Commutator estimates for normal operators in factors with applications to derivations

Abstract

For a normal measurable operator a affiliated with a von Neumann factor M we show that if M is infinite, then there is λ₀ ∈ ℂ so that for ε?> 0 there are (Formula presented.) with (Formula presented.). If M is finite, then there is λ₀ ∈ ℂ and u, v ∈ U(M) so that (Formula presented.). These bounds are optimal for infinite factors, II₁-factors and some I_n-factors. Furthermore, for finite factors applying ||.||₁-norms to the inequality provides estimates on the norm of the inner derivation (Formula presented.) associated to a. While by [3, Theorem 1.1] it is known for finite factors and self-adjoint (Formula presented.) that (Formula presented.), we present concrete examples of finite factors and normal operators a ∈ M for which this fails.