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Linear Maps on Standard Operator Algebras Characterized by Action on Zero Products

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Let $${\mathcal {A}}$$A be a unital standard algebra on a complex Banach space $${\mathcal {X}}$$X with dim$${\mathcal {X}}\ge 2$$X≥2. The main result of this paper is to characterize the linear… Click to show full abstract

Let $${\mathcal {A}}$$A be a unital standard algebra on a complex Banach space $${\mathcal {X}}$$X with dim$${\mathcal {X}}\ge 2$$X≥2. The main result of this paper is to characterize the linear maps $$\delta , \tau : {\mathcal {A}}\rightarrow B({\mathcal {X}})$$δ,τ:A→B(X) satisfying $$ A \tau ( B) + \delta ( A) B = 0$$Aτ(B)+δ(A)B=0 whenever $$A,B\in {\mathcal {A}}$$A,B∈A are such that $$AB=0$$AB=0. As application of our main result, we determine the linear map $$\delta : {\mathcal {A}}\rightarrow B({\mathcal {H}})$$δ:A→B(H) that has one of the following properties for $$A,B\in {\mathcal {A}}$$A,B∈A: if $$AB^{\star }=0$$AB⋆=0, then $$A\delta (B)^{\star }+\delta (A)B^{\star }=0$$Aδ(B)⋆+δ(A)B⋆=0, or if $$A^{\star }B=0$$A⋆B=0, then $$ A^{\star }\delta (B)+\delta (A)^{\star }B=0 $$A⋆δ(B)+δ(A)⋆B=0, where $${\mathcal {A}}$$A is a unital standard operator algebras on a Hilbert space $${\mathcal {H}}$$H such that $$ {\mathcal {A}} $$A is closed under the adjoint operation. We also provide other applications of the main result.

Keywords: linear maps; delta; standard operator; operator algebras; star

Journal Title: Bulletin of the Iranian Mathematical Society
Year Published: 2019

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