Truncation Approximations and Spectral Invariant Subalgebras in Uniform Roe Algebras of Discrete Groups

Abstract

In this paper we study band truncation approximations for operators in uniform Roe algebras of countable discrete groups. Under conditions on certain growth rates for discrete groups, we find large classes of dense subspaces of uniform Roe algebras whose elements can be approximated by their band truncations in the operator norm. We apply these results to construct a nested family of spectral invariant Banach algebras on discrete groups. For a group with polynomial growth, the intersection of these Banach algebras is a spectral invariant dense subalgebra of the uniform Roe algebra. For a group with subexponential growth, we show that the Wiener algebra of the group is a spectral invariant dense subalgebra of the uniform Roe algebra.

1 Introduction

Let $\mathbb {T}$ be the unit circle and $C(\mathbb {T})$ the algebra of continuous functions on $\mathbb {T}$. A well-known theorem of Fejér showed that for a function $f\in C(\mathbb {T})$, the Fourier expansion $\sum _{n=-\infty }^{\infty }\hat{f}(n)e^{int}$ may not uniformly converge to $f$, where $\displaystyle \hat{f}(n)=\frac{1}{2\pi }\int _0^{2\pi }f(t)e^{-int}dt$. This phenomenon can be viewed in the more general context of noncommutative topology. Indeed, it follows from the Fourier transform that $C(\mathbb {T})$ is isomorphic to the reduced $C^*$-algebra $C_r^*(\mathbb {Z})$ of the integer group $\mathbb {Z}$. Recall that any bounded linear operator $T$ on $\ell ^2(\mathbb {Z})$ admits a matrix form $T=[T(x, y)]_{x, y\in \mathbb {Z}}$, where $T(x, y)=\langle T\delta _y, \delta _x\rangle$. Under the Fourier transform, the partial sum $\sum _{n=-N}^{N}\hat{f}(n)e^{int}$ of the Fourier series of $f\in C(\mathbb {T})$ corresponds to the band truncation $T_N=[T_N(x, y)]$ of the operator $T\in C_r^*(\mathbb {Z})$ corresponding to $f$, where

$$\begin{aligned} T_N(x,y)=\left\{ \begin{array}{ll} T(x,y),&{} \quad \text {if } |x-y|\le N;\\ 0,&{} \quad \text {otherwise.} \end{array} \right. \end{aligned}$$

It reveals that there is a large amount of operators in $C_r^*(\mathbb {Z})$ which cannot be approximated by their band truncations in the operator norm. In general, let $G$ be a countable discrete group with a proper length function $\ell$. In [4], Chen and Fu investigated Fejér’s theorem in the context of the uniform Roe algebra $C^*_u(G)$, which contains the reduced group $C^*$-algebra $C_r^*(G)$ as a subalgebra. In particular, they proved that if $G$ contains a quasi-isometric copy of $\mathbb {Z}$, then there exist operators $T\in C^*_u(G)$ such that their band truncations $T_N$ do not converge to $T$ in the operator norm. They call this fact the non-commutative Fejér phenomenon. Since truncation is an operation that is essential for various areas of mathematics, it is a natural problem to find out those elements in $C^*_u(G)$ which can be approximated by their band truncations in the operator norm.

Recently, Bédos and Conti [1] studied twisted Fourier analysis and convergence of Fourier series on discrete groups. In particular, they introduced a notion of Haagerup content (H-content) and polynomial H-growth for discrete groups and showed that many groups with exponential growth have polynomial H-growth. They also studied certain notions of rapid decay which are generalizations of that in Jolissaint’s work. In [12], Jolissaint introduced the Rapid Decay property for groups and proved that the Schwartz space $H^{\infty }_{\ell }(G)$ is a spectral invariant dense subalgebra of the reduced group $C^*$-algebra $C^*_r(G)$ if $G$ has polynomial growth or if $G$ is the fundamental group of a negatively curved compact manifold. In [8], de la Harpe generalized Jolissaint’s theorem to all hyperbolic groups. The results of Jolissaint and de la Harpe play an important role in the work of Connes-Moscovici on the Novikov conjecture [7] and in the work of Lafforgue on the Baum-Connes conjecture [14]. Analogously, it is interesting to study the spectral invariant dense subalgebras of the uniform Roe algebra $C^*_u(G)$.

Inspired by the above works, we shall study band truncation approximations for operators in uniform Roe algebras of countable discrete groups. Under conditions for the H-growth rates of discrete groups, we find large classes of dense subspaces of uniform Roe algebras whose elements can be approximated by their band truncations in the operator norm (Theorem 3.13). We apply these results to construct a nested family of spectral invariant Banach algebras on discrete groups with polynomial H-growth (Theorem 4.5). By using a recent technical lemma of Bickel and Lindner [2], we show that for a group with polynomial growth, the intersection of these Banach algebras is a spectral invariant dense subalgebra of the uniform Roe algebra (Theorem 4.7). Furthermore, by using the same technical lemma of Bickel and Lindner [2], we are able to show (by a quite short proof) that, for a group with subexponential growth, the Wiener algebra of the group is a spectral invariant dense subalgebra of the uniform Roe algebra (Theorem 4.9). This result can be viewed as a generalization of a recent result by Fendler, Gröchenig and Leinert [9] which shows that the Wiener algebra of the group is a spectral invariant dense subalgebra of the uniform Roe algebra for groups with polynomial growth.

2 Preliminaries

Let $G$ be a countable discrete group. We can view $G$ as a metric space if we endow $G$ with a length function. Recall that a length function of $G$ is a map $\ell :G\rightarrow [0,\infty )$ satisfying

• (1) $\ell (g)=0$ iff $g=e$;

• (2) $\ell (g^{-1})=\ell (g)$ for all $g\in G$;

• (3) $\ell (gh)\le \ell (g)+\ell (h)$ for all $g,h\in G$.

If $G$ acts isometrically on a metric space $(X,d)$ and $x_0\in X$, then $\ell (g):=d(g\cdot x_0,x_0)$ gives a length function on $G$. If $G$ is finitely generated and $S$ is a finite generating set for $G$, then the word-length function $g\mapsto |g|_S$ with respect to the letters $S\cup S^{-1}$ gives a length function on $G$. A length function is called proper if for all $R>0$, $\ell ^{-1}([0,R])$ is finite. For a proper length function $\ell$ on $G$, let $d_{\ell }(x,y)=\ell (xy^{-1})$. Then $d_{\ell }$ is a metric on $G$ induced by $\ell$, which is right invariant in the sense that $d(xh,yh)=d(x,y)$ for all $h\in G$. Note that any countable discrete group admits a proper length function $\ell$ on $G$, which makes $(G,d_{\ell })$ a metric space with bounded geometry, i.e. for all $r>0$, there is a positive number $M$ such that any ball of radius $r$ in $G$ contains no more than $M$ elements.

Definition 2.1

Let $G$ be a countable group and let $\ell$ be a proper length function on $G$. For $r\in \mathbb {R}^+=[0, \infty )$, let $|B(e,r)|$ denote the number of elements in the ball $B(e,r)=\{g\in G:\ell (g)\le r\}$. Then we say

• (1) $G$ has polynomial growth with respect to $\ell$ if there exist $K,p>0$ such that

  $$\begin{aligned} |B(e,r)|\le K(1+r)^p \quad \text { for all }\quad r\in \mathbb {R}^+; \end{aligned}$$

• (2) $G$ has exponential growth with respect to $\ell$ if there exist $a>1$ such that

  $$\begin{aligned} |B(e,r)|\ge a^r \quad \text { for all }\quad r\in \mathbb {R}^+; \end{aligned}$$

• (3) $G$ has subexponential growth with respect to $\ell$, if for any $b>1$ there is some $r_0\in \mathbb {R}^+$ such that

  $$\begin{aligned} |B(e,r)|<b^r \quad \text {for all}\quad r\ge r_0. \end{aligned}$$

Remark 2.2

Every finitely generated group has at most exponential growth, i.e. for some $d>1$ we have $|B(e,r)|\le d^r$ for all $r\in \mathbb {R}^+$.

