Vector lattices admitting a positively homogeneous continuous function calculus

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 $\mathbf{x}=(x_1,\dots ,x_n)\in X^n$, where $X$ is an Archimedean vector lattice and $n\in \mathbb{N}$: - there is a vector lattice homomorphism ${\Phi }_{\mathbf{x}}:H_n\to X$ such that ${\Phi }_{\mathbf{x}}({\pi }_i^{(n)})=x_i$ $(i\in \{1,\dots ,n\})$, where $H_n$ denotes the vector lattice of positively homogeneous, continuous, real-valued functions defined on ${\mathbb{R}}^n$ and ${\pi }_i^{(n)}:{\mathbb{R}}^n\to \mathbb{R}$ is the $i^{\text{th}}$ coordinate projection; - there is a positive element $e\in X$ such that $e⩾|x_1|\vee \cdots \vee |x_n|$ and the norm $\Vert x{\Vert }_e=\inf \{\lambda \in [0,\infty ):|x|\le \lambda e\}$, 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 $x_1,\dots ,x_n$. 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.