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Approximation and convex decomposition by extremals in a $$C^*$$-algebra. (English) Zbl 0898.46051
For a unital $$C^*$$-algebra $${\mathfrak A}$$ the set $${\mathfrak E}$$ of extreme points of the unit ball $${\mathfrak A}_1$$ consists of precisely those elements $$V\in{\mathcal A}$$ which satisfy $$(I- VV^*){\mathfrak A}(I- V^*V)= 0$$. The authors continue their study [J. Reine Angew. Math. 469, 113-147 (1995; Zbl 0834.46041)] of the open set $${\mathfrak A}^{-1}_q= {\mathfrak A}^{-1}{\mathfrak E}{\mathfrak A}^{-1}$$ of quasi-invertible elements. Part of this article deals with spectral theory and functional calculus. The principal aim is to extend the theory of regular approximation and unitary decomposition to a setting where the group $${\mathfrak A}^{-1}$$ and the unitaries $${\mathfrak U}$$ are replaced by $${\mathfrak A}^{-1}_q$$ and $${\mathfrak E}$$, respectively. The results yield a complete description of the $$\lambda$$-function $$\lambda:{\mathfrak A}_1\to [0,1]$$, $\lambda(T)= \sup\{\lambda\in [0,1]\mid T=\lambda V+(1- \lambda)B\text{ with }V\in{\mathfrak E},\;B\in{\mathfrak A}_1\},$ which was introduced (for arbitrary Banach spaces) by R. M. Aron and R. H. Lohman [Pac. J. Math. 127, No. 2, 209-231 (1987; Zbl 0662.46020)].

##### MSC:
 46L05 General theory of $$C^*$$-algebras 47A60 Functional calculus for linear operators
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