Denote the $C^*$-algebra of all bounded linear operators on the Hilbert space $\ell ^2(G)$ by $\mathcal {B}(\ell ^2(G))$. For an operator $T\in \mathcal {B}(\ell ^2(G))$, let $\big [T(x,y)\big ]_{(x,y)\in G\times G}$ denote its matrix with respect to the canonical basis of $\ell ^2(G)$. We say that $T$ has finite propagation if there exists $R>0$ such that $T(x,y)=0$ if $d_{\ell }(x,y)>R$. The propagation of $T$ is defined to be the smallest such $R$, and denoted by prop($T$). The support of $T$ is defined to be

$$\begin{aligned} Supp (T)=\{(x,y)\in G\times G:T(x,y)\ne 0\}. \end{aligned}$$

Definition 2.3

Denote by $C_{u,alg}^*(G)$ the collection of all bounded linear operators with finite propagation on $\ell ^2(G)$, which is a *-subalgebra of $\mathcal {B}(\ell ^2(G))$. Its operator norm closure is called the uniform Roe algebra of $G$, denoted by $C_u^*(G)$, i.e.

$$\begin{aligned} C_u^*(G)=\overline{C_{u,alg}^*(G)}^{\Vert \cdot \Vert _{\mathcal {B}(\ell ^2(G))}}. \end{aligned}$$

Let $G$ be a discrete group. The complex group algebra $\mathbb {C}G$ consists of functions $f:G\rightarrow \mathbb {C}$ with finite support. The product in $\mathbb {C}G$ is induced by

$$\begin{aligned} f*g(t)=\sum _{s\in G}f(s)g(s^{-1}t) \end{aligned}$$

for all $f,g\in \mathbb {C}G$    and    $t\in G$. The left regular representation is given by

$$\begin{aligned} (\lambda (\gamma )\xi )(s)=\xi (\gamma ^{-1}s),\forall \gamma ,s\in G\text { and }\xi \in \ell ^{2}(G), \end{aligned}$$

so that $\lambda (f)\xi =f*\xi$ for all $f, \xi \in \mathbb {C}G$.

The reduced $C^*$-algebra of $G$, denoted by $C_r^*(G)$, is the norm closure of $\lambda (\mathbb {C}G)$ in $\mathcal {B}(\ell ^2(G))$, namely,

$$\begin{aligned} C_r^*(G)=\overline{\lambda (\mathbb {C}G)}^{\Vert \cdot \Vert _{\mathcal {B}(\ell ^2(G))}}, \end{aligned}$$

where

$$\begin{aligned} \Vert \lambda (f)\Vert _{\mathcal {B}(\ell ^2(G))}=\sup _{\Vert \xi \Vert _{\ell ^2(G)}=1}\Vert f*\xi \Vert _{\ell ^2(G)}. \end{aligned}$$

Similarly, the formula

$$\begin{aligned} (\rho (\gamma )\xi )(s)=\xi (s\gamma ),\forall \gamma ,s\in G \quad \text {and }\quad \xi \in \ell ^{2}(G) \end{aligned}$$

defines the right regular representation of $G$. Note that, for all $f,g\in \mathbb {C}G$,

$$\begin{aligned} (\rho (\check{f})g)(t)=\sum _{s\in G}\check{f}(s)g(ts)= \sum _{\mu \in G}g(\mu )\check{f}(t^{-1}\mu )= \sum _{\mu \in G}g(\mu )f(\mu ^{-1}t)=g*f(t), \end{aligned}$$

where $\check{f}(t)=f(t^{-1})$, $\forall t\in G$. Let $\check{J}$ be a unitary operator in $\mathcal {B}(\ell ^2(G))$ defined by $(\check{J}\xi )(s)=\xi (s^{-1})$, $\forall s\in G$. It is easy to check $\rho (f)=\check{J}\lambda (f)\check{J}$ for all $f\in \mathbb {C}G$. Hence, $\Vert \rho (f)\Vert _{\mathcal {B}(\ell ^2(G))}=\Vert \lambda (f)\Vert _{\mathcal {B}(\ell ^2(G))}$.

Let $G$ be a countable group with the right invariant metric $d_{\ell }$ given by a proper length function $\ell$. We shall always view the reduced group $C^*$-algebra $C_r^*(G)$ as a subalgebra of the uniform Roe $C_u^*(G)$ via the left regular representation of $G$ (cf.  [3]).

3 Truncation Approximations

In this section, we shall determine large classes of subspaces of uniform Roe algebras of countable groups whose elements can be approximated by their band truncations in the operator norm (Theorem 3.13).

Let $G$ be a countable discrete group with a proper length function $\ell$. Let $E$ be a finite subset of $G$. The tube of width $E$ is the subset $\mathrm {Tube}(E)$ in $G\times G$ given by

$$\begin{aligned} \mathrm {Tube}(E)=\left\{ (x,y)\in G\times G:xy^{-1}\in E\right\} . \end{aligned}$$

Denote by $\chi _E$ the characteristic function of $\mathrm {Tube}(E)$. For any $T \in \mathcal {B}(\ell ^2(G))$, the trunction $T_E$ of $T$ associated to $E$ is defined by $T_E=\chi _E\circ T$, where $\chi _E\circ T$ is the Schur product of $\chi _E$ and $T$ defined by $(\chi _E\circ T)(x,y)=\chi _E(x,y)T(x,y)$ for all $x,y\in G$, i.e.,

$$\begin{aligned} T_E(x,y)=\left\{ \begin{array}{ll} T(x,y), &{}\quad \text {if } (x,y)\in \mathrm {Tube}(E);\\ 0, &{} \quad \text {otherwise.} \end{array} \right. \end{aligned}$$

Especially, if $E$ is a single element subset $\{g\}$ of $G$, we call $T_{\{g\}}$ the g-th diagonal of $T$. Let $f:G\rightarrow [0, \infty )$ be the function defined by $f(g)=\sup _{xy^{-1}=g}|T(x,y)|$ for all $g\in G$. We call $f$ the dominating vector of $T$.

Definition 3.1

We say that an operator $T\in \mathcal {B}(\ell ^2(G))$ can be approximated by its band truncations if $\lim \limits _{r\rightarrow \infty }\Vert T-T_{B(e,r)}\Vert _{\mathcal {B}(\ell ^2(G))}=0$.

Remark 3.2

Note that $T\in C^*_{u,alg}(G)$ if and only if there is a finite subset $E$ of $G$ such that $Supp (T)\subseteq \mathrm {Tube}(E)$. Thus, the uniform Roe algebra $C^*_u(G)$ can be defined as the operator norm closure of bounded linear operators on $\ell ^2(G)$ supported on tubes of finite widths, without referring to a length function on $G$ in advance explicitly, see [3].

In [4] , Chen and Fu show that if $G$ contains a quasi-isometric copy of $\mathbb {Z}$, then there exist operators $T\in C_u^*(G)$ such that their band truncations $\{T_{B(e,r)}\}$ do not converge to $T$ in the operator norm as $r$ goes to infinity. They also find a class of operators in $C^*_u(G)$ which can be approximated by their truncations if $G$ has the property (MRD) (metric rapid decay). Note that $G$ has the property (MRD) if and only if $G$ has polynomial growth [6]. So an interesting question is that if $G$ does not have polynomial growth, can we decide which operator in $C_u^*(G)$ can be approximated by its band truncations? In this section we shall study this problem.

