Vector Lattices Admitting a Positively Homogeneous Continuous Function Calculus

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 x = (x₁,… ,xₙ)∈ Xⁿ, where X is an Archimedean vector lattice and n∈N: • there is a vector lattice homomorphism Φ ₓ Hₙ→ X such that equation*Φₓ(πᵢ⁽ⁿ⁾) = xᵢ (i∈\1,…,n\),equation*where Hₙ denotes the vector lattice of positively homogeneous, continuous, real-valued functions defined on Rⁿ and π ᵢ⁽ⁿ⁾Rⁿ→R is the i^th coordinate projection;• there is a positive element e∈ X such that e x₁ ⋯ xₙ and the normequation* xₑ = inf\ λ∈[0,∞)\:\: xλ e\,equation*defined for each x in the order ideal Iₑ of X generated by e, is complete when restricted to the closed sublattice of Iₑ generated by x₁,… ,xₙ. Moreover, we show that a vector space which admits a ‘sufficiently strong’ Hₙ-function calculus for each n∈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.