1 Introduction

By a classical result of Hulanicki [4], amenable groups can be characterized by the fact that their full and reduced group C\(^*\)-algebras coincide. In [2], Brown and Guentner obtained several far reaching generalizations of this fact by introducing a new C\(^*\)-completion of any discrete group G induced by an algebraic ideal \(\mathcal {D}\) of \(\ell _\infty (G)\). Namely, the corresponding group C\(^*\)-algebra, denoted by \(\textrm{C}^*_{\mathcal {D}}(G)\), is the completion of the group ring \(\mathbb {C}[G]\) with respect to the norm

$$\begin{aligned} \Vert x\Vert _{\mathcal {D}}=\sup \big \{\Vert \pi (x)\Vert :\pi \text{ is } \text{ a } \mathcal {D}\text{-representation }\big \}, \end{aligned}$$

where by a \(\mathcal {D}\)-representation we mean a unitary representation \(\pi \) of G on a Hilbert space H such that the matrix coefficient functions \(\pi _{\xi ,\eta }\) belong to \(\mathcal {D}\) for all \(\xi ,\eta \) from a dense subspace of H. Using this idea, Brown and Guentner provided new C\(^*\)-algebraic characterizations of a-T-menability and Kazhdan’s property (T) and, among other things, they showed that the equality \(\textrm{C}^*_{\ell ^p}(G)=\textrm{C}^*(G)\) is equivalent to G being amenable.

In this note, we consider the ideals of \(c_0(G)\) consisting of sequences with prescribed rate of convergence. Namely, for \(f\in c_0(G)\) and \(\varepsilon >0\), we set

$$\begin{aligned} \nu (f,\varepsilon )=\#\bigl \{s\in G:\vert {f(s)}\vert \ge \varepsilon \bigr \}, \end{aligned}$$

and define

$$\begin{aligned} \mathcal {I}_{(a_n)}=\big \{f\in c_0(G):\nu (f,\tfrac{1}{n})=O(a_n)\big \}. \end{aligned}$$

We show that the condition

$$\begin{aligned} \textrm{C}^*_{\mathcal {I}_{(a_n)}}(G)=\textrm{C}^*(G) \end{aligned}$$
(*)

is equivalent to (or implies) amenability, provided that \((a_n)\) does not grow too fast.

Amenability is strictly connected to the famous and widely studied stability property arising from a problem posed by Ulam [7] whether any quasimorphism can be uniformly approximated by a homomorphism, the problem first solved for commutative groups by Hyers [5]. We say that a group G has the Hyers–Ulam property provided that for every map \(\phi :G\rightarrow \mathbb {R}\) satisfying

$$\begin{aligned}\sup \bigl \{\vert {\phi (xy)-\phi (x)-\phi (y)}\vert :x,y\in G\bigr \}<\infty \end{aligned}$$

we have \(\textrm{dist}(\phi ,\textrm{Hom}(G,\mathbb {R}))<\infty \). It is known (see [6]) that every amenable group has the Hyers–Ulam property, but the converse is not true which is witnessed e.g. by the groups \(\textrm{SL}(n,\mathbb {Z})\) for \(n\ge 3\). Although there is an algebraic characterization of the Hyers–Ulam property, due to Bavard [1], no C\(^*\)-algebraic characterization is known.

Hence, a natural question concerning Ulam stability reads as follows: Is there an increasing sequence \((a_n)\subset \mathbb {R}_+\) such that for any discrete group G the following characterization holds true: G has the Hyers–Ulam property if and only if condition (\(*\)) holds true? Our result reduces the size of the set of possible candidates for \((a_n)\).

2 Results

In what follows, G stands for a general discrete group. We will need the following two results proved by Brown and Guentner.

Proposition 1

(see [2, Remark 2.5]) For any ideal \(\mathcal {D}\subset \ell ^\infty (G)\), \(\textrm{C}^*_{\mathcal {D}}(G)\) has a faithful \(\mathcal {D}\)-representation.

