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Vector Lattices Admitting a Positively Homogeneous Continuous Function Calculus
Quarterly Journal of Mathematics ( IF 0.6 ) Pub Date : 2020-01-25 , DOI: 10.1093/qmathj/haz031 Niels Jakob Laustsen 1 , Vladimir G Troitsky 2
中文翻译:
允许正均匀连续函数演算的矢量格
更新日期:2020-04-17
Quarterly Journal of Mathematics ( IF 0.6 ) Pub Date : 2020-01-25 , DOI: 10.1093/qmathj/haz031 Niels Jakob Laustsen 1 , Vladimir G Troitsky 2
Affiliation
Abstract
We characterize the Archimedean vector lattices that admit a positively homogeneous continuous function calculus by showing that the following two conditions are equivalent for each $n$-tuple $\boldsymbol{x} = (x_1,\ldots ,x_n)\in X^n$, where $X$ is an Archimedean vector lattice and $n\in{\mathbb{N}}$: - • there is a vector lattice homomorphism $\Phi _{\boldsymbol{x}}\colon H_n\to X$ such that $$\begin{equation*}\Phi_{\boldsymbol{x}}(\pi_i^{(n)}) = x_i\qquad (i\in\{1,\ldots,n\}),\end{equation*}$$where $H_n$ denotes the vector lattice of positively homogeneous, continuous, real-valued functions defined on ${\mathbb{R}}^n$ and $\pi _i^{(n)}\colon{\mathbb{R}}^n\to{\mathbb{R}}$ is the $i^{\text{}}$th coordinate projection;
- • there is a positive element $e\in X$ such that $e\geqslant \lvert x_1\rvert \vee \cdots \vee \lvert x_n\rvert$ and the norm$$\begin{equation*}\lVert x\rVert_e = \inf\bigl\{ \lambda\in[0,\infty)\:\colon\:\lvert x\rvert{\leqslant}\lambda e\bigr\},\end{equation*}$$
中文翻译:
允许正均匀连续函数演算的矢量格
摘要
我们通过证明以下两个条件对于每个$ n $ -tuple $ \ boldsymbol {x} =(x_1,\ ldots,x_n)\在X ^ n中是等价的,来表征准正连续函数演算的阿基米德向量格的特征$,其中$ X $是阿基米德向量格,$ n \ in {\ mathbb {N}} $: - •存在向量晶格同态$ \ Phi _ {\ boldsymbol {x}} \冒号H_n \ to X $,使得$$ \ begin {equation *} \ Phi _ {\ boldsymbol {x}}(\ pi_i ^ {( n)})= x_i \ qquad(i \ in \ {1,\ ldots,n \}),\ end {equation *} $$其中$ H_n $表示正齐次,连续,实值函数的向量格定义在$ {\ mathbb {R}} ^ n $和$ \ pi _i ^ {(n)} \冒号{\ mathbb {R}} ^ n \ to {\ mathbb {R}} $上的是$ i ^第{\ text {}}个坐标投影;
- •X $中有一个正元素$ e \,因此$ e \ geqslant \ lvert x_1 \ rvert \ vee \ cdots \ vee \ lvert x_n \ rvert $和范数$$ \ begin {equation *} \ lVert x \ rVert_e = \ inf \ bigl \ {\ lambda \ in [0,\ infty)\:\ colon \:\ lvert x \ rvert {\ leqslant} \ lambda e \ bigr \},\ end {equation *} $$