Representations of the Orlicz Figa-Talamanca Herz Algebras and Spectral subspaces

TL;DR

This paper characterizes non-degenerate *-representations of Orlicz Figa-Talamanca Herz algebras and explores spectral subspaces linked to Banach space representations, advancing understanding of their structure and spectral properties.

Contribution

It provides a new characterization of *-representations of A_ au(G) and B_ au(G), and analyzes spectral subspaces in this context, which was not previously established.

Findings

Characterization of non-degenerate *-representations of A_ au(G) and B_ au(G)

Analysis of spectral subspaces associated with Banach space representations

Enhanced understanding of the structure of Orlicz Figa-Talamanca Herz algebras

Abstract

Let G be a locally compact group. In this note, we characterise non-degenerate *-representations of A_\Phi(G) and B_\Phi(G). We also study spectral subspaces associated to a non-degenerate Banach space representation of A_\Phi(G).

Equations

$$u=\underset{n=1}{\sum ^{\infty }}{f}_n\ast \check{g_n},$$

$$\underset{n=1}{\sum ^{\infty }}{N}_{\Phi }(f_n)\Vert g_n{\Vert }_{\psi }<\infty .$$

$$L^2(G,d\mu )\cong \underset{\alpha \in \wedge }{\oplus }{L}^2(G,d{\mu }_{\alpha }),$$

$$\Vert \pi (u){\Vert }_{\mathcal{B}(\mathcal{H})}\le \Vert u{\Vert }_{\infty }.$$

$$T_{\phi }(u)=⟨\pi (u)\phi ,\phi ⟩,u\in C_0(G).$$

$$T_{\phi }(u)={\int }_{G}ud\mu .$$

$$⟨\pi (u)\phi ,\pi (u)\phi ⟩=$$

$$=$$

$$T(\pi (u)(\pi (v)\phi ))=$$

$$=$$

$$\{{\pi }_{\mu },L^2(G,d\mu )\}\cong \underset{\alpha \in \wedge }{\oplus }\{{\pi }_{{\mu }_{\alpha }},L^2(G,d{\mu }_{\alpha })\}\cong \underset{\alpha \in \wedge }{\oplus }\{{\pi }_{\alpha },{\mathcal{H}}_{\alpha }\}\cong \{\pi ,\mathcal{H}\}.$$

$$\sum _{\alpha \in \wedge }|{\mu }_{\alpha }(f)|=$$

$$\le$$

$$=$$

$$\le$$

$$\tilde{\pi }(u)\pi (v)=\pi (uv)$$

$$\tilde{\pi }(v)=\pi (v).$$

$$\widetilde{{\pi }_{\mu }}(u)({\pi }_{\mu }(v).1)={\pi }_{\mu }(uv).1.$$

$$\Vert \widetilde{{\pi }_{\mu }}(u)\left({\pi }_{\mu }(v).1\right){\Vert }_2^2=$$

$$=$$

$$\le$$

$$\widetilde{{\pi }_{\mu }}(v)({\pi }_{\mu }(u).1)={\pi }_{\mu }(vu).1={\pi }_{\mu }(v)\left({\pi }_{\mu }(u).1\right).$$

$$⟨\tilde{\pi }(u{)}^{\ast }\pi (v)\xi ,\eta ⟩$$

$$\pi (u)\pi {|}_{A_{\Phi }}(v)=\pi (u)\pi (v)=\pi (uv)=\pi {|}_{A_{\Phi }}(uv).$$

$$supp(T)=\{x\in G:u\in A_{\Phi }(G),u(x)\ne 0\Rightarrow u.T\ne 0\}.$$

$$⟨u,T_{x^{\ast },\phi }⟩:=⟨\pi (u)\phi ,x^{\ast }⟩\forall u\in A_{\Phi }(G).$$

$$X_E:=\{\phi \in X:supp(T_{x^{\ast },\phi })\subseteq E\forall x^{\ast }\in X^{\ast }\}.$$

$$T_{x^{\ast },{\phi }_1+\alpha {\phi }_2}=T_{x^{\ast },{\phi }_1}+\alpha T_{x^{\ast },{\phi }_2}.$$

$$T_{x^{\ast },\phi }=c{\delta }_x$$

$$c=c⟨u,{\delta }_x⟩=⟨u,c{\delta }_x⟩=⟨u,T_{x^{\ast },\phi }⟩=⟨\pi (u)\phi ,x^{\ast }⟩.$$