Theorem 2

([2, Thm. 3.2]) Let \(\mathcal {D}\subset \ell ^\infty (G)\) be a translation-invariant ideal. Then, we have \(\textrm{C}^*_{\mathcal {D}}(G)=\textrm{C}^*(G)\) if and only if there exists a sequence \((h_n)\subset \mathcal {D}\) of positive-definite functions converging pointwise to the constant one function.

Our main result reads as follows.

Theorem 3

(a) If \((a_n)=O(n^{2+\delta })\) for every \(\delta >0\), then

$$\begin{aligned} \textrm{C}^*_{\mathcal {I}_{(a_n)}}(G)=\textrm{C}^*_r(G) \end{aligned}$$

and hence condition (\(*\)) is equivalent to G being amenable.

(b) Suppose a sequence \((a_n)\subset \mathbb {R}_+\) is such that for some \(k>0\), we have

figure a

Then, condition (\(*\)) implies that G is amenable.

Lemma 4

Let \(f\in c_0(G)\) and \(p\ge 1\). We have \(f\in \ell ^{p}(G)\) if and only if the series

$$\begin{aligned} \sum _{n=1}^\infty \nu (f,\tfrac{1}{n})n^{-(p+1)} \end{aligned}$$
(2.1)

converges.

Proof

Let \(\Gamma _n=\{s\in G:\tfrac{1}{n}\le \vert {f(s)}\vert <\tfrac{1}{n-1}\}\) for \(n\in \mathbb {N}\) with the convention \(\tfrac{1}{0}=\infty \), and note that

$$\begin{aligned} \sum _{s\in G}\vert {f(s)}\vert ^p=\sum _{n=1}^\infty \sum _{s\in \Gamma _n}\vert {f(s)}\vert ^p. \end{aligned}$$

Since \(\vert {\Gamma _1}\vert =\nu (f,1)\) and \(\vert {\Gamma _n}\vert =\nu (f,\tfrac{1}{n})-\nu (f,\tfrac{1}{n-1})\) for \(n\ge 2\), we have

$$\begin{aligned} \sum _{s\in G}\vert {f(s)}\vert ^p\le \nu (f,1)\cdot \Vert f\Vert _\infty +\sum _{n=1}^\infty \big (\nu (f,\tfrac{1}{n+1})-\nu (f,\tfrac{1}{n})\big )\cdot n^{-p} \end{aligned}$$
(2.2)

and

$$\begin{aligned} \sum _{s\in G}\vert {f(s)}\vert ^p\ge \sum _{n=1}^\infty \big (\nu (f,\tfrac{1}{n+1})-\nu (f,\tfrac{1}{n})\big )\cdot (n+1)^{-p}. \end{aligned}$$
(2.3)

Denote \(d_n=\nu (f,\tfrac{1}{n+1})-\nu (f,\tfrac{1}{n})\); the series occurring in (2.2) is the limit of partial sums

$$\begin{aligned}{} & {} \lim _{N\rightarrow \infty }\sum _{n=1}^N d_nn^{-p}\\{} & {} \quad =\lim _{N\rightarrow \infty }\Bigg [\sum _{n=1}^{N-1}(d_1+\ldots +d_n)\big (n^{-p}-(n+1)^{-p}\big )+(d_1+\ldots +d_N)\cdot N^{-p}\Bigg ]. \end{aligned}$$

Since \(d_1+\ldots +d_N=\nu (f,\tfrac{1}{N+1})-\nu (f,1)\), the above limit exists if and only if the series

$$\begin{aligned} \sum _{n=1}^\infty \nu (f,\tfrac{1}{n+1})(n^{-p}-(n+1)^{-p}) \end{aligned}$$

converges. By Lagrange’s mean value theorem, we have \(n^{-p}-(n+1)^{-p}=p\,\theta _n^{-(p+1)}\) for some \(\theta _n\in (n,n+1)\), hence the above series converges if and only if (2.1) converges.

