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)
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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 In-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.