$$⟨\pi (v)\phi ,x^{\ast }⟩=$$

$$=$$

Full text

1. Introduction

Let $G$ be a locally compact group. It is well known that there is a one to one correspondence between the unitary representations of $G$ and the non-degenerate *-representations of $L^{1}(G)$ [3, Pg. 73]. Similarly, if $X$ is any locally compact Hausdorff space, then there is a one to one correspondence between the cyclic *-representations of $C_{0}(X)$ and positive bounded Borel measures on $G$ [6, Pg. 486]. The corresponding result for the Fourier algebra $A(G)$ of a locally compact group is due to Lau and Losert [7]. Recently, Guex [11] extended the result of Lau and Losert to Figà-Talamanca Herz algebras.

In [9], the authors have introduced and studied the $L^{\Phi}$-versions of the Figà-Talamanca Herz algebras. Here $L^{\Phi}$ denotes the Orlicz space corresponding to the Young function $\Phi.$ The space $A_{\Phi}(G)$ is defined as the space of all continuous functions $u,$ where $u$ is of the form

[TABLE]

where $f_{n}\in L^{\Phi}(G),$ $g_{n}\in L^{\Psi}(G),$ $(\Phi,\Psi)$ is a pair of complementary Young functions satisfying the $\Delta_{2}$-condition and

[TABLE]

It is shown in [9] that $A_{\Phi}(G)$ is a regular, tauberian, semisimple commutative Banach algebra with the Gelfand spectrum homeomorphic to $G.$ For further details about these algebras we refer to [9].

This paper has the modest aim of characterising the non-degenerate *-representations of $A_{\Phi}(G)$ in the spirit of [7]. This characterisation is given in Corollary 2.3. In section 3, we show that any non-degenerate *-representation of $A_{\Phi}(G)$ can be extended uniquely to a non-degenerate *-representation of $B_{\Phi}(G).$

Godement in his fundamental paper [4] on Wiener Tauberian theorems studied spectral subspaces associated to a certain Banach space representations. This result was extended to the Fourier algebra $A(G)$ by Parthasarathy and Prakash [8]. In Section 4, we also study spectral subspaces of $A_{\Phi}(G).$

We shall follow the notations as in [9]. For any undefined notations or definitions the reader is asked to refer the above paper.

2. Non-degenerate $\ast$-representations of $A_{\Phi}(G)$

In this section, motivated by the results of [7, 11], we describe all the non-degenerate *-representations of $A_{\Phi}(G).$ Throughout this section and the next, $\mathcal{H}$ will denote a Hilbert space.

Proposition 2.1**.**

Let $\mu$ be a positive measure (not necessarily bounded).

1. (i)

For each $u\in A_{\Phi}(G),$ the mapping $\pi_{\mu}(u):f\mapsto uf$ is a bounded linear operator on $L^{2}(G,d\mu).$ 2. (ii)

The mapping $u\mapsto\pi_{\mu}(u)$ defines a $\ast$-representation of $A_{\Phi}(G)$ on $\mathcal{B}(L^{2}(G,d\mu)).$ 3. (iii)

If $\mu$ is bounded, then $\pi_{\mu}$ is a cyclic representation of $A_{\Phi}(G)$ with the constant $1$ function as cyclic vector. 4. (iv)

If $\mu$ is arbitrary, then $\pi_{\mu}$ is non-degenerate.

Proof.

(i) and (ii) are just a routine check.

(iii) We show that the constant $1$ function is a cyclic vector. Since the measure $\mu$ is finite, the conclusion follows from the density of $A_{\Phi}(G)\cap C_{c}(G)$ in $C_{c}(G)$ with respect to the $L^{2}(G,d\mu)$-norm.

(iv) Let $\mu$ be an arbitrary measure on $G.$ By [2, Pg. 33, 2.2.7], it is enough to show that the representation $\pi_{\mu}$ is a direct sum of cyclic representations. By [1, INT IV.77] and [1, INT V.14, Proposition 4], it follows that

[TABLE]

where $\{\mu_{\alpha}\}_{\alpha\in\wedge}$ is a summable family of measures with pairwise disjoint support. Now the conclusion follows from (iii). ∎

In the next result, we characterise all cyclic *-representations.

