The similarity problem for Fourier algebras and corepresentations of group von Neumann algebras

Abstract

Let G be a locally compact group, and let A(G) and VN(G) be its Fourier algebra and group von Neumann algebra, respectively. In this paper we consider the similarity problem for A(G): Is every bounded representation of A(G) on a Hilbert space H similar to a * -representation? We show that the similarity problem for A(G) has a negative answer if and only if there is a bounded representation of A(G) which is not completely bounded. For groups with small invariant neighborhoods (i.e. SIN groups) we show that a representation π : A(G) → B(H ) is similar to a * -representation if and only if it is completely bounded. This, in particular, implies that corepresentations of VN(G) associated to non-degenerate completely bounded representations of A(G) are similar to unitary corepresentations. We also show that if G is a SIN, maximally almost periodic, or totally disconnected group, then a representation of A(G) is a * -representation if and only if it is a complete contraction. These results partially answer questions posed in Effros and Ruan (2003) [7] and Spronk (2002) [25].

Similarity degree of Fourier algebras

“…In fact, it is shown in [5] that for any locally compact G, either (i) or (ii) implies that π is similar to a * -homomorphism. A notable feature of their methods is that they gain a proof, [5,Corollary 10], that the Gelfand spectrum of A(G) is G, using Wendel's theorem characterizing M(G) as multipliers of L 1 (G).…”

“…(I) It follows from Theorem 2.5 (or from [5,Theorem 20]) that A(G) is isomorphic to a C*-algebra, hence has a bounded approximate identity. Thus, by Leptin's theorem [25], G is amenable.…”

“…Hence we wish to know if all completely bounded homomorphisms of A(G) enjoy the similarity property. This was the goal of the investigation of Brannan and the second named author [5] and was followed upon by the same two authors in conjunction with M. Daws [4]. They only gained a complete solution for small invariant neighbourhood (SIN) groups in the first paper, and for amenable groups containing open SIN subgroups in the second.…”

“…However, we also indicate some cases with completely bounded similarity degree at most 6 (Corollary 2.9); we suspect that completely bounded similarity degree at most 2 holds generally. Two ancillary representationsπ and π * are introduced in the article [5], which we mention in Corollary 2.10. Brannan and Samei showed that simultaneous complete boundedness for either of the pairs π andπ, or π and π * , characterizes the completely bounded similarity property for A(G).…”

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

“…By recent work of M. Brannan and the second author [6], the answer is positive if G is a SIN group, in particular, if it is either abelian, compact, or discrete. It is natural to ask if (for such G) one really requires complete boundedness of the homomorphism φ.…”