  Semidefinite Optimization has become a standard technique in the landscape of
Mathematical Programming that has many applications in finite dimensional
Quantum Information Theory. This paper presents a way for finite-dimensional
relaxations of general cone programs on $\mathcal{C}^\star$-algebras which have
structurally similar properties to ordinary cone programs, only putting the
notion of positivity at the core of optimization. We show that well-known
hierarchies for generalized problems like NPA but also Lasserre's hierarchy and
to some extend symmetry reductions of generic SDPs by de-Klerk et al. can be
considered from a general point of view of $\mathcal{C}^\star$-algebras in
combination to optimization problems.
