On the functorial properties of the p-analog of the Fourier-Stieltjes algebras and their homomorphisms

TL;DR

This paper explores the functorial properties of p-analog Fourier-Stieltjes algebras, generalizing existing theories to establish p-complete boundedness of certain maps and homomorphisms induced by piecewise affine maps.

Contribution

It extends the theory of p-analog Fourier-Stieltjes algebras by generalizing definitions and theorems, and proves p-complete boundedness of homomorphisms from piecewise affine maps.

Findings

Generalized definitions and theorems for p-analog Fourier-Stieltjes algebras.

Proved p-complete boundedness of certain algebra maps.

Established p-complete boundedness of homomorphisms from piecewise affine maps.

Abstract

In this paper, we follow two main goals. In the first attempt, we give some functorial properties of the $p$-analog of the Fourier-Stieltjes algebras in which we generalize some previously existed definitions and theorems in Arsac and Cowling's works, to utilize them to prove $p$-complete boundedness of some well-known maps on these algebras. In the second part, as an application of these generalizations, we prove $p$-completely boundedness of homomorphisms which are induced by continuous and proper piecewise affine maps that is a generalization of Ilie's work on Fig\`a-Talamanca-Herz algebras.

Equations

$$\displaystyle\Phi_{\alpha}u=\left\{\begin{array}[]{ll}u\circ\alpha&\text{on}\>Y\\ 0&\text{o.w.}\end{array}\right.,\quad u\in A(G),$$

$$\displaystyle\Phi_{\alpha}u=\left\{\begin{array}[]{ll}u\circ\alpha&\text{on}\>Y\\ 0&\text{o.w}\end{array}\right.,\quad u\in A_{p}(G),$$

$$\displaystyle\Phi_{\alpha}u=\left\{\begin{array}[]{ll}u\circ\alpha&\text{on}\>Y\\ 0&\text{o.w}\end{array}\right.,\quad u\in B_{p}(G).$$

$$\pi (f)=\int f(x)\pi (x)dx,f\in L_1(G),$$

$$⟨\pi (f)\xi ,\eta ⟩=\int f(x)⟨\pi (x)\xi ,\eta ⟩dx,\xi \in E,\eta \in E^{\ast },$$

$$\phi \pi (x){\phi }^{-1}=\rho (x),x\in G.$$

$$u(\cdot )=\sum _{n=1}^{\infty }⟨{\lambda }_p(\cdot ){\xi }_n,{\eta }_n⟩,$$

$$({\xi }_n{)}_{n\in \mathbb{N}}\subset L_p(G),({\eta }_n{)}_{n\in \mathbb{N}}\subset L_{p^{\prime }}(G),\text{and}\sum _{n=1}^{\infty }\Vert {\xi }_n\Vert \Vert {\eta }_n\Vert <\infty ,$$

$${\lambda }_p:G\to \mathcal{B}(L^p(G)),{\lambda }_p(x)\xi (y)=\xi (x^{-1}y),\xi \in L^p(G),x,y\in G.$$

$$\|u\|=\inf\Big{\{}\sum_{n=1}^{\infty}\|\xi_{n}\|\|\eta_{n}\|\>:\ u(\cdot)=\sum_{n=1}^{\infty}\langle\lambda_{p}(\cdot)\xi_{n},\eta_{n}\rangle\Big{\}},$$

$$u(\cdot )=⟨\pi (\cdot )\xi ,\eta ⟩,\xi \in E,\eta \in E^{\ast },\text{for some}(\pi ,E)\in {\text{Rep}}_p(G),$$

$$\|u\|=\inf\Big{\{}\sum_{n=1}^{\infty}\|\xi_{n}\|\|\eta_{n}\|\>:\ u(\cdot)=\sum_{n=1}^{\infty}\langle\pi_{n}(\cdot)\xi_{n},\eta_{n}\rangle\Big{\}},$$

$$({\pi }_n,E_n{)}_{n\in \mathbb{N}}\subset {\text{Cyc}}_p(G),\text{with}\sum _{n=1}^{\infty }\Vert {\xi }_n\Vert \Vert {\eta }_n\Vert <\infty ,$$

$$A_p(G)\subset B_p(G)\subset \mathcal{M}(A_p(G)).$$

$$A_p(G_0)\cong \{f\in A_p(G):\text{supp}(f)\subset G_0\},$$

$$\Vert f{\Vert }_{\pi }=\sup \{|⟨f,g⟩|:g\in B_p(G),\Vert g{\Vert }_{B_p(G)}\le 1\},f\in L_1(G).$$

$$⟨\pi (f),\varphi ⟩={\int }_{G}f(x)g(x)dx,f\in L_1(G).$$

$$\Vert \alpha u\beta {\Vert }_n\le \Vert \alpha {\Vert }_{\mathcal{B}(l_p^m,l_p^n)}\Vert u{\Vert }_m\Vert \beta {\Vert }_{\mathcal{B}(l_p^n,l_p^m)}.$$

$$\displaystyle\Omega_{0}(H)=\left\{Y\backslash\cup_{i=1}^{n}Y_{i}\>:\begin{array}[]{ll}&Y\>\text{is an open coset of }\>H,\\ &Y_{1},\ldots,Y_{n}\>\text{open subcosets of infinite index in}\>Y\\ \end{array}\right\}.$$

$$\displaystyle\Omega_{\text{am-}0}(H)=\left\{Y\backslash\cup_{i=1}^{n}Y_{i}\>:\begin{array}[]{ll}&Y\>\text{is an open coset of an open amenable subgroup of }\>H,\\ &Y_{1},\ldots,Y_{n}\>\text{open subcosets of infinite index in}\>Y\\ \end{array}\right\}.$$

$$\alpha (x{y}^{-1}z)=\alpha (x)\alpha (y{)}^{-1}\alpha (z),x,y,z\in Y,$$

$$\alpha {|}_{Y_i}={\alpha }_i{|}_{Y_i}.$$

$$\beta :H_0\subset H\to G,\beta (h)=\alpha (h_0{)}^{-1}\alpha (h_{0}h),h\in H_0.$$

$$1\le \Vert {\chi }_Y{\Vert }_{B_p(G)}\le 2^{m_Y},\text{with}{m}_Y=\inf \{m\in \mathbb{N}:Y=Y_0\backslash {\cup }_{i=1}^m{Y}_i\},$$

$$\displaystyle\chi_{Y}=\chi_{Y_{0}}-\big{(}\sum_{i=1}^{m}\chi_{Y_{i}}-\sum_{i,j=1}\chi_{Y_{i}\cap Y_{j}}+\sum_{i,j,k=1}\chi_{Y_{i}\cap Y_{j}\cap Y_{k}}+\cdots+(-1)^{m+1}\chi_{Y_{1}\cap Y_{2}\cap\ldots\cap Y_{m}}\big{)},$$

$$\Vert u{\Vert }_{A_{p,\pi }}=\inf \{\sum _{n=1}^{\infty }\Vert {\xi }_n\Vert \Vert {\eta }_n\Vert :u(\cdot )=⟨\pi (\cdot ){\xi }_n,{\eta }_n⟩,({\xi }_n{)}_{n\in \mathbb{N}}\subset E,({\eta }_n{)}_{n\in \mathbb{N}}\subset E^{\ast }\},u\in A_{p,\pi }.$$

$$\displaystyle\Psi_{p,\pi}:E^{*}\widehat{\otimes}E\rightarrow C_{b}(G),\qquad\Psi_{p,\pi}\bigg{(}\sum_{n=1}^{\infty}\eta_{n}\otimes\xi_{n}\bigg{)}=\sum_{n=1}^{\infty}\langle\pi(\cdot)\xi_{n},\eta_{n}\rangle,$$

$$\Vert \sum _{n=1}^{\infty }{\xi }_n\otimes {\eta }_n+\ker {\Psi }_{p,\pi }\Vert$$

$$=\Vert \sum _{n=1}^{\infty }⟨\pi (\cdot ){\xi }_n,{\eta }_n⟩{\Vert }_{A_{p,\pi }}.$$

$$\Vert u{\Vert }_{A_{p,\pi }}=\inf \{\sum _{n=1}^{\infty }\Vert x_n\Vert \Vert y_n\Vert :u(\cdot )=\sum _{n=1}^{\infty }⟨{\rho }_n(\cdot )x_n,y_n⟩\},$$

$$C:=\inf \{\sum _{n=1}^{\infty }\Vert x_n\Vert \Vert y_n\Vert :u(\cdot )=\sum _{n=1}^{\infty }⟨{\rho }_n(\cdot )x_n,y_n⟩,({\rho }_n,F_n{)}_{n\in \mathbb{N}}\subset {\text{Cyc}}_{p,\pi }(G)\}.$$

Full text

Mohammad Ali Ahmadpoor

Department of Pure Mathematics, Faculty of Mathematical Sciences, University of Guilan, Rasht, Iran.

email m-a-ahmadpoor@phd.guilan.ac.ir

Marzieh Shams Yousefi

Department of Pure Mathematics, Faculty of Mathematical Sciences, University of Guilan, Rasht, Iran.

email m.shams@guilan.ac.ir

MSC2010: 46L07, 43A30, 47L10

keywords: Completely bounded homomorphisms, Fourier-Stieltjes algebras, $QSL_{p}$-spaces, Piecewise affine maps

1 Introduction

Let $G$ be a locally compact group. The Fourier algebra, $A(G)$, and the Fourier-Stieltjes algebra, $B(G)$, on the locally compact group $G$, have been found by Eymard in 1964 [9]. The general form of special type of maps on the Fourier and Fourier-Stieltjes algebras has been studied extensively. For example, when $G$ is an Abelian topological group, $A(G)$ is nothing except $L_{1}(\widehat{G})$, where $\widehat{G}$ is the Pontrjagin dual group of $G$, and $B(G)$ is isometrically isomorphic to $M(\widehat{G})$, the measure algebra. In this case, Cohen in [3] and [4] studied homomorphisms from $L_{1}(G)$ to $M(H)$, for Abelian groups $G$ and $H$, and gave the general form of these maps, as the weighted maps by a piecewise affine map on the underlying groups.

