On the generators of separable unital C*-algebras

Abstract

Using the recent single-generation theorem for von Neumann $\mathrm{II}_1$ 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 $\partial \mathrm{T}(A)$ 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 $G_δ$ 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 $τ\in \partial \mathrm{T}(A)$, (a): the spectral distribution of $u$ with respect to $τ$ is atomless and has full support on $\mathbb T$, and (b): for every $a\in A$, $$\lim_{t \to 0}\sup_{p\in D_τ, 0<τ(p)\leq t} \| p\bigl(π_τ(a)-\mathbb E_{D_τ}(π_τ(a))\bigr)p\|=0, $$ where $M_τ=π_τ(A)''$, $D_τ=π_τ(u)''$, $\mathbb E_{D_τ}: M_τ\to D_τ$ is the trace-preserving conditional expectation, and $π_τ$ is the GNS representation associated to $τ$, then $A$ is singly generated. In particular, Conclusion (3) above implies that the reduced group C*-algebra $\mathrm{C}_r^*(\mathbb F_n)$ is singly generated for every $n\in\{2,3,\ldots,\infty\}$.