We say that the $C^*$-algebra $A$ generated by $a_1,...,a_n$ is universal subject to relations $R_1,...,R_m$ if for every $C^*$-algebra $B$ with elements $b_1,...,b_n$ satisfying relations $R_1,...,R_m$ there is $C^*$-epimorphism $\varphi: A \to B$ such that $\varphi(a_i)=b_i$. One of the basic examples is the $C^*$-algebra of complex valued function on three sphere $C(S^3)$ which is the universal commutative unital $C^*$-algebra generated by $a,b$ with relation $a^*a+b^*b=1$. My question is the following: what kind of relations can we impose on our $C^*$-algebra? In all examples which I saw the relations were algebraic and were of the form: $f(a_1,...,a_n,a_1^*,...,a_n^*)=0$ where $f$ was some polynomial. In particular do we admit:
quantification and referring to other elements not being the generators
order properties of $C^*$-algebras
functions which are no longer polynomials (continuous functions, Borel functions etc.)
If the answer is positive I would be grateful to know some (known in literature) examples of universal $C^*$-algebras arising in such a way.