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

TL;DR

The paper investigates the existence of non-completely bounded bounded representations of Fourier algebras for non-amenable groups and explores conditions under which restriction algebras are completely isomorphic to operator algebras.

Contribution

It demonstrates that for many non-amenable groups, bounded representations of $A(G)$ are not completely bounded and characterizes when restriction algebras are operator algebras.

Findings

Existence of bounded but not completely bounded representations for large classes of non-amenable groups.

Restriction algebra $A_G(E)$ is completely isomorphic to an operator algebra if and only if $E$ is finite when $G$ is virtually abelian.

Partial results on the structure of restriction algebras using a modified Helson set concept.

Abstract

We observe that for a large class of non-amenable groups $G$, one can find bounded representations of $A(G)$ on Hilbert space which are not completely bounded. We also consider restriction algebras obtained from $A(G)$, 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 Helson set which takes account of operator space structure. In particular, we show that if $G$ is virtually abelian, then the restriction algebra $A_G(E)$ is completely isomorphic to an operator algebra if and only if $E$ is finite.