On pseudo-amenability of Beurling algebras

TL;DR

This paper characterizes amenability and pseudo-amenability of Beurling algebras on specific semigroups and groups, providing equivalence conditions and extending the discussion to locally compact groups.

Contribution

It offers new characterizations and conditions for amenability and pseudo-amenability of Beurling algebras on semigroups and groups, including equivalence criteria.

Findings

Characterization of amenability and pseudo-amenability for specific semigroups.

Equivalence conditions for amenability of Beurling algebras.

Discussion of pseudo-amenability for locally compact groups.

Abstract

Amenability and pseudo-amenability of $ \ell^{1}(S,\omega) $ is characterized, where $S$ is a left (right) zero semigroup or it is a rectangular band semigroup. The equivalence conditions to amenability of $\ell^{1}(S,\omega)$ are provided, where $ S $ is a band semigroup. For a locally compact group $G$, pseudo-amenability of $ \ell^{1}(G,\omega) $ is also discussed.

Equations

$${\ell }^1(S,\omega )=\left\{f=\sum _{s\in S}f(s){\delta }_s:\Vert f{\Vert }_{\omega }=\sum _{s\in S}|f(s)|\omega (s)<\infty \right\}$$

$$\rho (f\ast g)={\delta }_{t_{\circ }}\otimes (f\ast g)={\phi }_S(f)({\delta }_{t_{\circ }}\otimes g)=(f\ast {\delta }_{t_0})\otimes g=f.({\delta }_{t_0}\otimes g)=f.\rho (g)$$

$${\pi }^{\ast }(F)(x,y)=F(xy)=\int {\chi }_K(xyt)f(t)dt.$$

$$⟨{\pi }^{\ast }(F),m_i⟩=⟨F,\pi (m_i)⟩=⟨{\chi }_K,\pi (m_i)f⟩⟶⟨{\chi }_K,f⟩=\int f.$$

$$\underset{i}{\lim }⟨{\pi }^{\ast }(F),m_i⟩\ne 0.(1)$$

$$|⟨{\pi }^{\ast }(F){\chi }_{B_r},m_i⟩|\le ||m_i||||F||r^{-1}{c}_1$$

$$\underset{r⟶\infty }{\lim }⟨{\pi }^{\ast }(F){\chi }_{B_r},m_i⟩=0.(2)$$

$$|⟨{\pi }^{\ast }(F){\chi }_{A_r},{\delta }_g.m_i.{\delta }_{g^{-1}}⟩|\le ||m_i||||F||r{c}_1{c}_2^2\frac{1}{\omega (g)\omega (g^{-1})}$$

$$|⟨{\pi }^{\ast }(F){\chi }_{A_r},m_i⟩|$$

$$\le ||{\pi }^{\ast }(F)||\underset{g\in G}{\sup }||m_i-{\delta }_g.m_i.{\delta }_{g^{-1}}||+||m_i||||F||r{c}_1{c}_2^2\frac{1}{\omega (g)\omega (g^{-1})}.(3)$$

$$|⟨{\pi }^{\ast }(F){\chi }_{A_r},m_i⟩|\le ||{\pi }^{\ast }(F)||\underset{g\in G}{\sup }||m_i-{\delta }_g.m_i.{\delta }_{g^{-1}}||.(4)$$

$$|⟨{\pi }^{\ast }(F),m_i⟩|\le ||{\pi }^{\ast }(F)||\underset{g\in G}{\sup }||m_i-{\delta }_g.m_i.{\delta }_{g^{-1}}||+|⟨{\pi }^{\ast }(F){\chi }_{B_r},m_i⟩|.$$

$$\underset{i}{\lim }⟨{\pi }^{\ast }(F),m_i⟩=0$$

$$\widetilde{m_i}(f):=\sup \{Re⟨m_i,\psi ⟩:0\le |\psi |\le f,\psi \in L^{\infty }(G\times G,{\omega }^{-1}\times {\omega }^{-1})\}.$$

$$\underset{i}{\lim }⟨\widetilde{m_i},{\pi }^{\ast }(\tilde{\omega })⟩$$

$$=c\underset{i}{\lim }Re⟨\pi (m_i).f,{\chi }_K⟩=cRe⟨f,{\chi }_K⟩=c>0.$$

$$A_g(x,y):=\frac{1}{2}(\log \frac{\omega (gx)\omega (g{y}^{-1})}{\omega (x)\omega (y^{-1})})F(x,y),(x,y\in G).$$

Full text

1. Introduction and Preliminaries

For a Banach algebra $\mathcal{A}$ the projective tensor product $\mathcal{A}\hat{\otimes}\mathcal{A}$ is a Banach $\mathcal{A}$-bimodule in a natural manner and the multiplication map $\pi:\mathcal{A}\hat{\otimes}\mathcal{A}\longrightarrow\mathcal{A}$ defined by $\pi(a\otimes b)=ab$ for $a,b\in\mathcal{A}$ is a Banach $\mathcal{A}$-bimodule homomorphism.

