Lifting algebraic elements in \(C^*\)-algebras (Q1891794)

Suppose \({\mathcal J}\) is a closed ideal in a unital \(C^*\)-algebra \({\mathcal A}\). The author reduces the problem of lifting algebraic elements from \({\mathcal A}/{\mathcal J}\) to \({\mathcal A}\) to the problem of lifting finite orthogonal families of projections. Such liftings are always possible when \({\mathcal A}\) has real rank zero. As application the author shows that for every \(\varepsilon >0\) and for every nonzero polynomial \(p\), there is a \(\delta>0\) such that whenever \(a\) is a \(C^*\)-element with \(|a|\leq 1\) and \(|p(a) |<\delta\), there is a \(b\) in \(C^*(a)\) such that \(p(b) =0\) and \(|b-a |< \varepsilon\).