By [2, 8], we know that $A(G)$ and $B(G)$ are operator spaces as the predual of a von Neumann algebra, and the dual of a $C^{*}$-algebra, respectively. Ilie in [17] and [18] studied the completely bounded homomorphisms from the Fourier to the Fourier-Stieltjes algebras. It is shown that for a continuous piecewise affine map $\alpha:Y\subset H\rightarrow G$, the homomorphism $\Phi_{\alpha}:A(G)\rightarrow B(H)$, defined through

[TABLE]

is completely bounded. Moreover, in the cases that $\alpha$ is an affine map and a homomorphism, the homomorphism $\Phi_{\alpha}$ is completely contractive and completely positive, respectively.

The Figà-Talamanca-Herz algebras were introduced by Figà-Talamanca for Abelian locally compact groups [11], and it is generalized for any locally compact group by Herz [15]. For $p\in(1,\infty)$, coefficient functions of the left regular representation of a locally compact group $G$ on $L_{p}(G)$ give us the Figà-Talamanca-Herz algebra $A_{p}(G)$, and we have $A_{2}(G)=A(G)$. Therefore, Figà-Talamanca-Herz algebras can be seen as the $p$-analog of the Fourier algebras.

Daws in [6] introduced the $p$-operator space structure, with an extensive application to $A_{p}(G)$, which generalizes the operator space structure of $A(G)$.

Oztop and Spronk in [21], and Ilie in [16] studied the $p$-completely bounded homomorphisms on the Figà-Talamanca-Herz algebras, using the $p$-operator space structure. In [16] it is shown that the map $\Phi_{\alpha}:A_{p}(G)\rightarrow A_{p}(H)$, defined via

[TABLE]

is a $p$-completely (bounded) contractive homomorphism for a continuous proper (piecewise) affine map $\alpha:Y\subset H\rightarrow G$ in the case that the locally compact group $H$ is amenable.

Runde in [23] found $p$-analog of the Fourier-Stieltjes algebras, $B_{p}(G)$. He used extensively the theory of $QSL_{p}$-spaces and representations on these spaces. Also, he gave the $p$-operator space structure of $B_{p}(G)$. More precisely, it is indicated that the space $B_{p}(G)$ is the dual space of the algebra of universal $p$-pseudofunctions $UPF_{p}(G)$, i.e. $B_{p}(G)=UPF_{p}(G)^{*}$. Therefore, by combining this result with the fact that for a concrete operator space like $UPF_{p}(G)$, we have $B_{p}(G)=\mathcal{CB}_{p}(UPF_{p}(G),\mathbb{C})$ [6], it can be concluded that $B_{p}(G)$ is a $p$-operator space as a dual of a $p$-operator space. The second author of this paper studied the $p$-analog of the Fourier-Stieltjes algebras on the inverse semigroups in [25].

In this paper, for a continuous proper map $\alpha:Y\subset H\rightarrow G$, we study the weighted maps $\Phi_{\alpha}:B_{p}(G)\rightarrow B_{p}(H)$ which is defined by

[TABLE]

We will show that when $\alpha$ is an affine map, $\Phi_{\alpha}$ is a $p$-complete contraction, and in the case that $\alpha$ is a piecewise affine map, it is $p$-completely bounded homomorphism. For this aim, we put amenability assumption on open subgroups of $H$. Our approach to the concept of $p$-operator space structure on the $p$-analog of the Fourier-Stieltjes algebra, is the $p$-operator structure that can be implemented on this space from its predual.

The paper is organized as follows: First we give required definitions and theorems about the $p$-analog of the Fourier-Stieltjes algebras and representations on $QSL_{p}$-spaces in Section 2. In Section 3, first we generalize Cohen-Host type idempotent theorem on the Fourier-Stieltjes algebras (see [24]) to the $p$-analog of the Fourier-Stieltjes algebras in Theorem 3.5. In addition, by Theorem 3.11 we give general form of the main theorem in [5]. For this aim, we need to give $p$-analog of some results in [1]. As a crucial theorem in this paper, we have Theorem 4.6 in Section 4 which will be applied in the next section. Final section, Section 5, is about generalization of Ilie’s results on homomorphisms of the Figà-Talamanca-Herz algebras in [16]. Indeed, we study $p$-completely boundedness of homomorphisms of the form (3).

2 Preliminaries

In this paper, $G$ and $H$ are locally compact groups, and for $p\in(1,\infty)$, the number $p^{\prime}$ is its complex conjugate, i.e. $1/p+1/p^{\prime}=1$. In the first step, we give essential notions and definitions on $QSL_{p}$-spaces, and representations of groups on such spaces. For more information one can see [23].

Definition 2.1**.**

A representation of a locally compact group $G$ is a pair $(\pi,E)$, where $E$ is a Banach space and $\pi$ is a group homomorphism from $G$ into the invertible isometries on $E$, that is continuous with respect to the given topology on $G$ and the strong operator topology on $\mathcal{B}(E)$.

Remark 2.2**.**

Every representation $(\pi,E)$ of a locally compact group $G$ induces a representation of the group algebra $L_{1}(G)$ on $E$, i.e. a contractive algebra homomorphism from $L_{1}(G)$ into $\mathcal{B}(E)$, which we shall denote likewise by $\pi$, through

[TABLE]

where the integral (4) converges with respect to the strong operator topology.

Definition 2.3**.**

Let $(\pi,E)$ and $(\rho,F)$ be representations of the locally compact group $G$. Then

$(\pi,E)$ and $(\rho,F)$ are called equivalent, if there exists an invertible isometry $\varphi:E\rightarrow F$ such that

[TABLE] 2. 2.

$(\rho,F)$ is said to be a subrepresentation of $(\pi,E)$, if $F$ is a closed subspace of $E$, and for every $x\in G$ we have $\pi(x){|_{F}}=\rho(x)$. 3. 3.

$(\rho,F)$ is said to be contained in $(\pi,E)$, if it is equivalent to a subrepresentation of $(\pi,E)$, and will be denoted by $(\rho,F)\subset(\pi,E)$.

Definition 2.4**.**

A Banach space is called an $L_{p}$-space if it is of the form $L_{p}(X)$ for some measure space $X$. 2. 2.

A Banach space is called a $QSL_{p}$-space if it is isometrically isomorphic to a quotient of a subspace of an $L_{p}$-space.

We denote by $\text{Rep}_{p}(G)$ the collection of all (equivalence classes) of representations of $G$ on a $QSL_{p}$-space.

Definition 2.5**.**

A representation of a Banach algebra $\mathcal{A}$ is a pair $(\pi,E)$, where $E$ is a Banach space, and $\pi$ is a contractive algebra homomorphism from $\mathcal{A}$ to $\mathcal{B}(E)$. We call $(\pi,E)$ isometric if $\pi$ is an isometry and essential if the linear span of $\{\pi(a)\xi\>:\ a\in\mathcal{A},\>\xi\in E\}$ is dense in $E$.

Remark 2.6**.**

If $G$ is a locally compact group and $(\pi,E)$ is a representation of $G$ in the sense of Definition 2.1, then (4) induces an essential representation of $L_{1}(G)$. Conversely, every essential representation of $L_{1}(G)$ arises in this fashion.

Definition 2.7**.**

A representation $(\pi,E)\in\text{Rep}_{p}(G)$ is called cyclic, if there exists $\xi_{0}\in E$ such that $\pi(L_{1}(G))\xi_{0}$ is dense in $E$. The set of cyclic representations of group $G$ on $QSL_{p}$-spaces is denoted by $\text{Cyc}_{p}(G)$. 2. 2.

A representation $(\pi,E)\in\text{Rep}_{p}(G)$ is called $p$-universal, if it contains every cyclic representation.

Remark 2.8**.**

We know that every $p$-universal representation of $G$, contains every cyclic representation of $G$ on a $QSL_{p}$-space, in the sense of equivalency. In Addition, every representation in $\text{Rep}_{p}(G)$ is contained in a $p$-universal representation. Actually, one could make a new $p$-universal representation by constructing a direct sum of an arbitrary representation with a $p$-universal representation. For more about representations of a locally compact group $G$ on $L_{p}$-spaces and $QSL_{p}$-spaces see [14].

Now we are ready to describe the Figà-Talamanca-Herz, and the $p$-analog of the Fourier-Stieltjes algebras.

