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On Maps Preserving Products

Published online by Cambridge University Press:  20 November 2018

M. A. Chebotar,
W.-F. Ke,
P.-H. Lee and
L.-S. Shiao
M. A. Chebotar
Affiliation:
Department of Mechanics and Mathematics, Tula State University, Tula, Russia e-mail: mchebotar@tula.net
W.-F. Ke
Affiliation:
Department of Mathematics, National Cheng Kung University, Tainan, Taiwan wfke@mail.ncku.edu.tw
P.-H. Lee
Affiliation:
Department of Mathematics, National Taiwan University, Taipei, Taiwan phlee@math.ntu.edu.tw
L.-S. Shiao
Affiliation:
Department of Mathematics, National Kaohsiung Normal University, Kaohsiung, Taiwan
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Abstract

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Maps preserving certain algebraic properties of elements are often studied in Functional Analysis and Linear Algebra. The goal of this paper is to discuss the relationships among these problems from the ring-theoretic point of view.

Keywords

16W2016N5016N60

Information

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 48 , Issue 3 , 01 September 2005, pp. 355 - 369
Copyright
Copyright © Canadian Mathematical Society 2005

References

[1] Ancochea, G., On semi-automorphisms of division algebras. Ann. of Math. (2) 48(1947), 147–153.Google Scholar
[2] Artin, E., Geometric Algebra. Interscience Publishers, New York, 1957.Google Scholar
[3] Beidar, K. I. and Chebotar, M. A., On functional identities and d-free subsets of rings, I. Comm. Algebra 28(2000), 3925–3951.Google Scholar
[4] Beidar, K. I. and Chebotar, M. A., On functional identities and d-free subsets of rings II. Comm. Algebra 28(2000), 3953–3972.Google Scholar
[5] Beidar, K. I., Martindale, W. S., and Mikhalev, A. V., Rings with Generalized Identities. Marcel Dekker, New York, 1996.Google Scholar
[6] Beidar, K. I., Mikhalev, A. V., and Chebotar, M. A., Functional identities in rings and their applications. RussianMath. Surveys 59(2004), 403–428.Google Scholar
[7] Brešar, M., Commuting traces of biadditive mappings, commutativity-preserving mappings and Lie mappings. Trans. Amer.Math. Soc. 335(1993), 525–546.Google Scholar
[8] Brešar, M., Functional identities: a survey. In: Algebra and its Applications, Contemp. Math., 259, Amer.Math. Soc., Providence, RI, 2000, pp. 93–109.Google Scholar
[9] Brešar, M. and Šemrl, P., Mappings which preserve idempotents, local automorphisms, and local derivations. Canad. J. Math. 45(1993), 483–496.Google Scholar
[10] Brešar, M. and Šemrl, P., On local automorphisms and mappings that preserve idempotents. Studia Math. 113(1995), 101–108.Google Scholar
[11] Chebotar, M. A., Ke, W.-F., and Lee, P. H., Maps characterized by action on zero products. Pacific J. Math. 216(2004), 217–228.Google Scholar
[12] Crist, R., Local automorphisms. Proc. Amer. Math. Soc. 128(2000), 1409–1414.Google Scholar
[13] Cui, J. and Hou, J., Linear maps on von Neumann algebras preserving zero products or TR-rank. Bull. Austral. Math. Soc. 65(2002), 79–91.Google Scholar
[14] Essannouni, H. and Kaidi, A., Le théorème de Hua pour les algèbres artiniennes simples. Linear Algebra Appl. 297(1999), 9–22.Google Scholar
[15] Font, J. J. and Hernandez, S., On separating maps between locally compact spaces. Arch. Math. (Basel) 63(1994), 158–165.Google Scholar
[16] Herstein, I. N., Noncommutative Rings. Carus Mathematical Monographs 15, Mathematical Association of Amererica, New York, 1968.Google Scholar
[17] Herstein, I. N., Rings with Involution. University of Chicago Press, Chicago, 1976.Google Scholar
[18] Hou, C. and Hou, S., Local automorphisms of nest algebras. Indian J. Pure Appl. Math. 32(2001), 1667–1678.Google Scholar
[19] Hua, L.-K., On the automorphisms of a field. Proc. Nat. Acad. Sci. U.S.A. 35(1949), 386–389.Google Scholar
[20] Jarosz, K., Automatic continuity of separating linear isomorphisms. Canad. Math. Bull. 33(1990), 139–144.Google Scholar
[21] Kim, S. O. and Kim, J. S., Local automorphisms and derivations on Mn. Proc. Amer. Math. Soc. 132(2004), 1389–1392.Google Scholar
[22] Larson, D., and Sourour, A., Local derivations and local automorphisms of B(X). In: Operator theory: operator algebras and applications, Proc. Sympos. Pure Math. 51, Part 2, Amer.Math. Soc., Providence, RI, 1990, pp. 187–194,Google Scholar
[23] Li, C.-K. and Pierce, S., Linear preserver problems. Amer. Math. Monthly 108(2001), 591–605.Google Scholar
[24] Li, C.-K. and Tsing, N.-K., Linear preserver problems: a brief introduction and some special techniques. In: Directions in Matrix Theory, Linear Algebra Appl. 162/164(1992), 217–235.Google Scholar
[25] Molnar, L., Local automorphisms of some quantum mechanical structures. Lett. Math. Phys. 58(2001), 91–100.Google Scholar
[26] Molnar, L., Local automorphisms of operator algebras on Banach spaces. Proc. Amer. Math. Soc. 131(2003), 1867–1874.Google Scholar
[27] Petek, T. and Šemrl, P., Adjacency preserving maps on matrices and operators. Proc. Roy. Soc. Edinburgh Sect. A. 132(2002), 661–684.Google Scholar
[28] Pierce, S., et al., A survey of linear preserver problems., Linear and Multilinear Algebra 33(1992), 1–129.Google Scholar
[29] Šemrl, P., Linear mappings preserving square-zero matrices. Bull. Austral. Math. Soc. 48(1993), 365–370.Google Scholar
[30] Šemrl, P., Local automorphisms and derivations on B(H). Proc. Amer.Math. Soc. 125(1997), 2677–2680.Google Scholar
[31] Scholz, E. and Timmermann, W., Local derivations, automorphisms and commutativity preserving maps on L + (D). Publ. Res. Inst.Math. Sci. 29(1993), 977–995.Google Scholar
[32] Wei, S.-Y. and Hou, S.-Z., Rank preserving maps on nest algebras. J. Operator Theory 39(1998), 207–217.Google Scholar
[33] Wong, W. J., Maps on simple algebras preserving zero products. I. The associative case. Pacific J. Math. 89(1980), 229–247.Google Scholar