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

Published online by Cambridge University Press:  17 April 2009

Cui Jianlian  and
Hou Jinchuan
Cui Jianlian
Affiliation:
Institute of Mathematics, Chinese Academy of Sciences, Beijing 100080, Peoples Republic of China Current address: Department of Applied Mathematics, Taiyuan University of Technology, Taiyuan 030024, Peoples Republic of China and Department of Mathematics, Shanxi Teachers University, Linfen 041004Peoples Republic of China e-mail: cuijl@dns.sxtu.edu.cn
Hou Jinchuan
Affiliation:
Department of Mathematics, Shanxi Teachers University, Linfen 041004, Peoples Republic of China Current address: Department of Mathematics, Shanxi University, Taiyuan 030000Peoples Republic of China e-mail: jhou@dns.sxtu.edu.cn
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Abstract

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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.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 65 , Issue 1 , February 2002 , pp. 79 - 91
Copyright
Copyright © Australian Mathematical Society 2002

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