Definition 2.9**.**

Figà-Talamanca-Herz algebra on the locally compact group $G$, which is denoted by $A_{p}(G)$, is the collection of functions $u:G\rightarrow\mathbb{C}$ of the form

[TABLE]

with

[TABLE]

where $\lambda_{p}$ is the left regular representation of $G$ on $L^{p}(G)$, defined as

[TABLE]

The norm of $A_{p}(G)$ is defined as

[TABLE]

where the infimum is taken over all expressions of $u$ in (5) with (6). With this norm and pointwise operations, $A_{p}(G)$ turns into a commutative regular Banach algebra.

Remark 2.10**.**

The $p$-analog of the Fourier-Stieltjes algebra has been studied, for example in [5], [13], [20] and [22], as the multiplier algebra of the Figà-Talamanca-Herz algebra. In this paper, we follow the construction of Runde in definition and notation (See [23]) which we swap indexes $p$ and $p^{\prime}$.

Definition 2.11**.**

The set of all functions of the form

[TABLE]

equipped with the norm

[TABLE]

where

[TABLE]

is denoted by $B_{p}(G)$, and is called the $p$-analog of the Fourier-Stieltjes algebra of the locally compact group $G$.

Remark 2.12**.**

By **[23, Lemma 4.6]**, the space $B_{p}(G)$ can be defined to be the set of all coefficient functions of a $p$-universal representation $(\pi,E)$, and the norm of an element $u\in B_{p}(G)$ is the infimum of all values $\sum_{n=1}^{\infty}\|\xi_{n}\|\|\eta_{n}\|<\infty$, which such vectors exist in the representation of $u$ as a coefficient function of $(\pi,E)$, i.e. $u(\cdot)=\sum_{n=1}^{\infty}\langle\pi(\cdot)\xi_{n},\eta_{n}\rangle$. 2. 2.

By **[23, Theorem 4.7]**, the space $B_{p}(G)$ equipped with the norm defined as above, and pointwise operations is a commutative unital Banach algebra, and by **[23, Corollary 5.3]**, by denoting multiplier algebra of $A_{p}(G)$ by $\mathcal{M}(A_{p}(G))$, we have the following contractive embeddings

[TABLE] 3. 3.

In **[24, Lemma 2.4]**, the following identification is shown for an open subgroup $G_{0}$ of a locally compact group $G$

[TABLE]

and through this fact, one can assume that functions in $A_{p}(G_{0})$ are restriction of functions in $A_{p}(G)$ to the open subgroup $G_{0}$.

Definition 2.13**.**

Let $(\pi,E)\in\text{Rep}_{p}(G)$.

For each $f\in L_{1}(G)$, let $\|f\|_{\pi}:=\|\pi(f)\|_{\mathcal{B}(E)}$, then $\|\cdot\|_{\pi}$ defines an algebra seminorm on $L_{1}(G)$. 2. 2.

By $PF_{p,\pi}(G)$, we mean the $p$-pseudofunctions associated with $(\pi,E)$, which is the closure of $\pi(L_{1}(G))$ in $\mathcal{B}(E)$. 3. 3.

If $(\pi,E)=(\lambda_{p},L_{p}(G))$, we denote $PF_{p,\lambda_{p}}(G)$ by $PF_{p}(G)$. 4. 4.

If $(\pi,E)$ is $p$-universal, we denote $PF_{p,\pi}(G)$ by $UPF_{p}(G)$, and call it the algebra of universal $p$-pseudofunctions.

Remark 2.14**.**

For $p=2$, the algebra $PF_{p}(G)$ is the reduced group $C^{*}$-algebra, and $UPF_{p}(G)$ is the full group $C^{*}$-algebra of $G$. 2. 2.

If $(\rho,F)\in\text{Rep}_{p}(G)$ is such that $(\pi,E)$ contains every cyclic subrepresentation of $(\rho,F)$, then $\|\cdot\|_{\rho}\leq\|\cdot\|_{\pi}$ holds. In particular, the definition of $UPF_{p}(G)$ is independent of a particular $p$-universal representation. 3. 3.

With $\langle\cdot,\cdot\rangle$ denoting $L_{1}(G)-L_{\infty}(G)$ duality, and with $(\pi,E)$ a $p$-universal representation of $G$, we have

[TABLE]

Next lemma states that $B_{p}(G)$ is a dual space.

Lemma 2.15**.**

[23, Lemma 6.5]** Let $(\pi,E)\in\text{Rep}_{p}(G)$. Then, for each $\phi\in PF_{p,\pi}(G)^{*}$, there is a unique $g\in B_{p}(G)$, with $\|g\|_{B_{p}(G)}\leq\|\phi\|$ such that

[TABLE]

Moreover, if $(\pi,E)$ is $p$-universal, we have $\|g\|_{B_{p}(G)}=\|\phi\|$.

The $p$-operator space structure which is used in this paper is Daws’ approach for $A_{p}(G)$ [6]. A concrete $p$-operator space is a closed subspace of $\mathcal{B}(E)$, for some $QSL_{p}$-space $E$. In this case for each $n\in\mathbb{N}$ one can define a norm $\|\cdot\|_{n}$ on $\mathbb{M}_{n}(X)=\mathbb{M}_{n}\otimes X$ by identifying $\mathbb{M}_{n}(X)$ with a subspace of $\mathcal{B}(l_{p}^{n}\otimes_{p}E$). So, we have the family of norms \Big{(}\|\cdot\|_{n}\Big{)}_{n\in\mathbb{N}} satisfying:

1. $\mathcal{D}_{\infty}:$

For $u\in\mathbb{M}_{n}(X)$ and $v\in\mathbb{M}_{m}(X)$, we have that $\|u\oplus v\|_{n+m}=\max\{\|u\|_{n},\|v\|_{m}\}$. Here $u\oplus v\in\mathbb{M}_{n+m}(X)$ has block representation $\begin{pmatrix}u&0\\ 0&v\end{pmatrix}.$ 2. $\mathcal{M}_{p}:$

For every $u\in\mathbb{M}_{m}(X)$ and $\alpha\in\mathbb{M}_{n,m}$, $\beta\in\mathbb{M}_{m,n}$ we have that

[TABLE]

Definition 2.16**.**

A linear operator $\Psi:X\rightarrow Y$ between two $p$-operator spaces is called $p$-completely bounded, if $\|\Psi\|_{\text{p-cb}}=\sup_{n\in\mathbb{N}}\|\Psi^{(n)}\|<\infty$, and $p$-completely contractive if $\|\Psi\|_{\text{p-cb}}=\sup_{n\in\mathbb{N}}\|\Psi^{(n)}\|\leq 1$, where $\Psi^{(n)}:\mathbb{M}_{n}(X)\rightarrow\mathbb{M}_{n}(Y)$ is defined in the natural way.

Theorem 2.17**.**

[6, Theorem 4.3]** Let $X$ be a $p$-operator space. There exists a $p$-complete isometry $\varphi:X^{*}\rightarrow\mathcal{B}(l_{p}(I))$ for some index set $I$.

Lemma 2.18**.**

[6, Lemma 4.5]** If $\Psi:X\rightarrow Y$ is $p$-completely bounded map between two operator spaces $X$ and $Y$, then $\Psi^{*}:Y^{*}\rightarrow X^{*}$ is $p$-completely bounded, with $\|\Psi^{*}\|_{\text{p-cb}}\leq\|\Psi\|_{\text{p-cb}}$.

Remark 2.19**.**

It should be noticed that, converse of Lemma 2.18 is not necessarily true, unless $X$ be a closed subspace of $\mathcal{B}(E)$, for some $L_{p}$-space $E$. 2. 2.

In the case of $p$-analog of the Fourier-Stieltjes algebras, from duality $B_{p}(G)=UPF_{p}(G)^{*}$, and the fact that $UPF_{p}(G)\subset\mathcal{B}(E)$, for $p$-universal representation $(\pi,E)$, by Theorem 2.17, we can induce $p$-operator space structure from predual $UPF_{p}(G)$ on $B_{p}(G)$ through identification $B_{p}(G)=\mathcal{CB}_{p}(UPF_{p}(G),\mathbb{C})$. 3. 3.

In comparison to **[16]**, because of above explanations, a major difference in our work is that we need to study predual of some crucial $p$-completely bounded maps (See Theorem 4.6), instead of their duals.

In Section 5, we will study the homomorphisms on the $p$-analog of the Fourier-Stieltjes algebras induced by the continuous map $\alpha:Y\subset H\rightarrow G$, in the cases that $\alpha$ is homomorphism, affine and piecewise affine map, and $Y$ in the coset ring of $H$. So, we give some preliminaries here.

For a locally compact topological group $H$, let $\Omega_{0}(H)$ denote the ring of subsets which generated by open cosets of $H$. By [16] we have

[TABLE]

Moreover, for a set $Y\subset H$, by $\text{Aff}(Y)$ we mean the smallest coset containing $Y$, and if $Y=Y_{0}\backslash\cup_{i=1}^{n}Y_{i}\in\Omega_{0}(H)$, then $\text{Aff}(Y)=Y_{0}$. Similarly, let us denote by $\Omega_{\text{am-}0}(H)$ the ring of open cosets of open amenable subgroups of $H$, i.e.

