On the Nonexistence of Nontrivial Involutive n-Homomorphisms of C*-algebras

Details On the Nonexistence of Nontrivial Involutive n-Homomorphisms of C*-algebras On the Nonexistence of Nontrivial Involutive n-Homomorphisms of C*-algebras

2007-04-06

arxiv.org/abs/0704.0910v3

Mathematics

Operator Algebras

13 pages; final version to appear in Transactions of the AMS

Scientific paper

An n-homomorphism between algebras is a linear map $\phi : A \to B$ such that $\phi(a_1 ... a_n) = \phi(a_1)... \phi(a_n)$ 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 $\phi$ is just an ordinary *-homomorphism. If n >= 3 is odd, then $\phi$ is a difference of two orthogonal *-homomorphisms. Thus, there are no nontrivial *-linear n-homomorphisms between C*-algebras.

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