Weak almost periodicity of ${\displaystyle L_1}$ contractions and coboundaries of non-singular transformations

It is well known that a weakly almost periodic operator T in a Banach space is mean ergodic, and in the complex case, also λT is mean ergodic for every |λ|=1. We prove that a positive contraction on ${\displaystyle L_1}$ is weakly almost periodic if (and only if) it is mean ergodic. An example shows that without positivity the result is false. In order to construct a contraction T on a complex ${\displaystyle L_1}$ such that λT is mean ergodic whenever |λ|=1, but T is not weakly almost periodic, we prove the following: Let τ be an invertible weakly mixing non-singular transformation of a separable atomless probability space. Then there exists a complex function ${\displaystyle \phi \in L_{\infty }}$ with |φ(x)|=1 a.e. such that for every λ ∈ℂ with |λ|=1 the function ⨍ ≡ 0 is the only solution of the equation ⨍(τx)=λφ(x)⨍(x). Moreover, the set of such functions φ is residual in the set of all complex unimodular measurable functions (with the ${\displaystyle L_1}$ topology)