Equivalent characterization of right (left) centralizers or centralizers on Banach algebras

Subject Areas : Functional analysis

H.  Ghahramani ¹ (Department of Mathematics, Faculty of Science, University of Kurdistan, P.O. Box 416, Sanandaj, Kurdistan, Iran)
Gh.  Moradkhani ² (Department of Mathematics, Faculty of Science, University of Kurdistan, P.O. Box 416, Sanandaj, Kurdistan, Iran)
S.  Sattari ³ (Department of Mathematics, Faculty of Science, University of Kurdistan, P.O. Box 416, Sanandaj, Kurdistan, Iran)

Received: 2023-08-07 Accepted : 2023-10-13 Published : 2023-10-01

Keywords: ‎Left centralizer‎, ‎right centralizer‎, ‎centralizer‎, ‎Banach algebra‎, ‎Banach $ \star $-algebra,

Abstract :

‎Let $ \mathcal{A} $ be a unital Banach algebra‎, ‎$ w\in \mathcal{A}$‎, ‎and $ \gamma‎ : ‎\mathcal{A} \to \mathcal{A} $ is a continuous linear map‎. ‎We show that $\gamma$ satisfies $a\gamma(b)=\gamma(w)$ ($\gamma(a)b=\gamma(w)$) whenever $a,b\in \mathcal{A}$ with $ab=w$ and $w$ is a left (right) separating point in $\mathcal{A}$ if and only if $\gamma$ is a right (left) centralizer‎. ‎Also‎, ‎we prove that $\gamma$ satisfies $a\gamma(b)=\gamma(a)b=\gamma(w)$ whenever $a,b\in \mathcal{A}$ with $ab=w$ and $w$ is a left or right separating point in $\mathcal{A}$ if and only if $\gamma$ is a centralizer‎. ‎We also provide some applications of the obtained results for characterization of a continuous linear map $\gamma:\mathcal{A}\rightarrow \mathcal{A}$ on a unital Banach $*$-algebra $\mathcal{A}$ satisfying $a\gamma(b)^{*}=\gamma(w^{*})^{*}$ ($\gamma(a)^{*}b=\gamma(w^{*})^{*}$) whenever $a,b\in \mathcal{A}$ with $ab^{*}=w$ ($a^{*}b=w$) and $w$ is a left (right) separating point‎, ‎or $\gamma$ satisfying $a\gamma(b)^{*}=\gamma(c)^{*}d=\gamma(w^{*})^{*}$ whenever $a,b,c,d\in \mathcal{A}$ with $ab^{*}=c^{*}d =w$ and $w$ is a left or right separating point‎.

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