Characterization of certain traces on von Neumann algebras

Abstract

Consider a unital $C^*$-algebra $\mathcal{A}$. Let $n\geq 2$ and let $P_1, \ldots , P_n$ be projections in $\mathcal{A}$ such that $P_1 + \ldots +P_n=I$. We costruct $\mathcal{P}_n\colon \mathcal{A}\to \mathcal{A}$ 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: {\rm (i)} $\varphi$ is a trace; {\rm (ii)} $\varphi((A^{m/2}B^mA^{m/2} )^k)\leq\varphi ((A^{k/2}B^kA^{k/2})^m)$ for all $A, B\in\mathcal{A}^+$ and some numbers $k,m \in\mathbb{R}$ such that $k)m)0$; {\rm (iii)} $\varphi (|\mathcal{P}_n(A)|)\leq\varphi (|A|)$ 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.