On the generators of separable unital C*-algebras

Using the recent single-generation theorem for von Neumann ${\mathrm{I}\mathrm{I}}_1$ factors with separable predual, we prove the following results on the generators of a separable unital C*-algebra. Let $A$ be a separable unital C*-algebra, and suppose that $\mathrm{\partial }\mathrm{T}(A)$ is nonempty, compact, and zero-dimensional. Then (1) If $A$ has strict comparison of positive elements by tracial states, then $A$ can be generated by two ordinary elements. (2) If $A$ has real rank zero and $A$ has strict comparison of projections by tracial states, then $A$ is singly generated. In this case, its single generators form a norm-dense $G_{\delta }$ subset of $A$. (3) If $A$ has strict comparison of positive elements by tracial states and there is a unitary $u\in A$ such that for every $\tau \in \mathrm{\partial }\mathrm{T}(A)$, (a): the spectral distribution of $u$ with respect to $\tau$ is atomless and has full support on $\mathbb{T}$, and (b): for every $a\in A$, $$\underset{t\to 0}{\lim }\underset{p\in D_{\tau },0<\tau (p)\le t}{\sup }\mathrm{\parallel }p({\pi }_{\tau }(a)-{\mathbb{E}}_{D_{\tau }}({\pi }_{\tau }(a)))p\mathrm{\parallel }=0,$$ where $M_{\tau }={\pi }_{\tau }(A{)}^{\mathrm{\prime }\mathrm{\prime }}$, $D_{\tau }={\pi }_{\tau }(u{)}^{\mathrm{\prime }\mathrm{\prime }}$, ${\mathbb{E}}_{D_{\tau }}:M_{\tau }\to D_{\tau }$ is the trace-preserving conditional expectation, and ${\pi }_{\tau }$ is the GNS representation associated to $\tau$, then $A$ is singly generated. In particular, Conclusion (3) above implies that the reduced group C*-algebra ${\mathrm{C}}_r^{\ast }({\mathbb{F}}_n)$ is singly generated for every $n\in \{2,3,\dots ,\mathrm{\infty }\}$.