A. M. Bikchentaev, “Metrics on projections of the von neumann algebra associated with tracial functionals”, Sibirsk. Mat. Zh., 2019, Volume 60, Number 6,Pages <nobr>1223

This article is cited in 6 papers

Metrics on projections of the von neumann algebra associated with tracial functionals

A. M. Bikchentaev

Institute of Mathematics and Mechanics, Kazan (Volga Region) Federal University

Abstract: Let П† be a positive functional on a von Neumann algebra $\mathscr{A}$ and let $\mathscr{A}^{pr}$ be the projection lattice in $\mathscr{A}$. Given $P,Q \in \mathscr{A}^{pr}$, put ПЃ$_{П†}$(P, Q) = П†(в€ЈP в€’ Qв€Ј) and d$_{П†}$(P, Q) = П†(P в€Ё Q в€’ P в€§ Q). Then ПЃ$_{П†}$(P, Q) в‰¤ d$_{П†}$(P, Q) and ПЃ$_{П†}$(P, Q) = d$_{П†}$(P, Q) provided that PQ = QP. The mapping ПЃ$_{П†}$ (or d$_{П†}$) meets the triangle inequality if and only if П† is a tracial functional. If П„ is a faithful tracial functional then ПЃ$_{П„}$ and d$_{П„}$ are metrics on $\mathscr{A}^{pr}$. Moreover, if П„ is normal then ($\mathscr{A}^{pr}$, ПЃ$_{П„}$) and ($\mathscr{A}^{pr}$, d$_{П„}$) are complete metric spaces. Convergences with respect to ПЃ$_{П„}$ and d$_{П„}$ are equivalent if and only if $\mathscr{A}$ is abelian; in this case ПЃ$_{П„}$ = d$_{П„}$. We give one more criterion for commutativity of $\mathscr{A}$ in terms of inequalities.

UDC: 517.98

Received: 06.04.2018
Revised: 19.12.2018
Accepted: 24.07.2019

DOI: 10.33048/smzh.2019.60.603