Advances in the theory of unitary rank and regular approximation (Q1112298)

In a unital \(C^*\)-algebra A let \(\alpha\) (T) denote the distance from T to the group of invertible elements. The author proves: (1) The distance of a non-invertible \(T\in A\) to the unitary group is given by \(\max \{\alpha (T)+1,\| T\| -1\}.\) (2) The group of invertible elements is dense in A iff the convex hull of the unitary group coincides with the unit ball in A. [This confirms a conjecture by \textit{A. Robertson}, Bull. Lond. Math. Soc. 6, 333-335 (1974; Zbl 0291.46042)]. The main result of this paper which uses (1) and proves (2) relates \(\alpha\) (T) with the unitary rank u(T), i.e., the minimal length of expressions for T by convex combinations of unitaries.