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An Amir–Cambern theorem for subspaces of Banach lattice-valued continuous functions
Banach Journal of Mathematical Analysis ( IF 1.1 ) Pub Date : 2021-01-18 , DOI: 10.1007/s43037-020-00112-8
Jakub Rondoš , Jiří Spurný

For \(i=1,2\), let \(E_i\) be a reflexive Banach lattice over \({\mathbb {R}}\) with a certain parameter \(\lambda ^+(E_i)>1\), let \(K_i\) be a locally compact (Hausdorff) topological space and let \({\mathcal {H}}_i\) be a closed subspace of \({\mathcal {C}}_0(K_i, E_i)\) such that each point of the Choquet boundary \({\text {Ch}}_{{\mathcal {H}}_i} K_i\) of \({\mathcal {H}}_i\) is a weak peak point. We show that if there exists an isomorphism \(T:{\mathcal {H}}_1\rightarrow {\mathcal {H}}_2\) with \(\left\| T\right\| \cdot \left\| T^{-1}\right\| <\min \lbrace \lambda ^+(E_1), \lambda ^+(E_2) \rbrace\) such that T and \(T^{-1}\) preserve positivity, then \({\text {Ch}}_{{\mathcal {H}}_1} K_1\) is homeomorphic to \({\text {Ch}}_{{\mathcal {H}}_2} K_2\).



中文翻译:

Banach格值连续函数子空间的Amir-Cambern定理

对于\(i = 1,2 \),令\(E_i \)\({\ mathbb {R}} \)上具有特定参数\(\ lambda ^ +(E_i)> 1 \ ),令\(K_i \)为局部紧凑(Hausdorff)拓扑空间,令\({\ mathcal {H}} _ i \)\({\ mathcal {C}} _​​ 0(K_i,E_i )\),这样\({\ mathcal {H}} _ i \)的Choquet边界\({\ text {Ch}} _ {{\ mathcal {H}} _ i} K_i \)的每个点都是弱的高峰点。我们证明如果存在同构\(T:{\ mathcal {H}} _ 1 \ rightarrow {\ mathcal {H}} _ 2 \)\(\ left \ | T \ right \ | \ cdot \ left \ | T ^ {-1} \ right \ | <\ min \ lbrace \ lambda ^ +(E_1),\ lambda ^ +(E_2)\ rbrace \ ),使得T\(T ^ {-1} \)保持正数,则\({\ text {Ch}} _ {{\ mathcal {H}} _ 1} K_1 \)\({\ text {Ch}} _ {{\ mathcal {H}} _ 2} K_2 \)

更新日期:2021-01-18
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