On the nonexistence of nontrivial involutive -homomorphisms of
-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
Full-text PDF Free Access
Abstract | References | Similar Articles | Additional Information
Abstract: An -homomorphism between algebras is a linear map
such that
for all elements
Every homomorphism is an
-homomorphism for all
, but the converse is false, in general. Hejazian et al. (2005) ask: Is every
-preserving
-homomorphism between
-algebras continuous? We answer their question in the affirmative, but the even and odd
arguments are surprisingly disjoint. We then use these results to prove stronger ones: If
is even, then
is just an ordinary
-homomorphism. If
is odd, then
is a difference of two orthogonal
-homomorphisms. Thus, there are no nontrivial
-linear
-homomorphisms between
-algebras.
- 1. 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
- 2.
J. Bračič and S. Moslehian, On Automatic Continuity of
-Homomorphisms on Banach Algebras, to appear in Bull. Malays. Math. Sci. Soc. arXiv: math.FA/0611287.
- 3. Paul J. Cohen, Factorization in group algebras, Duke Math. J. 26 (1959), 199–205. MR 104982
- 4. Shalom Feigelstock, Rings whose additive endomorphisms are 𝑁-multiplicative, Bull. Austral. Math. Soc. 39 (1989), no. 1, 11–14. MR 976254, https://doi.org/10.1017/S0004972700027921
- 5. S. Feigelstock, Rings whose additive endomorphisms are 𝑛-multiplicative. II, Period. Math. Hungar. 25 (1992), no. 1, 21–26. MR 1200838, https://doi.org/10.1007/BF02454380
- 6. Lawrence A. Harris, A generalization of 𝐶*-algebras, Proc. London Math. Soc. (3) 42 (1981), no. 2, 331–361. MR 607306, https://doi.org/10.1112/plms/s3-42.2.331
- 7. S. Hejazian, M. Mirzavaziri, and M. S. Moslehian, 𝑛-homomorphisms, Bull. Iranian Math. Soc. 31 (2005), no. 1, 13–23, 88 (English, with English and Arabic summaries). MR 2228453
- 8. 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
- 9. W. Forrest Stinespring, Positive functions on 𝐶*-algebras, Proc. Amer. Math. Soc. 6 (1955), 211–216. MR 69403, 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
DOI:
https://doi.org/10.1090/S0002-9947-08-04648-5
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.