This is not a complete answer, but just an example of relation we can't impose to a $C^*$-algebra.

Let $\mathcal{A}$ be a $C^*$-algebra and let $a, b \in \mathcal{A}$, then the relation $ab-ba = 1$ is impossible :

Lemma : If $ab-ba = 1$ then $ab^n-b^na = nb^{n-1}$.
Proof by induction : for $n=1$ it's ok.
Now if it's true for $n$, then $ab^{n+1}-b^{n+1}a = ab^nb -b^{n+1}a = (b^na+ nb^{n-1})b-b^{n+1}a $ $ = b^nab+nb^n-b^{n+1}a = b^n(1+ba)+nb^n-b^{n+1}a = (n+1)b^n $. $\square$

Corollary : The relation $ab-ba = 1$ is impossible in a Banach algebra.
Proof : If it's possible, then $ab^n-b^na = nb^{n-1}$. Next $\Vert nb^{n-1} \Vert = n \Vert b^{n-1} \Vert = \Vert ab^n-b^na \Vert $ $ \le \Vert ab^n \Vert + \Vert b^na \Vert \le \Vert ab \Vert \Vert b^{n-1} \Vert + \Vert b^{n-1} \Vert \Vert ba \Vert$.
Conclusion $\Vert ab \Vert + \Vert ba \Vert \ge n$ $\forall n$, contradiction. $\square$

Remark : This relation is realized by unbounded operators.
For example let the Hilbert space $H = l^2(\mathbb{Z})$, $a: e_n \to ne_{n-1}$ and $b: e_n \to e_{n+1}$

This is not a complete answer, just a well-known example of relation we can't impose to a $C^*$-algebra.

Let $\mathcal{A}$ be a $C^*$-algebra, then, the relation $ab-ba = 1$ is impossible for $a, b \in \mathcal{A}$ :

Lemma : If $ab-ba = 1$ then $ab^n-b^na = nb^{n-1}$.
Proof by induction : for $n=1$ it's ok.
Now if it's true for $n$, then $ab^{n+1}-b^{n+1}a = ab^nb -b^{n+1}a = (b^na+ nb^{n-1})b-b^{n+1}a $ $ = b^nab+nb^n-b^{n+1}a = b^n(1+ba)+nb^n-b^{n+1}a = (n+1)b^n $. $\square$

Corollary : The relation $ab-ba = 1$ is impossible in a Banach algebra.
Proof : If it's possible, then $ab^n-b^na = nb^{n-1}$. Next $\Vert nb^{n-1} \Vert = n \Vert b^{n-1} \Vert = \Vert ab^n-b^na \Vert $ $ \le \Vert ab^n \Vert + \Vert b^na \Vert \le \Vert ab \Vert \Vert b^{n-1} \Vert + \Vert b^{n-1} \Vert \Vert ba \Vert$.
Conclusion $\Vert ab \Vert + \Vert ba \Vert \ge n$ $\forall n$, contradiction. $\square$

Remark : This (Heisenberg) relation is realized by unbounded operators.
For example let the Hilbert space $H = l^2(\mathbb{Z})$, $a: e_n \to ne_{n-1}$ and $b: e_n \to e_{n+1}$