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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

On the nonexistence of nontrivial involutive $n$-homomorphisms of $C^{\star }$-algebras


Authors: Efton Park and Jody Trout
Journal: Trans. Amer. Math. Soc. 361 (2009), 1949-1961
MSC (2000): Primary 46L05; Secondary 47B99, 47L30
DOI: https://doi.org/10.1090/S0002-9947-08-04648-5
Published electronically: October 22, 2008
MathSciNet review: 2465825
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Abstract: An $n$-homomorphism between algebras is a linear map $\phi : A \to B$ such that $\phi (a_1 \cdots a_n) = \phi (a_1)\cdots \phi (a_n)$ for all elements $a_1, \dots , a_n \in A.$ Every homomorphism is an $n$-homomorphism for all $n \geq 2$, but the converse is false, in general. Hejazian et al. (2005) ask: Is every $*$-preserving $n$-homomorphism between $C^{\star }$-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 $n \geq 3$ is odd, then $\phi$ is a difference of two orthogonal $*$-homomorphisms. Thus, there are no nontrivial $*$-linear $n$-homomorphisms between $C^{\star }$-algebras.


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Additional Information

Efton Park
Affiliation: Department of Mathematics, Texas Christian University, Box 298900, Fort Worth, Texas 76129
Email: e.park@tcu.edu

Jody Trout
Affiliation: Department of Mathematics, Dartmouth College, 6188 Kemeny Hall, Hanover, New Hampshire 03755
Email: jody.trout@dartmouth.edu

Received by editor(s): April 6, 2007
Published electronically: October 22, 2008
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.