We have proved that the convergence of series (2.1) implies that \(f\in \ell ^p(G)\). The converse implication is proved in a similar fashion by using estimate (2.3) instead of (2.2). \(\square \)

Proof of Theorem 3

(a) Suppose that \((a_n)=O(n^{2+\delta })\) for each \(\delta >0\). Then for any \(f\in \mathcal {I}_{(a_n)}\) and any \(\delta >0\) there is \(C_\delta >0\) such that

$$\begin{aligned} \nu (f,\tfrac{1}{n}) n^{-(p+1)}\le C_\delta \cdot n^{-p+1+\delta }\quad (n\in \mathbb {N}). \end{aligned}$$

Therefore, series (2.1) converges for every \(p>2\) and hence Lemma 4 implies that

$$\begin{aligned} \mathcal {I}_{(a_n)}\subseteq \bigcap _{\varepsilon >0}\ell ^{2+\varepsilon }(G). \end{aligned}$$
(2.4)

By the Cowling–Haagerup–Howe theorem [3], if \(\pi :G\rightarrow \mathcal {B}(H)\) is a unitary representation of G with a cyclic vector \(v\in H\) such that \(\pi _{v,v}\in \bigcap _{\varepsilon >0}\ell ^{2+\varepsilon }(G)\), then \(\pi \) is weakly contained in the regular representation \(\lambda \), i.e. \(\Vert \pi (x)\Vert \le \Vert \lambda (x)\Vert \) for each \(x\in G\).

Now, for any fixed \(x\in \textrm{C}^*_{\mathcal {I}_{(a_n)}}(G)\) we use Proposition 1 to

pick a cyclic \(\mathcal {I}_{(a_n)}\)-representation \(\pi \) with \(\pi (x)\ne 0\) (the restriction of a faithful \(\mathcal {I}_{(a_n)}\)-representation to a cyclic subspace). Then, as explained above, inclusion (2.4) implies that \(\pi \) is weakly contained in the regular representation. Therefore, x is not in the kernel of the canonical

map \(\textrm{C}^*_{\mathcal {I}_{(a_n)}}(G)\rightarrow \textrm{C}^*_r(G)\), which proves that \(\textrm{C}^*_{\mathcal {I}_{(a_n)}}(G)=\textrm{C}^*_r(G)\).

(b) This is essentially [2, Remark 2.13] by Brown and Guentner. Notice that condition (\(**\)) says that for any \(f\in \mathcal {I}_{(a_n)}\) we have \(f^k\in \ell ^1(G)\). Indeed, let \(C>0\) be such that \(\nu (f,\tfrac{1}{n})\le Ca_n\). Then, the inequality \(\vert {f(x)}\vert ^k\ge n^{-k}\) holds true for at most \(Ca_n\) elements \(x\in G\), hence \(\Vert f^k\Vert _1\le \nu (f,1)\cdot \Vert f\Vert _\infty +C\sum _{n\ge 2}a_n n^{-k}<\infty \).

Now, by Theorem 2, condition (\(*\)) implies that there exists a sequence \((h_n)\subset \mathcal {I}_{(a_n)}\) of positive-definite functions converging pointwise to the constant one function. In view of (\(**\)), we have \((h_n^k)\subset \ell ^1(G)\); if \(f_n\subset c_{00}(G)\) approximates the square root of \(h_n^k\) in \(\textrm{C}^*_r(G)\), then \(h_n^k\) is approximated by the finitely supported positive-definite functions \(f_n^*f_n\). This yields \(\textrm{C}^*_r(G)=\textrm{C}^*(G)\), i.e. G is amenable. \(\square \)

We conclude our note with a corollary which shows that if there is any ideal of the form \(\mathcal {I}_{(a_n)}\) characterizing the Hyers–Ulam property for discrete groups, then \((a_n)\) must grow quite rapidly. This follows from Theorem 3 and the fact that the Hyers–Ulam property is weaker than amenability.

Corollary 5

If there exists a sequence \((a_n)\subset \mathbb {R}_+\) such that condition (\(*\)) characterizes the Hyers–Ulam property, then \((a_n)\) grows faster than any polynomial.