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 $A{♯}_{r}B$, 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 $A{♯}_{r}B$ for such quantities of $r$, such as we show that $A{♯}_{r}B$ 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.