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 | References | Similar Articles | Additional Information

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.

- B. Blackadar,
*Operator algebras*, Encyclopaedia of Mathematical Sciences, vol. 122, Springer-Verlag, Berlin, 2006. Theory of $C^*$-algebras and von Neumann algebras; Operator Algebras and Non-commutative Geometry, III. MR**2188261** - J. BraΔiΔ and S. Moslehian,
*On Automatic Continuity of $3$-Homomorphisms on Banach Algebras*, to appear in Bull. Malays. Math. Sci. Soc. arXiv: math.FA/0611287. - Paul J. Cohen,
*Factorization in group algebras*, Duke Math. J.**26**(1959), 199β205. MR**104982** - Shalom Feigelstock,
*Rings whose additive endomorphisms are $N$-multiplicative*, Bull. Austral. Math. Soc.**39**(1989), no. 1, 11β14. MR**976254**, DOI https://doi.org/10.1017/S0004972700027921 - S. Feigelstock,
*Rings whose additive endomorphisms are $n$-multiplicative. II*, Period. Math. Hungar.**25**(1992), no. 1, 21β26. MR**1200838**, DOI https://doi.org/10.1007/BF02454380 - Lawrence A. Harris,
*A generalization of $C^{\ast } $-algebras*, Proc. London Math. Soc. (3)**42**(1981), no. 2, 331β361. MR**607306**, DOI https://doi.org/10.1112/plms/s3-42.2.331 - S. Hejazian, M. Mirzavaziri, and M. S. Moslehian,
*$n$-homomorphisms*, Bull. Iranian Math. Soc.**31**(2005), no. 1, 13β23, 88 (English, with English and Arabic summaries). MR**2228453** - Iain Raeburn and Dana P. Williams,
*Morita equivalence and continuous-trace $C^*$-algebras*, Mathematical Surveys and Monographs, vol. 60, American Mathematical Society, Providence, RI, 1998. MR**1634408** - W. Forrest Stinespring,
*Positive functions on $C^*$-algebras*, Proc. Amer. Math. Soc.**6**(1955), 211β216. MR**69403**, DOI https://doi.org/10.1090/S0002-9939-1955-0069403-4

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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.