Amenability for Banach algebras introduced by B. E. Johnson [9]. Let $\mathcal{A}$ be a Banach algebra and $E$ be a Banach $\mathcal{A}$-bimodule. A continuous linear operator $D:\mathcal{A}\longrightarrow E$ is a derivation if it satisfies $D(ab)=D(a)\ .\ b+a\ .\ D(b)$ for all $a,b\in\mathcal{A}$. Given $x\in E$, the inner derivation $ad_{x}:\mathcal{A}\longrightarrow E$ is defined by $ad_{x}(a)=a\ .\ x-x\ .\ a$. A Banach algebra $\mathcal{A}$ is amenable if for every Banach $\mathcal{A}$-bimodule $E$, every derivation from $\mathcal{A}$ into $E^{*}$, the dual of $E$, is inner.

An approximate diagonal for a Banach algebra $\mathcal{A}$ is a net $(m_{i})_{i}$ in $\mathcal{A}\hat{\otimes}\mathcal{A}$ such that $a\ .\ m_{i}-m_{i}\ .\ a\longrightarrow 0$ and $a\pi(m_{i})\longrightarrow a$, for each $a\in\mathcal{A}$. The concept of pseudo-amenability introduced by F. Ghahramani and Y. Zhang in [5]. A Banach algebra $\mathcal{A}$ is pseudo-amenable if it has an approximate diagonal. It is well-known that amenability of $\mathcal{A}$ is equivalent to the existence of a bounded approximate diagonal.

The notions of biprojectivity and biflatness of Banach algebras introduced by Helemski$\breve{i}$ in [7]. A Banach algebra $\mathcal{A}$ is biprojective if there is a bounded $\mathcal{A}$-bimodule homomorphism $\rho:\mathcal{A}\longrightarrow\mathcal{A}\hat{\otimes}\mathcal{A}$ such that $\pi o\rho=I_{\mathcal{A}}$, where $I_{\mathcal{A}}$ is the identity map on $\mathcal{A}$. We say that $\mathcal{A}$ is biflat if there is a bounded $\mathcal{A}$-bimodule homomorphism $\rho:\mathcal{A}\longrightarrow(\mathcal{A}\hat{\otimes}\mathcal{A})^{**}$ such that $\pi^{**}o\rho=k_{\mathcal{A}}$, where $k_{\mathcal{A}}:\mathcal{A}\longrightarrow\mathcal{A}^{**}$ is the natural embedding of $\mathcal{A}$ into its second dual.

Let $S$ be a semigroup. A continuous function $\omega:S\longrightarrow(0,\infty)$ is a weight on $S$ if $\omega(st)\leq\omega(s)\omega(t)$, for all $s,t\in S$. Then it is standard that

[TABLE]

is a Banach algebra with the convolution product $\delta_{s}\ast\delta_{t}=\delta_{st}$. These algebras are called Beurling algebras.

In this note, we study the earlier mentioned properties of Banach algebras for Beurling algebras. Firstly in section 2, we characterize amenability and pseudo-amenability of $\ell^{1}(S,\omega)$, for some certain class of semigroups. Let $S$ be a left or right zero semigroup. We prove that pseudo-amenability of $\ell^{1}(S,\omega)$ is equivalent to it’s amenability and these equivalent conditions imply that $S$ is singleton. We show that the same result holds for $\ell^{1}(S,\omega)$, whenever $S$ is a rectangular band semigroup and $\omega$ is separable. Further, we investigate biprojectivity of $\ell^{1}(S,\omega)$ whenever $S$ is either left (right) zero semigroup or a rectangular band semigroup. For a band semigroup $S$, we show that amenability of $\ell^{1}(S,\omega)$ is equivalent to that of $\ell^{1}(S)$ and these are equivalent to $S$ being a finite semilattice.

