A note on strong (skew) η-Lie products preserving maps on some algebras

ABSTRACT

Let $\mathcal{A}$ be an arbitrary unital *-algebra over the real or complex field $\mathbb{F}$, and let $\eta \in \mathbb{F}$ with $\eta \ne 1$. Assume that $\mathrm{\Phi }:\mathcal{A}⟶\mathcal{A}$ is a map. It is shown that, if $\mathrm{\Phi }$ satisfies $\mathrm{\Phi }(A)\mathrm{\Phi }(P)-\eta \mathrm{\Phi }(P)\mathrm{\Phi }{(A)}^{\ast }=AP-\eta P{A}^{\ast },$ for each $A\in \mathcal{A}$ and projection $P\in \}{P}_1,I-P_1\{,$ then $\mathrm{\Phi }{(I)}^2=I$ and $\mathrm{\Phi }(A)=A\mathrm{\Phi }(I)$ for all $A\in \mathcal{A}$. Also, if $\mathcal{A}$ is a nest algebra on Hilbert space $\mathcal{H}$ and $\mathrm{\Phi }$ is a surjective map that satisfies $\mathrm{\Phi }(A)\mathrm{\Phi }(P)-\mathrm{\Phi }(P)\mathrm{\Phi }(A)=AP-PA,$ for each $A\in \mathcal{A}$ and all idempotents P in $\mathcal{A}$, then we characterize the concrete form of $\mathrm{\Phi }$ on $\mathcal{A}$.

KEYWORDS:

• Preserving problems
• skew product maps
• $C^{\ast }-$algebras
• nest algebra

AMS SUBJECT CLASSIFICATIONS:

• 46J10
• 47B48
• 47L35

Notes

No potential conflict of interest was reported by the authors.