In general, suppose $C^*_{u,alg}(G)$ admits a new norm $\Vert \cdot \Vert ^{\prime }$, and let $\mathcal {L}$ be the completion of $C^*_{u,alg}(G)$ respect to the new norm $\Vert \cdot \Vert ^{\prime }$. If $\lim \limits _{r\rightarrow \infty }\Vert T-T_{B(e,r)}\Vert ^{\prime }=0$ for all $T\in \mathcal {L}$ and the inclusion map $\pi :(C^*_{u,alg}(G),\Vert \cdot \Vert ^{\prime })\rightarrow (C^*_u(G),\Vert \cdot \Vert _{\mathcal {B}(\ell ^2(G))})$ is continuous, then, by identifying it as its image, $\mathcal {L}$ is a dense subspace of $C^*_u(G)$ whose elements can be approximated by their band truncations with respect to the operator norm $\Vert \cdot \Vert _{\mathcal {B}(\ell ^2(G))}$.

Specifically, we shall consider weighted spaces of functions on $G\times G$. Let $\kappa :G\times G\rightarrow [1,+\infty )$ be a (weight) function and define

$$\begin{aligned} \mathcal {L}_{\kappa }=\left\{ T: G\times G\rightarrow \mathbb {C}: \Vert T\Vert _{\kappa }<\infty \right\} , \end{aligned}$$

where

$$\begin{aligned} \Vert T\Vert _{\kappa }=\left[ \sum _{z\in G}\left[ \sup _{\{x,y:xy^{-1}=z\}}|T(x,y)|\kappa (x,y)\right] ^2\right] ^{\frac{1}{2}}. \end{aligned}$$

Clearly $C^*_{u,alg}(G)\subseteq \mathcal {L}_{\kappa }$, and $\mathcal {L}_{\gamma }\subseteq \mathcal {L}_{\kappa }$ for two weights $\gamma , \kappa :G\times G\rightarrow [1,+\infty )$ with $\kappa \le \gamma .$

Proposition 3.3

$\mathcal {L}_{\kappa }$ is a Banach space with respect to the norm $\Vert \cdot \Vert _{\kappa }$.

Proof

Let $\{T_n\}$ be a Cauchy sequence in $\mathcal {L}_{\kappa }$. For all $\varepsilon >0$, $z\in G$ and $x,y\in G$ with $xy^{-1}=z$, then there is an $N>0$ such that for all $n,n^{\prime }\ge N$ we have

$$\begin{aligned} |T_n(x,y)-T_{n^{\prime }}(x,y)|^2&\le \sum _{z\in G}\sup _{xy^{-1}=z}|T_n(x,y)-T_{n^{\prime }}(x,y)|^2\kappa (x,y)^2\nonumber \\&= \Vert T_n-T_{n^{\prime }}\Vert _{\kappa }^2<\varepsilon . \end{aligned}$$

(3.1)

Then $\{T_n(x,y)\}$ is a Cauchy sequence. Let $T(x,y)=\lim \limits _{n\rightarrow \infty }T_n(x,y)$ for all $x,y\in G$. For each $r>0$, by inequality (3.1) we get

$$\begin{aligned} \sum _{z\in B(e,r)}\sup _{xy^{-1}=z}|T_n(x,y)-T_{n^{\prime }}(x,y)|^2\kappa (x,y)^2 \le \varepsilon . \end{aligned}$$

Let $n^{\prime }\rightarrow \infty$, we get

$$\begin{aligned} \sum _{z\in B(e,r)}\sup _{xy^{-1}=z}|T_n(x,y)-T(x,y)|^2\kappa (x,y)^2 \le \varepsilon . \end{aligned}$$

and let $r\rightarrow \infty$, we have

$$\begin{aligned} \sum _{z\in G}\sup _{xy^{-1}=z}|T_n(x,y)-T(x,y)|^2\kappa (x,y)^2 \le \varepsilon . \end{aligned}$$

This implies that $T_n-T\in \mathcal {L}_{\kappa }$ for $n\ge N$. Note that $\mathcal {L}_{\kappa }$ is a linear space, so $T\in \mathcal {L}_{\kappa }$. Therefore

$$\begin{aligned} \Vert T_n-T\Vert ^2_{\kappa }=\sum _{z\in G}\sup _{xy^{-1}=z}|T_n(x,y)-T(x,y)|^2\kappa (x,y)^2 \le \varepsilon . \end{aligned}$$

Hence $\mathcal {L}_{\kappa }$ is a Banach space. $\square$

The following concept is introduced by Bédos and Conti in [1]. Let $E$ be any non-empty finite subset of a countable discrete group $G$. The Haagerup content of $E$ is defined to be

$$\begin{aligned} C(E)=\sup \{\Vert \lambda (f)\Vert _{\mathcal {B}(\ell ^2(G))}:f\in \mathbb {C}G, Supp (f)\subseteq E,\Vert f\Vert _{\ell ^2(G)}\le 1\}, \end{aligned}$$

where $\lambda$ is the left regular representation of $G$ on $\ell ^2(G)$.

Analogously, we may introduce the following quantity for $E$:

$$\begin{aligned} B(E)=\sup \{\Vert T\Vert _{\mathcal {B}(\ell ^2(G))}:T\in \mathcal {B}(\ell ^2(G)), Supp (T)\subseteq \mathrm {Tube}(E) \text { and }\Vert T\Vert ^{\prime }\le 1\}, \end{aligned}$$

where

$$\begin{aligned} \Vert T\Vert '=\left[ \sum _{z\in G}\left[ \sup _{\{x,y:xy^{-1}=z\}}|T(x,y)|\right] ^2 \right] ^{\frac{1}{2}}. \end{aligned}$$

We call $B(E)$ the band Haagerup content of $E$.

Proposition 3.4

For any finite subset $E$ of $G$, $B(E)=C(E)$.

Proof

It is clear that $C(E)\le B(E)$.

For the converse inequality, let $T\in \mathcal {B}(\ell ^2(G))$ with $\Vert T\Vert ^{\prime }\le 1$ and $Supp (T)\subseteq \mathrm {Tube}(E)$. Let $f$ be the dominating vector of $T$. Then we have $f\in \ell ^2(G)$ with norm less than $1$ and $Supp (f)\subseteq E$. Set $T^{\prime }=\lambda (f)$. Note that $\displaystyle T^{\prime }(x,y)=\sup \nolimits _{mn^{-1}=xy^{-1}}|T(m,n)|$ for all $x,y\in G$. Then for any unit vector $\xi \in \ell ^2(G)$, we have

$$\begin{aligned} \Vert T\xi \Vert _{\ell ^2(G)}^2&= \sum _{x\in G}\left| \sum _{y\in G}T(x,y)\xi (y)\right| ^2 \le \sum _{x\in G}\left[ \sum _{y\in G}|T(x,y)||\xi (y)|\right] ^2\\&\le \sum _{x\in G}\left[ \sum _{y\in G}T^{\prime }(x,y)|\xi (y)|\right] ^2\le \Vert T^{\prime }\Vert ^2_{\mathcal {B}(\ell ^2(G))}\\ \end{aligned}$$

So we have $\Vert T\Vert _{\mathcal {B}(\ell ^2(G))}\le \Vert T^{\prime }\Vert _{\mathcal {B}(\ell ^2(G))}$. Hence $B(E)=C(E)$. $\square$

We have the following properties (cf. [1]):

• (1) $C(E)\le C(F)$, if $E\subseteq F$.

• (2) $C(E\cup F)\le C(E)+C(F)$, whenever $E$ and $F$ are pairwise disjoint.

The following notions are introduced by Bédos and Conti in [1].

Definition 3.5

(cf. [1]) Let $G$ be a countable discrete group with a proper length function $\ell$. We say that

• (1) $G$ has polynomial H-growth if there are $K,p>0$ such that $C(B(e,r))\le K(1+r)^p, \text {for all } r\in \mathbb {R}^+$,

• (2) and $G$ has subexponential H-growth if for any $b>1$, there exists some $r_0\in \mathbb {R}^+$ such that $C(B(e,r))<b^r$ for all $r\ge r_0$.

The following lemma is easily proven.

Lemma 3.6

(cf. [1]) Let $G$ be a countable discrete group with a proper length function $\ell$. For each $n\in \mathbb {N}$, let $E_n=\{g\in G:n\le \ell (g)<n+1\}$. Let $c(n)=C(E_n)$ if $E_n$ is nonempty, and $c(n)=0$ otherwise. Then

• (1) $G$ has polynomial H-growth if and only if there exist constants $K,p>0$ such that $c(n)\le K(1+n)^p$ for all $n\ge 0$.