[TABLE]

Definition 2.20**.**

Let $\alpha:Y\subset H\rightarrow G$ be a map.

The map $\alpha$ is called an affine map on an open coset $Y$ of an open subgroup $H_{0}$, if

[TABLE] 2. 2.

The map $\alpha$ is called a piecewise affine map if

1. (a)

there are pairwise disjoint $Y_{i}\in\Omega_{0}(H)$, for $i=1,\ldots,n$, such that $Y=\cup_{i=1}^{n}Y_{i}$, 2. (b)

there are affine maps $\displaystyle{\alpha_{i}:\text{Aff}(Y_{i})\subset H\rightarrow G}$, for $i=1,\ldots,n$, such that

[TABLE]

Definition 2.21**.**

If $X$ and $Y$ are locally compact spaces, then a map $\alpha:Y\rightarrow X$ is called proper, if $\alpha^{-1}(K)$ is compact subset of $Y$, for every compact subset $K$ of $X$.

Proposition 2.22**.**

[7, Proposition 4]** Let $\alpha:H\rightarrow G$ be a continuous group homomorphism. Then $\alpha$ is proper if and only if the bijective homomorphism $\tilde{\alpha}:H/{\ker\alpha}\rightarrow\alpha(H)=G_{0}$, is a topological group isomorphism, and $\ker\alpha$ is compact.

Remark 2.23**.**

Proposition 2.22 implies that every continuous proper homomorphism is automatically a closed map. Therefore, $\alpha(H)$ is a closed subgroup of $G$. Also, $\ker\alpha$ is a compact normal subgroup of $H$. 2. 2.

It is well-known that $\tilde{\alpha}$ is a group isomorphism, if and only if $\alpha$ is an open homomorphism into $\alpha(H)$, with the relative topology. 3. 3.

[17, Remark 2.2]** If $Y=h_{0}H_{0}$ is an open coset of an open subgroup $H_{0}\subset H$, and $\alpha:Y\subset H\rightarrow G$ is an affine map, then there exists a group homomorphism $\beta$ associated to $\alpha$ such that

[TABLE] 4. 4.

It is clear that, $\alpha$ is a proper affine map, if and only if $\beta$ is a proper homomorphism. 5. 5.

[16, Lemma 8]** Let $Y\in\Omega_{0}(H)$, and $\alpha:\text{Aff}(Y)\rightarrow G$ be an affine map such that $\alpha|_{Y}$ is proper, then $\alpha$ is proper.

3 Some generalizations

In order to being prepared for Theorem 3.5 which is a generalization of [24, Theorem 1.5], we need some elementary definitions and facts which we give them in the following.

Definition 3.1**.**

A Banach space $(E,\|\cdot\|)$ is said to be uniformly convex if for every $0<\epsilon\leq 2$ there is $\delta>0$ so that for any two vectors $x$ and $y$ in $E$ with $\|x\|=\|y\|=1$, the condition $\|x-y\|\geq\epsilon$ implies that $\|\frac{x+y}{2}\|\leq 1-\delta$. Intuitively, the center of a line segment inside the unit ball must lie deep inside the unit ball unless the segment is short. 2. 2.

A Banach space $E$ is said to be smooth if for each $\xi\in E\backslash\{0\}$ there exists a unique $\eta\in E^{*}$ such that $\|\eta\|=1$ and $\langle\xi,\eta\rangle=\|\xi\|$.

Remark 3.2**.**

It is worthwhile to note that by Definition 3.1, every closed subspace of a uniformly convex Banach space is again a uniformly convex Banach space.

Now we state an immensely important theorem about a quotient space which can be found in [19].

Theorem 3.3**.**

[19, Theorem 2.4.18]** Let $E$ be a uniformly convex Banach space and $F$ be a closed linear subspace of $E$. Then the quotient space $E/F$ is uniformly convex Banach space.

Now we can conclude the following statement.

Corollary 3.4**.**

Every $QSL_{p}$-space $E$ is uniformly convex and smooth.

Proof.

Uniformly convexity of $QSL_{p}$-space $E$ can be derived from Remark 3.2 and Theorem 3.3. Since $E$ is uniformly convex, by [10, Lemma 8.4(i) and Theorem 9.10] it is concluded that $E^{*}$ is smooth, but $E^{*}$ is a $QSL_{p^{\prime}}$-space so is uniformly convex, and then $E^{**}$ is smooth, but $E=E^{**}$ so $E$ is smooth. ∎

Theorem 3.5**.**

For a subset $C\subset G$ following statements are equivalent.

$C$* is a left open coset,* 2. 2.

$\chi_{C}\in B(G)$* with $\|\chi_{C}\|_{B(G)}=1$,* 3. 3.

$\chi_{C}\neq 0$* is a normalized coefficient function of a representation $(\pi,E)$ where $E$ or $E^{*}$ is smooth,* 4. 4.

$\chi_{C}\in B_{p}(G)$* with $\|\chi_{C}\|_{B_{p}(G)}=1$.*

Proof.

Equivalency of the first three statements have been proved in [24, Theorem 1.5]. We demonstrate (2)$\Rightarrow$(4)$\Rightarrow$(3). Let (2) hold. Then from the fact that $B(G)\subset B_{p}(G)$ and this embedding is a contraction, we have $\chi_{C}\in B_{p}(G)$ with $\|\chi_{C}\|_{B_{p}(G)}\leq 1$, which by inequality $\|\cdot\|_{C_{b}(G)}\leq\|\cdot\|_{B_{p}(G)}$, we have $\|\chi_{C}\|_{B_{p}(G)}=1$ which shows (2) implies (4).

Now let $\chi_{C}\in B_{p}(G)$ with $\|\chi_{C}\|_{B_{p}(G)}=1$. So, by Definition 2.11, the function $\chi_{C}$ is a normalized coefficient function of an isometric group representation on a $QSL_{p}$-space, which is smooth by Corollary 3.4 that is (3). ∎

Corollary 3.6**.**

Let $G$ be a locally compact group and $Y\in\Omega_{0}(G)$, then we have $\chi_{Y}\in B_{p}(G)$. Moreover, we have

[TABLE]

where for $i=0,1,\ldots,m$ sets $Y_{i}$, are as (13).

Proof.

Since $Y\in\Omega_{0}(G)$, then by (13), there exist open coset $Y_{0}$ and open subcosets $Y_{i}\subset Y_{0}$, for $i=1,\ldots,m$ and $m\in\mathbb{N}$ such that $Y=Y_{0}\backslash\cup_{i=1}^{m}Y_{i}$. By Theorem 3.5-(4), we have $\chi_{Y_{i}}\in B_{p}(G)$, with $\|\chi_{Y_{i}}\|_{B_{p}(G)}=1$, for $i=0,1,\ldots,m$. On the other hand, since

[TABLE]

then we have $\|\chi_{Y}\|_{B_{p}(G)}\leq 2^{m}$, and by taking infimum on all possible decomposition of $Y$ as (13) relation (15) holds. ∎

In the sequel, we will give some extensions of results in [1]. For a unitary representation $(\pi,\mathcal{H}_{\pi})$ with Hilbert space $\mathcal{H}_{\pi}$, the $\pi$-Fourier space has been defined to be closed linear span of the set of the coefficient functions of the representation $(\pi,\mathcal{H}_{\pi})$, and is denoted by $A_{\pi}$ with the norm in usual way. Moreover, $\pi$-Fourier-Stieltjes algebra, $B_{\pi}$, for such representation is defined to be $w^{*}$-closure of $A_{\pi}$. Additionally, if we let $C^{*}_{\pi}(G)$ be the $C^{*}$-algebra associated with $\pi$, we have $B_{\pi}=C^{*}_{\pi}(G)^{*}$. Here we introduce $p$-generalization of these results.

Definition 3.7**.**

For a representation $(\pi,E)\in\text{Rep}_{p}(G)$, we define the $p$-analog of the $\pi$-Fourier space, $A_{p,\pi}$, to be closed linear span of the collection of the coefficient functions of representation $(\pi,E)$ equipped with the norm

[TABLE]

Remark 3.8**.**

Let $(\pi,E)\in\text{Rep}_{p}(G)$. Consider the following map,

[TABLE]

so we can identify coimage of $\Psi_{p,\pi}$ with Banach space $E^{*}\widehat{\otimes}E/\ker{\Psi_{p,\pi}}$, which implies that the norm on coimage is the quotient norm i.e.,

[TABLE]

So, one can identify $A_{p,\pi}$ with coimage of $\Psi_{p,\pi}$, or equivalently with the quotient space $E^{*}\widehat{\otimes}E/\ker\Psi_{p,\pi}$. 2. 2.

Since we have $A_{p,\pi}\cong E^{*}\widehat{\otimes}E/\ker\Psi_{p,\pi}$, then the space $A_{p,\pi}$ is a Banach space.

