MAPS PRESERVING SOME MULTIPLICATIVE STRUCTURES ON STANDARD JORDAN OPERATOR ALGEBRAS
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초록
Let $\mathcal A$ be a unital real standard Jordan operator algebra acting on a Hilbert space $H$ of dimension at least 2. We show that every bijection $\phi$ on $\mathcal A$ satisfying $\phi(A^2\circ B)=\phi(A)^2\circ \phi(B)$ is of the form $\phi=\varepsilon\psi$ where $\psi$ is an automorphism on $\mathcal A$ and $\varepsilon \in \{-1, 1\}$. As a consequence if $\mathcal A$ is the real algebra of all self-adjoint operators on a Hilbert space $H$, then there exists a unitary or conjugate unitary operator $U$ on $H$ such that $\phi(A)=\varepsilon UAU^{*}$ for all $A\in \mathcal A$.
Abstract
Let $\mathcal A$ be a unital real standard Jordan operator algebra acting on a Hilbert space $H$ of dimension at least 2. We show that every bijection $\phi$ on $\mathcal A$ satisfying $\phi(A^2\circ B)=\phi(A)^2\circ \phi(B)$ is of the form $\phi=\varepsilon\psi$ where $\psi$ is an automorphism on $\mathcal A$ and $\varepsilon \in \{-1, 1\}$. As a consequence if $\mathcal A$ is the real algebra of all self-adjoint operators on a Hilbert space $H$, then there exists a unitary or conjugate unitary operator $U$ on $H$ such that $\phi(A)=\varepsilon UAU^{*}$ for all $A\in \mathcal A$.
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