• (2) $G$ has subexponential H-growth if and only if for any $b>1$, there exists some $n_0\in \mathbb {N}$ such that $c(n)<b^n$ whenever $n\ge n_0$.

Remark 3.7

Many groups of exponential growth have polynomial H-growth. For example, the free group $\mathbb {F}_n$ has polynomial H-growth with $c(n)\le n+1$ for all $n\ge 0$ [1, 11]. More generally, Gromov hyperbolic groups have polynomial H-growth with $c(n)\le K(1+n)$ for some $K>0$ (cf. [1, 12]).

The following estimate of Haagerup content is due to [1].

Proposition 3.8

([1]) For any finite subset $E\subset G$, $1\le C(E)\le |E|^{1/2}.$ If $G$ is amenable, then $C(E)=|E|^{\frac{1}{2}}$.

Remark 3.9

It follows from the above result that a group with polynomial growth has polynomial H-growth and a group with subexponential growth has subexponential H-growth. In general, let $G$ be a finitely generated group. Then $G$ has at most exponential growth. Hence, $G$ has at most exponential H-growth, i.e. $c(n)\le a^n$ for some $a>1$.

Definition 3.10

Let $G$ be a countable discrete group and $\kappa : G\times G\rightarrow [1, \infty )$ a (weight) function. We say that $G$ is band $\kappa$-decaying if the inclusion

$$\begin{aligned} \pi :(C^*_{u,alg}(G),\Vert \cdot \Vert _{\kappa })&\rightarrow (C^*_u(G),\Vert \cdot \Vert _{\mathcal {B}(\ell ^2(G))})\\ T&\mapsto T \end{aligned}$$

is continuous.

Lemma 3.11

(cf. [1]) Assume that $G$ is countably infinite and $\{E_n\}_{n=0}^{\infty }$ is a partition of $G$ into finite subsets. Set $c_n=C(E_n),n\ge 0$. Pick $d_n\ge 1$ for each $n$ such that $\sum _{n=0}^{\infty } \left( \frac{c_n}{d_n}\right) ^2<\infty$. Define $\kappa :G\times G\rightarrow [1,\infty )$ by $\kappa =\sum _{n=0}^{\infty }d_n\chi _{E_n}$. Then $G$ is band $\kappa$-decaying.

Proof

For simplicity, denote $\chi _n=\chi _{E_n},n\ge 0$, the characteristic function of the Tube ($E_n$). For $T\in C^*_{u,alg}(G)$, we have

$$\begin{aligned} \Vert \pi (T)\Vert _{\mathcal {B}(\ell ^2(G))}&= \Vert \sum _{n=0}^{\infty }\pi (T\circ \chi _n)\Vert _{\mathcal {B}(\ell ^2(G))}\le \sum _{n=0}^{\infty }\Vert \pi (T\circ \chi _n)\Vert _{\mathcal {B}(\ell ^2(G))}\\&= \sum _{n=0}^{\infty }\Vert T\circ \chi _n\Vert _{\mathcal {B}(\ell ^2(G))}\le \sum _{n=0}^{\infty }c_n\Vert T\circ \chi _n\Vert ^{\prime } =\sum _{n=0}^{\infty }\frac{c_n}{d_n}d_n\Vert T\circ \chi _n\Vert ^{\prime }\\&\le \left( \sum _{n=0}^{\infty }\left( \frac{c_n}{d_n}\right) ^2\right) ^{\frac{1}{2}}\left( \sum _{n=0}^{\infty }d_n^2\Vert T\circ \chi _n\Vert ^{\prime }2\right) ^{\frac{1}{2}}\\&= C\Vert T\Vert _{\kappa }, \end{aligned}$$

where $C=\left( \sum _{n=0}^{\infty }\left( \frac{c_n}{d_n}\right) ^2\right) ^{\frac{1}{2}}$. Hence $G$ is $\kappa$-decaying. $\square$

We have the following result.

Theorem 3.12

(cf. [1]) Let $G$ be a countable infinite group with a proper length function $\ell$.

• (1) If $G$ has polynomial H-growth, then there is $s_0>0$ such that $G$ is band $(1+d_{\ell })^{s_0}$-decaying.

• (2) If $G$ has subexponential H-growth, then $G$ is band $a^{d_{\ell }}$-decaying for all $a>1$.

• (3) In general, for any finitely generated group $G$, then there is $a_0>0$ such that $G$ is band $a^{d_{\ell }}$-decaying for all $a\ge a_0$.

Proof

For each $n\ge 0$, let $E_n=\{g\in G:n\le \ell (g)<n+1\}$. Then $\{E_n\}_{n=0}^{\infty }$ is a partition of $G$ into finite nonempty subsets. Let $c(n)=C(E_n)$.

• (1) By the definition of polynomial H-growth, we have $c_n\le K(1+n)^p,\forall n\in \mathbb {N}$, for some $K,p>0$. Choose $s_0>p+\frac{1}{2}$. Then we have

  $$\begin{aligned} \sum _{n=0}^{\infty }\left( \frac{c_n}{(1+n)^{s_0}}\right) ^2\le \sum _{n=0}^{\infty }K^2\left( \frac{(1+n)^p}{(1+n)^{s_0}}\right) ^2 =K^2\sum _{n=0}^{\infty }\frac{1}{(1+n)^{2(s_0-p)}}<\infty . \end{aligned}$$

  Let $\kappa =\sum _{n=0}^{\infty }(1+n)^{s_0}\chi _{E_n}$. By lemma 3.11 we conclude that $G$ is band $\kappa$-decaying. Now, as $\kappa (g,h)\le (1+\ell (gh^{-1}))^{s_0}$, this implies that $G$ is band $(1+d_{\ell })^{s_0}$-decaying. Now the proof of (1) is complete.

• (2) Suppose $G$ has subexponential H-growth. For any $a>1$, choose $1<b<a$. Then there is $n_0>0$ such that $c_n<b^n$ for $n\ge n_0$. We have

  $$\begin{aligned} \sum _{n=n_0}^{\infty }\left( \frac{c_n}{a^n}\right) ^2\le \sum _{n=n_0}^{\infty }\left( \frac{b^n}{a^n}\right) ^2= \sum _{n=n_0}^{\infty }\left( \frac{b^2}{a^2}\right) ^n<\infty . \end{aligned}$$

  Let $\gamma =\sum _{n=0}^{\infty }a^n\chi _{E_n}$. Then $G$ is band $\gamma$-decaying. Note that $\gamma (g,h)<a^{\ell (gh^{-1})}$. Hence, $G$ is band $a^{d_{\ell }}$-decaying. The proof of (2) is complete.

• (3) If $G$ is finitely generated, by Proposition 3.8 and Remark 3.9 there is $b>1$ such that $c_n\le b^n$ for all $n\in \mathbb {N}$. Take $a_0>b$. Then for any $a\ge a_0$ we have

  $$\begin{aligned} \sum _{n=0}^{\infty } \left( \frac{c_n}{a^n}\right) ^2\le \sum _{n=0}^{\infty } \left( \frac{b^n}{a^n}\right) ^2\le \sum _{n=0}^{\infty } \left( \frac{b}{a}\right) ^{2n}< \infty . \end{aligned}$$

  Let $\varphi =\sum _{n=0}^{\infty }a^n\chi _{E_n}$. Then $G$ is band $\varphi$-decaying. Note $\varphi (g,h)\le a^{\ell (gh^{-1})}$. Hence $G$ is band $a^{d_{\ell }}$-decaying. The proof of (3) is complete.$\square$

The following is the main result of this section.

Theorem 3.13

Let $G$ be a countable infinite group with a proper length function $\ell$.

