# Fourier and Fourier-Stieltjes Algebra of Fell Bundles over discrete groups

Massoud Amini and Mohammad Reza Ghanei  Department of Mathematics, Khansar
Campus, University of Isfahan, Iran<EMAIL_ADDRESS>,
<EMAIL_ADDRESS>Faculty of Mathematical Sciences
Tarbiat Modares University
Tehran 14115-134
Iran
School of Mathematics Institute for Research in Fundamental Sciences (IPM)
Tehran 19395-5746
Iran<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract.

For a Fell bundle $\mathcal{B}=\left\\{B_{s}\right\\}_{s\in G}$ over a
discrete group $G$, we use representations theory of $\mathcal{B}$ to
construct the Fourier and Fourier-Stieltjes spaces $A(\mathcal{B})$ and
$B(\mathcal{B})$ of $\mathcal{B}$. When $\mathcal{B}$ is saturated we show
$B(\mathcal{B})$ is canonically isomorphic to the dual space of the cross
sectional $C^{*}$-algebra $C^{*}(\mathcal{B})$ of $\mathcal{B}$. When there is
a compatible family of co-multiplications on the fibers we show that
$B(\mathcal{B})$ and $A(\mathcal{B})$ are Banach algebras. This holds in
particular if either the fiber $B_{e}$ at identity is a Hopf $C^{*}$-algebra
or $\mathcal{B}$ is the Fell bundle of a $C^{*}$-dynamical system. When
$A(\mathcal{B})$ is a Banach algebra with bounded approximate identity, we
show that $B(\mathcal{B})$ is the multiplier algebra of $A(\mathcal{B})$. We
prove a Leptin type theorem by showing that amenability of $G$ implies the
existence of bounded approximate identity for $A(\mathcal{B})$ for bundles
coming from a $C^{*}$-dynamical system $(A,G,\gamma)$. The converse is left as
an open problem.

###### Key words and phrases:

Fell bundle, Fourier algebra, Fourier-Stieltjes algebra, action, coaction

###### 2010 Mathematics Subject Classification:

Primary 46L05; Secondary 46M20

##  1\. Introduction

The Fourier algebra $A(G)$ and Fourier-Stieltjes algebra $B(G)$ of a locally
compact group $G$ are defined and studied by Pierre Eymard in [4] and play an
important role in modern theory of abstract harmonic analysis (see, for
instance [7]). The Fourier-Stieltjes algebra is the collection of all
coefficient functions of continuous unitary representations of $G$, whereas
the Fourier algebra $A(G)$ consists of coefficient functions of the left
regular representation. The Fourier-Stieltjes algebra $B(G)$ admits a natural
norm making it into a commutative Banach $*$-algebra under pointwise
multiplication. The Fourier algebra $A(G)$ as a closed ideal of $B(G)$. When
$G$ is amenable, $B(G)$ is the multiplier algebra of $A(G)$; for more details,
see [7]

Fell bundles were introduced in the late 60’s [5] by James Fell to provide a
framework for understanding and extending George Mackey’s theory of induced
representations [8]. We refer the reader to [6] for the basic theory of Fell
bundles and to [3] for a concise treatment and of this and other important
concepts.

