Distance to normal elements in 𝐶*-algebras of real rank zero

Abstract

We obtain an order sharp estimate for the distance from a given bounded operator A A on a Hilbert space to the set of normal operators in terms of ‖ [ A , A ∗ ] ‖ \|[A,A^*]\| and the distance to the set of invertible operators. A slightly modified estimate holds in a general C ∗ C^* -algebra of real rank zero.