Abstract
Let H be an infinite dimensional complex Hilbert space. Denote by B(H) the algebra of all bounded linear operators on H, and by I(H) the set of all idempotents in B(H). Suppose that Φ is a surjective map from B(H) onto itself. If for every λ ∈ -1,1,2,3, and A, B ∈ B(H),A-λB ∈I(H) ⇔ Φ(A) -λΦ(B) ∈I(H, then Φ is a Jordan ring automorphism, i.e. there exists a continuous invertible linear or conjugate linear operator T on H such that Φ(A) = TAT -1 for all A ∈ B(H), or Φ(A) = TA*T -1 for all A ∈ B(H); if, in addition, A-iB ∈I(H)⇔ Φ(A)-iΦ(B) ∈I(H), here i is the imaginary unit, then Φ is either an automorphism or an anti-automorphism.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Hua, L. K., Geometry of matrices I, generalizations of von staudt’s theorem, Trans. Amer. Math. Soc., 1945, 57: 441–481.
Hua, L. K., Geometry of matrices II, Study of involutions in the geometry of symmetric matrices, Trans. Amer. Math. Soc., 1947, 61: 193–228.
Hua, L. K., Geometry of matrices III, Fundamental theorems in the geometry of symmetric matrices, Trans. Amer. Math. Soc., 1947, 61: 229–255.
Hua, L. K., Geometry of symmetric matrices over any field with characteristic other than two, Ann. math., 1949, 50: 8–31.
Bai, Z., Hou, J., Numerical radius distance preserving maps on B(H), Proc Amer. Math. Soc., 2004, 132: 1453–1461.
Cui, J., Hou, J., Non-linear numerical radius isometries on atomic nest algebras and diagonal algebras, J. Funct. Anal., 2004, 206: 414–448.
Li, C. K., Šemrl, P., Numerical radius isometries, Lin. Multiplin. Alg., 2002, 50: 307–314.
Frobenius, G., Uber die darstellung der endlichen Gruppen durch lineare Substitionen, Sitzungsber. Deutsch. Akad. Wiss. Berlin, 1897, 994–1015.
Li, C. K., Tsing, N. K., Linear preserver problems: a brief introduction and some special techniques, Lin. Alg. Appl., 1992, 162-164: 217–235.
Hou, J., Cui, J., Introduction to Linear Maps on Operator Algebras, Beijing: Science Press, 2002.
Šemrl, P., Hua’s fundamental theorems of the geometry of matrices and related results, Lin. Alg. Appl., 2003, 361: 161–179.
Dolinar, G., Maps on matrix algebras preserving idempotents, Lin. Alg. Appl., 2003, 371: 287–333.
Ovchinnikov, P. G., Automorphisms of the poset of skew projections, J. Func Anal., 1993, 115: 184–189.
Pearcy, C., Topping, D., Sums of samll numbers of idempotents, Mich. Math. J., 1967, 14: 453–465.
Dolinar, G., Maps on β(H) preserving idempotents, Lin. Multilin. Alg., 2004, 52: 335–347.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Jianlian, C., Jinchuan, H. An algebraic invariant for Jordan automorphisms on β(H): The set of idempotents. Sci. China Ser. A-Math. 48, 1585–1596 (2005). https://doi.org/10.1360/03ys0298
Received:
Accepted:
Issue Date:
DOI: https://doi.org/10.1360/03ys0298