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

Abstract

We observe that for a large class of non-amenable groups G G , one can find bounded representations of A ( G ) {\operatorname {A}}(G) on a Hilbert space which are not completely bounded. We also consider restriction algebras obtained from A ( G ) {\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 G is virtually abelian and E E is a closed subset, the restriction algebra A G ( E ) {\operatorname {A}} _G(E) is completely isomorphic to an operator algebra if and only if E E is finite.