Bulletin of the Australian Mathematical Society

Research Article

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

Cui Jianliana1a2 and Hou Jinchuana3a4

a1 Institute of Mathematics, Chinese Academy of Sciences, Beijing 100080, Peoples Republic of China

a2 Current address: Department of Applied Mathematics, Taiyuan University of Technology, Taiyuan 030024, Peoples Republic of China and Department of Mathematics, Shanxi Teachers University, Linfen 041004 Peoples Republic of China e-mail: cuijl@dns.sxtu.edu.cn

a3 Department of Mathematics, Shanxi Teachers University, Linfen 041004, Peoples Republic of China

a4 Current address: Department of Mathematics, Shanxi University, Taiyuan 030000 Peoples Republic of China e-mail: jhou@dns.sxtu.edu.cn

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 xs2133 of a finite von Neumann algebra xs211B into xs211B can be extended to an algebraic homomorphism from the subalgebra generated by xs2133 into xs211B; 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.

(Received May 14 2001)