Characterizing centralizer maps and Jordan centralizer maps through zero products
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Characterizing centralizer maps and Jordan centralizer maps through zero products Characterizing centralizer maps and Jordan... M. A. Bahmani1 · F. Ghomanjani2 Received: 29 May 2019 / Accepted: 24 September 2019 © Springer-Verlag Italia S.r.l., part of Springer Nature 2019
Abstract Let A be an algebra and M be an A-bimodule. The main results show, under some conditions, that centralizer maps or Jordan centralizer maps φ : A −→ M through the action on zero products are centralizer maps. These results are applied to some algebras such as C ∗ -algebras and the algebras generated by idempotents. Keywords Centralizer · Jordan centralizer · Zero centralizer map Mathematics Subject Classification 47B47 · 15A86 · 47A07 · 47B49
1 Introduction Throughout this paper all algebras and vector spaces will be over the complex field C. Let A be an algebra and M be an A-bimodule. In an algebra A, we can define the Jordan product by a◦b = ab+ba for all a, b ∈ A. Also, for a ∈ A and m ∈ M, we use a◦m to denote the Jordan product am + ma of a and m. In this paper, R(A) denotes the subalgebra of A generated by all idempotents in A. Let φ : A −→ M be a linear map. Recall that φ is said to be a right (left) centralizer if φ(ab) = aφ(b)(φ(ab) = φ(a)b) for all a, b ∈ A. It is called a centralizer if φ is both a right centralizer and a left centralizer. We say that φ is a Jordan centralizer if φ(a ◦ b) = a ◦ φ(b) = b ◦ φ(a) for all a, b ∈ A. In case A be an unital algebra and M be an unital A-bimodule, φ is centralizer if and only if φ(a) = φ(1)a = aφ(1) for all a ∈ A. Clearly, each centralizer is a Jordan centralizer. The converse is, in general, not true (see [3, Example 2.6]). A linear map φ is said to zero centralizer map if φ(a)b = 0 = aφ(b) whenever ab = 0. Obviously, a centralizer map is a zero centralizer map. One of the main purpose of this paper is to answer the converse this problem on the unital zero product determined algebras. In [7], Qi and Hou showed that under some conditions, a zero centralizer map
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F. Ghomanjani [email protected]; [email protected]
1
Department of Pure Mathematics, Ferdowsi University of Mashhad, Mashhad, Iran
2
Department of Mathematics, Kashmar Higher Education Institute, Kashmar, Iran
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M. A. Bahmani, F. Ghomanjani
on a triangular algebra is a centralizer map. In this article, we also consider the following condition on a linear map φ from a algebra A into a A-bimodule M: a, b ∈ A, a ◦ b = 0 ⇒ a ◦ φ(b) = 0.
(d)
In [2], Ghahramani showed that under some conditions, if φ be a linear map on a triangular algebra such that satisfying (d), then φ is a centralizer map. In this paper, we show that on the unital zero Jordan product determined algebras the maps with the condition (d) are centralizer maps.
2 Main results Our aim is to study a centralizer map and a Jordan cetralizer map through zero products. We give conditions under which they are centralizer maps. Let us start with a definition that introduced by Ghahramani in [4]. Let A be an algebra and M be an A-bimodule.We
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