Finally in section 3, we investigate pseudo-amenability of $L^{1}(G,\omega)$ where $G$ is a locally compact group and $\omega$ is a weight on $G$. We prove that pseudo-amenability of $L^{1}(G,\omega)$ implies amenability of $G$, and under a certain condition it implies diagonally boundedness of $\omega$. Next, if $L^{1}(G,\omega)$ is pseudo-amenable we may obtain a character $\varphi$ on $G$ for which $\varphi\leq\omega$.

2. Amenability and pseudo-amenability of $\ell^{1}(S,\omega)$

A semigroup $S$ is a *left zero semigroup * if $st=s$, and it is a *right zero semigroup * if $st=t$ for each $s,t\in S$. Then for $f,g\in\ell^{1}(S,\omega)$, it is obvious that $f*g=\varphi_{S}(f)g$ if $S$ is a right zero semigroup, and $f*g=\varphi_{S}(g)f$ if $S$ is a left zero semigroup, where $\varphi_{S}$ is the augmentation character on $\ell^{1}(S,\omega)$.

We extend somewhat the obtained results for $\ell^{1}(S)$ in [2,3] to the weighted case $\ell^{1}(S,\omega)$.

Proposition 2.1**.**

Suppose that $S$ is a right (left) zero semigroup and $\omega$ be a weight on $S$. Then $\ell^{1}(S,\omega)$ is biprojective.

Proof. We only give the proof in the case $S$ is a right zero semigroup. Define $\rho:\ell^{1}(S,\omega)\longrightarrow\ell^{1}(S,\omega)\otimes\ell^{1}(S,\omega)$ by $\rho(f)=\delta_{t_{\circ}}\otimes f,$, where $t_{0}$ is an arbitrary element $S$. Then for each $f,g\in\ell^{1}(S,\omega)$ we have

[TABLE]

and similarly $\rho(f\ast g)=\rho(f)\ .\ g.$ Further, $\pi\rho$ is the identity map on $\ell^{1}(S,\omega)$, as required. ∎

Remark 2.2**.**

It is known that every biprojective Banach algebra is biflat. Hence Proposition 2.1 shows that for every right or left zero semigroup $S$, $\ell^{1}(S,\omega)$ is biflat.

Given two semigroups $S_{1}$ and $S_{2}$, we say that a weight $\omega$ on $S:=S_{1}\times S_{2}$ is separable if there exist two weights $\omega_{1}$ and $\omega_{2}$ on $S_{1}$ and $S_{2}$, respectively such that $\omega=\omega_{1}\otimes\omega_{2}$. It is easy to verify that $\ell^{1}(S,\omega)\cong\ell^{1}(S_{1},\omega_{1})\hat{\otimes}\ell^{1}(S_{2},\omega_{2})$.

Let $S$ be a semigroup and let $E(S)=\{p\in S:p^{2}=p\}$. We say that $S$ is a band semigroup if $S=E(S)$. A band semigroup $S$ satisfying $sts=s$, for each $s,t\in S$ is called a rectangular band semigroup. For a rectangular band semigroup $S$, it is known that $S\simeq L\times R$, where $L$ and $R$ are left and right zero semigroups, respectively [8, Theorem 1.1.3].

Proposition 2.3**.**

Let $S$ be a rectangular band semigroup and $\omega$ be a separable weight on $S$. Then $\ell^{1}(S,\omega)$ is biprojective, and so it is biflat.

Proof. In view of earlier argument, it follows From Proposition 2.1, and then from [10, Proposition 2.4].

Theorem 2.4**.**

Let $S$ be a rectangular band semigroup and $\omega$ be a weight on $S$. Then $\ell^{1}(S,\omega)$ is amenable if and only if $S$ singleton.

Proof. From [11, Theorem 3.6], $\ell^{1}(S)$ is amenable. Then it is immediate by [2, Theorem 3.3]. ∎

For a semigroup $S$, we denote by $S^{op}$ the semigroup whose underlying space is $S$ but whose multiplication is the multiplication in $S$ reversed.

Proposition 2.5**.**

Let $S$ be a right (left) zero semigroup and $\omega$ be a weight on $S$. Then $\ell^{1}(S,\omega)$ is amenable if and only if $S$ is singleton.

