Characterization of linear mappings on (Banach) ⋆-algebras by similar properties to derivations
Keyword(s):
AbstractLet 𝓐 be a ⋆-algebra, δ : 𝓐 → 𝓐 be a linear map, and z ∈ 𝓐 be fixed. We consider the condition that δ satisfies xδ(y)⋆ + δ(x)y⋆ = δ(z) (x⋆δ(y) + δ(x)⋆y = δ(z)) whenever xy⋆ = z (x⋆y = z), and under several conditions on 𝓐, δ and z we characterize the structure of δ. In particular, we prove that if 𝓐 is a Banach ⋆-algebra, δ is a continuous linear map, and z is a left (right) separating point of 𝓐, then δ is a Jordan derivation. Our proof is based on complex variable techniques. Also, we describe a linear map δ satisfying the above conditions with z = 0 on two classes of ⋆-algebras: zero product determined algebras and standard operator algebras.
1993 ◽
Vol 45
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pp. 483-496
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2006 ◽
Vol 134
(11)
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pp. 3255-3256
2018 ◽
Vol 11
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pp. 1850022
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2018 ◽
Vol 168
(2)
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pp. 345-360
2018 ◽
Vol 11
(02)
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pp. 1850021
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2018 ◽
Vol 17
(09)
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pp. 1850169
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2003 ◽
Vol 7
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pp. 605-613
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