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- Publisher
- Springer Journals
- Copyright
- Copyright © Tusi Mathematical Research Group (TMRG) 2021
- ISSN
- 2639-7390
- eISSN
- 2008-8752
- DOI
- 10.1007/s43034-021-00123-y
- Publisher site
- See Article on Publisher Site
Abstract
Let A\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\mathcal {A}}$$\end{document} be an algebra. In this paper, we consider the problem of determining a linear map ψ\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$\psi $$\end{document} on A\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\mathcal {A}}$$\end{document} satisfying a,b∈A\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$a,b\in {\mathcal {A}}$$\end{document}, ab=0⟹ψ([a,b])=[ψ(a),b](C1)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$ab=0 \Longrightarrow \psi ([a,b])=[\psi (a),b] \, (C1) $$\end{document} or ab=0⟹ψ([a,b])=[a,ψ(b)](C2)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$ab=0 \Longrightarrow \psi ([a,b])=[a,\psi (b)] \, (C2)$$\end{document}. We first compare linear maps satisfying (C1) or (C2), commuting linear maps, and Lie centralizers with a variety of examples. In fact, we see that linear maps satisfying (C1), (C2) and commuting linear maps are different classes of each other. Then, we introduce a class of operator algebras on Banach spaces such that if A\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\mathcal {A}}$$\end{document} is in this class, then any linear map on A\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\mathcal {A}}$$\end{document} satisfying (C1) (or (C2)) is a commuting linear map. As an application of these results, we characterize Lie centralizers and linear maps satisfying (C1) (or (C2)) on nest algebras.
Journal
Annals of Functional Analysis – Springer Journals
Published: Mar 11, 2021