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Linear maps on von Neumann algebras preserving zero products on tr-rank

  • Cui Jianlian (a1) (a2) and Hou Jinchuan (a3) (a4)
Abstract

In this paper, we give some characterisations of homomorphisms on von Neumann algebras by linear preservers. We prove that a bounded linear surjective map from a von Neumann algebra onto another is zero-product preserving if and only if it is a homomorphism multiplied by an invertible element in the centre of the image algebra. By introducing the notion of tr-rank of the elements in finite von Neumann algebras, we show that a unital linear map from a linear subspace ℳ of a finite von Neumann algebra ℛ into ℛ can be extended to an algebraic homomorphism from the subalgebra generated by ℳ into ℛ; and a unital self-adjoint linear map from a finite von Neumann algebra onto itself is completely tr-rank preserving if and only if it is a spatial *-automorphism.

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[2] L.G. Brown and G.K. Pedersen , ‘C*-algebras of real rank zero’, J. Funct. Anal. 99 (1991), 131149.

[3] W. Chooi and M. Lim , ‘Linear preserves on triangular matrices’, Linear Algebra Appl. 269 (1998), 241255.

[4] J. Cui , J. Hou and B. Li , ‘Linear preservers on upper triangular operator matrix algebras’, Linear Algebra Appl. 336 (2001), 2950.

[8] J. Hou , ‘Multiplicative maps on ℬ(X)’, Sci. China Ser. A 41 (1998), 337345.

[12] M. Lim , ‘Rank and tensor rank preservers’, Linear and Multilinear Algebra 33 (1992), 721.

[13] L. Molnar , ‘Some linear preserver problems on ℬ(H) concerning rank and corank’, Linear Algebra Appl. 286 (1999), 311321.

[14] L. Molnar and P. Semrl , ‘Some linear preserver problems on upper triangular matrices’, Linear and Multilinear Algebra 45 (1998), 189206.

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Bulletin of the Australian Mathematical Society
  • ISSN: 0004-9727
  • EISSN: 1755-1633
  • URL: /core/journals/bulletin-of-the-australian-mathematical-society
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