• (1) Suppose $G$ has polynomial H-growth. Let $\kappa _s=(1+d_{\ell })^s$. Then there is $s_0>0$ such that all elements in the dense subspace $\mathcal {L}_{\kappa _s}$ of $C_u^*(G)$ for any $s\ge s_0$ can be approximated by their band truncations in the operator norm.

• (2) Suppose $G$ has subexponential H-growth. Let $\kappa _a=a^{d_{\ell }}$. Then all elements in the dense subspace $\mathcal {L}_{\kappa _a}$ of $C_u^*(G)$ for any $a>1$ can be approximated by their band truncations in the operator norm.

• (3) In general, let $G$ be a finitely generated group and let $\kappa _a=a^{d_{\ell }}$. Then there is $a_0>1$ such that all elements in the dense subspace $\mathcal {L}_{\kappa _a}$ of $C_u^*(G)$ for any $a\ge a_0$ can be approximated by their band truncations in the operator norm.

Proof

It is obvious that all elements in $\mathcal {L}_{\kappa _s}$ can be approximated by their band truncations in the norm $\Vert \cdot \Vert _{\kappa _s}$. It follows from Theorem 3.12 that the operator norm on $\mathcal {L}_{\kappa _s}$ is dominated by the norm $\Vert \cdot \Vert _{\kappa _s}$ for $s\ge s_0$ for some $s_0>0$. Hence, the statement (1) holds. The other statements follow similarly. $\square$

4 Spectral Invariant Subalgebras

In this section, we shall use the results in Sect. 3, together with a technical lemma by Bickel and Lindner [2], to construct a class of spectral invariant subalgebras of the uniform Roe algebras of discrete groups. Moreover, We employ this technical lemma to show that, for a countable group with subexponential growth, the Wiener algebra is a spectral invariant dense subalgebra of the uniform Roe algebra.

To begin with, we recall the notion of rapid decay (RD) property for groups introduced by Jolissaint [12]. Let $G$ be a countable group with a proper length function $\ell$. For any $s\ge 0$, the Sobolev space of $G$ of order $s$ with respect to $\ell$ is defined as follows:

$$\begin{aligned} H_{\ell }^s(G)=\left\{ \varphi \in \ell ^2(G):\sum _{g\in G}|\varphi (g)|^2(1+\ell (g))^{2s}<\infty \right\} . \end{aligned}$$

The space of rapidly decreasing functions on $G$ is $H_{\ell }^{\infty }(G)=\bigcap _{s\ge 0}H^s_{\ell }(G)$. A group $G$ is said to have the property (RD) if there exists a proper length function $\ell$ on $G$ such that $H_{\ell }^{\infty }(G)$ is contained in $C_r^*(G)$. That is, any function in $H_{\ell }^{\infty }(G)$ defines a bounded operator in $C_r^*(G)$ via the left convolution on $\ell ^2(G)$.

Analogously, we shall introduce the notation of band rapid decay for groups. To do so, let

$$\begin{aligned} H_{\ell ,B}^s(G)=\left\{ T:G\times G\rightarrow \mathbb {C} \Bigg | \sum _{z\in G}\left[ \sup _{\{x,y:x^{-1}y=z\}}|T(x,y)|\right] ^2(1+\ell (z))^{2s}<\infty \right\} . \end{aligned}$$

(Here, the subscript $B$ in $H_{\ell ,B}^s(G)$ refers to band.) For $T\in H^s_{\ell ,B}(G)$, let

$$\begin{aligned} \Vert T\Vert _s=\sqrt{\sum _{z\in G}\left[ \sup _{\{x,y:xy^{-1}=z\}}|T(x,y)|\right] ^2(1+\ell (z))^{2s}}. \end{aligned}$$

Then

$$\begin{aligned} H_{\ell ,B}^{\infty }(G):=\bigcap _{s=0}^{\infty } H_{\ell ,B}^s(G) \end{aligned}$$

is a Fréchet space with the topology generated by the norms $\Vert \cdot \Vert _s$, ($s\ge 0$).

In the following, we also view functions in $H^s_{\ell ,B}(G)$ as bounded linear operators acting on $\ell ^2(G)$ via convolution. That is, for any function $T$ in $H^s_{\ell ,B}(G)$, $T:\ell ^2(G)\rightarrow \ell ^2(G)$ defined by

$$\begin{aligned} T\xi (x)=\sum _{y\in G}T(x,y)\xi (y) \end{aligned}$$

is a bounded operator.

Definition 4.1

(cf. [12]) Let $G$ be a countable discrete group with a proper length function $\ell$. Then $G$ is said to have band rapid decay if $H_{\ell ,B}^{\infty }(G)\subseteq C^*_u(G)$.

Proposition 4.2

Assume that $G$ has polynomial H-growth with respect to $\ell$. Then $G$ has band rapid decay.

Proof

From Theorem 3.13 (1), there exists an $s_0$ such that $H_{\ell ,B}^s(G)\subseteq C_u^*(G)$ for $s\ge s_0$. Hence $G$ has band rapid decay.

Proposition 4.3

Assume that $G$ has polynomial H-growth with respect to $\ell$. Then there exists $s_0>0$ such that $H^s_{\ell ,B}(G)$ is a Banach algebra for all $s\ge s_0$.

Proof

The closedness under addition is clear. By proposition 4.2, there is an $s_0$ such that $H^s_{\ell ,B}(G)\subseteq C^*_u(G)$ for $s\ge s_0$.

To show the closedness under multiplication, let $A,B\in H^s_{\ell ,B}(G)$. We will show in the following $C:=AB\in H^s_{\ell ,B}(G)$. Let $a$, $b$ and $c$ be the dominating vectors of $A,B$ and $C$, respectively. For any $x,y\in G$, let $z=xy^{-1}$. Then we have

$$\begin{aligned} |C(x,y)|&= \left| \sum _{h\in G} A(x,h)B(h,y)\right| \\&\le \sum _{h\in G} |A(x,h)\Vert B(h,y)|\\&\le \sum _{h\in G} a(xh^{-1})b(hy^{-1})\\&\le \sum _{\mu \in G} a(\mu )b(\mu ^{-1}xy^{-1})\\&\le \sum _{\mu \in G} a(\mu )b(\mu ^{-1}z)\\&= a*b(z). \end{aligned}$$

By definition

$$\begin{aligned} c(z)=\sup \{|C(x,y)|:xy^{-1}=z\}. \end{aligned}$$

We get

$$\begin{aligned} c(z)\le a*b(z). \end{aligned}$$

For $\gamma \in G$ and $s\ge 1$, we have

$$\begin{aligned} |a*b(\gamma )(1+\ell (\gamma ))^s|&\le \sum _{\mu \in G}a(\mu )b(\mu ^{-1}\gamma )(1+\ell (\mu )+\ell (\mu ^{-1}\gamma ))^s\\&\le 2^s\left( \sum _{\mu \in G}a(\mu )b(\mu ^{-1}\gamma )(1+\ell (\mu ))^s\right) \\&\quad +\,2^s\left( \sum _{\mu \in G}a(\mu )b(\mu ^{-1}\gamma )(1+\ell (\mu ^{-1}\gamma ))^s\right) \\&= 2^s(a(1+\ell )^s*b)(\gamma )+2^s(a*b(1+\ell )^s)(\gamma ). \end{aligned}$$

Summing over $G$, we get

$$\begin{aligned} \Vert (1+\ell )^sa*b\Vert _{\ell ^2(G)}^2&\le 2^{2s+1}\left( \Vert a(1+\ell )^s*b\Vert ^2_{\ell ^2(G)}+\Vert a*b(1+\ell )^s\Vert ^2_{\ell ^2(G)}\right) \nonumber \\&\le 2^{2s+1}\left( \Vert a(1+\ell )^s\Vert ^2_{\ell ^2(G)}\Vert \rho (\check{b})\Vert _{\mathcal {B}(\ell ^2(G))}^2\right) \nonumber \\&\quad +\,2^{2s+1}\left( \Vert \lambda (a)\Vert _{\mathcal {B}(\ell ^2(G))}^2\Vert b(1+\ell )^s\Vert ^2_{\ell ^2(G)}\right) \nonumber \\&= 2^{2s+1}\left( \Vert \lambda (\check{b})\Vert _{\mathcal {B}(\ell ^2(G))}^2\Vert A\Vert _s^2+\Vert \lambda (a)\Vert _{\mathcal {B}(\ell ^2(G))}^2\Vert B\Vert _s^2\right) .\nonumber \\ \end{aligned}$$

