Are all $*$-algebras with matrix representations projective?

Question

If a complex unital $*$-algebra $\mathscr{A}$ has a (bounded) matrix representation (i.e. a unital $*$-homomorphism) $$\psi:\mathscr{A}\rightarrow M_d(\mathbb{C}),$$ such that $\|\psi(a)\|_{\infty}<+\infty$ for all $a\in \mathscr{A}$, does $\psi$ necessarily factor through the $C^*$-enveloping algebra of $\mathscr{A}$, and does this mean every such $\mathscr{A}$ with this property is a projective object in the category of unital $*$-algebras?

Answer 1

This is trivial from the universal property of the $C^*$-enveloping algebra: $M_d(\mathbb{C})$ is a $C^*$-algebra, so any homomorphism to it factors through the $C^*$-enveloping algebra.

This doesn't have anything to do with being projective, though. For a simple example, let $\mathscr{A}=M_2(\mathbb{C})\times M_3(\mathbb{C})$. This has matrix representations, but it is not projective: there is an epimorphism $\mathscr{A}\to M_2(\mathbb{C})$ (the first projection) which does not split since there are no homomorphisms $M_2(\mathbb{C})\to M_3(\mathbb{C})$.