Roughly speaking, a Fell bundle (also known as a $C^{*}$-algebraic bundle)
over a discrete group $G$ is a collection $\mathcal{B}$ of closed subspaces of
a $C^{*}$-algebra indexed by $G$, such that multiplication and involution in
fibers follow the group rules. Now, just as in group $C^{*}$-algebras, one
could use representation theory of the $*$-algebra of finitely supported
sections of the bundle to associate full and reduced cross sectional
$C^{*}$-algebras $C^{*}(\mathcal{B})$ and $C^{*}_{r}(\mathcal{B})$. For the
trivial bundle (with fibers $\mathbb{C}$ everywhere) these are the same as
full and reduced group $C^{*}$-algebras $C^{*}(G)$ and $C^{*}_{r}(G)$. Now
since the Fourier-Stieltjes algebra $B(G)$ as a Banach space is nothing but
the dual space of the full group $C^{*}$-algebra $C^{*}(G)$, it is natural to
expect that “coefficients” of representations of a Fell bundle $\mathcal{B}$
form a vector space, whose completion (in an appropriate norm) is identified
with the dual space of $C^{*}(\mathcal{B})$. Moreover, in the group case,
positive definite functions on the group canonically associate with positive
functionals on $C^{*}(G)$, and so the next thing to look at would be the order
structure on the positive cone of $C^{*}(\mathcal{B})$ and the corresponding
positive functionals. Indeed, Fell and Doran already initiated studying
properties of the positive cone of a Fell bundle in their original monograph
[6] and gave a “module structure” on (the linear span of) positive definite
functionals on the bundle over (the linear span of) positive definite
functions on the base group.

The other problem to deal with is whether the coefficients of representations
of $\mathcal{B}$ form an algebra, or equivalently, whether the product of two
positive definite functionals could be defined, and happens to be a positive
functional (or at least inside their linear span). In the group case, the
algebra structure on the dual $B(G)$ of full group $C^{*}$-algebra $C^{*}(G)$
comes from the co-algebra structure on $C^{*}(G)$ given by the canonical co-
product $\Delta:C^{*}(G)\rightarrow C^{*}(G)\otimes C^{*}(G)$. Such a co-
algebra structure is not available for general Fell bundles, and could be
expected only for Fell bundles associated to group actions on $C^{*}$-bi-
algebras.

In this paper we define and study Fourier and Fourier-Stieltjes spaces of a
Fell bundle. We are particular interested in the spacial case where these are
Banach algebras. In this latter case, we investigate the existence of a
bounded approximate identity in the Fourier algebra of a Fell bundle and
relate it to identifying its multiplier algebra. We study the special cases of
bundles associated to group actions in somewhat more details. Similar
construction is done by Bedos and Conti in [2].

Most of the material of this paper could be extended (without considerable
difficulty) to Fell bundles over locally compact groups, but we feel that at
this stage (of developing a new theory), this level of generality is not
necessary. Moreover, certain natural settings in which Fell bundles are
motivated and discussed are only developed in the discrete case (see for
instance [11]).

The paper is organized as follows. In Section 2, we give a brief overview of
Fell bundles over discrete groups, fixing basic notations used in the rest of
this paper. In Section 3, we define the Fourier-Stieltjes Space
$B(\mathcal{B})$ of a Fell bundle $\mathcal{B}$ over a discrete group $G$. We
show that $B(\mathcal{B})$ is a dual Banach space, and when $\mathcal{B}$ is
saturated, we observe that $B(\mathcal{B})$ is canonically isomorphic to the
dual of cross sectional $C^{*}$-algebra $C^{*}(\mathcal{B})$ of $\mathcal{B}$.
In Section 4, we find general conditions under which $B(\mathcal{B})$ and
$A(\mathcal{B})$ are Banach algebras, and determine when the multiplier
algebra of $A(\mathcal{B})$ is $B(\mathcal{B})$.

##  2\. Preliminaries

Throughout this paper we work with a discrete group $G$, with unit element
$e$. We denote the set of all linear continuous maps between Banach spaces $X$
and $Y$ by $\mathcal{L}(X,Y)$. We denote $\mathcal{L}(X,X)$ by
$\mathcal{L}(X)$.

######  Definition 2.1.

A Fell bundle (or a $C^{*}$-algebraic bundle) over a discrete group $G$ is a
collection $\mathcal{B}=\left\\{B_{s}\right\\}_{s\in G}$ of closed subspaces
of a $C^{*}$-algebra $B$, indexed by $G$, satisfying $B_{s}B_{t}\subseteq
B_{st}$ and $B_{s}^{*}=B_{s^{-1}}$, for $s,t\in G$. The disjoint union of
fibers is called the total space of $\mathcal{B}$ (again denoted by
$\mathcal{B}$). A section of $\mathcal{B}$ is a map $y$ from $G$ to the total
space of $\mathcal{B}$ with $y_{s}:=y(s)\in B_{s}$, for $s\in G$.

