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$$\begin{equation*}\lVert x\rVert_e = \inf\bigl\{ \lambda\in[0,\infty)\:\colon\:\lvert x\rvert{\leqslant}\lambda e\bigr\},\end{equation*}$$

defined for each $x$ in the order ideal $I_e$ of $X$ generated by $e$, is complete when restricted to the closed sublattice of $I_e$ generated by $x_1,\ldots ,x_n$.Moreover, we show that a vector space which admits a ‘sufficiently strong’ $H_n$-function calculus for each $n\in{\mathbb{N}}$ is automatically a vector lattice, and we explore the situation in the non-Archimedean case by showing that some non-Archimedean vector lattices admit a positively homogeneous continuous function calculus, while others do not.

中文翻译：

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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$$ \ begin {equation *} \ lVert x \ rVert_e = \ inf \ bigl \ {\ lambda \ in [0，\ infty）\：\ colon \：\ lvert x \ rvert {\ leqslant} \ lambda e \ bigr \}，\ end {equation *} $$

当限制为由$ x_1，\ ldots，x_n $生成的$ I_e $的闭合子格时，对于由$ e $生成的$ X $的理想$ I_e $的每个$ x $定义的定义是完整的。一个向量空间，它为每个$ n \ in {\ mathbb {N}} $承认一个“足够强”的$ H_n $函数演算，它自动是一个向量晶格，我们通过显示以下内容来探索非阿基米德案例中的情况一些非阿基米德向量格接受一个正齐次的连续函数演算，而另一些则不然。