It is known that the $C^*$-algebra $\mathcal U$ generated by bilateral shift $\ell^2 (\mathbb Z) \ni e_k \mapsto e_{k+1} \in \ell ^2(\mathbb Z)$, is a universal $C^*$ algebra generated by unitary: for every $a$ in a $C^*$-algebra $A$ which satisfies $aa^*=a^*a=\mathbb 1$ there is a $C^*$-algebra morphism $\mathcal U\rightarrow C^*(a)$.
It isn't true, though, in general - if we consider some generators $\{x_1,\dots,x_n\}$ which satisfies relations expressed with noncommutative polynomial in $2n$ variables $p$: $p(x_1,\dots,x_n,x_1^*,\dots,x_n^*)=0$, there may not be any $C^*$-algebra with above universal property.
Is there any nice condition on $p$ which would say when such $C^*$-algebra exists? What if we consider more polynomials, not just one?