Transformations preserving the norm of means between positive cones of general and commutative $C^*$-algebras

In this paper, we consider a (nonlinear) transformation $\Phi$ of invertible positive elements in $C^*$-algebras which preserves the norm of any of the three fundamental means of positive elements; namely, $\|\Phi(A)\mm \Phi(B)\| = \|A\mm B\|$, where $\mm$ stands for the arithmetic mean $A\nabla B=(A+B)/2$, the geometric mean $A\#B=A^{1/2}(A^{-1/2}BA^{-1/2})^{1/2}A^{1/2}$, or the harmonic mean $A!B=2(A^{-1} + B^{-1})^{-1}$. Assuming that $\Phi$ is surjective and preserves either the norm of the arithmetic mean or the norm of the geometric mean, we show that $\Phi$ extends to a Jordan $*$-isomorphism between the underlying full algebras. If $\Phi$ is surjective and preserves the norm of the harmonic mean, then we obtain the same conclusion in the special cases where the underlying algebras are $AW^*$-algebras or commutative $C^*$-algebras. In the commutative case, for a transformation $T: F(\mathrm{X})\subset C_0(\mathrm{X})_+\rightarrow C_0(\mathrm{Y})_+$, we can relax the surjectivity assumption and show that $T$ is a generalized composition operator if $T$ preserves the norm of the (arithmetic, geometric, harmonic, or in general any power) mean of any finite collection of positive functions, provided that the domain $F(\mathrm{X})$ contains sufficiently many elements to peak on compact $G_\delta$ sets. When the image $T(F(\mathrm{X}))$ also contains sufficiently many elements to peak on compact $G_\delta$ sets, $T$ extends to an algebra $*$-isomorphism between the underlying full function algebras.