TY - JOUR

T1 - Vector lattices admitting a positively homogeneous continuous function calculus

AU - Laustsen, Niels

AU - Troitsky, Vladimir G.

N1 - 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

PY - 2020/3/1

Y1 - 2020/3/1

N2 - 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 functioncalculus, while others do not.

AB - 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 functioncalculus, while others do not.

KW - vector lattice

KW - positively homogeneous continuous function calculus

KW - uniform completeness

U2 - 10.1093/qmathj/haz031

DO - 10.1093/qmathj/haz031

M3 - Journal article

VL - 71

SP - 281

EP - 294

JO - The Quarterly Journal of Mathematics

JF - The Quarterly Journal of Mathematics

SN - 0033-5606

IS - 1

ER -