Zero product determined matrix algebras

✍ Scribed by Matej Brešar; Mateja Grašič; Juana Sánchez Ortega

• Publisher: Elsevier Science
• Year: 2009
• Tongue: English
• Weight: 166 KB
• Volume: 430
• Category: Article
• ISSN: 0024-3795
• DOI: 10.1016/j.laa.2007.11.018

✦ Synopsis

Let A be an algebra over a commutative unital ring C. We say that A is zero product determined if for every C-module X and every bilinear map {•, •} : A × A → X the following holds: if {x, y} = 0 whenever xy = 0, then there exists a linear operator T such that {x, y} = T (xy) for all x, y ∈ A. If we replace in this definition the ordinary product by the Lie (resp. Jordan) product, then we say that A is zero Lie (resp. Jordan) product determined. We show that the matrix algebra M n (B), n 2, where B is any unital algebra, is always zero product determined, and under some technical restrictions it is also zero Jordan product determined. The bulk of the paper is devoted to the problem whether M n (B) is zero Lie product determined. We show that this does not hold true for all unital algebras B. However, if B is zero Lie product determined, then so is M n (B).