Nonlinear disjointness/supplement preservers of nonnegative continuous functions

Lei Li; Ching Jou Liao; Luoyi Shi; Liguang Wang; Ngai Ching Wong

doi:10.1016/j.jmaa.2023.127483

Nonlinear disjointness/supplement preservers of nonnegative continuous functions

Lei Li
, Ching Jou Liao
, Luoyi Shi
, Liguang Wang
, Ngai Ching Wong^*

^*Corresponding author for this work

Department of Mathematics

Research output: Contribution to journal › Journal article › peer-review

3 Citations (Scopus)

Abstract

Let F(X),F(Y) be sufficiently large sets of nonnegative continuous real-valued functions defined on completely regular spaces X,Y, respectively. Let Φ:F(X)→F(Y) be a surjective map satisfying that f∨g>0⟺Φ(f)∨Φ(g)>0,∀f,g∈F(X). In many cases, we show that there is a homeomorphism τ:Y→X such that Φ(f)(y)≠0⟺f(τ(y))≠0,∀f∈F(X),∀y∈Y. Assume X,Y are locally compact Hausdorff (resp. separable and metrizable) and Φ:C₀(X)₊→C₀(Y)₊ (resp. Φ:C^b(X)₊→C^b(Y)₊) is a surjective map. We show that Φ preserves the norms of infima, i.e., ‖Φ(f)∧Φ(g)‖=‖f∧g‖,∀f,g∈C₀(X)₊ (resp.C^b(X)₊), if and only if there is a homeomorphism τ:Y→X such that Φ(f)(y)=f(τ(y)),∀f∈C₀(X)₊ (resp.C^b(X)₊), ∀y∈Y.

Original language	English
Article number	127483
Journal	Journal of Mathematical Analysis and Applications
Volume	528
Issue number	1
Early online date	7 Jun 2023
DOIs	https://doi.org/10.1016/j.jmaa.2023.127483
Publication status	Published - 1 Dec 2023

User-Defined Keywords

Continuous functions
Cozero sets
Disjointness preservers
Kaplansky theorem
Order isomorphism

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10.1016/j.jmaa.2023.127483

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@article{3ffae63e9655413f9ef980226355444e,

title = "Nonlinear disjointness/supplement preservers of nonnegative continuous functions",

abstract = "Let F(X),F(Y) be sufficiently large sets of nonnegative continuous real-valued functions defined on completely regular spaces X,Y, respectively. Let Φ:F(X)→F(Y) be a surjective map satisfying that f∨g>0⟺Φ(f)∨Φ(g)>0,∀f,g∈F(X). In many cases, we show that there is a homeomorphism τ:Y→X such that Φ(f)(y)≠0⟺f(τ(y))≠0,∀f∈F(X),∀y∈Y. Assume X,Y are locally compact Hausdorff (resp. separable and metrizable) and Φ:C0(X)+→C0(Y)+ (resp. Φ:Cb(X)+→Cb(Y)+) is a surjective map. We show that Φ preserves the norms of infima, i.e., ‖Φ(f)∧Φ(g)‖=‖f∧g‖,∀f,g∈C0(X)+ (resp.Cb(X)+), if and only if there is a homeomorphism τ:Y→X such that Φ(f)(y)=f(τ(y)),∀f∈C0(X)+ (resp.Cb(X)+), ∀y∈Y.",

keywords = "Continuous functions, Cozero sets, Disjointness preservers, Kaplansky theorem, Order isomorphism",

author = "Lei Li and Liao, \{Ching Jou\} and Luoyi Shi and Liguang Wang and Wong, \{Ngai Ching\}",

note = "Li is partly supported by NSF of China ( 12171251 ). Wang is partly supported by NSF of China ( 11871303 , 11971463 ) and NSF of Shandong Province ( ZR2020MA008 ). Wong is partly supported by MOST of Taiwan ( 110-2115-M-110-002-MY2 ). Publisher Copyright: {\textcopyright} 2023 Elsevier Inc.",

year = "2023",

month = dec,

day = "1",

doi = "10.1016/j.jmaa.2023.127483",

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journal = "Journal of Mathematical Analysis and Applications",

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T1 - Nonlinear disjointness/supplement preservers of nonnegative continuous functions

AU - Li, Lei

AU - Liao, Ching Jou

AU - Shi, Luoyi

AU - Wang, Liguang

AU - Wong, Ngai Ching

N1 - Li is partly supported by NSF of China ( 12171251 ). Wang is partly supported by NSF of China ( 11871303 , 11971463 ) and NSF of Shandong Province ( ZR2020MA008 ). Wong is partly supported by MOST of Taiwan ( 110-2115-M-110-002-MY2 ). Publisher Copyright: © 2023 Elsevier Inc.

PY - 2023/12/1

Y1 - 2023/12/1

N2 - Let F(X),F(Y) be sufficiently large sets of nonnegative continuous real-valued functions defined on completely regular spaces X,Y, respectively. Let Φ:F(X)→F(Y) be a surjective map satisfying that f∨g>0⟺Φ(f)∨Φ(g)>0,∀f,g∈F(X). In many cases, we show that there is a homeomorphism τ:Y→X such that Φ(f)(y)≠0⟺f(τ(y))≠0,∀f∈F(X),∀y∈Y. Assume X,Y are locally compact Hausdorff (resp. separable and metrizable) and Φ:C0(X)+→C0(Y)+ (resp. Φ:Cb(X)+→Cb(Y)+) is a surjective map. We show that Φ preserves the norms of infima, i.e., ‖Φ(f)∧Φ(g)‖=‖f∧g‖,∀f,g∈C0(X)+ (resp.Cb(X)+), if and only if there is a homeomorphism τ:Y→X such that Φ(f)(y)=f(τ(y)),∀f∈C0(X)+ (resp.Cb(X)+), ∀y∈Y.

AB - Let F(X),F(Y) be sufficiently large sets of nonnegative continuous real-valued functions defined on completely regular spaces X,Y, respectively. Let Φ:F(X)→F(Y) be a surjective map satisfying that f∨g>0⟺Φ(f)∨Φ(g)>0,∀f,g∈F(X). In many cases, we show that there is a homeomorphism τ:Y→X such that Φ(f)(y)≠0⟺f(τ(y))≠0,∀f∈F(X),∀y∈Y. Assume X,Y are locally compact Hausdorff (resp. separable and metrizable) and Φ:C0(X)+→C0(Y)+ (resp. Φ:Cb(X)+→Cb(Y)+) is a surjective map. We show that Φ preserves the norms of infima, i.e., ‖Φ(f)∧Φ(g)‖=‖f∧g‖,∀f,g∈C0(X)+ (resp.Cb(X)+), if and only if there is a homeomorphism τ:Y→X such that Φ(f)(y)=f(τ(y)),∀f∈C0(X)+ (resp.Cb(X)+), ∀y∈Y.

KW - Continuous functions

KW - Cozero sets

KW - Disjointness preservers

KW - Kaplansky theorem

KW - Order isomorphism

UR - https://www.scopus.com/pages/publications/85162922700

U2 - 10.1016/j.jmaa.2023.127483

DO - 10.1016/j.jmaa.2023.127483

M3 - Journal article

AN - SCOPUS:85162922700

SN - 0022-247X

VL - 528

JO - Journal of Mathematical Analysis and Applications

JF - Journal of Mathematical Analysis and Applications

IS - 1

M1 - 127483

ER -

Nonlinear disjointness/supplement preservers of nonnegative continuous functions

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