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Rights statement: This is a pre-copy-editing, author-produced PDF of an article accepted for publication in The Quarterly Journal of Mathematics following peer review. The definitive publisher-authenticated version Niels Jakob Laustsen, Vladimir G Troitsky, Vector Lattices Admitting a Positively Homogeneous Continuous Function Calculus, The Quarterly Journal of Mathematics, Volume 71, Issue 1, March 2020, Pages 281–294 is available online at: https://academic.oup.com/qjmath/article-abstract/71/1/281/5709796

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## Vector lattices admitting a positively homogeneous continuous function calculus

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Journal publication date 1/03/2020 The Quarterly Journal of Mathematics 1 71 14 281–294 Published English

### 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.

### Bibliographic note

This is a pre-copy-editing, author-produced PDF of an article accepted for publication in The Quarterly Journal of Mathematics following peer review. The definitive publisher-authenticated version Niels Jakob Laustsen, Vladimir G Troitsky, Vector Lattices Admitting a Positively Homogeneous Continuous Function Calculus, The Quarterly Journal of Mathematics, Volume 71, Issue 1, March 2020, Pages 281–294 is available online at: https://academic.oup.com/qjmath/article-abstract/71/1/281/5709796