysis of general bounded linear operators on $\ell _\infty /c_0$. We present many examples, introduce and investigate several classes of operators, for some of them we obtain satisfactory representations and for others give examples showing that this is impossible. In particular, we show that there are automorphisms of $\ell _\infty /c_0$ which cannot be lifted to operators on $\ell _\infty $, and assuming OCA+MA we show that every automorphism $T$ of $\ell _\infty /c_0$ with no fountains or with no funnels is locally induced by a bijection, i.e., $T_{B,A}$ is induced by a bijection from some infinite $B\subseteq \mathbb {N}$ to some infinite $A\subseteq \mathbb {N}$. This additional set-theoretic assumption is necessary as we show that the Continuum Hypothesis implies the existence of counterexamples of diverse flavours. However, many basic problems, some of which are listed in the last section, remain open.

Détail

DOI : 10.4064/fm117-1-2016

Keywords: investigate banach space automorphisms ell infty rightarrow ell infty focusing possibility representing their fragments form ell infty rightarrow ell infty subseteq mathbb infinite means linear operators ell infty ell infty infinite times b matrices continuous maps * beta setminus * bijections leads analysis general bounded linear operators ell infty present many examples introduce investigate several classes operators obtain satisfactory representations others examples showing impossible particular there automorphisms ell infty which cannot lifted operators ell infty assuming oca every automorphism ell infty fountains funnels locally induced bijection induced bijection infinite subseteq mathbb infinite subseteq mathbb additional set theoretic assumption necessary continuum hypothesis implies existence counterexamples diverse flavours however many basic problems which listed section remain