Linear maps on operator algebras that preserve elements annihilated by a polynomial

Hou, Jinchuan; Hou, Shengzhao

doi:10.1090/S0002-9939-02-06362-1

Linear maps on operator algebras that preserve elements annihilated by a polynomial
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by Jinchuan Hou and Shengzhao Hou PDF

Proc. Amer. Math. Soc. 130 (2002), 2383-2395 Request permission

Abstract:

In this paper some purely algebraic results are given concerning linear maps on algebras which preserve elements annihilated by a polynomial of degree greater than 1 and with no repeated roots and applied to linear maps on operator algebras such as standard operator algebras, von Neumann algebras and Banach algebras. Several results are obtained that characterize such linear maps in terms of homomorphisms, anti-homomorphisms, or, at least, Jordan homomorphisms.

References

Matej Brešar and Peter emrl, Linear maps preserving the spectral radius, J. Funct. Anal. 142 (1996), no. 2, 360–368. MR 1423038, DOI 10.1006/jfan.1996.0153
Matej Brešar and Peter emrl, On local automorphisms and mappings that preserve idempotents, Studia Math. 113 (1995), no. 2, 101–108. MR 1318418, DOI 10.4064/sm-113-2-101-108
Matej Brešar and Peter emrl, Mappings which preserve idempotents, local automorphisms, and local derivations, Canad. J. Math. 45 (1993), no. 3, 483–496. MR 1222512, DOI 10.4153/CJM-1993-025-4
Gin Hor Chan and Ming Huat Lim, Linear preservers on powers of matrices, Linear Algebra Appl. 162/164 (1992), 615–626. Directions in matrix theory (Auburn, AL, 1990). MR 1148420, DOI 10.1016/0024-3795(92)90396-R
G. Frobenius, Uber die Darstellung der endlichen Gruppen durch lineare Substitutionen, Sitzungsber. Deutsch. Akad. Wiss. Berlin (1897), 994-1015.
The Hadwin Lunch Bunch, Local multiplications on algebras spanned by idempotents, Linear and Multilinear Algebra 37 (1994), no. 4, 259–263. MR 1310968, DOI 10.1080/03081089408818328
Jin Chuan Hou, Rank-preserving linear maps on $B(X)$, Sci. China Ser. A 32 (1989), no. 8, 929–940. MR 1055310
Ralph Howard, Linear maps that preserve matrices annihilated by a polynomial, Linear Algebra Appl. 30 (1980), 167–176. MR 568789, DOI 10.1016/0024-3795(80)90192-5
Ali A. Jafarian and A. R. Sourour, Spectrum-preserving linear maps, J. Funct. Anal. 66 (1986), no. 2, 255–261. MR 832991, DOI 10.1016/0022-1236(86)90073-X
S. Kantor, Theorie der Aquivalenz von linearen $\infty$ Scharen bilinearer Formen, Sitzungsber. Munchener Akad., (1897), 367-381.
Chi-Kwong Li and Nam-Kiu Tsing, Linear preserver problems: a brief introduction and some special techniques, Linear Algebra Appl. 162/164 (1992), 217–235. Directions in matrix theory (Auburn, AL, 1990). MR 1148401, DOI 10.1016/0024-3795(92)90377-M
Marvin Marcus, All linear operators leaving the unitary group invariant, Duke Math. J. 26 (1959), 155–163. MR 101241
Marvin Marcus, Linear operations on matrices, Amer. Math. Monthly 69 (1962), 837–847. MR 147491, DOI 10.2307/2311230
Matjaž Omladič, On operators preserving commutativity, J. Funct. Anal. 66 (1986), no. 1, 105–122. MR 829380, DOI 10.1016/0022-1236(86)90084-4
Matjaž Omladič and Peter emrl, Linear mappings that preserve potent operators, Proc. Amer. Math. Soc. 123 (1995), no. 4, 1069–1074. MR 1254849, DOI 10.1090/S0002-9939-1995-1254849-4
Peter emrl, Linear mappings that preserve operators annihilated by a polynomial, J. Operator Theory 36 (1996), no. 1, 45–58. MR 1417185