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- Title
Lateral order on complex vector lattices and narrow operators.
- Authors
Dzhusoeva, Nonna; Huang, Jinghao; Pliev, Marat; Sukochev, Fedor
- Abstract
In this paper, we continue investigation of the lateral order on vector lattices started in [25]. We consider the complexification EC$E_{\mathbb {C}}$ of a real vector lattice E and introduce the lateral order on EC$E_{\mathbb {C}}$. Our first main result asserts that the set of all fragments Fv$\mathfrak {F}_v$ of an element v∈EC$v\in E_{\mathbb {C}}$ of the complexification of an uniformly complete vector lattice E is a Boolean algebra. Then, we study narrow operators defined on the complexification EC$E_{\mathbb {C}}$ of a vector lattice E, extending the results of articles [22, 27, 28] to the setting of operators defined on complex vector lattices. We prove that every order‐to‐norm continuous linear operator T:EC→X$\mathcal {T}: E_{\mathbb {C}} \rightarrow X$ from the complexification EC$E_{\mathbb {C}}$ of an atomless Dedekind complete vector lattice E to a finite‐dimensional Banach space X is strictly narrow. Then, we prove that every C‐compact order‐to‐norm continuous linear operator T$\mathcal {T}$ from EC$E_{\mathbb {C}}$ to a Banach space X is narrow. We also show that every regular order‐no‐norm continuous linear operator from EC$E_{\mathbb {C}}$ to a complex Banach lattice (ℓp(D)C$(\ell _p(\mathcal {D})_{\mathbb {C}}$ is narrow. Finally, in the last part of the paper we investigate narrow operators taking values in symmetric ideals of compact operators.
- Subjects
RIESZ spaces; BIVECTORS; COMPACT operators; BOOLEAN algebra; BANACH spaces; BANACH lattices
- Publication
Mathematische Nachrichten, 2023, Vol 296, Issue 11, p5157
- ISSN
0025-584X
- Publication type
Article
- DOI
10.1002/mana.202200415