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Let $${\mathcal {R}}$$ R be a unital prime $$*$$ ∗ -ring containing a nontrivial symmetric idempotent. For $$A,B\in {\mathcal {R}}$$ A , B ∈ R , the skew commutator and 2-skew commutator are defined, respectively, by $${}_*[A,B]=AB-BA^*$$ ∗ [ A , B ] = A B - B A ∗ and $${}_*[A,B]_2= {{}_*[A, {{}_*[A,B]}]}$$ ∗ [ A , B ] 2 = ∗ [ A , ∗ [ A , B ] ] . Let $$\Phi :{\mathcal {R}} \rightarrow {\mathcal {R}}$$ Φ : R → R be a surjective map. We show that (1) $$\Phi $$ Φ satisfies $${}_*[\Phi (A),\Phi (B)] = {{}_*[A,B] }$$ ∗ [ Φ ( A ) , Φ ( B ) ] = ∗ [ A , B ] for all $$A, B\in {\mathcal {R}}$$ A , B ∈ R if and only if there exists $$\lambda \in \{-1,1\}$$ λ ∈ { - 1 , 1 } such that $$\Phi (A)=\lambda A$$ Φ ( A ) = λ A for all $$A\in {\mathcal {R}}$$ A ∈ R ; (2) $$\Phi $$ Φ satisfies $${}_*[\Phi (A),\Phi (B)]_2= {{}_*[A,B]_2}$$ ∗ [ Φ ( A ) , Φ ( B ) ] 2 = ∗ [ A , B ] 2 for all $$A, B\in {\mathcal {R}}$$ A , B ∈ R if and only if there exists $$\lambda \in {\mathcal {C}}_S $$ λ ∈ C S with $$\lambda ^{3} = I$$ λ 3 = I such that $$\Phi (A) = \lambda A $$ Φ ( A ) = λ A for all $$A \in {\mathcal {R}}$$ A ∈ R , where I is the unit of $${\mathcal {R}}$$ R and $${\mathcal {C}}_S $$ C S is the symmetric extend centroid of $${\mathcal {R}}$$ R . This is then applied to prime $$\hbox {C}^*$$ C ∗ -algebras, factor von Neumann algebras and indefinite self-adjoint standard operator algebras to get a complete invariant for the identity map and to symmetric standard operator algebras as well as matrix algebras.
Bulletin of the Malaysian Mathematical Sciences Society – Springer Journals
Published: Feb 1, 2017
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