Transformations preserving the norm of means between positive cones of general and commutative $C^{\ast }$-algebras

In this paper, we consider a (nonlinear) transformation $\mathrm{\Phi }$ of invertible positive elements in $C^{\ast }$-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$\mm$standsforthearithmeticmean$A\nabla B=(A+B)/2$, the geometric mean $A\mathrm{\#}B=A^{1\mathrm{/}2}(A^{-1\mathrm{/}2}B{A}^{-1\mathrm{/}2}{)}^{1\mathrm{/}2}{A}^{1\mathrm{/}2}$, or the harmonic mean $A!B=2(A^{-1}+B^{-1}{)}^{-1}$. Assuming that $\mathrm{\Phi }$ is surjective and preserves either the norm of the arithmetic mean or the norm of the geometric mean, we show that $\mathrm{\Phi }$ extends to a Jordan $\ast$-isomorphism between the underlying full algebras. If $\mathrm{\Phi }$ 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 $A{W}^{\ast }$-algebras or commutative $C^{\ast }$-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 $\ast$-isomorphism between the underlying full function algebras.