Theorem 2.2**.**

Let $\{\pi,\mathcal{H}\}$ be a cyclic $\ast$-representation of $A_{\Phi}(G).$ Then there exists a bounded positive measure $\mu\in M(G)$ such that $\pi$ is unitarily equivalent to the representation $\{\pi_{\mu},L^{2}(G,d\mu)\}$ given in Proposition 2.1.

Proof.

Let $u\in A_{\Phi}(G).$ Then, by [10, Pg. 22], it follows that $\|\pi(u)\|_{sp}\leq\|u\|_{sp}.$ By [9, Theorem 3.4], $A_{\Phi}(G)$ is a commutative Banach algebra and hence the spectral norm and the operator norm for $\pi(u)$ coincides. Further, as $A_{\Phi}(G)$ is semi-simple, $\|u\|_{sp}=\|u\|_{\infty}.$ Thus,

[TABLE]

As a consequence of this inequality and the fact that $A_{\Phi}(G)$ is dense in $C_{0}(G)$, it follows that $\pi$ extends to a *-representation of $C_{0}(G)$ on $\mathcal{H},$ still denoted as $\pi.$ Note that $\pi$ is a cyclic -representation of the C-algebra $C_{0}(G).$ Let $\varphi$ be the cyclic vector of the representation $\{\pi,C_{0}(G)\}.$ Define $T_{\varphi}:C_{0}(G)\rightarrow\mathbb{C}$ as

[TABLE]

It is clear that $T_{\varphi}$ is a positive linear functional on $C_{0}(G)$ and hence, by Riesz representation theorem, there exists a bounded positive measure $\mu\in M(G)$ such that

[TABLE]

Let $\pi_{\mu}$ denote the cyclic *-representation of $A_{\Phi}(G)$ on $L^{2}(G,d\mu),$ given by Proposition 2.1.

We now claim that the representations $\pi$ and $\pi_{\mu}$ of $A_{\Phi}(G)$ are unitarily equivalent. Since $\varphi$ is a cyclic vector, in order to prove the above claim, it is enough to show that the correspondence $\pi(u)\varphi\mapsto u.1$ is an isometry and commutes with $\pi$ and $\pi_{\mu}.$ Note that the above correspondence is well-defined by (1). Let $T$ denote the above well-defined correspondence.

We now show that $T$ is an isometry. Let $u\in A_{\Phi}(G).$ Then

[TABLE]

Finally, we show that $T$ intertwines with $\pi$ and $\pi_{\mu}.$ Let $u\in A_{\Phi}(G).$ Then, for $v\in A_{\phi}(G),$ we have,

[TABLE]

Here is the main result of this section, describing all the non-degenerate Hilbert space representations of $A_{\Phi}(G).$

Corollary 2.3**.**

*If $\{\pi,\mathcal{H}\}$ is any non-degenerate -representation of $A_{\Phi}(G)$ then $\pi$ is unitarily equivalent to $\{\pi_{\mu},L^{2}(G,d\mu)\}$ for some positive measure $\mu.$

Proof.

Let $\{\pi,\mathcal{H}\}$ be a non-degenerate *-representation of $A_{\Phi}(G).$ By [2, Proposition 2.2.7], $\pi$ is a direct sum of cyclic *-representations $\{\pi_{\alpha},\mathcal{H}_{\alpha}\}_{\alpha\in\wedge}.$ For each $\alpha\in\wedge,$ by Theorem 2.2, there exists a bounded positive measure $\mu_{\alpha}$ such that the representations $\{\pi_{\alpha},\mathcal{H}_{\alpha}\}$ and $\{\pi_{\mu_{\alpha}},L^{2}(G,d\mu_{\alpha})\}$ are unitarily equivalent.

Suppose that the family $\{\mu_{\alpha}\}_{\alpha\in\wedge}$ is summable. Let $\mu=\underset{\alpha\in\wedge}{\sum}\mu_{\alpha}.$ Then $\mu$ will be a positive measure and

[TABLE]

Thus, we are done if we can show that $\{\mu_{\alpha}\}_{\alpha\in\wedge}$ is a summable family. Let $f:G\rightarrow\mathbb{C}$ be a continuous function with compact support. Then,

[TABLE]

which follows from the boundedness of $\mu_{\alpha}$’s and the fact that $\underset{\alpha\in\wedge}{\sum}\left(\int_{G}\left|f\right|^{2}\ d\mu_{\alpha}\right)^{1/2}$ is finite. ∎