A Fell bundle $\mathcal{B}=\left\\{B_{s}\right\\}_{s\in G}$ is said to be
saturated if the linear span $B_{s}B_{t}$ of all multiples of elements of
$B_{s}$ in that of $B_{t}$ is dense in $B_{st}$, for each $s,t\in G$ [6,
VIII.2.8]. For elementary examples of saturated bundles, see [6,
VIII.3.15-16].

The set of all finitely supported (respectively, countably summable) sections
of $\mathcal{B}$ is denoted by $c_{c}(\mathcal{B})$ (respectively, by
$\ell^{1}(\mathcal{B})$). Taking supremum over all $C^{*}$-seminorms on
$c_{c}(\mathcal{B})$ one gets a $C^{*}$-seminorm $\|.\|_{\rm max}$ on
$c_{c}(\mathcal{B})$, and the completion of the quotient by norm zero elements
is a $C^{*}$-algebra $C^{*}(\mathcal{B})$, called the cross sectional
$C^{*}$-algebra of $\mathcal{B}$. By a similar construction using the regular
representation of $\mathcal{B}$, one could construct the reduced cross
sectional $C^{*}$-algebra $C^{*}_{r}(\mathcal{B})$ of $\mathcal{B}$; for more
details see, [3, 17.6].

######  Example 2.2.

Let us exemplify the above notions in certain basic cases.

$(i)$ Let $G$ be a discrete group. Consider the trivial bundle over $G$, i.e.,
$\mathcal{B}=G\times\mathbb{C}=\\{B_{s}\\}_{s\in G}$, where
$B_{s}=\\{(s,z)|z\in\mathbb{C}\\}$, with operations $(s,z)(t,w)=(st,zw)$,
$(s,z)^{*}=(s^{-1},\bar{z})$ and norm $\|(s,z)\|=|z|$. Then canonically,
$c_{c}(\mathcal{B})\cong c_{c}(G)$, $\ell^{1}(\mathcal{B})\cong\ell^{1}(G)$,
$C^{*}_{r}(\mathcal{B})\cong C^{*}_{r}(G)$, and $C^{*}(\mathcal{B})\cong
C^{*}(G)$.

$(ii)$ Let $(A,G,\gamma)$ be a $C^{*}$-dynamical system. For the fibers
$B_{s}=\\{s\\}\times A$, with operations $(s,a)(t,b)=(st,a\gamma_{s}(b))$,
$(s,a)^{*}=(s^{-1},\gamma_{s^{-1}}(a^{*}))$, and norm $\|(s,a)\|=\|a\|$,
$c_{c}(\mathcal{B})\cong c_{c}(G,A)$,
$\ell^{1}(\mathcal{B})\cong\ell^{1}(G,A)$, $C^{*}_{r}(\mathcal{B})\cong
A\rtimes_{r}G$, and $C^{*}(\mathcal{B})\cong A\rtimes G$.

$(iii)$ A coaction of discrete group $G$ on a $C^{*}$-algebra $A$ is an
injective non-degenerate homomorphism $\delta:A\rightarrow$ $A\otimes
C^{*}(G)$ such that
$(\delta\otimes\mathrm{id})\circ\delta=\left(\mathrm{id}\otimes\delta_{G}\right)\circ\delta$,
where $\delta_{G}:C^{*}(G)\rightarrow C^{*}(G)\otimes C^{*}(G)$ is the
canonical co-product defined by $\delta_{G}(u_{s})=u_{s}\otimes u_{s}$, $s\in
G$, where $u$ is the universal representation of $G$. With the spectral
subspaces $A_{s}:=\\{a\in A:\delta(a)=a\otimes s\\}$ as fibers,
$\mathcal{A}=\\{A_{s}\\}$ is a Fell bundle with $A\cong C^{*}(\mathcal{A})$.
Conversely, for each Fell bundle $\mathcal{A}=\\{A_{s}\\}$, $G$ coacts on
$C^{*}(\mathcal{A})$ via

