Characterizations of Centralizable Mappings on Algebras of Locally Measurable Operators
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Acta Mathematica Sinica, English Series Springer-Verlag GmbH Germany & The Editorial Office of AMS 2020
Characterizations of Centralizable Mappings on Algebras of Locally Measurable Operators Jun HE Department of Mathematics, Anhui Polytechnic University, Wuhu 241000, P. R. China E-mail : [email protected]
Guang Yu AN Department of Mathematics, Shaanxi University of Science and Technology, Xi’an 710021, P. R. China E-mail : [email protected]
Jian Kui LI Department of Mathematics, East China University of Science and Technology, Shanghai 200237, P. R. China E-mail : [email protected]
Wen Hua QIAN1) School of Mathematical Sciences, Chongqing Normal University, Chongqing 401331, P. R. China E-mail : [email protected] Abstract A linear mapping φ from an algebra A into its bimodule M is called a centralizable mapping at G ∈ A if φ(AB) = φ(A)B = Aφ(B) for each A and B in A with AB = G. In this paper, we prove that if M is a von Neumann algebra without direct summands of type I1 and type II, A is a ∗-subalgebra with M ⊆ A ⊆ LS(M) and G is a fixed element in A, then every continuous (with respect to the local measure topology t(M)) centralizable mapping at G from A into M is a centralizer. Keywords
Centralizable mapping, centralizer, von Neumann algebra, locally measurable operator
MR(2010) Subject Classification
1
47B47, 46L57, 47L60
Introduction
Let A be an associative algebra over the complex field C, M be an A-bimodule, and L(A, M) be the set of all linear mappings from A into M. If A = M, then denote L(A, A) by L(A). A linear mapping φ in L(A, M) is called a centralizer if φ(AB) = φ(A)B = Aφ(B) for each A Received August 27, 2019, revised December 2, 2019, accepted March 26, 2020 This paper was partially supported by National Natural Science Foundation of China (Grant Nos. 11801005, 11801342, 11801004, 11871021, 11801050), The first author was also partially supported by a Startup Fundation of Anhui Polytechnic University (Grant No. 2017YQQ017), The second author was also partially supported by Shaanxi Provincial Education Department (Grant No. 19JK0130), The fourth author was also partially supported by Research Foundation of Chongqing Educational Committee (Grant No. KJQN2018000538) 1) Corresponding author
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and B in A. In particular, if A is a unital algebra with a unit element I, then φ is a centralizer if and only if φ(A) = φ(I)A = Aφ(I) for every A in A. Let G be a fixed element in A. A linear mapping φ in L(A, M) is called a centralizable mapping at G if φ(AB) = φ(A)B = Aφ(B) for each A and B in A with AB = G. Moreover, we say that G is a full-centralizable point of L(A, M) if every centralizable mapping at G from A into M is a centralizer. Suppose that R is a prime ring with a nontrival idempotent, in [5], Breˇsar shows that zero is a full-centralizable point of L(R); and in [12], Qi shows that every nontrival idempotent in R is a full-centralizable point of L(R). In [16], Xu, An and Hou prove that if H is a Hilbert space with dim H ≥ 2, then every element G in B(H) is a full-centra
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