  Since their inception in the 30's by von Neumann, operator algebras have been
used in shedding light in many mathematical theories. Classification results
for self-adjoint and non-self-adjoint operator algebras manifest this approach,
but a clear connection between the two was sought since their emergence in the
late 60's.
  We connect these seemingly separate type of results by uncovering a hierarchy
of classification for non-self-adjoint operator algebras and $C^*$-algebras
with additional $C^*$-algebraic structure. Our approach naturally applies to
algebras arising from $C^*$-correspondences to resolve self-adjoint and
non-self-adjoint isomorphism problems in the literature. We apply our strategy
to completely elucidate this newly found hierarchy for operator algebras
arising from directed graphs.