Proof. Suppose that $S$ is a left zero semigroup, and that $\ell^{1}(S,\omega)$ is amenable. Then $S^{op}$ is a right zero semigroup. It is readily seen that $S\times S^{op}$ is a rectangular band semigroup, and $\ell^{1}(S^{op},\omega)$ is amenable. Hence $\ell^{1}(S,\omega)\hat{\otimes}\ell^{1}(S^{op},\omega)\cong\ell^{1}(S\times S^{op},\omega\otimes\omega)$ is amenable. Now, by Theorem 2.4, $S$ is singleton. ∎

Let $\mathcal{A}$ be Banach algebra, $\mathcal{I}$ be a semilattice (i.e., $\mathcal{I}$ is a commutative band semigroup) and $\{\mathcal{A}_{\alpha}:\alpha\in\mathcal{I}\}$ be a collection of closed subalgebras of $\mathcal{A}$. Then $\mathcal{A}$ is $\ell^{1}$-graded of $\mathcal{A}_{\alpha}$’s over the semilattice $\mathcal{I}$, denoted by $\mathcal{A}=\bigoplus_{\alpha\in\mathcal{I}}^{\ell^{1}}\mathcal{A}_{\alpha}$, if it is $\ell^{1}$-directsum of $\mathcal{A}_{\alpha}$’s as Banach space such that $\mathcal{A}_{\alpha}\mathcal{A}_{\beta}\subseteq\mathcal{A}_{\alpha\beta}$, for each $\alpha,\beta\in\mathcal{I}$.

Suppose that $S^{1}$ is the unitization of a semigroup $S$. An equivalence relation $\tau$ on $S$ is defined by $s\tau t\Longleftrightarrow S^{1}sS^{1}=S^{1}tS^{1}$, for all $s,t\in S$. If $S$ is a band semigroup, then by [8, Theorem 4.4.1], $S=\bigcup_{\alpha\in\mathcal{I}}S_{\alpha}$ is a semilattice of rectangular band semigroups, where $\mathcal{I}=\frac{S}{\tau}$ and for each $\alpha=[s]\in\mathcal{I}$, $S_{\alpha}=[s]$.

Theorem 2.6**.**

Let $S$ be a band semigroup and $\omega$ be a weight on $S$. Then the following are equivalent:

$(i)$ $\ell^{1}(S,\omega)$ is amenable.

$(ii)$ $S$ is finite and each $\tau-$class is singleton.

$(iii)$ $\ell^{1}(S)$ is amenable.

$(iv)$ $S$ is a finite semilattice.

Proof: The implications $(ii)$ to $(iv)$ are equivalent [2, Theorem 3.5]. We establish $(i)\longrightarrow(ii)$ and $(iv)\longrightarrow(i)$.

$(i)\longrightarrow(ii)$ If $\ell^{1}(S,\omega)$ is amenable, then $E(S)=S$ is finite and so $\mathcal{I}=\frac{S}{\tau}$ is a finite semilattice. Hence $\ell^{1}(S,\omega)\cong\bigoplus_{\alpha\in\mathcal{I}}^{\ell^{1}}\ell^{1}(S_{\alpha},\omega_{\alpha})$, where $\omega_{\alpha}=\omega|_{S_{\alpha}}$. Then by [6, Proposition 3.1], each $\ell^{1}(S_{\alpha},\omega_{\alpha})$ is amenable. Now by Theorem 2.4, $S_{\alpha}$ is singleton for each $\alpha\in\mathcal{I}$, as required.

$(iv)\longrightarrow(i)$ In this case $\ell^{1}(S,\omega)\cong\ell^{1}(S)$, and $\ell^{1}(S)$ is amenable. ∎

Theorem 2.7**.**

Let $S$ be a rectangular band semigroup, and let $\omega$ be a separable weight on $S$. Then $\ell^{1}(S,\omega)$ is pseudo-amenable if and only if $S$ is singleton.