Proposition 3.9**.**

Let $(\pi,E)\in\text{Rep}_{p}(G)$ and $\text{Cyc}_{p,\pi}(G)=\{(\rho,F)\in\text{Cyc}_{p}(G)\ \&\ (\rho,F)\subset(\pi,E)\}$. Then for $u(\cdot)=\sum_{n=1}^{\infty}\langle\pi(\cdot)\xi_{n},\eta_{n}\rangle\in A_{p,\pi}$ we have

[TABLE]

where the infimum is taken on all representations of $u$ in which $((\rho_{n},F_{n}))_{n\in\mathbb{N}}\subset\text{Cyc}_{p,\pi}(G)$ with $(x_{n})_{n\in\mathbb{N}}\subset F_{n}$ and $(y_{n})_{n\in\mathbb{N}}\subset F^{*}_{n}$.

Proof.

Let

[TABLE]

Assume that $u(\cdot)=\sum_{n=1}^{\infty}\langle\pi(\cdot)\xi_{n},\eta_{n}\rangle$ with $\sum_{n=1}^{\infty}\|\xi_{n}\|\|\eta_{n}\|<\infty$. For each $n\in\mathbb{N}$ we may put

[TABLE]

then we have

[TABLE]

with $C\leq\sum_{n=1}^{\infty}\|x_{n}\|\|y_{n}\|\leq\sum_{n=1}^{\infty}\|\xi_{n}\|\|\eta_{n}\|$. Since $(\xi)_{n\in\mathbb{N}}\subset E$ and $(\eta)_{n\in\mathbb{N}}\subset E^{*}$ are arbitrary in the representing of $u$, we have $C\leq\|u\|_{A_{p,\pi}}$.

For the inverse inequality, let $\epsilon>0$ is given. Then there exist $((\rho_{n},F_{n}))_{n\in\mathbb{N}}\subset\text{Cyc}_{p,\pi}(G)$, $(x_{n})_{n\in\mathbb{N}}\subset F_{n}$, $(y_{n})_{n\in\mathbb{N}}\subset F_{n}^{*}$, and for each $n\in\mathbb{N}$, we have $(\rho_{n},F_{n})\subset(\pi,E)$ such that

[TABLE]

Now for each $n\in\mathbb{N}$, by applying Hahn-Banach theorem extend each $y_{n}\in F_{n}^{*}$ to the $\eta_{n}\in E^{*}$ such that $\|\eta_{n}\|=\|y_{n}\|$. Therefore,

[TABLE]

and it means $\|u\|_{A_{p,\pi}}\leq C$.

∎

For a representation $(\pi,E)\in\text{Rep}_{p}(G)$, by \big{(}\pi^{\infty},l_{p}(\mathbb{N},E)\big{)} we denote the representation

[TABLE]

Similarly, for a free ultrafilter $\mathcal{U}$, and ultrapower of the space $l_{p}(\mathbb{N},E)$, by \big{(}(\pi^{\infty})_{\mathcal{U}},l_{p}(\mathbb{N},E\big{)}_{\mathcal{U}}), we mean the representation

[TABLE]

Proposition 3.10**.**

For each $(\pi,E)\in\text{Rep}_{p}(G)$, there exists a free ultrafilter $\mathcal{U}$, such that by restricting $(\pi^{\infty})_{\mathcal{U}}$ to the subspace

[TABLE]

the representation $((\pi^{\infty})_{\mathcal{U}},F)$ is weak-weak∗ continuous, essential and isometric representation of $PF_{p,\pi}(G)$, and we have $PF_{p,\pi}(G)^{*}=\overline{A_{p,(\pi^{\infty})_{\mathcal{U}}}{}}^{w^{*}}$.

Proof.

By [23, Lemma 6.5], there exists a free ultrafilter $\mathcal{U}$ such that canonical representation of $PF_{p,\pi}(G)$ on $l_{p}(\mathbb{N},E)_{\mathcal{U}}$ is weak-weak∗ continuous and isometric, and by restricting this representation to the subspace $F$, it is essential so. Indeed, this representation is an essential representation of $L_{1}(G)$ as following

[TABLE]

Moreover, this representation comes from a representation of $G$ which we still denote it by $((\pi^{\infty})_{\mathcal{U}},F)$. We need to notice that the space $F$ is a $QSL_{p}$-space, therefore, it is super-reflexive, and we have

[TABLE]

Since $(\pi^{\infty})_{\mathcal{U}}:PF_{p,\pi}(G)\rightarrow\mathcal{B}(F)$ is weak-weak∗ continuous and isometric, then $(\pi^{\infty})_{\mathcal{U}}^{*}$ restricted to $F^{*}\widehat{\otimes}F$ is a quotient map onto $PF_{p,\pi}(G)^{*}$, so we have

[TABLE]

Indeed, the restricted map $(\pi^{\infty})_{\mathcal{U}}^{*}:F^{*}\widehat{\otimes}F\rightarrow PF_{p,\pi}(G)^{*}$ is of the type of maps in Remark 3.8-(1), $\Psi_{p,(\pi^{\infty})_{\mathcal{U}}}$.We note that $\ker(\pi^{\infty})_{\mathcal{U}}^{*}$ is weak∗ closed. Additionally, since $(\pi^{\infty})_{\mathcal{U}}$ is one-to-one, then $\mathcal{R}((\pi^{\infty})_{\mathcal{U}}^{*})$, range of $(\pi^{\infty})_{\mathcal{U}}^{*}$, is weak∗ dense in $PF_{p,\pi}(G)^{*}$, so we have

[TABLE]

∎

Next theorem is a generalization of [5, Theorem 4], in which for a compact subset $K\subset G$, by $A_{p,(\pi^{\infty})_{\mathcal{U}}}|_{K}$ we mean the restriction of functions in $A_{p.(\pi^{\infty})_{\mathcal{U}}}$ to $K$. It is evident that for a function $u\in A_{p,(\pi^{\infty})_{\mathcal{U}}}$ and compact set $K\subset G$ we have $\|u|_{K}\|_{A_{p,(\pi^{\infty})_{\mathcal{U}}}|_{K}}\leq\|u\|_{A_{p,(\pi^{\infty})_{\mathcal{U}}}}$, where the norm $\|u|_{K}\|_{A_{p,(\pi^{\infty})_{\mathcal{U}}}|_{K}}$ is naturally defined to be the infimum of all possible expressions of restricted function $u|_{K}$ as a coefficient function of representation $((\pi^{\infty})_{\mathcal{U}},F)$.

Theorem 3.11**.**

Let $(\pi,E)\in\text{Rep}_{p}(G)$, then a function $w\in L_{\infty}(G)$ belongs to $PF_{p,\pi}(G)^{*}$ with $\|w\|\leq C$ if and only if $w|_{K}\in A_{p,(\pi^{\infty})_{\mathcal{U}}}|_{K}$ with $\|w|_{K}\|_{A_{p,(\pi^{\infty})_{\mathcal{U}}}|_{K}}\leq C$, for every compact subset $K$ of $G$.

Proof.

1. First, we assume that $w|_{K}\in A_{p,(\pi^{\infty})_{\mathcal{U}}}|_{K}$, with $\|w|_{K}\|_{A_{p,(\pi^{\infty})_{\mathcal{U}}}|_{K}}\leq C$, for all compact subset $K\subset G$. Let $f\in L_{1}(G)$ with compact support $K_{f}$, then

[TABLE]

Since $w|_{K_{f}}\in A_{p,(\pi^{\infty})_{\mathcal{U}}}|_{K_{f}}$, then $w|_{K_{f}}(\cdot)=\sum_{n=1}^{\infty}\langle(\pi^{\infty})_{\mathcal{U}}(\cdot)\xi_{n}^{f},\eta_{n}^{f}\rangle$ with $(\xi_{n}^{f})_{n=1}^{\infty}\subset F$ and $(\eta_{n}^{f})_{n=1}^{\infty}\subset F^{*}$, so we have

[TABLE]

Consequently, we have $|\langle\pi(f),w\rangle|\leq C\|f\|_{\pi}$, and since compact support functions are dense in $PF_{p,\pi}(G)$, so we have $w\in PF_{p,\pi}(G)^{*}$, and $\|w\|\leq C$. 2. Now let $w\in PF_{p,\pi}(G)^{*}=\overline{A_{p,(\pi^{\infty})_{\mathcal{U}}}{}}^{w^{*}}$. It follows from Proposition 3.10 and [23, Lemma 6.5] that there exists a unique $u\in B_{p}(G)$ which is a coefficient function of the representation $((\pi^{\infty})_{\mathcal{U}},F)$ such that for every $\epsilon>0$ there are vectors $(\xi_{n})_{n\in\mathbb{N}}\subset F$ and $(\eta_{n})_{n\in\mathbb{N}}\subset F^{*}$ so that

[TABLE]

and

[TABLE]

which (17) and (18) mean that $w$ and $u$ are equal as functionals on $PF_{p,\pi}(G)$, and by Hahn-Banach theorem we have $w=u$ with $\|w\|=\|u\|$. Therefore, the restriction of $w$ to every compact subset $K\subset G$ means the restriction of $u$ to $K$, and we have

[TABLE]

∎

Remark 3.12**.**

We follow **[1]** in notation, and denote $\overline{A_{p,(\pi^{\infty})_{\mathcal{U}}}{}}^{w^{*}}$ by $B_{p,\pi}$, and we call it $p$-analog of the $\pi$-Fourier-Stieltjes algebra, which by Proposition 3.10 is the dual space of the space of $p$-pseudofunctions associated with $(\pi,E)\in\text{Rep}_{p}(G)$, i.e. the dual space of $PF_{p,\pi}(G)$ through following duality

[TABLE]

and as we expect that, we have

[TABLE] 2. 2.

