$L^2$--eta--invariants and their approximation by unitary eta--invariants

Cochran, Orr and Teichner introduced $L^2$–eta–invariants to detect highly non–trivial examples of non slice knots. Using a recent theorem by L"uck and Schick we show that their metabelian $L^2$–eta–invariants can be viewed as the limit of finite dimensional unitary representations. We recall a ribbon obstruction theorem proved by the author using finite dimensional unitary eta–invariants. We show that if for a knot $K$ this ribbon obstruction vanishes then the metabelian $L^2$–eta–invariant vanishes too. The converse has been shown by the author not to be true.