Vector Lattices Admitting a Positively Homogeneous Continuous Function Calculus

DOI10.1093/QMATHJ/HAZ031zbMATH Open1460.46004arXiv1901.07522OpenAlexW3003357096MaRDI QIDQ5108048

Vladimir G. Troitsky, Niels Jakob Laustsen

Publication date: 29 April 2020

Published in: The Quarterly Journal of Mathematics (Search for Journal in Brave)

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

, where

X

is an Archimedean vector lattice and

n i n m a t h b b N

: - there is a vector lattice homomorphism

such that

(i i n 1, l d o t s, n)

, where

H_{n}

denotes the vector lattice of positively homogeneous, continuous, real-valued functions defined on

m a t h b b R^{n}

and

p i_{i}^{(n)} c o l o n m a t h b b R^{n} o m a t h b b R

is the

i^{e x t t h}

coordinate projection; - there is a positive element

e i n X

such that

e g e q s l a n t l v e r t x_{1} v e r t v e e c d o t s v e e l v e r t x_{n} v e r t

and the norm

, 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}, l d o t s, x_{n}

. Moreover, we show that a vector space which admits a `sufficiently strong'

H_{n}

-function calculus for each

n i n m a t h b b 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.

Full work available at URL: https://arxiv.org/abs/1901.07522

zbMATH Keywords

functional calculus Archimedean vector lattice

Mathematics Subject Classification ID

Ordered topological linear spaces, vector lattices (46A40)