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  Volume 7, Issue 1, Article 36
Characterizations of Tracial Property via Inequalities

    Authors: Takashi Sano, Takeshi Yatsu,  
    Keywords: Trace; Inequality; Non matrix monotone function of order 2; Non matrix convex function of order $2$; Bessis-Moussa-Villani conjecture.  
    Date Received: 13/10/05  
    Date Accepted: 08/12/05  
    Subject Codes:

15A42, 47A63, 15A60, 47A30.

    Editors: Frank Hansen,  

In this article, we give characterizations of a tracial property for a positive linear functional via inequalities; we have necessary and sufficient conditions for a faithful positive linear functional $ varphi$ to be a positive scalar multiple of the trace by inequalities: for a non matrix monotone, increasing function $ f$,

$displaystyle X leqq Y Rightarrow varphi ( f(X) ) leqq varphi ( f(Y) ) $
is considered. Also for a non matrix convex, convex function $ f$,
$displaystyle varphi left(f left(frac{X + Y}{2}right) right) leqq varphi left( frac{f(X) + f(Y)}{2} right) $
is studied. We also show that suppose
$displaystyle 0 leqq varphi left( p_{m,k} left( X, Y right) right) $
for all $ X,Y geqq O,$ then $ varphi$ should be a positive scalar multiple of the trace. Here, $ p_{m,k}(X,Y)$ is the coefficient of $ t^k$ in the polynomial $ left( X + t Y right)^m$ and $ 1 leqq k leqq m - 1$.

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