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 $\boldsymbol{x} = (x_1,\ldots,x_n)\in X^n$, 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 $\Phi_{\boldsymbol{x}}(\pi_i^{(n)})=x_i$ $(i\in\{1,\ldots,n\})$, 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 $\lVert x\rVert_e = \inf\bigl\{\lambda\in[0,\infty)\:\colon\:\lvert x\rvert\le\lambda e\bigr\}$, 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.