## Advances in the theory of unitary rank and regular approximation.(English)Zbl 0659.46052

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 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.
Reviewer: H.Schröder

### MSC:

 46L05 General theory of $$C^*$$-algebras

Zbl 0291.46042
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