For my Master's, I wrote a paper which arrived at the structure theorem of finite-dimensional C*-algebras under purely algebraic hypotheses on a finite-dimensional *-algebra via geometry of projections. We studied two algebraic properties endowed to a *-algebra which allowed for the theorem to be obtained. From here we applied this structure theorem and the theory of projections to study the relative structure theory of an inclusion of two such *-algebras.  You can read the paper here.