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AUTOMATIC CONTINUITY OF n-HOMOMORPHISMS BETWEEN TOPOLOGICAL ALGEBRAS

Part of: Topological algebras, normed rings and algebras, Banach algebras

Published online by Cambridge University Press:  01 April 2011

TAHER G. HONARY  and
H. SHAYANPOUR
TAHER G. HONARY*
Affiliation:
Faculty of Mathematical Sciences and Computer, Tarbiat Moallem University, Tehran 1561836314, Iran (email: honary@tmu.ac.ir)
H. SHAYANPOUR
Affiliation:
Faculty of Mathematical Sciences and Computer, Tarbiat Moallem University, Tehran 1561836314, Iran (email: h.shayanpour@yahoo.com)
*
✉For correspondence; e-mail: honary@tmu.ac.ir
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Abstract

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A map θ:A→B between algebras A and B is called n-multiplicative if θ(a1a2⋯an)=θ(a1) θ(a2)⋯θ(an) for all elements a1,a2,…,an∈A. If θ is also linear then it is called an n-homomorphism. This notion is an extension of a homomorphism. We obtain some results on automatic continuity of n-homomorphisms between certain topological algebras, as well as Banach algebras. The main results are extensions of Johnson’s theorem to surjective n-homomorphisms on topological algebras, a theorem due to C. E. Rickart in 1950 to dense range n-homomorphisms on topological algebras and two theorems due to E. Park and J. Trout in 2009 to * -preserving n-homomorphisms on lmc * -algebras.

Keywords

automatic continuityn-homomorphismtopological algebraslmc algebrasQ-algebrasn-involution*-preserving n-homomorphisms

MSC classification

Secondary: 46H40: Automatic continuity 46H05: General theory of topological algebras
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 83 , Issue 3 , June 2011 , pp. 389 - 400
Copyright
Copyright © Australian Mathematical Publishing Association Inc. 2011

References

[1]Bračič, J. and Moslehian, S., ‘On automatic continuity of 3-homomorphisms on Banach algebras’, Bull. Malays. Math. Sci. Soc. (2) 30(2) (2007), 195–200.Google Scholar
[2]Dales, H. G., Banach Algebras and Automatic Continuity, London Mathematical Society Monograph, 24 (Clarendon Press, Oxford, 2000).Google Scholar
[3]Dixon, P. G., ‘Automatic continuity of positive functionals on topological involution algebras’, Bull. Aust. Math. Soc. 23 (1981), 265–281.CrossRefGoogle Scholar
[4]Fragoulopoulou, M., ‘Automatic continuity of *-morphisms between nonnormed topological *-algebras’, Pacific J. Math. 147(1) (1991).CrossRefGoogle Scholar
[5]Fragoulopoulou, M., ‘Uniqueness of topology for semisimple LFQ-algebras’, Proc. Amer. Math. Soc. 117(4) (1993), 963–969.Google Scholar
[6]Fragoulopoulou, M., Topological Algebras with Involution (Elsevier, Amsterdam, 2005).Google Scholar
[7]Hejazian, M., Mirzavaziri, M. and Moslehian, M. S., ‘n-Homomorphism’, Bull. Iranian Math. Soc. 31(1) (2005), 13–23.Google Scholar
[8]Honary, T. G. and Najafi Tavani, M., ‘Upper semicontinuity of the spectrum function and automatic continuity in topological Q-algebras’, Note Mat. 28(2) (2008).Google Scholar
[9]Mallios, A., Topological Algebras, Selected Topics (North-Holland, Amsterdam, 1986).Google Scholar
[10]Park, E. and Trout, J., ‘On the nonexistence of nontrivial involutive n-homomorphisms of C*-algebras’, Trans. Amer. Math. Soc. 361 (2009), 1949–1961.CrossRefGoogle Scholar
[11]Ransford, T. J., ‘A short proof of Johnson’s uniqueness-of-norm theorem’, Bull. Lond. Math. Soc. 21 (1989), 487–488.CrossRefGoogle Scholar
[12]Sinclair, A. M., Automatic Continuity of Linear Operators, London Mathematical Society Lecture Notes Series, 21 (Cambridge University Press, Cambridge, 1976).CrossRefGoogle Scholar