Universal \(C^*\)-algebras of some properties

Let (P) be a property of \(C^*\) $C^{\ast }$ -algebras which may be satiesfied or not, and \(\mathscr {S}(P)\) $\mathcal{S}(P)$ be the set of separable \(C^*\) $C^{\ast }$ -algebras which satiesfies (P). We give a sufficient condition for (P) to admit a universal separable element \(A\in \mathscr {S}(P)\) $A\in \mathcal{S}(P)$ in the sense that for any \(B\in \mathscr {S}(P)\) $B\in \mathcal{S}(P)$ , there exists a surjective \(*\) $\ast$ -homomorphism \(\pi :A\rightarrow B\) $\pi :A\to B$ , and use the sufficient condition to show that when (P) is “unital with stable rank n”, “the small projection property” or “unital with stable exponential lenght b”, the sufficient condition is satisfied and hence there exists a corresponding universal \(C^*\) $C^{\ast }$ -algebra. We also give a stronger condition for property (P), which additionally implies that the set of corresponding universal \(C^*\) $C^{\ast }$ -algebras is uncountable, and use it to show that the set of universal unital separable \(C^*\) $C^{\ast }$ -algebras of stable rank n is uncountable as an example.