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Transactions of the American Mathematical Society
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
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

DOI: http://dx.doi.org/10.1090/S0002-9947-08-04648-5
PII: S 0002-9947(08)04648-5
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.