$\delta:C^{*}(\mathcal{A})\rightarrow C^{*}(\mathcal{A})\otimes C^{*}(G);\ \
\delta(a)=a\otimes\delta_{s},$

for $a\in A_{s}$, where $\delta_{s}\in c_{c}(G)$ is regarded as an element in
$C^{*}(G)$ (cf., [11, 3.3]).

Given sections $y$ and $z$ in $c_{c}(\mathcal{B})$, we define their
convolution by

$(y\star z)_{s}=\sum_{t\in G}y_{t}z_{t^{-1}s}\in B_{s},$

and the involution by $y_{s}^{*}=\left(y_{s^{-1}}\right)^{*}$, for $s\in G.$
The space $c_{c}(\mathcal{B})$ is an associative $*$-algebra under these
operations.

######  Definition 2.3.

A representation $\pi$ of a Fell bundle
$\mathcal{B}=\left\\{B_{s}\right\\}_{s\in G}$ on a Hilbert space $H_{\pi}$ is
a map $\pi:\mathcal{B}\rightarrow\mathcal{L}(H_{\pi})$ such that, for each
$s\in G$, the map

$\pi_{s}:=\pi(s):B_{s}\rightarrow\mathcal{L}(H_{\pi})$

is linear on each fiber, and for all $s,t\in G,b\in B_{s}$ and $c\in B_{t}$,

($i$) $\pi_{s}(b)\pi_{t}(c)=\pi_{st}(bc)$,

($ii$) $\pi_{s}(b)^{*}=\pi_{s^{-1}}\left(b^{*}\right)$.

We always assume that $\pi:\mathcal{B}\rightarrow\mathcal{L}(H_{\pi})$ is
continuous, that is, for every $s\in G$ and $\xi\in H_{\pi}$, the map $b\in
B_{s}\mapsto\pi_{s}(b)\xi$ is norm-continuous. Since,

$\|\pi_{s}(b)\|^{2}=\|\pi_{s^{-1}}(b^{*})\pi_{s}(b)\|=\|\pi_{e}(b^{*}b)\|\leq\|b^{*}b\|=\|b\|^{2},$

we get, $\|\pi_{s}(b)\|\leq\|b\|$.

######  Example 2.4.

There are basic examples of representations.

$(i)$ For the trivial bundle $\mathcal{B}=G\times\mathbb{C}$, each
representation $u$ of $G$ gives a representation $\Pi=\\{\pi_{s}\\}$ of the
trivial bundle, via $\pi_{s}(s,z)=u_{s}$. Conversely, given a representation
$\Pi=\\{\pi_{s}\\}_{s\in G}$ of the trivial bundle, $u_{s}=\pi_{s}(s,1)$
defines a representation of $G$.

$(ii)$ Given a representation $\Pi$ of the Fell bundle $\mathcal{B}$ of a
$C^{*}$-dynamical system $(A,G,\gamma)$, $\pi(a)=\Pi_{e}(a,e)$ defines a
representation of $A$, and when $A$ is unital, $u(s)=\Pi_{s}(s,1)$ gives a
representation of $G$. Conversely, each covariant representation $(\pi,u)$ of
$(A,G,\gamma)$ defines a representation $\Pi_{s}(a)=\pi(a)u_{s}$, for $s\in
G,a\in A$.

