Similarity degree of Fourier algebras

Abstract

We show that for a locally compact group G, amongst a class which contains amenable and small invariant neighbourhood groups, its Fourier algebra A(G) satisfies a completely bounded version Pisier's similarity property with similarity degree at most 2. Specifically, any completely bounded homomorphism π:A(G)→B(H) admits an invertible S in B(H) for which ‖S‖‖S−1‖≤‖π‖cb2 and S−1π(⋅)S extends to a ⁎-representation of the C*-algebra C0(G). This significantly improves some results due to Brannan and Samei (2010) [5] and Brannan, Daws and Samei (2013) [4]. We also note that A(G) has completely bounded similarity degree 1 if and only if it is completely isomorphic to an operator algebra if and only if G is finite.