I'm trying to prove that the universal $C^\ast$ algebra $C^{\ast}(\{ a \} \rvert \{a=a^\ast\})$ does not exists:
I was thinking of using the maps: $$f_n:a \mapsto n+\frac{1}{1+x^2} \in B(\mathbb{R}). $$ $$\text{Since } \ ||f_n(a)||=n+1, \ \text{ we would have that:}$$ $$p(a)\geqslant n+1, \ \forall \ n \in \mathbb{N},$$ where $p$ is the seminorm given by the supremum of the norm of bounded operators on Hilbert spaces. Thus we can't take the universal $C^\ast$ algebra, as the seminorm of $a$ is not bounded.