(4.1)

Let $\kappa =(1+\ell )^s$. Since the dominating vectors of $\lambda (a)$ and $A$ are the same vector $a$, we have $\lambda (a)\in H^s_{\ell ,B}(G)$. Note that the inclusion $\pi :(C_{u,alg}^*(G),\Vert \cdot \Vert _{\kappa })\rightarrow (C^*_u(G),\Vert \cdot \Vert _{\mathcal {B}(\ell ^2(G))})$ is continuous, there exists a constant $C>0$ such that $\Vert \lambda (a)\Vert _{\mathcal {B}(\ell ^2(G))}\le C\Vert \lambda (a)\Vert _{\kappa }=C\Vert \lambda (a)\Vert _{s}=C\Vert A\Vert _s$. Since

$$\begin{aligned} \displaystyle \Vert \lambda (\check{b})\Vert _s=\sqrt{\sum _{\mathfrak {g}\in G}\check{b}(g)(1+\ell (g))^{2s}}=\sqrt{\sum _{\mathfrak {g}\in G}b(g^{-1})(1+\ell (g^{-1}))^{2s}}=\Vert \lambda (b)\Vert _s, \end{aligned}$$

we have $\lambda (\check{b})\in H_{\ell ,B}^s(G)$. Similarly, we get

$$\begin{aligned} \Vert \lambda (\check{b})\Vert _{\mathcal {B}(\ell ^2(G))}\le C\Vert \lambda (\check{b})\Vert _s=C\Vert \lambda (b)\Vert _s=C\Vert B\Vert _s. \end{aligned}$$

Therefore,

$$\begin{aligned} \Vert (1+\ell )^sa*b\Vert _{\ell ^2(G)}^2\le 2^{2s+2}C^2\Vert A\Vert _s^2\Vert B\Vert _s^2=:C_0^2\Vert A\Vert _s^2\Vert B\Vert _s^2, \end{aligned}$$

i.e.,

$$\begin{aligned} \Vert AB\Vert _s^2=\Vert C\Vert _s^2\le \Vert (1+\ell )^sa*b\Vert _{\ell ^2(G)}^2\le C_0^2\Vert A\Vert _s^2\Vert B\Vert _s^2. \end{aligned}$$

This completes the proof of closedness under the multiplication. Let $\Vert \cdot \Vert _s^{\prime }=C_0\Vert \cdot \Vert _s$. Then the norms $\Vert \cdot \Vert _s^{\prime }$ and $\Vert \cdot \Vert _s$ are equivalent and $\Vert AB\Vert _s^{\prime }\le \Vert A\Vert _s^{\prime }\Vert B\Vert _s^{\prime }$. Hence $\Vert \cdot \Vert _s^{\prime }$ is a Banach algebra norm. $\square$

Let $B$ be a Banach algebra, and $A$ its dense subalgebra. If $A$ has no unit, let $\tilde{A}$ be $A$ with unit adjoined, and let $\tilde{B}$ be $B$ with the same unit adjoined (even if $B$ is already unital, we adjoin a new one).

Definition 4.4

Let $A$ be a dense subalgebra of a Banach algebra $B$. We say that $A$ is spectral invariant in $B$ if the invertible elements of $\tilde{A}$ are precisely those elements of $\tilde{A}$ which are invertible in $\tilde{B}$.

Now we characterize the spectral invariant property among the nested family of $\{H^s_{\ell ,B}(G)\}_{s\ge 0}$. Inspired by a result of Lafforgue in [13], we have the following:

Theorem 4.5

Assume that $G$ has polynomial H-growth with respect to $\ell$. Then there exists $s_0>0$ such that for each $t\in [s_0, s)$, $H^s_{\ell ,B}(G)$ is spectral invariant in $H^t_{\ell ,B}(G)$.

Proof

Since $G$ has polynomial H-growth, by Proposition 4.2, there exists $s_0>0$ such that $\{H^s_{\ell ,B}\}_{s\ge s_0}$ are Banach subalgebras of $C^*_u(G)$. Fix $s>s_0$ and $t\in [s_0,s)$. For any $x\in H^s_{\ell ,B}(G)$, let $\rho _s(x)$ and $\rho _t(x)$ be its spectral radius in $H^s_{\ell ,B}(G)$ and $H^t_{\ell ,B}(G)$, respectively.

First, we show that if $\rho _s(x)=\rho _t(x)$ for all $x\in H_{\ell ,B}^s(G)$, then $H^s_{\ell ,B}(G)$ is spectral invariant in $H^t_{\ell ,B}(G)$. Indeed, assume $x\in H^s_{\ell ,B}(G)$ has an inverse $x^{-1}\in H^t_{\ell ,B}(G)$. Since $H^s_{\ell ,B}(G)$ is dense in $H^t_{\ell ,B}(G)$, there is a sequence $\{x_n\}_{n=1}^{\infty }\subset H_{\ell ,B}^s(G)$ such that $\Vert x_{n}-x^{-1}\Vert _t\rightarrow 0$. Then $\Vert 1-xx_n\Vert _t\rightarrow 0$. Fix $n_0$ such that $\Vert 1-xx_{n_0}\Vert _t<1$. Then $\rho _s(1-xx_{n_0})=\rho _t(1-xx_{n_0})<1$. Therefore $xx_{n_0}=1-(1-xx_{n_0})$ is invertible in $H^s_{\ell ,B}(G)$. Choose $y\in H_{\ell ,B}^s(G)$ with $xx_{n_0}y=1$. Then $x^{-1}=x_{n_0}y\in H^s_{\ell ,B}(G)$.

Therefore we only have to prove that, for any $A\in H^s_{\ell ,B}(G)$,

$$\begin{aligned} \lim _{n\rightarrow \infty }\Vert A^n\Vert _s^{\frac{1}{n}}=\lim _{n\rightarrow \infty }\Vert A^n\Vert _t^{\frac{1}{n}}. \end{aligned}$$

It is clear $\lim _{n\rightarrow \infty }\Vert A^n\Vert _s^{\frac{1}{n}}\ge \lim _{n\rightarrow \infty }\Vert A^n\Vert _t^{\frac{1}{n}}.$ Let $a$ and $b$ be the dominating vectors of $A$ and $A^n$, respectively. For any $x,y\in G$, let $z=xy^{-1}$. We get

$$\begin{aligned} |A^n(x,y)|&= \left| \,\sum _{h_1,\cdots ,h_{n-1}\in G} A(x,h_1)A(h_1,h_2)\cdots A(h_{n-1},y)\right| \\&\le \sum _{h_1,\cdots ,h_{n-1}\in G} |A(x,h_1)\Vert A(h_1,h_2)|\cdots |A(h_{n-1},y)|\\&\le \sum _{h_1,\cdots ,h_{n-1}\in G}a(xh_1^{-1})a(h_1h_2^{-1})\cdots a(h_{n-1}y^{-1})\\&\le \sum _{\tiny \begin{array}{l} z_1,\cdots ,z_n\in G\\ z_1\cdots z_n=z\\ \end{array}} a(z_1)a(z_2)\cdots a(z_n).\\ \end{aligned}$$

So we have

$$\begin{aligned} b(z)\le \sum _{\tiny \begin{array}{l} z_1,\cdots ,z_n\in G\\ z_1\cdots z_n=z\\ \end{array}} a(z_1)a(z_2)\cdots a(z_n). \end{aligned}$$