Proof. There is a left zero semigroup $L$ and a right zero semigroup $R$, and there are weights $\omega_{L}$ and $\omega_{R}$ on $L$ and $R$, respectively such that $S\cong L\times R$ and $\omega=\omega_{L}\otimes\omega_{R}$. We have $\ell^{1}(S,\omega)\cong\ell^{1}(L,\omega_{L})\hat{\otimes}\ell^{1}(R,\omega_{R})$. Hence the map $\theta:\ell^{1}(S,\omega)\longrightarrow\ell^{1}(L,\omega_{L})$ defined by $\theta(f\otimes g)=\varphi_{R}(g)f$ for $f\in\ell^{1}(L,\omega_{L})$ and $g\in\ell^{1}(R,\omega_{R})$, is an epimorphism of Banach algebras, whereas $\varphi_{R}$ is the augmentation character on $\ell^{1}(R,\omega_{R})$. Whence $\ell^{1}(L,\omega_{L})$ has left and right approximate identity. Therefore $L$ is singleton, because it is left zero semigroup. Similarly $R$ is singleton, so is $S$. ∎

Corollary 2.8**.**

Let $S$ be a right (left) zero semigroup and $\omega$ be a weight on $S$. Then the following are equivalent:

$(i)$ $\ell^{1}(S,\omega)$ is pseudo-amenable.

$(ii)$ $S$ is singleton.

$(iii)$ $\ell^{1}(S,\omega)$ is amenable.

Proof. The implication $(ii)\longleftrightarrow(iii)$ is Proposition 2.5. For $(i)\longrightarrow(ii)$, we apply Theorem 2.7 for the rectangular band semigroup $S\times S^{op}$ with $\omega_{L}=\omega_{R}=\omega$. ∎

The following is a combination of Theorems 2.4 and 2.7. Notice that in Theorem 2.4, we need not $\omega$ to be separable.

Corollary 2.9**.**

Let $S$ be a rectangular band semigroup, and let $\omega$ be a separable weight on $S$. Then the following are equivalent:

$(i)$ $\ell^{1}(S,\omega)$ is pseudo-amenable.

$(ii)$ $S$ is singleton.

$(iii)$ $\ell^{1}(S,\omega)$ is amenable.

For the left cancellative semigroups we have the following.

Theorem 2.10**.**

Suppose that $S$ is a left cancellative semigroup and $\omega$ is a weight on $S$. If $\ell^{1}(S,\omega)$ is pseudo-amenable, then $S$ is a group.

proof: This is a more or less verbatim of the proof of [3, Theorem 3.6 $(i)\longrightarrow(ii)$]. ∎

3. Pseudo-amenability of $L^{1}(G,\omega)$

Throughout $G$ is a locally compact group and $\omega$ is a weight on $G$. The weight $\omega$ is diagonally bounded if $\sup_{g\in G}\omega(g)\omega(g^{-1})<\infty$. It seems to be a right conjecture that $L^{1}(G,\omega)$ will fail to be pseudo-amenable whenever $\omega$ is not diagonally bounded. Although we are not able to prove (or disprove) the conjecture, the following is a weaker result.

The proofs in this section owe much to those of [4, Section 8].

Theorem 3.1**.**

Suppose that $L^{1}(G,\omega)$ is pseudo-amenable for which there is an approximate diagonal $(m_{i})_{i}$ such that $m_{i}-\delta_{g}\ .\ m_{i}\ .\ \delta_{g^{-1}}\longrightarrow 0$ uniformly on $G$. Then $\omega$ is diagonally bounded.

Proof. We follow the standard argument in [4, Proposition 8.7]. Choose $f\in L^{1}(G,\omega)$ such that $K:=suppf$ is compact and $\int f\neq 0$. Putting $F:=f\ .\ \chi_{K}\in L^{\infty}(G,\omega^{-1})$, we see that $\pi^{*}(F)\in L^{\infty}(G\times G,\omega^{-1}\times\omega^{-1})$ with

[TABLE]

Let $(m_{i})_{i}\subseteq L^{1}(G\times G,\omega\times\omega)$ be an approximate diagonal for $L^{1}(G,\omega)$ such that $\delta_{g}\ .\ m_{i}\ .\ \delta_{g^{-1}}-m_{i}\longrightarrow 0$ uniformly on $G$, and $\pi(m_{i})f-f\longrightarrow 0$. Then for each $i$

[TABLE]

Consequently

[TABLE]

We define $E:=KK^{-1}$, and $A:=\{(x,y)\in G\times G\ :\ xy\in E\}.$ For $r>0$, we define $A_{r}:=\{(x,y)\in A\ :\ \omega(x)\omega(y)<r\},$ and $B_{r}:=\{(x,y)\in A\ :\ \omega(x)\omega(y)\geq r\}$. Obviously, $\pi^{*}(F)\chi_{A_{r}}$ and $\pi^{*}(F)\chi_{B_{r}}$ both are in $L^{\infty}(G\times G,\omega^{-1}\times\omega^{-1})$, and $\pi^{*}(F)=\pi^{*}(F)\chi_{A}=\pi^{*}(F)\chi_{A_{r}}+\pi^{*}(F)\chi_{B_{r}}$. For every $i$, it is easy to see that

