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Additivity of Jordan multiplicative maps on Jordan operator algebras. (English) Zbl 1107.46047
Let $H$ be a complex Hilbert space with $\dim(H)>1$ and let $B(H)_{sa}$ denote the Jordan algebra of all self-adjoint bounded linear operators on $H$. The paper under review is concerned with the bijective maps $\Phi: B(H)_{sa}\rightarrow B(H)_{sa}$ which satisfy one of the following conditions: (i) $\Phi(ABA)=\Phi(A)\Phi(B)\Phi(A)$; (ii) $\Phi(1/2(AB+BA))=1/2(\Phi(A)\Phi(B)+\Phi(B)\Phi(A))$; (iii) $\Phi(AB+BA)=\Phi(A)\Phi(B)+\Phi(B)\Phi(A)$; for all $A,B\in B(H)_{sa}$. The authors show that $\Phi$ is necessarily of the form $\Phi(A)=\varepsilon UAU^*$ for some unitary or conjugate unitary operator $U$ and $\varepsilon=\pm 1$ in the first case and $\varepsilon=1$ in the remaining cases.

46L70Nonassociative selfadjoint operator algebras
47B49Transformers, preservers (operators on spaces of operators)