On automorphisms of the Banach space ${\ell }_{\infty }/c₀$

Piotr Koszmider, and Cristóbal Rodríguez-Porras. "On automorphisms of the Banach space $ℓ_{∞}/c₀$." Fundamenta Mathematicae 235.1 (2016): 49-99. <http://eudml.org/doc/286425>.

@article{PiotrKoszmider2016,
abstract = {We investigate Banach space automorphisms $T: ℓ_\{∞\}/c₀ → ℓ_\{∞\}/c₀$ focusing on the possibility of representing their fragments of the form $T_\{B,A\}: ℓ_\{∞\}(A)/c₀(A) → ℓ_\{∞\}(B)/c₀(B)$ for A,B ⊆ ℕ infinite by means of linear operators from $ℓ_\{∞\}(A)$ into $ℓ_\{∞\}(B)$, infinite A×B-matrices, continuous maps from B* = βB∖B into A*, or bijections from B to A. This leads to the analysis of general bounded linear operators on $ℓ_\{∞\}/c₀$. 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 $ℓ_\{∞\}/c₀$ which cannot be lifted to operators on $ℓ_\{∞\}$, and assuming OCA+MA we show that every automorphism T of $ℓ_\{∞\}/c₀$ 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 ⊆ ℕ to some infinite A ⊆ 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.},
author = {Piotr Koszmider, Cristóbal Rodríguez-Porras},
journal = {Fundamenta Mathematicae},
keywords = {Banach space; Čech-Stone compactification of the integers; automorphism; continuum hypothesis; open colouring axiom},
language = {eng},
number = {1},
pages = {49-99},
title = {On automorphisms of the Banach space $ℓ_\{∞\}/c₀$},
url = {http://eudml.org/doc/286425},
volume = {235},
year = {2016},
}

TY - JOUR
AU - Piotr Koszmider
AU - Cristóbal Rodríguez-Porras
TI - On automorphisms of the Banach space $ℓ_{∞}/c₀$
JO - Fundamenta Mathematicae
PY - 2016
VL - 235
IS - 1
SP - 49
EP - 99
AB - We investigate Banach space automorphisms $T: ℓ_{∞}/c₀ → ℓ_{∞}/c₀$ focusing on the possibility of representing their fragments of the form $T_{B,A}: ℓ_{∞}(A)/c₀(A) → ℓ_{∞}(B)/c₀(B)$ for A,B ⊆ ℕ infinite by means of linear operators from $ℓ_{∞}(A)$ into $ℓ_{∞}(B)$, infinite A×B-matrices, continuous maps from B* = βB∖B into A*, or bijections from B to A. This leads to the analysis of general bounded linear operators on $ℓ_{∞}/c₀$. 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 $ℓ_{∞}/c₀$ which cannot be lifted to operators on $ℓ_{∞}$, and assuming OCA+MA we show that every automorphism T of $ℓ_{∞}/c₀$ 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 ⊆ ℕ to some infinite A ⊆ 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.
LA - eng
KW - Banach space; Čech-Stone compactification of the integers; automorphism; continuum hypothesis; open colouring axiom
UR - http://eudml.org/doc/286425
ER -