  Let $\Omega\subset \mathbb C^n$ be a bounded domain, and let $f$ be a
real-valued function defined on the whole topological boundary $\partial
\Omega$. The aim of this paper is to find a characterization of the functions
$f$ which can be extended to the inside to a $m$-subharmonic function under
suitable assumptions on $\Omega$. We shall do so by using a function algebraic
approach with focus on $m$-subharmonic functions defined on compact sets. We
end this note with some remarks on approximation of $m$-subharmonic functions.
