Abstract
Let $\mathfrak{U}$ be a $C^\ast$-algebra and let $T$ be an element in $\mathfrak{U}$. Let $u(T)$ denote the least integer $n$ such that $T$ is a convex combination of $n$ unitaries in $\mathfrak{U}$. Set $u(T)-\infty$ if $T$ is not expressible as such a convex combination. Let $\alpha(T)$ denote the distance from $T$ to $\mathfrak{U}_{\mathrm{inv}}$, the group of invertible elements in $\mathfrak{U}$. We show that if $T$ is a non-invertible element of $(\mathfrak{U})_1 (= \{T\in \mathfrak{U}: \Vert T\Vert \le 1\})$, the unit ball in $\mathfrak{U}$, then $u(T)= \infty$ if $\alpha(T)=1$. Moreover, if $\alpha(T)<1$ and $\beta = 2(1-\alpha(T))^{-1}$ then $u(T)$ lies in $[\beta;\beta+1]$ (so that $u(T)$ is, in effect, a function of $\alpha(T))$. We show that $\mathfrak{U}_{\mathrm{inv}}$ is dense in $\mathfrak{U}$ (that is, the topological stable rank of $\mathfrak{U}$ is $1$; cf. [17]) if and only if $\mathrm{co}\mathscr{U}(\mathfrak{U}) = (\mathfrak{U})_1$, where $\mathrm{co}\mathscr{U}(\mathfrak{U})$ is the convex hull of $\mathscr{U}(\mathfrak{U})$, the group unitary elements in $\mathfrak{U}$. This establishes a conjecture of A. G. Robertson [18]. We study formulas for the distance to the invertibles, and prove aong other results that \[ \mathrm{dist}(T ,\mathscr{U}(\mathfrak{U})) = \max\{\alpha(T) + 1, \Vert T\Vert -1\} \] for each non-invertible $T$ in $\mathfrak{U}$.
