Recently, several papers have been published concerning preservers of norms of means on function algebras and operator algebras. In our approach to the study of those transformations and in the proofs of our results the key point was an order determining property of the norms of the considered means. Motivated by that, in a current paper we have initiated the investigation of the mentioned property in a general setting. The aim of this talk is to report on the obtained results concerning the question of when the operator norm of a Kubo-Ando mean $\sigma$ determines the order on the positive definite cone of an operator algebra. More precisely, the question asks when, for any given pair $A,B$ of positive definite elements of the underlying algebra, we have that $A\leq B$ holds if and only if $\| A\sigma X\|\leq \| B\sigma X\|$ is valid for all positive definite elements $X$. Our study is not completed, besides presenting results we formulate open problems.