It is obvious that $B_{p,\pi}\subset B_{p}(G)$ is a contractive inclusion for every $(\pi,E)\in\text{Rep}_{p}(G)$, and if $(\pi,E)$ is a $p$-universal representation it will become an isometric isomorphism. 3. 3.

It is valuable to note that if $\mathcal{V}$ is another free ultrafilter as it is described in Proposition 3.10, then we have

[TABLE]

So, our definition is independent of choosing suitable free ultrafilter, therefore, it is well-defined. 4. 4.

For a locally compact group $G$ we have the following relations

[TABLE]

and all inclusions will become equalities in the case that $G$ is amenable (See **[23, Theorem 6.6 and Theorem 6.7]**).

4 Applicable $p$-completely bounded homomorphisms on $B_{p}(G)$

In the following, we study completely boundedness of special type of maps on the $p$-analog of the Fourier-Stieltjes algebras. To provide requirements of forthcoming propositions, Theorem 4.6 plays a critical role. For this aim, we give next lemma that is a kind of application of Proposition 3.10.

Let $G_{0}\subset G$, be any subset, and $u:G_{0}\rightarrow\mathbb{C}$ be a function. By $u^{\circ}$ we mean

[TABLE]

Lemma 4.1**.**

Let $G_{0}$ be an open subgroup of the locally compact group $G$, and $(\pi,E)$ denote the $p$-universal representation of $G$. Then $(\pi_{G_{0}},E)$, restriction of $\pi$ to $G_{0}$, is a representation of $G_{0}$, and the restriction mapping of functions in $B_{p}(G)$ to $G_{0}$, is a contractive linear homomorphism into $B_{p,\pi_{G_{0}}}\subset B_{p}(G_{0})$. Moreover, we have the following contractive inclusions

[TABLE]

Proof.

Let us define

[TABLE]

which obviously implies that $(\pi_{G_{0}},E)\in\text{Rep}_{p}(G_{0})$, and by Remark 2.8 it is contained in a $p$-universal representation of $G_{0}$, namely $(\rho,F)$, and we have (up to an isometry)

[TABLE]

Consequently, through Remark 3.12-(2), for a function $u\in B_{p}(G)$, it can be obtained that

[TABLE]

For the inclusion $B_{p,\lambda_{p,G_{0}}}\subset B_{p,\pi_{G_{0}}}$, let $u\in A_{p}(G_{0})$, then by Remark 2.12-(3), we have $u^{\circ}\in A_{p}(G)\subset B_{p}(G)$. Since $B_{p,\pi_{G_{0}}}$ is the collection of functions in $B_{p}(G)$ restricted to $G_{0}$, and since $u=(u^{\circ})|_{G_{0}}$, then we have $u\in B_{p,\pi_{G_{0}}}$, and it means that $A_{p}(G_{0})\subset B_{p,\pi_{G_{0}}}$. Therefore, via Remark 3.12-(4), we have $\overline{A_{p}(G_{0}){}}^{w^{*}}=B_{p,\lambda_{p,G_{0}}}\subset B_{p,\pi_{G_{0}}}$. Additionally, one can reach to this inclusion by utilizing Theorem 3.11. ∎

Lemma 4.2**.**

Let $G_{0}$ be an open subgroup of the locally compact group $G$, and $u\in\mathcal{M}(A_{p}(G_{0}))$. Then we have $u^{\circ}\in\mathcal{M}(A_{p}(G))$ with $\|u^{\circ}\|_{\mathcal{M}(A_{p}(G))}=\|u\|_{\mathcal{M}(A_{p}(G_{0}))}$.

Proof.

Let $u\in\mathcal{M}(A_{p}(G_{0}))$ and $v\in A_{p}(G)$. By the relation $u^{\circ}\cdot v=(u\cdot v|_{G_{0}})^{\circ}$, it can be concluded that $u^{\circ}\in\mathcal{M}(A_{p}(G))$, and obviously we have $\|u^{\circ}\|_{\mathcal{M}(A_{p}(G))}=\|u\|_{\mathcal{M}(A_{p}(G_{0}))}$. ∎

Proposition 4.3**.**

Let $G$ be a locally compact group and $G_{0}$ be its open subgroup. Then

for every $u\in B_{p}(G_{0})$, we have $u^{\circ}\in\mathcal{M}(A_{p}(G))$, 2. 2.

if $G_{0}$ is also an amenable subgroup, then for every $u\in B_{p}(G_{0})$, we have $u^{\circ}\in B_{p}(G)$.

Proof.

This part can be concluded by the inclusions in Remark 3.12-(4) and Lemma 4.2. 2. 2.

Since $G_{0}$ is amenable, then by ”Moreover” part in the Lemma 4.1, and equalities in Remark 3.12-(4) we have the result.

∎

As an immediate consequence of Proposition 4.3, we have the next corollary.

Corollary 4.4**.**

Let $G$ and $H$ be locally compact groups, and $\alpha:Y=\cup_{k=1}^{n}Y_{k}\subset H\rightarrow G$ be a continuous piecewise affine map with disjoint $Y_{k}\in\Omega_{\text{am-}0}(H)$, for $k=1,\ldots,n$. Then $u\in B_{p}(G)$ implies $(u\circ\alpha)^{\circ}\in B_{p}(H)$.

Proof.

First of all, we note that similar to the case of the Fourier-Stieltjes algebras, a continuous homomorphism $\beta:H\rightarrow G$, between two locally compact groups $G$ and $H$, induces a homomorphism from $B_{p}(G)$ into $B_{p}(H)$, by taking $u\in B_{p}(G)$ to the function $u\circ\beta\in B_{p}(H)$, see [18].

Now we divide our proof into two steps.

First, we let $\alpha:Y=y_{0}H_{0}\rightarrow G$ be a continuous affine map, and $\beta:H_{0}\rightarrow G$ be the homomorphism associated with $\alpha$, as it is explained in Remark 2.23-(3), for an open amenable subgroup $H_{0}$ of $H$. As we initially explained, the map $u\mapsto u\circ\beta$ is an algebra homomorphism from $B_{p}(G)$ into $B_{p}(H_{0})$. Consider the following translation maps

[TABLE]

then by the following relation, and applying Proposition 4.3, we have the result

[TABLE]

Now let $\alpha:Y\subset H\rightarrow G$ be a continuous piecewise affine map, so by our assumption of amenability, and similar to the Definition 2.20, there exist pairwise disjoint sets $Y_{k}\in\Omega_{\text{am-}0}(H)$, for $k=1,\ldots,n$ with $n\in\mathbb{N}$, and affine maps $\alpha_{k}:\text{Aff}(Y_{k})\subset H\rightarrow G$ such that $Y=\cup_{k=1}^{n}Y_{k}$, and $\alpha_{k}|_{Y_{k}}=\alpha|_{Y_{k}}$. By previous step, we know that $(u\circ\alpha_{k})^{\circ}\in B_{p}(H)$, and since

[TABLE]

we have the result via Corollary 3.6, and the fact that $B_{p}(H)$ is a Banach algebra. ∎

Remark 4.5**.**

For an open amenable subgroup $G_{0}$ of the locally compact group $G$, by Proposition 4.3, we can say that the space $B_{p}(G_{0})$ is the space of functions which are restriction of functions in $B_{p}(G)$, those are equal to zero outside of $G_{0}$. Therefore, while we are working on the $p$-analog of the Fourier-Stieltjes algebras, we may assume that the $p$-universal representation of an open amenable subgroup $G_{0}$ of $G$ is the restriction of the $p$-universal representation of $G$ to $G_{0}$. 2. 2.

For an open amenable subgroup $G_{0}$ of $G$, by Lemma 4.1 and Proposition 4.3, the restriction mapping from $B_{p}(G)$ to $B_{p}(G_{0})$ is surjective. 3. 3.

In the case that $G$ is amenable, Proposition 4.3 can be concluded directly from Lemma 4.2, via the isometric identification $B_{p}(G)=\mathcal{M}(A_{p}(G))$.

Next theorem is our first main result of this paper, and it will be applied to give the results on weighted homomorphisms on the $p$-analog of the Fourier-Stieltjes algebras. For more clarification, we need to introduce the notion of the $p$-tensor product $E\tilde{\otimes}_{p}F$ of two $QSL_{p}$-spaces $E$ and $F$, that is defined in [23]. In fact, Runde introduced the norm $\|\cdot\|_{p}$ on the algebraic tensor product $E\otimes F$ which benefits from pivotal properties. As an important property of the norm $\|\cdot\|_{p}$, is the fact that the completion $E\tilde{\otimes}_{p}F$ of $E\otimes F$ with respect to $\|\cdot\|_{p}$ is a $QSL_{p}$-space. Furthermore, for two representations $(\pi,E)$ and $(\rho,F)$ of the locally compact group $G$ in $\text{Rep}_{p}(G)$, the representation $(\pi\otimes\rho,E\tilde{\otimes}_{p}F)$ is well-defined and belongs to $\text{Rep}_{p}(G)$. As a result, for two functions $u(\cdot)=\langle\pi(\cdot)\xi,\eta\rangle$ and $v(\cdot)=\langle\rho(\cdot)\xi^{\prime},\eta^{\prime}\rangle$, the pointwise product of them is a coefficient function of the representation $(\pi\otimes\rho,E\tilde{\otimes}_{p}F)$, i.e. $u\cdot v(\cdot)=\langle(\pi(\cdot)\otimes\rho(\cdot))(\xi\otimes\xi^{\prime}),\eta\otimes\eta^{\prime}\rangle$. For more details on $p$-tensor product $\tilde{\otimes}_{p}$ see [23, Theorem 3.1 and Corollary 3.2].