3. Non-degenerate $\ast$-representations of $B_{\Phi}(G)$

In this section, we show that the non-degenerate representations described in the previous section can be extended uniquely to $B_{\Phi}(G).$

Theorem 3.1**.**

*Let $\{\pi,\mathcal{H}\}$ be a non-degenerate -representation of $A_{\Phi}(G).$

1. (i)

For each $u\in B_{\Phi}(G),$ there exists a unique operator $\widetilde{\pi}(u)\in\mathcal{B}(\mathcal{H})$ such that, $\forall\ v\in A_{\Phi}(G),$

[TABLE]

and

[TABLE] 2. (ii)

*The mapping $u\mapsto\widetilde{\pi}(u)$ defines a non-degenerate -representation of $B_{\Phi}(G)$ on $\mathcal{H}.$

Proof.

(i) Let $\pi$ be a non-degenerate *-representation of $A_{\Phi}(G).$ By [2, Proposition 2.2.7], $\pi$ is a direct sum of cyclic *-representations, say $\{\pi_{\alpha},\mathcal{H}_{\alpha}\}_{\alpha\in\wedge}.$ If we can prove (i) for each of these $\pi_{\alpha}$’s, then the argument for $\pi$ is similar to the one given in Corollary 2.3. Thus, in order to prove this, we assume that the representation $\pi$ is cyclic. Since $\pi$ is a cyclic *-representation, by Theorem 2.2, $\pi$ is unitarily equivalent to $\pi_{\mu},$ for some bounded positive measure $\mu.$ So, without loss of generality, let us assume that the non-degenerate *-representation of $A_{\Phi}(G)$ is $\pi_{\mu}$ for some bounded positive measure $\mu.$

Let $u\in B_{\Phi}(G).$ By Proposition 2.1, the space $\mathcal{K}:=span\{\pi_{\mu}(v).1:v\in A_{\Phi}(G)\}$ is dense in $L^{2}(G,d\mu).$ Define $\widetilde{\pi_{\mu}}(u):\mathcal{K}\rightarrow L^{2}(G,d\mu)$ as

[TABLE]

It is clear that $\widetilde{\pi_{\mu}}(u)$ is linear. We now claim that $\widetilde{\pi_{\mu}}(u)$ is bounded. Let $v\in A_{\Phi}(G).$ Then

[TABLE]

Thus, $\widetilde{\pi_{\mu}}(u)$ extends to a bounded linear operator on $L^{2}(G,d\mu),$ still denoted $\widetilde{\pi_{\mu}}(u).$ Further, it is clear that, for $u\in B_{\Phi}(G)$ and $v\in A_{\Phi}(G),$ $\widetilde{\pi_{\mu}}(u)\pi_{\mu}(v)=\pi_{\mu}(uv).$ Now, let $v\in A_{\Phi}(G).$ Then, for $u\in A_{\Phi}(G),$

[TABLE]

Again, as $\mathcal{K}$ is dense in $L^{2}(G,d\mu),$ it follows that $\widetilde{\pi_{\mu}}(v)=\pi_{\mu}(v)$ for all $v\in A_{\Phi}(G).$

Finally, uniqueness follows from condition (2).

(ii) Non-degeneracy of $\widetilde{\pi}$ follows from the fact that $\pi$ is non-degenerate. Further, homomorphism property of $\widetilde{\pi}$ follows from (2). Now, we show that $\widetilde{\pi}$ preserves involution. Let $u\in B_{\Phi}(G).$ Then, for $v\in A_{\Phi}(G)$ and $\xi,\eta\in\mathcal{H},$ we have

[TABLE]

Since the representation $\pi$ is non-degenerate, the space $\{\pi(u)\xi:u\in A_{\Phi}(G),\xi\in\mathcal{H}\}$ is dense in $\mathcal{H}.$ Thus, it follows that $\widetilde{\pi}(u)^{*}=\widetilde{\pi}(\overline{u})$ for all $u\in B_{\Phi}(G).$ ∎

The following corollary is the converse of the above theorem.

Corollary 3.2**.**

*Let $\{\pi,\mathcal{H}\}$ be a -representation of $B_{\Phi}(G)$ such that $\pi|_{A_{\Phi}}$ is non-degenerate. Then, $\widetilde{\pi|_{A_{\Phi}}}=\pi$ and $\pi$ is non-degenerate.

