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Adjoining inverses to noncommutative Banach algebras and extensions of operators. (English) Zbl 0677.46031
Summary: We exhibit an example of a Banach algebra A with an element \(u\in A\) such that u is left invertible in an extension \(B\supset A\), u is right invertible in another extension \(B'\supset A\) and u is invertible in no extension \(C\supset A\). This solves some problems of W. Żelazko [On ideal theory in Banach and topological algebras (1984; Zbl 0556.46030), Problems 2.8 and 2.9] and shows that Arens’ characterization of permanently singular elements is not true in noncommutative Banach algebras. Further, two problems of B. Bollobás [Lect. Notes Math. 1043, 210 (1984)] are solved and the following is proved: If T is a bounded operator on a Banach space X then there exists a Banach space \(Y\supset X\) and \(S\in B(Y)\) such that \(S| X=T\) and \(\sigma (S)=\{\lambda:\quad \inf \{\| (T-\lambda)x\|:\quad x\in X,\quad \| x\| =1\}=0\}.\)

46H05 General theory of topological algebras
47A20 Dilations, extensions, compressions of linear operators
Zbl 0556.46030
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