Note that if $z=z_1\cdots z_n$, then

$$\begin{aligned} \left( 1+\ell (z)\right) ^{s-t} \le n^{s-t}\left( \left( 1+\ell (z_1)\right) ^{s-t}+\cdots + \left( 1+\ell (z_n)\right) ^{s-t}\right) . \end{aligned}$$

Hence

$$\begin{aligned} \Vert A^n\Vert _s&= \Vert z\mapsto (1+\ell (z))^s b(z)\Vert _{\ell ^2(G)}\\&= \Vert z\mapsto (1+\ell (z))^{s-t} b(z)\Vert _t\\&\le n^{s-t}\sum _{k=1}^n\left\| z\mapsto \sum _{\tiny \begin{array}{l} z_1,\cdots ,z_n\in G\\ z_1\cdots z_n=z\\ \end{array}}(1+\ell (z_k))^{s-t}a(z_1)\cdots a(z_k)\cdots a(z_n)\right\| _t \\&\le n^{s-t}\sum _{k=1}^n\left\| \left[ (1+\ell )^{s-t}a\right] *a\cdots *a\right\| _t\\&\le n^{s-t}\sum _{k=1}^n C^{n-1}\left\| \left[ (1+\ell )^{s-t}a\right] \right\| _t\Vert a\Vert _t\cdots \Vert a\Vert _t\\&= n^{s-t+1}C^{n-1}\Vert A\Vert _s\Vert A\Vert _t^{n-1}, \end{aligned}$$

where $C$ is a constant such that $\Vert AB\Vert _t\le C\Vert A\Vert _t\Vert B\Vert _t$ for any $A,B\in H^t_{\ell ,B}(G)$. So we have

$$\begin{aligned} \lim _{n\rightarrow \infty }\Vert A^n\Vert _s^{\frac{1}{n}}\le C\Vert A\Vert _t \end{aligned}$$

Replacing $A$ with $A^p$ in above inequality, and letting $p$ go to infinity, we get $\lim _{n\rightarrow \infty }\Vert A^n\Vert _s^{\frac{1}{n}}\le \lim _{n\rightarrow \infty }\Vert A^n\Vert _t^{\frac{1}{n}}.$ It follows that Theorem 4.5 holds. $\square$

In the following, we are going to prove that, for a discrete group $G$ with polynomial growth, the algebra $H_{\ell ,B}^{\infty }(G)$ is spectral invariant in $C^*_u(G)$. We will employ a technical lemma from [2]. To state this lemma, we need some notations. For each $k\ge 0$, denote

$$\begin{aligned} \mathcal {B}_k=\left\{ T\in C^*_{u,alg}(G): \mathrm {prop}(T)\le k\right\} . \end{aligned}$$

For $g\in G$, let $d_g$ be the norm of g-th diagonal of $T\in C^*_{u,alg}(G)$, that is

$$\begin{aligned} d_g=\sup \{|T(x,y)|:x,y\in G, xy^{-1}=g\}. \end{aligned}$$

Note that

$$\begin{aligned} \Vert T\Vert _s^2=\sum _{z\in G}d_z^2(1+\ell (z))^{2s}, \quad \text {for}\quad T\in H_{\ell ,B}^s(G). \end{aligned}$$

Clearly, we have $\mathcal {B}_k \subset \mathcal {B}_{k+1}$, $C^*_{u,alg}(G)=\bigcup _{k\ge 0}\mathcal {B}_k$ and

$$\begin{aligned} T\in C^*_{u}(G)\text { iff } 0=\mathrm {dist}(T,C^*_{u,alg}(G))=\mathrm {dist}\left( T,\bigcup _{k=0}^{\infty }\mathcal {B}_k \right) =\lim _{k\rightarrow \infty }\mathrm {dist}(T,\mathcal {B}_k ). \end{aligned}$$

where $\mathrm {dist}(T,\mathcal {L}):=\inf _{B\in \mathcal {L}}\Vert T-B\Vert _{\mathcal {B}(\ell ^2(G))}$, for an operator $T\in \mathcal {B}(\ell ^2(G))$ from a subset $\mathcal {L}\subset \mathcal {B}(\ell ^2(G))$. Note that if $d(x,y)=\ell (xy^{-1})>k$ and $T(x,y)$ is a nonzero matrix entry of $T$, then clearly $T(x,y)$ is still a nonzero matrix entry of $T-B$ for all $B\in \mathcal {B}_k$, so that $\Vert T-B\Vert _{\mathcal {B}(\ell ^2(G))}\ge |T(x,y)|$. Consequently,

$$\begin{aligned} \mathrm {dist}(T,\mathcal {B}_k)=\inf _{B\in \mathcal {B}_k}\Vert T-B\Vert _{\mathcal {B}(\ell ^2(G))}\ge |T(x,y)|,\quad \text {for}\quad d(x,y)>k. \end{aligned}$$

(4.2)

holds. Hence, we have

$$\begin{aligned} \mathrm {dist}(T,\mathcal {B}_k)\ge \sup \{|T(x,y)|:x,y\in G,xy^{-1}=g\}=d_g,\quad \text {for}\quad \ell (g)>k. \end{aligned}$$

Suppose $A\in C^*_{u,alg}(G)$ is invertible in $\mathcal {B}(\ell ^2(G))$. Let $\displaystyle M=\Vert A\Vert _{\mathcal {B}(\ell ^2(G))}$, and let $m=\frac{1}{\Vert A^{-1}\Vert _{\mathcal {B}(\ell ^2(G))}}$, and $v=M/m=\Vert A\Vert _{\mathcal {B}(\ell ^2(G))}\Vert A^{-1}\Vert _{\mathcal {B}(\ell ^2(G))}$, the condition number of $A$.

The following technical lemma which is proved in [2] by Bickel and Lindner for the case $G=\mathbb {Z}$ is important in the following of this section. It plainly holds for all countable discrete groups.

Lemma 4.6

(cf. [2]) Suppose $A\in \mathcal {B}_k$ for some $k\in \mathbb {N}$ is invertible in $\mathcal {B}(\ell ^2(G))$. Define $M,m$ and $v$ as above. Then for every $n\in \mathbb {N}$,

$$\begin{aligned} \mathrm {dist}(A^{-1},\mathcal {B}_{n\cdot 3k})\le \frac{M}{m^2}\left( \frac{M^2-m^2}{M^2+m^2}\right) ^{n+1}=\frac{v^2}{M}\left( \frac{v^2-1}{v^2+1}\right) ^{n+1} \end{aligned}$$

(4.3)

Theorem 4.7

Let $G$ be a countable discrete group with a proper length function $\ell$. If $G$ has polynomial growth, then the algebra $H^{\infty }_{\ell ,B}(G)$ is spectral invariant in $C^*_u(G)$.

Proof

By Proposition 4.3, there is $s_0>0$ such that $H_{\ell ,B}^s(G)$ is Banach algebra for $s\ge s_0$. So it suffices to prove that, if $G$ has polynomial growth, then the algebra $H_{\ell ,B}^{s}(G)$ is spectral invariant in ${\mathcal {B}(\ell ^2(G))}$ for each $s\ge s_0$.