Proof.

Let $u\in B_{\Phi}(G)$ and $v\in A_{\Phi}(G).$ Then

[TABLE]

Thus, by Theorem 3.1, it follows that $\widetilde{\pi|_{A_{\Phi}}}=\pi.$ Again by Theorem 3.1, $\widetilde{\pi|_{A_{\Phi}}}$ is non-degenerate and hence it follows that the representation $\pi$ is non-degenerate. ∎

4. Spectral subspaces

In this section, we study the spectral subspaces associated to a non-degenerate Banach space representation of $A_{\Phi}(G).$ Our main aim in this section is to prove Corollary 4.9. Most of the ideas of this section are taken from [8].

Definition 4.1**.**

Let $T\in PM_{\Psi}(G).$ Then the support of $T$ is defined as

[TABLE]

Here we recall some of the properties of the support of $T$ in the form of a Lemma [5, Pg. 101].

Lemma 4.2**.**

---

1. (i)

If $T_{1},T_{2}\in PM_{\Psi}(G)$ then $supp(T_{1}+T_{2})\subseteq supp(T_{1})\cup supp(T_{2}).$ 2. (ii)

If $u\in A_{\Phi}(G)$ and $T\in PM_{\Psi}(G)$ then $supp(u.T)\subseteq supp(u)\cap supp(T).$ 3. (iii)

If $c\in\mathbb{C}$ and $T\in PM_{\Psi}(G)$ then $supp(cT)\subseteq supp(T).$ 4. (iv)

Let $T\in PM_{\Psi}(G)$ and let $E$ be a closed subset of $G.$ If a net $\{T_{\alpha}\}\subset PM_{\Psi}(G)$ converges weakly to $T$ with $supp(T_{\alpha})\subset E$ for all $\alpha,$ then $supp(T)\subset E.$

Let $X$ be a Banach space and let $\pi$ be an algebra representation of $A_{\Phi}(G)$ on $X.$ For $\varphi\in X$ and $x^{*}\in X^{*},$ define $T_{x^{*},\varphi}:A_{\Phi}(G)\rightarrow\mathbb{C}$ as

[TABLE]

We say that the representation $\pi$ is continuous if $T_{x^{*},\varphi}$ is a continuous linear functional on $A_{\Phi}(G)$ for each $\varphi\in X$ and $x^{*}\in X^{*}.$ It follows from uniform boundedness principle that the linear map $\pi:A_{\Phi}(G)\rightarrow\mathcal{B}(X)$ is norm continuous.

From now onwards, $X$ will denote a Banach space and $\pi$ an algebra representation of $A_{\Phi}(G)$ on $X.$

Let $E$ be a closed subset of $G.$ Define

[TABLE]

Remark 4.3**.**

An immediate consequence of the above definition is that, if $E=G$ then $X_{E}=X.$

Lemma 4.4**.**

The set $X_{E}$ is a closed $\pi$-invariant subspace of $X.$

Proof.

Note that for any $x^{*}\in X^{*},$ $\varphi_{1},\varphi_{2}\in X_{E}$ and $\alpha\in\mathbb{C},$ we have

[TABLE]

Thus, it follows from (i) and (iii) of Lemma 4.2 that $X_{E}$ is a linear space. Further, closedness of $X_{E}$ is an immediate consequence of (iv) from Lemma 4.2. Again, note that, for any $u\in A_{\Phi}(G),\varphi\in X$ and $x^{*}\in X^{*},$ we have $T_{x^{*},\pi(u)\varphi}=u.T_{x^{*},\varphi}$ and hence the invariance of $X_{E}$ under $\pi$ follows from (ii) of Lemma 4.2. ∎

The subspace $X_{E}$ is called as the spectral subspace associated with the representation $\pi$ and the closed set $E.$

Lemma 4.5**.**

Let $\pi$ be a non-degenerate representation of $A_{\Phi}(G).$

1. (i)

The space $X_{\emptyset}=\{0\}.$ 2. (ii)

If $\{E_{i}\}$ is an arbitrary collection of closed subsets of $G,$ then $X_{\underset{i}{\cap}E_{i}}=\underset{i}{\cap}X_{E_{i}}.$

Proof.

(i) is an easy consequence of the non-degeneracy of $\pi,$ while (ii) is trivial. ∎

The following is an immediate corollary of Remark 4.3 and Lemma 4.5.

