Bilinear forms on matrix algebras vanishing on zero products of xy and yx

详细信息    查看全文

• 作者：M. Tamer Ko艧an^a ; ^{mtkosan@gyte.edu.tr" class="auth_mail} ; Tsiu-Kwen Lee^b ; ¹ ; ^{tklee@math.ntu.edu.tw" class="auth_mail} ; Yiqiang Zhou^c ; ^{zhou@mun.ca" class="auth_mail}
• 关键词：15A03 ; 15A04 ; 15A99 ; 16N60
• 刊名：Linear Algebra and its Applications
• 出版年：15 July 2014
• 年：2014
• 卷：453
• 期：Complete
• 页码：110-124
• 全文大小：298 K

文摘

Let D   be a division algebra finite-dimensional over its center C  , 惟:=M_m(D)$惟:={\mathrm{M}}_m(D)$, the m×m$m\times m$ matrix algebra over D  , and V   be a vector space over C  . We characterize all n  -linear forms on 惟   in terms of reduced traces and elementary operators. For m>1$m>1$, it is proved that a bilinear form B:惟×惟→V$B:惟\times 惟\to V$ vanishes on zero products of xy   and yx   if and only if there exist linear maps g,h:惟→V$g,h:惟\to V$ such that B(x,y)=g(xy)+h(yx)$B(x,y)=g(xy)+h(yx)$ for all x,y∈惟$x,y\in 惟$. As an application, a bilinear form B   is completely characterized if B(x,y)=0$B(x,y)=0$ whenever x,y∈惟$x,y\in 惟$ satisfy xy+尉yx=0$xy+尉yx=0$, where 尉 is a fixed nonzero element in C.