Characterization of Certain Traces on von Neumann Algebras

Abstract

Consider a unital $C^*$-algebra $\mathcal {A}$. Let $n\ge 2$ and let $P_1, \cdots , P_n$ be projections in $\mathcal {A}$ such that $P_1 + \ldots +P_n=I$. We costruct $\mathcal {P}_n:\mathcal {A}\rightarrow \mathcal {A}$ being a block projection operator given by the formula $\mathcal {P}_n(X)=\sum _{k=1}^n P_kXP_k$ for all $X\in \mathcal {A}$. For a weight $\varphi$ on a von Neumann algebra $\mathcal {A}$, we prove that $\varphi$ is a trace if and only if $\varphi (\mathcal {P}_2(A))=\varphi (A)$ for all $A\in \mathcal {A}^+$. We also prove that if $\mathcal {A}$ is a von Neumann algebra then for a normal semifinite weight $\varphi$ on $\mathcal {A}$ the following conditions are equivalent: (i) $\varphi$ is a trace; (ii) $\varphi ((A^{m/2}B^mA^{m/2} )^k)\le \varphi ((A^{k/2}B^kA^{k/2})^m)$ for all $A, B\in \mathcal {A}^+$ and some numbers $k,m \in \mathbb {R}$ such that $k>m>0$; (iii) $\varphi (|\mathcal {P}_n(A)|)\le \varphi (|A|)$ for all $A\in \mathcal {A}$ and for all projections $P_1, \ldots , P_n\in \mathcal {A}$. As a consequence, we obtain a criterions for commutativity of von Neumann algebras and $C^*$-algebras.