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.

We prove that, for a finite-dimensional real normed space V, every bounded mean zero function f ∈ L_∞([0, 1]; V) can be written in the form f = g ◦ T − g for some g ∈ L_∞([0, 1]; V) and some ergodic invertible measure preserving transformation T of [0, 1]. Our method moreover allows us to choose g, for any given ε > 0, to be such that ∥g∥_∞ ⩽ (S_V + ε)∥f∥_∞, where S_V is the Steinitz constant corresponding to V.

In 1984, Kwapien announced that every mean zero function f 2 L1[0; 1] can be written as a coboundary f = g o T -g for some g 2 L1[0; 1] and some measure preserving transformation T of [0; 1]. Whereas Kwapien's original proof holds for continuous functions, there is a serious gap in the proof for functions with discontinuities. In this article we fill in this gap and establish Kwapien's result in full generality. Our method also allows us to improve the original result by showing that for any given ϵ > 0 the function g can be chosen to satisfy ∥g∥1 ≤ (1 + ϵ)∥f∥1.