Corollary 4.6**.**

There exists a smallest closed non-empty set $E$ of $G$ such that $X_{E}=X.$

Proposition 4.7**.**

Let $K_{1}$ and $K_{2}$ be disjoint compact subsets of $G.$ Then $X_{K_{1}\cup K_{2}}=X_{K_{1}}\oplus X_{K_{2}}.$

Proof.

The proof of this follows exactly as given in [8, Proposition 2 (iii)]. ∎

Theorem 4.8**.**

Let $\pi$ be a non-degenerate representation of $A_{\Phi}(G)$ such that the only spectral subspaces are the trivial subspaces. Then there exists $x\in G$ such that $X_{\{x\}}=X.$

Proof.

Choose a smallest non-empty closed set $E$ such that $X_{E}=X,$ which is possible by Corollary 4.6. Suppose there exists $x,y\in E$ such that $x\neq y.$ As $G$ is locally compact and Hausdorff, there exists an open set $U$ and a compact set $K$ such that $x\in U\subset K$ and $y\notin K.$ Since $A_{\Phi}(G)$ is regular, there exists $u\in A_{\Phi}(G)$ such that $u=1$ on $U$ and $supp(u)\subset K.$

Let $v\in A_{\Phi}(G)$ be arbitrary. Let $v_{1}=v-uv$ and $v_{2}=uv$ so that $v=v_{1}+v_{2}.$ Let $V=\{z\in G:v_{1}(z)\neq 0\}.$ The choice of $u$ tells us that $x\notin\overline{V}.$ Again, using the regularity of $A_{\Phi}(G),$ choose a function $w\in A_{\Phi}(G)$ such that $w=1$ on some open set $W$ containing $x$ and $supp(w)\cap V=\emptyset.$ Further, it is clear that $v_{1}w=0.$

We now claim that $\pi(v)=0.$ Let $\varphi\in X$ and $x^{*}\in X^{*}.$ If $z\in W,$ then $w(z)=1$ and hence $T_{x^{*},\pi(v_{1}w)\varphi}=0$ as $T_{x^{*},\pi(v_{1}w)\varphi}=w.T_{x^{*},\pi(v_{1})\varphi}.$ Thus $supp(T_{x^{*},\pi(v_{1})\varphi})\subset W^{c}.$ Therefore, using the non-degeneracy of $\pi,$ it follows that, if $\pi(v_{1})\varphi\neq 0$ then $X_{W^{c}}=X$ and hence, by the choice of the set $E,$ it follows that $E$ is a subset of $W^{c}.$ On the other hand, $x\notin W^{c}$ and $x\in E$ and hence $E$ is not a subset of $W^{c}.$ Therefore, $\pi(v_{1})=0.$ Similarly, one can show that $\pi(v_{2})=0.$ Thus $\pi(v)=0.$ Since $v$ is arbitrary, it follows that $\pi(v)=0$ for all $v\in A_{\Phi}(G),$ which is a contradiction. Thus the set $E$ is a singleton. ∎

Corollary 4.9**.**

Let $\pi$ be a non-degenerate representation of $A_{\Phi}(G)$ such that the only spectral subspaces are the trivial subspaces. Then $\pi$ is a character.

Proof.

By Theorem 4.8, there exists $x\in G$ such that $X_{\{x\}}=X,$ i.e., $supp(T_{x^{*},\varphi})\subset\{x\}$ for all $\varphi\in X$ and $x^{*}\in X^{*}.$ As singletons are sets of spectral synthesis for $A_{\Phi}(G)$ [9, Theorem 3.6 (i)], it follows that

[TABLE]

for some $c\in\mathbb{C}.$ Let $u\in A_{\Phi}(G)$ such that $u(x)=1.$ Then

[TABLE]

We now claim that $\pi$ is a character. Let $v\in A_{\Phi}(G).$ Then, for $\varphi\in X$ and $x^{*}\in X^{*},$ we have

[TABLE]

Since $\varphi$ and $x^{*}$ are arbitrary, it follows that $\pi(v)=u(x)\pi(u).$ Now

[TABLE]

i.e., $\pi(u)$ is a projection. As $\pi$ is non-degenerate, it follows that $\pi(u)$ is the identity operator $I$ on $X.$ Thus

[TABLE]

i.e., $\pi$ is a character. ∎

Acknowledgement

The first author would like to thank the University Grants Commission, India, for research grant.