On bilinear maps determined by rank one idempotents

• Published in: Linear algebra and its applications Vol. 432; no. 2; pp. 738 - 743
• Main Authors: Alaminos, J., Brešar, M., Extremera, J., Villena, A.R.
• Format: Journal Article
• Language: English
• Published: Amsterdam Elsevier Inc 15.01.2010; Elsevier
• Subjects: Algebra; Bilinear map; Exact sciences and technology; Field theory and polynomials; Linear and multilinear algebra, matrix theory; Linear map; Linear preserver problem; Mathematics; Matrix algebra; Rank one idempotent; Sciences and techniques of general use; Zero product; Bilinear map; Linear map; Matrix algebra; Zero product; 15A99; 15A27; 15A04; Linear preserver problem; 15A03; Rank one idempotent; Vector space; Preservation; Idempotent matrix
• ISSN: 0024-3795; 1873-1856
• DOI: 10.1016/j.laa.2009.09.017

Let M n , n ⩾ 2 , be the algebra of all n × n matrices over a field F of characteristic not 2 , and let Φ be a bilinear map from M n × M n into an arbitrary vector space X over F . Our main result states that if ϕ ( e , f ) = 0 whenever e and f are orthogonal rank one idempotents, then there exist linear maps Φ 1 , Φ 2 : M n → X such that ϕ ( a , b ) = Φ 1 ( ab ) + Φ 2 ( ba ) for all a , b ∈ M n . This is applicable to some linear preserver problems.