Theorem 4.6**.**

Let $p\in(1,\infty)$ and $G$ be a locally compact group. Then we have the following statements:

For any $(\pi_{p},E_{p})\in\text{Rep}_{p}(G)$, the identity map $I:B_{p,\pi_{p}}\rightarrow B_{p}(G)$ is a $p$-completely contractive map. 2. 2.

For an open subgroup $G_{0}$ of $G$, the restriction map $R_{G_{0}}:B_{p}(G)\rightarrow B_{p}(G_{0})$, is a $p$-completely contractive homomorphism. 3. 3.

For an element $a\in G$, the translation map $L_{a}:B_{p}(G)\rightarrow B_{p}(G)$, defined through $L_{a}(u)={}_{a}u$, where ${}_{a}u(x)=u(ax)$, for $x\in G$, is a $p$-completely contractive map. 4. 4.

For a closed normal subgroup $G_{1}$ of $G$, let $q:G\rightarrow G/G_{1}$ be the canonical quotient map. Then the homomorphism $\Phi_{q}:B_{p}(G/G_{1})\rightarrow B_{p}(G)$, with $\Phi_{q}(u)=u\circ q$, is a $p$-completely contractive homomorphism. 5. 5.

For an open amenable subgroup $G_{2}$ of $G$, the extension map $E_{G_{2}}:B_{p}(G_{2})\rightarrow B_{p}(G)$ is a $p$-completely contractive homomorphism. 6. 6.

For an open coset $Y$ of an open subgroup $G_{2}$ of $G$, the map $M_{Y}:B_{p}(G)\rightarrow B_{p}(G)$, with $M_{Y}(u)=u\cdot\chi_{Y}$, is $p$-completely contractive homomorphism. More generally, for a set $Y\in\Omega_{0}(G)$, the map $M_{Y}$ is a $p$-completely bounded homomorphism.

Proof.

We want to prove that for each $(\pi_{p},E_{p})\in\text{Rep}_{p}(G)$, the following map is a $p$-complete contraction.

[TABLE]

Let $(\pi,E)$ be a $p$-universal representation of $G$ that contains the representation $(\pi_{p},E_{p})$. Following relations hold between $(\pi_{p},E_{p})$, and $(\pi,E)$.

[TABLE]

Since $\pi_{p}(f)=\pi(f)|_{E_{p}}$, then $\|\pi_{p}(f)\|\leq\|\pi(f)\|$. Additionally, the map $I$ is weak∗-weak∗ continuous, and it is a contraction by [23, Theorem 6.6-$(i)$]. Define

[TABLE]

then ${}_{*}I$ is the predual of the map (19). Because, we have $\langle\pi(f),I(u)\rangle=\langle\pi_{p}(f),u\rangle$, for every $f\in L_{1}(G)$ and $u\in B_{p,\pi_{p}}$. Following calculations indicate that ${}_{*}I$ is a $p$-complete contraction: for each $n\in\mathbb{N}$, and $(\pi_{p}(f_{ij}))\in\mathbb{M}_{n}(PF_{p,\pi_{p}}(G))$ we have

[TABLE]

so we have $\|(\pi_{p}(f_{ij}))\|_{n}\leq\|(\pi(f_{ij}))\|_{n}$, and by this, it is concluded that

[TABLE] 2. 2.

Let $G_{0}\subset G$ be an open subgroup and consider the following map:

[TABLE]

Let $(\pi,E)$ be a $p$-universal representation of $G$, and $(\pi_{G_{0}},E)$ be the restriction of $(\pi,E)$ to $G_{0}$. Their liftings are related as following

[TABLE]

In addition, range of the map $R_{G_{0}}$ is the space $B_{p,\pi_{G_{0}}}\subset B_{p}(G_{0})$, as it is described in Lemma 4.1. This map is weak∗-weak∗ continuous by the relation (20), and

[TABLE]

So, we may define ${}_{*}R_{G_{0}}$ as following

[TABLE]

We have

[TABLE]

Therefore, $({}_{*}R_{G_{0}})^{*}=R_{G_{0}}$. Additionally, by (20) we have ${}_{*}R_{G_{0}}(\pi_{G_{0}}(f))=\pi(f^{\circ})=\pi_{G_{0}}(f)$, so ${}_{*}R_{G_{0}}$ is an identity map which is $p$-completely isometric

[TABLE]

therefore, $\|R_{G_{0}}\|_{\text{p-cb}}=\|({}_{*}R_{G_{0}})^{*}\|_{\text{p-cb}}\leq\|{}_{*}R_{G_{0}}\|_{\text{p-cb}}=1$. 3. 3.

Now we want to prove that for $a\in G$, the following map is a $p$-complete isometry

[TABLE]

Predual of the map $L_{a}$ is as following

[TABLE]

and it is clearly $p$-completely contractive, and consequently, this is true for $L_{a}$. On the other hand, the map $L_{a}$ has the inverse $L_{a^{-1}}$, and similar to $L_{a}$, it is $p$-completely contractive which makes $L_{a}$ to be $p$-completely isometric map. 4. 4.

Let $G_{1}\subset G$ be a closed normal subgroup. Let

[TABLE]

be the canonical quotient map, and

[TABLE]

Let $(\rho,F)$ be a $p$-universal representation of $G/G_{1}$. Then obviously we have $(\rho\circ q,F)\in\text{Rep}_{p}(G)$, and this representation is contained in a $p$-universal representation $(\pi,E)$ of $G$ which implies that (up to an isometry)

[TABLE]

Let us define closed subspace $K$ of $E$, which itself is a $QSL_{p}$-space, through

[TABLE]

and consider the representation of $G$, for which an element $x\in G$ goes to the restriction of $\pi(x)$ to $K$, so we can induce the following representation for $G/G_{1}$

[TABLE]

by the definition of $K$, we have $F\subset K$, therefore, $(\rho,F)\subset(\tilde{\pi},K)$. It is obtained that every $p$-universal representation of $G/G_{1}$, like $(\rho,F)$, is contained in a representation of $G/G_{1}$, as we described, $(\tilde{\pi},K)$, that is induced by the $p$-universal representation $(\pi,E)$ of $G$. So, we can work by $(\tilde{\pi},K)$ as the $p$-universal representation of $G/G_{1}$.

Now, let us consider the following map

[TABLE]

which is at least a contractive isomorphism into the subalgebra of $B_{p}(G)$, of functions which are constant on each coset of $G_{1}$. For functions $f\in L_{1}(G)$, and $u\in B_{p}(G/G_{1})$, we have

[TABLE]

where the map $P:L_{1}(G)\rightarrow L_{1}(G/G_{1})$ is defined [12]:

[TABLE]

This implies that the map $\Phi_{q}$ is weak∗-weak∗ continuous, and by this we define the predual map ${}_{*}\Phi_{q}$, as following:

[TABLE]

which by (21) we have $({}_{*}\Phi_{q})^{*}=\Phi_{q}$. For a function $v:G\rightarrow\mathbb{C}$ that is constant on the cosets of $G_{1}$, by $\tilde{v}$ we denote

[TABLE]

We need to note that for $f\in L_{1}(G)$, $\xi\in K$ and $\eta\in K^{*}$, we have

[TABLE]

so we have $\tilde{\pi}\circ q(f)=\tilde{\pi}(Pf)$, which means that the predual map ${}_{*}\Phi_{q}$ is an identity map that is $p$-completely isometric map via the following computation

[TABLE]

Therefore, we have $\|\Phi_{q}\|_{\text{p-cb}}\leq 1$. 5. 5.

