Quotients of Fourier algebras, and representations which are not completely bounded

Abstract:

We observe that for a large class of non-amenable groups $G$, one can find bounded representations of ${\operatorname {A}}(G)$ on a Hilbert space which are not completely bounded. We also consider restriction algebras obtained from ${\operatorname {A}}(G)$, equipped with the natural operator space structure, and ask whether such algebras can be completely isomorphic to operator algebras. Partial results are obtained using a modified notion of the Helson set which takes into account operator space structure. In particular, we show that when $G$ is virtually abelian and $E$ is a closed subset, the restriction algebra ${\operatorname {A}} _G(E)$ is completely isomorphic to an operator algebra if and only if $E$ is finite.