2015/11/11 by Lee, Hun Hee, Samei, Ebrahim, Spronk, Nico
#43A07 #46L07 #FOS: Mathematics #Functional Analysis (math.FA) #Operator Algebras (math.OA) #Primary 46K10 #Secondary 43A30
paper · doi:10.48550/arxiv.1511.03423
We show that for a locally compact group G, amongst a class which contains amenable and small invariant neighbourhood groups, that 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 (J. Funct. Anal. 259, 2010) and Brannan, Daws and Samei (Münster J. Math 6, 2013). 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.