Let $G_{2}\subset G$, be an open amenable subgroup, and $u\in B_{p}(G_{2})$. Since by Proposition 4.3 we have $u^{\circ}\in B_{p}(G)$, then we are allowed to define

[TABLE]

Let $(\pi,E)$ be a $p$-universal representation of $G$. We denote the restriction of $(\pi,E)$ to $G_{2}$ by $(\pi_{G_{2}},E)$ which is a $p$-universal representation of $G_{2}$ via Remark 4.5-(1). We note that by the relation

[TABLE]

the map $E_{G_{2}}$ is weak∗-weak∗ continuous, so we define the predual map ${}_{*}E_{G_{2}}$, as following:

[TABLE]

which by (22) we have $({}_{*}E_{G_{2}})^{*}=E_{G_{2}}$. We need to take notice of the fact that since $\chi_{G_{2}}\in B_{p}(G)$, via Theorem 3.5-(3), $\chi_{G_{2}}$ is a normalized coefficient function of $(\pi,E)$, i.e. there are $\xi_{\chi}\in E$, and $\eta_{\chi}\in E^{*}$ with $\|\xi_{\chi}\|=\|\eta_{\chi}\|=1$ so that $\chi_{G_{2}}(\cdot)=\langle\pi(\cdot)\xi_{\chi},\eta_{\chi}\rangle$. Also, for $g\in L_{1}(G_{2})$, and $\xi\in E$, and $\eta\in E^{*}$, we have

[TABLE]

and for $f\in L_{1}(G)$, $\xi\in E$, and $\eta\in E^{*}$ we have

[TABLE]

On the other hand,

[TABLE]

Therefore, by combining last equality with (23), we have

[TABLE]

Additionally, since $(\pi,E)$ is a $p$-universal representation, and we have

[TABLE]

thus $(\pi\otimes\pi,E\tilde{\otimes}_{p}E)$ can be assumed as a $p$-universal of $G$. Let

[TABLE]

then via (24) we have

[TABLE]

and since norm of $UPF_{p}(G)$ is independent of choosing $p$-universal representation then we have $\|{}_{*}E_{G_{2}}\|_{\text{p-cb}}\leq 1$, which implies that $\|E_{G_{2}}\|_{\text{p-cb}}\leq 1$. 6. 6.

By Corollary 3.6, the map $M_{Y}:B_{p}(G)\rightarrow B_{p}(G)$ with $M_{Y}(u)=u\cdot\chi_{Y}$ is well-defined, and

[TABLE]

On the other hand, by the following relation this map is weak∗-weak∗ continuous

[TABLE]

So, one may define its predual map as following

[TABLE]

and by (25) we have $({}_{*}M_{Y})^{*}=M_{Y}$.

Step 1: For proving the claim, first we let $Y$ be an open coset itself. By Theorem 3.5-(3), the function $\chi_{Y}$ is a normalized coefficient function of representation $(\pi,E)$ which means that there are elements $\xi_{Y}\in E$, and $\eta_{Y}\in E^{*}$ with $\|\xi_{Y}\|=\|\eta_{Y}\|=1$ such that

[TABLE]

So, for a matrix $(\pi(f_{ij}))\in\mathbb{M}_{n}(UPF_{p}(G))$, we have

[TABLE]

By these computations, we obtain that the map ${}_{*}M_{Y}$ is a $p$-complete contraction. Therefore, we have $\|M_{Y}\|_{\text{p-cb}}\leq 1$. Note that in the above calculations, we used a relation similar to (24) and an argument about independence of choosing $p$-universal representation.

Step 2: Now for $Y=Y_{0}\backslash\cup_{i=1}^{m}Y_{i}\in\Omega_{0}(G)$, from (16) we have,

[TABLE]

Therefore, we have $\|M_{Y}\|_{\text{p-cb}}\leq 2^{m_{Y}}$.

∎

Remark 4.7**.**

The importance of Theorem 4.6-(1) is that while we are working with maps with ranges as subspaces of the $p$-analog of the Fourier-Stieltjes algebras, we just need to restrict ourselves to their ranges, as what we have done in the rest of Theorem 4.6. 2. 2.

In the proof of Theorem 4.6-(6), if $Y=y_{2}G_{2}$, for an open amenable subgroup $G_{2}$ of $G$, and some $y_{2}\in G$, then by Theorem 4.6-(2)-(3)-(5) we can find out that the map $M_{Y}$ is a $p$-complete contraction through the following relation

[TABLE]

5 $p$-Completely homomorphisms on $B_{p}(G)$ induced by proper piecewise affine maps

As an application of previous sections, we are ready to study on homomorphisms $\Phi_{\alpha}:B_{p}(G)\rightarrow B_{p}(H)$ of the form

[TABLE]

for the proper continuous piecewise affine map $\alpha:Y\subset H\rightarrow G$ with $Y=\cup_{i=1}^{n}Y_{i}$ and $Y_{i}\in\Omega_{\text{am-}0}(H)$, which are pairwise disjoint, for $i=1,\ldots,n$. We will give some results in the sequel. For our aim we need the following lemma. For general form of this lemma, see [16, Lemma 1], and related references there.

Lemma 5.1**.**

Let $G$ and $H$ be locally compact groups and $\alpha:H\rightarrow G$ be a proper homomorphism that is onto, then there is a constant $c_{\alpha}>0$, such that

[TABLE]

Proposition 5.2**.**

Let $G$ and $H$ be locally compact groups and $\alpha:H\rightarrow G$ be a proper continuous group homomorphism. Then the homomorphism $\Phi_{\alpha}:B_{p}(G)\rightarrow B_{p}(H)$, of the form $\Phi_{\alpha}(u)=u\circ\alpha$, is well-defined and $p$-completely contractive homomorphism.

Proof.

Let $(\pi,E)$ be a $p$-universal representation of $G$. Obviously, $(\pi\circ\alpha,E)\in\text{Rep}_{p}(H)$, and $\Phi_{\alpha}$ is a contractive map so that its range is the subspace of $B_{p}(H)$ of functions which are coefficient functions of the representation $(\pi\circ\alpha,E)$. We will divide our proof into two steps.

1. Step 1:

First, we suppose that $\alpha:H\rightarrow G$ is a continuous isomorphism. In this case, $(\pi\circ\alpha,E)$ is a $p$-universal representation of $H$, and by Lemma 5.1, for every $f\in L_{1}(H)$ and $u\in B_{p}(G)$, we have

[TABLE]

Consequently, the map $\Phi_{\alpha}$ is weak∗-weak∗ continuous, and we define

[TABLE]

According to the above relation, we have $({}_{*}\Phi_{\alpha})^{*}=\Phi_{\alpha}$. On the other hand, for every $\xi\in E$ and $\eta\in E^{*}$, we have

[TABLE]

which means $\pi\circ\alpha(f)=c_{\alpha}\pi(f\circ\alpha^{-1})$. Consequently, ${}_{*}\Phi_{\alpha}$ is an identity map, so is a $p$-complete isometry

[TABLE]

Therefore, $\|\Phi_{\alpha}\|_{\text{p-cb}}\leq\|{}_{*}\Phi_{\alpha}\|_{\text{p-cb}}=1$. 2. Step 2:

Now let $\alpha:H\rightarrow G$ be any proper continuous homomorphism. Let $G_{0}=\alpha(H)$, and $N=\ker\alpha$. Let us define

[TABLE]

then by Proposition 2.22, the map $\tilde{\alpha}$ is a continuous isomorphism, $N$ is a compact normal subgroup of $H$, and $G_{0}$ is an open subgroup of $G$. Therefore, $\alpha=\tilde{\alpha}\circ q$. By Step 1, the map $\Phi_{\tilde{\alpha}}$ is $p$-completely contractive, and because of the following composition, $\Phi_{\alpha}$ is $p$-completely contractive, via Theorem 4.6-(2)-(4).

[TABLE]

∎

For the next proposition, we have to put the amenability assumption on the subgroups of $H$, because of Proposition 4.3.

Proposition 5.3**.**

Let $G$ and $H$ be two locally compact groups, $Y$ be an open coset of an open amenable subgroup of $H$, and $\alpha:Y\subset H\rightarrow G$ be a continuous proper affine map. Then the map $\Phi_{\alpha}:B_{p}(G)\rightarrow B_{p}(H)$, defined as

[TABLE]

is $p$-completely contractive. More generally, if $\alpha$ is a continuous proper piecewise affine map, and $Y=\cup_{i=1}^{n}Y_{i}$, where disjoint sets $Y_{i}$ belong to $\Omega_{\text{am-}0}(H)$, then the map $\Phi_{\alpha}$ is $p$-completely bounded.

Proof.

Let $\alpha:Y=y_{0}H_{0}\rightarrow G$ be a continuous proper affine map on the open coset $Y=y_{0}H_{0}$, and $H_{0}$ be an open amenable subgroup of $H$, for which by Remark 2.23-(3), there exists a continuous group homomorphism $\beta:H_{0}\subset H\rightarrow G$ associated to $\alpha$ such that

[TABLE]

which is proper via Remark 2.23-(4). Now consider the following composition

[TABLE]

then by Proposition 5.2, and Theorem 4.6-(3)-(5) the map $\Phi_{\alpha}$ is $p$-completely contractive homomorphism.

Next, we consider the piecewise affine case. Let the map $\alpha:Y\subset H\rightarrow G$ be a continuous piecewise affine map. Then for some $n\in\mathbb{N}$, and $i=1,\ldots,n$, there are disjoint sets $Y_{i}\in\Omega_{\text{am-}0}(H)$, such that $Y=\cup_{i=1}^{n}Y_{i}$, and $\alpha_{i}:{Aff(Y_{i})}\rightarrow G$ which are affine maps, and $\alpha_{i}|_{Y_{i}}=\alpha|_{Y_{i}}$. Additionally, by Remark 2.23-(5), each affine map $\alpha_{i}$ is proper. Therefore, by considering

[TABLE]

and through the above computations for the maps $\Phi_{\alpha_{i}}$, we have

[TABLE]

where $m_{Y_{i}}$ is the corresponding number to each $Y_{i}$, as it is in Theorem 4.6-(6). ∎