Riesz representation theorems for vector lattices and Banach lattices of regular operators

Author(s):
Jeu, M.F.E. de (0000-0002-7295-8969); Jiang, X. (0000-0002-7295-8969)
Published
2026
Publication type
Article / Letter to editor

Abstract

For a non-empty locally compact Hausdorff space X and a Dedekind complete normal

vector lattice E, we show that the vector lattice of norm to order bounded operators

from Cc(X) or C0(X) into E is isomorphic to the vector lattice of E-valued regular

Borel measures on X. When E is an order continuous Banach lattice, the isomorphism is an isometric isomorphism between Banach lattices. When X is compact, every regular operator from C(X) into E is norm to order bounded. For some spaces E, such as KB-spaces or the regular operators on a KB-space, every regular operator from C0(X) into E is norm to order bounded. Additional results are obtained for the whole space of regular operators from Cc(X) into an order continuous Banach lattice. As a preparation, vector lattices and Banach lattices, resp. cones, of measures with values in a Dedekind complete vector lattice E, resp. in the extended positive cone of E, are investigated, as well as vector and Banach lattices of norm to order bounded operators. When E is the real numbers, our results specialise to the well-known Riesz representation theorems for the order and norm duals of Cc(X) and C0(X).

Vector-valued measureOrder integralVector latticeBanach latticeNorm to order bounded operatorRiesz representation theorem
Date issued
2026-03-23
Language
English
Publisher
Springer Science and Business Media LLC
Is part of
Banach Journal of Mathematical Analysis (2026), No. 2, 26
Advanced publication
Yes
hdl
1887/5006239
doi
10.1007/s43037-026-00486-1
URL
10.1007/s43037-026-00486-1
URL
https://link.springer.com/article/10.1007/s43037-026-00486-1