Vector lattices admitting a positively homogeneous continuous function calculus

Laustsen, Niels and Troitsky, Vladimir G. (2020) Vector lattices admitting a positively homogeneous continuous function calculus. The Quarterly Journal of Mathematics, 71 (1). 281–294. ISSN 0033-5606

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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 x = (x_1,...,x_n)∈X^n, where X is an Archimedean vector lattice and n∈ N : (i) there is a vector lattice homomorphism Φ_x: H_n→X such that Φ_x(π_i) = x_i for each i∈{1,...,n}, where H_n denotes the vector lattice of positively homogeneous, continuous, real-valued functions defined on R ^n and π_i: R ^n→ R is the i'th coordinate projection; (ii) there is a positive element e∈X such that e≤max{|x_1|,...,|x_n|} and the norm ||x||_e = inf{λ∈[0,∞) : |x|≤λe}, 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,...,x_n. Moreover, we show that a vector space which admits a `sufficiently strong' H_n-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.