• (1) We prove the theorem first in the case of $T\in C^*_{u,alg}(G)\subset H_{\ell ,B}^{s}(G)$. Then $T\in \mathcal {B}_k$ for some $k\in \mathbb {N}$. By inequality (4.3) of lemma 4.6, we have

  $$\begin{aligned} \mathrm {dist}(T^{-1},\mathcal {B}_{n\cdot 3k})\le t_n:=\frac{v^2}{M}\left( \frac{v^2-1}{v^2+1}\right) ^{n+1}=:cr^{n+1} \end{aligned}$$

  for every $n\in \mathbb {N}$, where $M=\Vert T\Vert _{\mathcal {B}(\ell ^2(G))}$, $m:=1/\Vert T^{-1}\Vert _{\mathcal {B}(\ell ^2(G))}$ and $v=M/m=\Vert T\Vert _{\mathcal {B}(\ell ^2(G))}\Vert T^{-1}\Vert _{\mathcal {B}(\ell ^2(G))}$. For every $z\in G$, let $\displaystyle d_z=\sup _{xy^{-1}=z}|T^{-1}(x,y)|$. Let

  $$\begin{aligned} E_n=\{g\in G:(n-1)\cdot 3k<\ell (g)\le n\cdot 3k\} \end{aligned}$$

  for $n=1,2,3,\cdots .$ By the previous inequality and (4.2) we get that

  $$\begin{aligned} \begin{array}{lll} d_z\le t_0=cr &{}\quad \hbox {for} &{}\quad z\in E_1,\\ d_z\le t_1=cr^2 &{}\quad \hbox {for} &{}\quad z\in E_2,\\ d_z\le t_2=cr^3 &{}\quad \hbox {for} &{}\quad z\in E_3,\\ &{}\vdots &{} \end{array} \end{aligned}$$

  Since $G$ has polynomial growth, by definition 2.1 there are constants $K,p>0$ such that

  $$\begin{aligned} |E_n|\le |B(e,n\cdot 3k)|<K(1+n\cdot 3k)^p. \end{aligned}$$

  Summing up, we have

  $$\begin{aligned} \Vert T^{-1}\Vert _s^2&= \sum _{z\in G}d_z^2(1+\ell (z))^{2s}\\&= d^2_e +\sum _{n=1}^{\infty } \sum _{z\in E_n}d_z^2(1+\ell (z))^{2s}\\&\le d^2_e +\sum _{n=1}^{\infty }K(1+n\cdot 3k)^pt_{n-1}^2(1+n\cdot 3k)^{2s}\\&= d^2_e+ Kc^2\sum _{n=1}^{\infty }(1+n\cdot 3k)^{p+2s}r^{2n}<+\infty , \end{aligned}$$

  For the positive series $\sum _{n=1}^{\infty }(1+n\cdot 3k)^{p+2s}r^{2n}$, by the D’Alembert ratio test we have

  $$\begin{aligned} \lim _{n\rightarrow \infty }\frac{(1+(n+1)\cdot 3k)^{p+2s}r^{2(n+1)}}{(1+n\cdot 3k)^{p+2s}r^{2n}}=r^2<1. \end{aligned}$$

  Hence we have $T^{-1}\in H_{\ell ,B}^{s}(G)$.

• (2) Now let $T\in H_{\ell ,B}^{s}(G)$ be invertible and take $T_1,T_2,\cdots ,\in C_{u,alg}^*$ such that $\Vert T-T_n\Vert _s\rightarrow 0$ as $n\rightarrow \infty$. Since $(H_{\ell ,B}^{s}(G),\Vert \cdot \Vert _s)$ is a Banach algebra we know that for sufficiently large $n$, $T_n$ is also invertible and $\Vert T^{-1}-T_n^{-1}\Vert _s\rightarrow 0$ as $n\rightarrow \infty$. The theorem follows from part (1) and the norm closedness of $(H_{\ell ,B}^{s}(G),\Vert \cdot \Vert _s)$.$\square$

Remark 4.8

The proof of the above result depends on the concrete norm estimates given by the technical lemma of Bickel and Lindner. In general, even if $G$ has subexponential growth, we don’t know whether $H^{\infty }_{\ell ,B}(G)$ is a subalgebra of $C^*_u(G)$.

Finally, let us consider the Wiener algebra $\mathcal {W}$ of $G$. Precisely,

$$\begin{aligned} \mathcal {W}:=\left\{ T:G\times G\rightarrow \mathbb {C}:\sum _{z\in G}\sup _{\{x,y:xy^{-1}=z\}}|T(x,y)|< \infty \right\} . \end{aligned}$$

For $T\in \mathcal {W}$, let

$$\begin{aligned} \Vert T\Vert _{\mathcal {W}}=\sum _{z\in G}\sup _{\{x,y:xy^{-1}=z\}}|T(x,y)|. \end{aligned}$$

Note that for each $T\in C^*_{u,alg}(G)$, we have $\Vert T\Vert _{\mathcal {B}(\ell ^2(G))}\le \Vert T\Vert _{\mathcal {W}}$. Hence $\mathcal {W}$ is a dense Banach subalgebra of $C^*_u(G)$ via left convolutions on $\ell ^2(G)$.

In [9], Fendler, Gröchenig, Leinert showed that if $G$ is amenable and rigidly symmetric, then $\mathcal {W}$ is a spectral invariant subalgebra of ${\mathcal {B}(\ell ^2(G))}$. The main examples of amenable and rigidly symmetric groups are groups with polynomial growth. However, by using the above technical lemma of Bickel and Lindner, we are able to give a short proof for the spectral invariance of the Wiener algebras of groups with subexponential growth inside the uniform Roe algebra $C^*_u(G)$. Although a group $G$ with subexponential growth is always amenable, it is not clear whether such a group $G$ is rigidly symmetric as well. Our result is as follows:

Theorem 4.9

Let $G$ be a countable discrete group with a proper length function $\ell$. If $G$ has subexponential growth, then $\mathcal {W}$ is a spectral invariant subalgebra of $C^*_u(G)$.

Proof

First we prove the theorem for the case $T\in C^*_{u,alg}(G)\subset \mathcal {W}$, $T\in \mathcal {B}_k$ for some $k\in \mathbb {N}$. Let $M$, $m$ and $v$ be the constants defined in lemma 4.6, then, by lemma 4.6’s inequality (4.3) we have

$$\begin{aligned} \mathrm {dist}(T^{-1},\mathcal {B}_{n\cdot 3k})\le t_n:=\frac{v^2}{M}\left( \frac{v^2-1}{v^2+1}\right) ^{n+1}=:cr^{n+1} \end{aligned}$$

for every $n\in \mathbb {N}$. For every $z\in G$, let $\displaystyle d_z=\sup _{xy^{-1}=z}|T^{-1}(x,y)|$. Let $E_n=\{g\in G:(n-1)\cdot 3k<\ell (g)\le n\cdot 3k\}$ for $n=1,2,3,\cdots .$ By the previous inequality and (4.2) we get that

$$\begin{aligned} \begin{array}{lll} d_z\le t_0=cr &{}\quad \hbox {for}&{}\quad z\in E_1,\\ d_z\le t_1=cr^2 &{}\quad \hbox {for} &{}\quad z\in E_2,\\ d_z\le t_2=cr^3 &{}\quad \hbox {for} &{}\quad z\in E_3,\\ \vdots \end{array} \end{aligned}$$

Since $G$ has subexponential growth, by the definition 2.1, choose $b>1$ such that $r^{\prime }:=b^{3k}r<1$. There is $n_0>0$ such that

$$\begin{aligned} |E_n|\le |B(e,n\cdot 3k)|<b^{n\cdot 3k}\quad \text {for all}\quad n>n_0 \end{aligned}$$

Summing up, we have

$$\begin{aligned} \Vert T^{-1}\Vert _{\mathcal {W}}&= \sum _{z\in G}d_z\\&= d_e+ \sum _{n\le \frac{n_0}{3k}} \sum _{z\in E_n}d_z+\sum _{n>\frac{n_0}{3k}} \sum _{z\in E_n}d_z\\&\le d_e+ \sum _{n\le \frac{n_0}{3k}} \sum _{z\in E_n}d_z + \sum _{n>\frac{n_0}{3k}} b^{n\cdot 3k}t_{n-1}\\&= d_e+ \sum _{n<\frac{n_0}{3k}} \sum _{z\in E_n}d_z + c\sum _{n>\frac{n_0}{3k}} b^{n\cdot 3k}r^{n}\\&= d_e+ \sum _{n\le \frac{n_0}{3k}} \sum _{z\in E_n}d_z +c\sum _{n>\frac{n_0}{3k}} {r^{\prime }}^n\\&< \infty . \end{aligned}$$

Hence we have $T^{-1}\in \mathcal {W}$.

For the case $T\in \mathcal {W}$, the theorem follows from the norm closedness of $\mathcal {W}$. $\square$