Let L be an additive map between (real or complex) matrix algebras sending $n\times n$ Hermitian idempotent matrices to $m\times m$ Hermitian idempotent matrices. We show that there are nonnegative integers $p,q$ with $n(p+q)=r\le m$ and an $m\times m$ unitary matrix U such that$L(A)=U[(I_p\otimes A)\oplus (I_q\otimes A^t)\oplus 0_{m-r}]U^{⁎},\text{for any }n\times n\text{ Hermitian }A\text{ with rational trace}.$ We also extend this result to the (complex) von Neumann algebra setting, and provide a supplement to the Dye-Bunce-Wright Theorem asserting that every additive map of Hermitian idempotents extends to a Jordan ⁎-homomorphism.

中文翻译：

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算子代数之间的加法 Hermitian 幂等保护器

令L是（实数或复数）矩阵代数发送之间的加法映射$n\times n$ Hermitian 幂等矩阵 $米\times 米$厄米幂等矩阵。我们证明存在非负整数$磷,q$ 和 $n(磷+q)=r\le 米$ 和 $米\times 米$酉矩阵U使得$升(\mathrm{一种})=你[({\mathrm{一世}}_{磷}\otimes \mathrm{一种})\oplus ({\mathrm{一世}}_q\otimes {\mathrm{一种}}^{吨})\oplus 0_{米-r}]{你}^{⁎},\text{对于任何 }n\times n\text{ 埃尔米特 }\mathrm{一种}\text{ 有理性的痕迹}.$ 我们还将这个结果扩展到（复杂的）von Neumann 代数设置，并提供对 Dye-Bunce-Wright 定理的补充，该定理断言 Hermitian 幂等的每个加法映射都扩展到 Jordan ⁎-同态。