2019/01/22 by Niels Jakob Laustsen, Laustsen, Niels Jakob, Vladimir G. Troitsky +1
Mathematics · #46A40 #Advanced Topology and Set Theory #FOS: Mathematics #Functional Analysis (math.FA) #Mathematical and Theoretical Analysis #advanced mathematical theories
paper · pdf · doi:10.48550/arxiv.1901.07522
openalex publication_date 2019/01/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We characterize the Archimedean vector lattices that admit a positively\nhomogeneous continuous function calculus by showing that the following two\nconditions are equivalent for each n-tuple boldsymbolx =\n(x1,\…,xn)\∈ Xn, where X is an Archimedean vector lattice and\nn\∈ mathbb N:\n - there is a vector lattice homomorphism \Φ_ boldsymbolx colon Hn\→\nX such that \Φ_ boldsymbolx(\πi(n))=xi (i\∈ 1,\…,n ),\nwhere Hn denotes the vector lattice of positively homogeneous, continuous,\nreal-valued functions defined on mathbb Rn and \πi(n) colon mathbb\nRn\→ mathbb R is the i\th coordinate projection;\n - there is a positive element e\∈ X such that e geqslant\|\nx1\| vee\⋯ vee\| xn\| and the norm \‖ x\‖e =\n\inf \\λ\∈[0,\∞) : colon :\| x\|\≤\λ e \,\ndefined for each x in the order ideal Ie of X generated by e, is\ncomplete when restricted to the closed sublattice of Ie generated by\nx1,\…,xn.\n Moreover, we show that a vector space which admits a `sufficiently strong'\nHn-function calculus for each n\∈ mathbb N is automatically a vector\nlattice, and we explore the situation in the non-Archimedean case by showing\nthat some non-Archimedean vector lattices admit a positively homogeneous\ncontinuous function calculus, while others do not.\n