In contrast to rings or algebras, where one can consider quotients by free rings to construct universal objects, C*-algebras must be realizable as algebras of bounded operators on a Hilbert space by the Gelfand-Naimark-Segal construction and the relations must prescribe a uniform bound on the norm of each generator.
This means that depending on the generators and relations, a universal C*-algebra may not exist.
One is, as previously mentioned, due to the non-existence of free C*-algebras, not every set of relations defines a C*-algebra.
The basic motivation behind the following definitions is that we will define relations as the category of their representations.
[1] Alternatively, one can use a more concrete characterization of universal C*-algebras that more closely resembles the construction in abstract algebra.
A representation of (G, R) on a Hilbert space H is a function ρ from X to the algebra of bounded operators on H such that
Then is finite and defines a seminorm satisfying the C*-norm condition on the free algebra on X.