A generalized spectral radius formula and Olsen’s question

Let A be a C∗-algebra and I be a closed ideal in A. For x ∈ A, its image in A/I is denoted by ˙x, and its spectral radius is denoted by r(x). We prove that max{r(x), ˙x } = inf (1 + i)−1x(1 + i) (where the infimum is taken over all i ∈ I such that 1 + i is invertible), which generalizes the spectral radius formula of Murphy and West. Moreover if r(x) < ˙x then the infimum is attained. A similar result is proved for a commuting family of elements of a C∗-algebra. Using this we give a partial answer to an open question of C. Olsen: if p is a polynomial then for “almost every” operator T ∈ B(H) there is a compact perturbation T +K of T such that p(T +K) = p(T ) e. © 2011 Elsevier Inc. All rights reserved.