  Whether the global well-posedness of strong solutions of $n$-dimensional
compressible isentropic magnetohydrodynamic (MHD for short) equations without
magnetic diffusion holds true or not remains an challenging open problem, even
for the small initial data. In recent years, stared from the pioneer work by Wu
and Wu [Adv. Math. 310 (2017), 759--888], much more attention has been paid to
the system when the magnetic field near an equilibrium state (the background
magnetic field for short). In particular, when the background magnetic field
satisfies the Diophantine condition (see (1.3) for details), Wu and Zhai [Math.
Models Methods Appl. Sci. 33 (2023), no. 13, 2629--2656] established the decay
estimates and asymptotic stability for smooth solutions of the 3D compressible
isentropic MHD system without magnetic diffusion in $H^{4r+7}(\mathbb{T}^3)$
with $r>2$ by exploiting a wave structure. In this paper, a new dissipative
mechanism is found out and applied so that we can improve the spaces where the
decay estimates and asymptotic stability of solutions are taking place by Wu
and Zhai. More precisely, we establish the decay estimates of solutions in
$H^{r+1}(\mathbb{T}^n)$ and asymptotic stability result in
$H^{\left(3r+3\right)^+}(\mathbb{T}^n)$ for any dimensional periodic domain
$\mathbb{T}^n$ with $n\geq 2$ and $r>n-1$. Our results provide an approach for
establishing the decay estimates and asymptotic stability in the Sobolev spaces
with much lower regularity and uniform dimension, which can be used to study
many other related models such as the compressible non-isentropic MHD system
without magnetic diffusion and so on.
