A generalized spectral radius formula and Olsenʼs 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 , and its spectral radius is denoted by r(x). We prove that (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 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.