Geodesic
Commutativity Preserving Maps of Factors
Canadian journal of mathematics, Tome 40 (1988) no. 1, pp. 248-256

Voir la notice de l'article provenant de la source Cambridge University Press

By a von Neumann algebra M we mean a weakly closed, self-adjoint algebra of operators on a Hilbert space which contains I, the identity operator. A factor is a von Neumann algebra whose centre consists of scalar multiples of I.In all that follows φ:M → N will be a one to one, *-linear map from the von Neumann factor M onto the von Neumann algebra N such that both φ and φ −1 preserve commutativity. Our main result states that if M is not of type I 2 then where is an isomorphism or an antiisomorphism, c is a non-zero scalar, and λ is a *-linear map from M into ZN , the centre of N.Our interest in this problem was aroused by several recent results. In [1], Choi, Jafarian, and Radjavi proved that if S is the real linear space of n × n matrices over any algebraically closed field, n ≧ 3, and ψ a linear operator on S which preserves commuting pairs of matrices, then either ψ(S) is commutative or there exists a unitary matrix U such that for all A in S. They proved an analogous result for the collection of all bounded self-adjoint operators on an infinite dimensional Hilbert space when ψ is one to one. Subsequently, Omladic [7] proved that if ψ:L(X) → L(X) is a bijective linear operator preserving commuting pairs of operators where X is a non-trivial Banach space, then where U is a bounded invertible operator on X and A′ is the adjoint of A. 
Miers, C. Robert. Commutativity Preserving Maps of Factors. Canadian journal of mathematics, Tome 40 (1988) no. 1, pp. 248-256. doi: 10.4153/CJM-1988-011-1
@article{10_4153_CJM_1988_011_1,
     author = {Miers, C. Robert},
     title = {Commutativity {Preserving} {Maps} of {Factors}},
     journal = {Canadian journal of mathematics},
     pages = {248--256},
     year = {1988},
     volume = {40},
     number = {1},
     doi = {10.4153/CJM-1988-011-1},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/CJM-1988-011-1/}
}
TY  - JOUR
AU  - Miers, C. Robert
TI  - Commutativity Preserving Maps of Factors
JO  - Canadian journal of mathematics
PY  - 1988
SP  - 248
EP  - 256
VL  - 40
IS  - 1
UR  - http://geodesic.mathdoc.fr/articles/10.4153/CJM-1988-011-1/
DO  - 10.4153/CJM-1988-011-1
ID  - 10_4153_CJM_1988_011_1
ER  - 
%0 Journal Article
%A Miers, C. Robert
%T Commutativity Preserving Maps of Factors
%J Canadian journal of mathematics
%D 1988
%P 248-256
%V 40
%N 1
%U http://geodesic.mathdoc.fr/articles/10.4153/CJM-1988-011-1/
%R 10.4153/CJM-1988-011-1
%F 10_4153_CJM_1988_011_1

[1] 1. Choi, M. D., Jafarian, A. A. and Radjavi, H., Linear maps preserving commutativity, Linear Algebra and Its Applications. 87 (1987), 227–242. Google Scholar

[2] 2. Dixmier, J., Les algèbres d'operateurs dans l'espace Hilbertien, Cahiers Scientifiques. 25 (Gauthier-Villars, Paris, 1969). Google Scholar

[3] 3. Dye, H. A., On the geometry of projections in certain operator algebras, Ann. of Math.. 61 (1955), 73–89. Google Scholar

[4] 4. Fack, T. and Harpe, P. de la, Sommes de commutateurs dans les algèbres de von Neumann finies continues, Ann. Inst. Fourier. 30 (1980), 49–73. Google Scholar

[5] 5. Kadison, R. V., Normalcy in operator algebras, Duke Math. J.. 29 (1962), 459–464. Google Scholar

[6] 6. Miers, C. R., Lie isomorphisms of factors, Trans. Amer. Math. Soc.. 147 (1970), 55–63. Google Scholar

[7] 7. Omladic, M., On operators preserving commutativity, J. Func. Anal.. 66 (1986), 105–122. Google Scholar

[8] 8. Pearcy, C. and Topping, D., Sums of small numbers of idempotents, Mich. Math. J.. 14 (1967), 453–465. Google Scholar

Cité par Sources :