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Let $G$ be a locally compact group and $R$ be any family of representations of $G$. Let $A_R(G)$ be the closed linear span in Fourier–Stieltjes algebras $B(G)$ of the coefficient functions of all representations in $R$. Let $F$ denote the family of all finite-dimensional representations of $G$. Then $AF(G) = B(G) ∩ AP(G)$. see "Operator amenability of Fourier–Stieltjes algebras", Volker Runde and Nico Spronk.

Question. Is $A_F(G)$ an ideal in $B(G)$?

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    $\begingroup$ No, for the following elementary reason: $F$ contains the trivial representation, and so $A_F(G)$ contains the constant function $1$. (This does not need the result of Runde and Spronk which you quote) $\endgroup$
    – Yemon Choi
    Commented Feb 6, 2018 at 16:14
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    $\begingroup$ Actually, the complement of that, consisting of coefficients of weakly mixing representations, is an ideal. $\endgroup$
    – Mateusz Wasilewski
    Commented Feb 6, 2018 at 17:31

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