A Note on Dominant Contractions of Jordan Algebras

In the paper we consider two positive contractions $T,S:L^{1}(A,τ)\longrightarrow L^{1}(A,τ)$ 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 $n\geq n_{0}.$

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