A Note on Dominant Contractions of Jordan Algebras

TL;DR

This paper investigates the behavior of dominant contractions on Jordan algebras, establishing conditions under which the difference of their powers remains below one, extending understanding of operator inequalities in this algebraic context.

Contribution

It proves that if two positive contractions satisfy a certain norm condition at some power, then this inequality holds for all higher powers in the setting of semi-finite JBW-algebras.

Findings

If ^{n_0}-T^{n_0} <1, then ^{n}-T^{n} <1 for all n .

The result extends known inequalities to the setting of Jordan algebras.

Provides a criterion for the stability of inequalities under iteration of contractions.

Abstract

In the paper we consider two positive contractions $T,S:L^{1}(A,\tau)\longrightarrow L^{1}(A,\tau)$ such that $T\leq S$, here $(A,\t)$ is a semi-finite $JBW$-algebra. If there is an $n_{0}\in\mathbb{N}$ such that $\|S^{n_{0}}-T^{n_{0}}\|<1$. Then we prove that $\|S^{n}-T^{n}\|<1$ holds for every $n\geq n_{0}.$