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Journal Article

# Advances in the Theory of Unitary Rank and Regular Approximation

Mikael Rordam
Annals of Mathematics
Second Series, Vol. 128, No. 1 (Jul., 1988), pp. 153-172
DOI: 10.2307/1971465
Stable URL: http://www.jstor.org/stable/1971465
Page Count: 20

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## Abstract

Let U be a C*-algebra and let T be an element in U. Let u(T) denote the least integer n such that T is a convex combination of n unitaries in U. Set u(T) = ∞ if T is not expressible as such a convex combination. Let α(T) denote the distance from T to $\mathcal{U}_{\operatorname {inv}}$, the group of invertible elements in U. We show that if T is a non-invertible element of $(U)_1 (= \lbraceT \in U: \VertT\Vert\leq 1})$, the unit ball in U, then u(T) = ∞ if α(T) = 1. Moreover, if $\alpha(T) < 1$ and β = 2(1 - α(T))-1, then u(T) lies in [ β; β + 1] (so that u(T) is, in effect, a function of α(T)). We show that $\mathcal{U}_\operatorname {inv}$ is dense in U (that is, the topological stable rank of U is 1; cf. $\lbrack17\rbrack) if and only if$\operatorname{co} U(\mathcal{U}) = (\mathcal{U})_1, where \operatorname{co} U(\mathcal{U}) is the convex hull of U(\mathcal{U}), the group unitary elements in {\mathcal{U}. This establishes a conjecture of A. G. Robertson \lbrack18\rbrack$. We study formulas for the distance to the invertibles, and prove among other results that$\operatorname{dist}(T, U(\mathcal{U}) = \operatorname{max}\lbrace\alpha(T) + 1, \Vert T \Vert - 1}$for each non-invertible$T in \mathcal{U}.

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