关于C* 代数的非平凡对合n-同态的不存在性
On the Nonexistence of Nontrivial Involutive n-Homomorphisms of C*-algebras
摘要：
代数之间的n-同态是一个线性映射$ϕ: A \to B$使得$ϕ(a_1 ... a_n) = ϕ(a_1)... ϕ(a_n)$对所有元素$a_1, >..., a_n \in A.$每个同态都是n-同态，对所有n >= 2，但一般情况下反之亦然。Hejazian等人[7]提出了一个问题：C* 代数之间的每一个保 * n-同态是连续的吗？我们肯定地回答他们的问题，但偶数和奇数n的论点令人惊讶地不相交。然后我们用这些结果证明了更强的结果：如果n >2是偶数，那么$ϕ$只是一个普通的 *-同态。如果n >= 3是奇数，则$ϕ$是两个正交 *-同态的差。因此，C* 代数之间不存在非平凡的 *-线性n-同态。
英文摘要：
An n-homomorphism between algebras is a linear map $ϕ: A \to B$ such that $ϕ(a_1 ... a_n) = ϕ(a_1)... ϕ(a_n)$ for all elements $a_1, >..., a_n \in A.$ Every homomorphism is an n-homomorphism, for all n >= 2, but the converse is false, in general. Hejazian et al. [7] ask: Is every *-preserving n-homomorphism between C*-algebras continuous? We answer their question in the affirmative, but the even and odd n arguments are surprisingly disjoint. We then use these results to prove stronger ones: If n >2 is even, then $ϕ$ is just an ordinary *-homomorphism. If n >= 3 is odd, then $ϕ$ is a difference of two orthogonal *-homomorphisms. Thus, there are no nontrivial *-linear n-homomorphisms between C*-algebras.
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