Extension and approximation of $m$-subharmonic functions

Let $\mathrm{\Omega }\subset {\mathbb{C}}^n$ be a bounded domain, and let $f$ be a real-valued function defined on the whole topological boundary $\partial \Omega$\mathrm{.}Theaimofthispaperistofindacharacterizationofthefunctions$f$whichcanbeextendedtotheinsidetoa$m$-subharmonic function under suitable assumptions on $\mathrm{\Omega }$. 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.