An extension of the weighted geometric mean in unital $JB$-algebras

Abstract

Let $\mathcal{A}$ be a unital $JB$-algebra and $A,~B\in\mathcal{A}$, we extend the weighted geometric mean $A\sharp_r B$, from $r\in [0,1]$ to $r\in (-1, 0)\cup(1, 2)$. We will notice that many results will be reversed when the domain of $r$ change from $[0,1]$ to $(-1,0)$ or $(1,2)$. We investigate some property of $A\sharp_r B$ for such quantities of $r$, such as we show that $A\sharp_r B$ is separately operator convex with respect to $A, B$. We also introduce the Heinz and Heron means for unital $JB$-algebras and give some famous inequalities involving them.