[TABLE]

where $c_{1}:=\sup_{t\in E}\omega(t)$. Hence

[TABLE]

Next, for every $g\in G$, $r>0$, and $i$, we obtain

[TABLE]

where $c_{2}:=\sup_{t\in E^{-1}}\omega(t)$. Therefore

[TABLE]

Towards a contradiction, we assume that $\omega$ is not diagonally bounded. Then there is a sequence $(g_{n})_{n}$ in $G$ such that $\lim_{n}\omega(g_{n})\omega(g_{n}^{-1})=\infty\ .$ Whence, it follows from $(3)$ that for each $i$ and $r>0$

[TABLE]

Hence

[TABLE]

Putting $(2)$ and $(4)$ together, we may see that

[TABLE]

contradicting $(1)$. ∎

Theorem 3.2**.**

Suppose that $L^{1}(G,\omega)$ is pseudo-amenable, and that $\omega$ is bounded away from 0. Then $G$ is amenable.

Proof. Since $L^{1}(G,\omega)$ is unital, pseudo-amenability and approximate amenability are the same [5, Proposition 3.2]. Now, it is immediate by [4, Proposition 8.1]. ∎

We conclude by the following which is an analogue of [4, Proposition 8.9].

Proposition 3.3**.**

Let $L^{1}(G,\omega)$ be pseudo-amenable. Then there is a continuous positive character $\varphi$ on $G$ such that $\varphi\leq\omega$.

Proof. Suppose that $(m_{i})_{i}\subseteq L^{1}(G\times G,\omega\times\omega)$ be an approximate diagonal for $L^{1}(G,\omega)$. For each $i$ and $f\in L^{\infty}(G\times G,\omega^{-1}\times\omega^{-1})^{+}$ we define

[TABLE]

Then $\widetilde{m_{i}}\neq 0$ on $L^{\infty}(G\times G,\omega^{-1}\times\omega^{-1})^{+}$ and we may extend $\widetilde{m_{i}}$ to a bounded linear functional on $L^{\infty}(G\times G,\omega^{-1}\times\omega^{-1})$ in the obvious manner. It is readily seen that $\widetilde{m_{i}}\neq 0$, $\langle\widetilde{m_{i}},f\rangle\geq 0$, and $\delta_{g^{-1}}\ .\ \widetilde{m_{i}}\ .\ \delta_{g}-\widetilde{m_{i}}\longrightarrow 0$, for every $f\in L^{\infty}(G\times G,\omega^{-1}\times\omega^{-1})^{+}$ and $g\in G$.

Putting $\widetilde{\omega}(x):=\sup_{g\in G}\omega(g^{-1}xg)$, $x\in G$. Then $\widetilde{\omega}\in L^{\infty}(G,\omega^{-1})$, $\widetilde{\omega}(xy)=\widetilde{\omega}(yx)$, $\pi^{*}(\widetilde{\omega})\in L^{\infty}(G\times G,\omega^{-1}\times\omega^{-1})$, and $\delta_{g}\ .\ \pi^{*}(\widetilde{\omega})\ .\ \delta_{g^{-1}}=\pi^{*}(\widetilde{\omega})$.

Take $f\in C_{c}(G)^{+}$ with $\int f=1$, and then $h:=f\ .\ \chi_{K}$, where $K:=suppf$. One may see that $h$ is continuous, and there is $c>0$ such that $\pi^{*}(\widetilde{\omega})\geq c\pi^{*}(h)$. Hence

[TABLE]

Therefore there is $i_{0}$ for which $\langle\widetilde{m_{i_{0}}},\pi^{*}(\widetilde{\omega})\rangle>0$. Set $F:=\langle\widetilde{m_{i_{0}}},\pi^{*}(\widetilde{\omega})\rangle^{-1}\pi^{*}(\widetilde{\omega})$, and for $g\in G$ we put

[TABLE]

Finally, for each $g\in G$, we define $\varphi(g):=\exp\langle\widetilde{m_{i_{0}}},A_{g}\rangle.$ A similar argument used in [4, Proposition 8.9], shows that $\varphi$ is the desired character on $G$. ∎