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

Abstract

We observe that for a large class of non-amenable groups G, one can find bounded representations of A(G) on a 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 the Helson set which takes into account operator space structure. In particular, we show that when G is virtually abelian and E is a closed subset, the restriction algebra A G (E) is completely isomorphic to an operator algebra if and only if E is finite.

Similarity degree of Fourier algebras

“…It is proved by Choi and the second named author [7], that if G contains a discrete copy of a free group, then there are bounded homomorphisms π : A(G) → B(H) which are not completely bounded. In particular, such homomorphisms cannot be similar to * -homomorphisms.…”

Realization of compact spaces as cb-Helson sets

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Dedicated to Anthony To-Ming Lau, with thanks for all his work on behalf of the international community in Abstract Harmonic Analysis.

AbstractWe show that, given a compact Hausdorff space Ω, there is a compact group G and a homeomorphic embedding of Ω into G, such that the restriction map A(G) → C(Ω) is a complete quotient map of operator spaces. In particular, this shows that there exist compact groups which contain infinite cb-Helson subsets, answering a question raised in [4]. A negative result from [4] is also improved.

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“…The notion of a cb-Helson subset of a locally compact group was introduced in [4], in connection with the study of quotients of Fourier algebras. For instance, the following result can be obtained by an easy modification of the proof of [4,Theorem B]. Theorem 1.1 (Corollary of work in [4]).…”

“…For instance, the following result can be obtained by an easy modification of the proof of [4,Theorem B]. Theorem 1.1 (Corollary of work in [4]). Let G be a SIN group and let J be a closed ideal in the Fourier algebra A(G).…”

“…Theorem 1.1 (Corollary of work in [4]). Let G be a SIN group and let J be a closed ideal in the Fourier algebra A(G).…”