The similarity problem for Fourier algebras and corepresentations of group von Neumann algebras (Q992797)

The paper under review deals with the similarity problem for the Fourier algebra \({\mathcal A}(G)\) of a locally compact group \(G\). Recall that for a general involutive Banach algebra \({\mathcal A}\) the similarity problem asks if every bounded representation of \({\mathcal A}\) as operators on a Hilbert space \({\mathcal H}\) is similar to a \(\star\)-representation. The similarity problem for \(C^*\)-algebras is known as the Kadison Similarity Problem, which in its full generality is still open. The authors consider the similarity problem for the Fourier algebra \({\mathcal A}(G)\), which may be considered as the dual version of the similarity problem for \(L^1(G)\) from the point of view of locally compact quantum groups. Effros and Ruan used this point of view to define Hopf algebraic structures on \({\mathcal A}(G)\) and asked for a similarity problem in this setting, i.e. whether every completely contractive representation of \({\mathcal A}(G)\) on a Hilbert space is in fact a \(\ast\)-representation. The second question the paper addresses is as follows, whether every completely bounded representation of \({\mathcal A}(G)\) on a Hilbert space is similar to a completely contractive representation. The main results give partial affirmative answers to both of these questions. The authors obtain an analogous characterization for Fourier algebras to that of Haagerup's for \(C^*\)-algebras: the similarity problem for the Fourier algebra \({\mathcal A}(G)\) has a negative answer if and only if there is a bounded representation of \({\mathcal A}(G)\) which is not completely bounded. Furthermore, it is shown that there is a close connection between the similarity problem for \({\mathcal A}(G)\) and the invertibility of corepresentations of \(VN(G)\).