Far Eastern Mathematical Journal

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Maps on the Algebras of Measurable Operators Preserving Zero and Jordan Zero-products

I. M. Juraev

2012, issue 2, P.

In this paper, we prove that a continuous linear surjection from an algebra of measurable operators onto another one preserves zero products (resp. zero Jordan products) if and only if it is a non-zero scalar multiple of a homomorphism (resp. of an Jordan homomorphism).

von Neumann algebras, measurable operator, topology of convergence in measure, trace, homomorphism, Jordan homomorphism

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[1] S. K. Berberian, “The regular ring of finite $AW^?$-algebra”, Ann. Math, 65 (1957), 224–242.
[2] M. A. Chebotar, Wen-Fong Ke, Pjek-Hwee Lee, “Maps characterized by action on zero products”, Pacific J.Math., 216:2 (2004), 217–228.
[3] M. A. Chebotar, Wen-Fong Ke, Pjek-Hwee Lee, Ruibin Zhand, “On maps preserving zero Jordan products”, Monatsh. Math., 149 (2006), 91–101.
[4] J. Cui, J. Hou, “Linear maps on von Neumann algebras preserving zero products or TR-rank”, Bull. Austral. Math. Soc., 65 (2002), 79–91.
[5] J. Cui, J. Hou, “Characterizations of nest algebra automorphisms”, Chinese Ann. Math. (to appear).
[6] J. Hou, M. Gao, “Additive mappings on $B(H)$ that preserve zero products”, Kexue Tongbao (Chinese), 43 (1998), 2388–2392.
[7] M. A. Muratov, V. I. Chilin, Algebry izmerimykh i lokal'no izmerimykh operatorov, t. 69, Pratsi In-tu matematiki NAN Ukraini, Kiiv, 2007, 390 s.
[8] P. Semrl, “Linear mappings preserving square-zero matrices”, Bull. Austral. Math. Soc., 48 (1993), 365–370.
[9] L. Zhao, J. Hou, “Jordan zero product preserving additive maps on operator algebras”, J. Math. Anal. Appl., 314 (2006), 689–700.

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