For a representation $\pi$ of a Fell bundle $\mathcal{B}$, we consider the
coefficient functions

$\pi_{\xi,\eta,s}:B_{s}\rightarrow\mathbb{C};\quad\pi_{\xi,\eta,s}(b)=\left\langle\pi_{s}(b)\xi,\eta\right\rangle,$

for $\xi,\eta\in H_{\pi}$ and $s\in G$. Two representations $\pi$ and $\sigma$
of $\mathcal{B}$ are equivalent if there exists a unitary
$U:\mathcal{H}_{\pi}\rightarrow\mathcal{H}_{\sigma}$ such that
$U\pi_{s}(b)\xi=\sigma_{s}(b)U\xi$, for $s\in G,b\in B_{s}$, and $\xi\in
H_{\pi}$. We write $\pi\sim\sigma$ in this case and call $U$ an intertwining
operator for $\pi$ and $\sigma$. We denote the collection of equivalence
classes of representations of $\mathcal{B}$ by $\Sigma_{c}(\mathcal{B})$.

For $s$ in $G$, we denote by $j_{s}$ the natural inclusion of $B_{s}$ in
$C_{c}(\mathcal{B})$, namely the map $j_{s}:B_{s}\rightarrow
C_{c}(\mathcal{B}),$ defined by

$\left.j_{s}(b)\right|_{h}=\begin{cases}b,&\text{ if }h=s\\\ 0,&\text{
otherwise. }\end{cases}$

The inclusion $j_{e}$ at identity is a $*$-homomorphism through which we may
view $B_{e}$ as a subalgebra of $c_{c}(\mathcal{B})$. This also makes
$c_{c}(\mathcal{B})$ a right $B_{e}$-module with a $B_{e}$-valued inner-
product

$\langle y,z\rangle=\sum_{s\in G}y_{s}^{*}z_{s}.$

We denote by $\ell^{2}(\mathcal{B})$ the right Hilbert $B_{e}$-module obtained
by completing $c_{c}(\mathcal{B})$ under the norm $\|\cdot\|_{2}$ coming from
the above inner-product.

We may regard $j_{s}$ as a map from $B_{s}$ to $\ell^{2}(\mathcal{B})$, which
is then an isometry. Given $b\in B_{s}$, the map

$\lambda_{s}(b):y\in c_{c}(\mathcal{B})\longmapsto j_{s}(b)\star y\in
c_{c}(\mathcal{B}),$

is continuous in the Hilbert module norm and extends to a bounded operator on
$\ell^{2}(\mathcal{B})$, still denoted by $\lambda_{s}(b)$, such that,
$\left\|\lambda_{s}(b)\right\|=\|b\|$, and

$\lambda_{s}(b)(j_{t}(c))=j_{st}(bc),\
\left\langle\lambda_{s}(b)\xi,\eta\right\rangle=\left\langle\xi,\lambda_{s-1}\left(b^{*}\right)\eta\right\rangle\
(t\in G,c\in B_{t},\xi,\eta\in\ell^{2}(\mathcal{B})).$

The representation $\lambda=\left\\{\lambda_{s}\right\\}_{s\in G}$ is called
the regular representation of $\mathcal{B}$.

##  3\. Fourier and Fourier-Stieltjes spaces of Fell bundles

Given a Fell bundle $\mathcal{B}=\left\\{B_{s}\right\\}_{s\in G}$ over a
discrete group $G$, the $\ell^{1}$-cross sectional algebra
$\ell^{1}(\mathcal{B})$, consists of the $\ell^{1}$-cross sections of
$\mathcal{B}$, with operations

$f*g(t)=\sum_{s\in G}f(s)g(s^{-1}t),\ f^{*}(t)=f(t^{-1})^{*},\quad(t\in
G,f,g\in\ell^{1}(\mathcal{B}))$

and norm $\|f\|_{1}=\sum_{s\in G}\left\|f_{s}\right\|$. To each
$\pi\in\Sigma_{c}(\mathcal{B})$, one could associate an algebra representation
$\tilde{\pi}:\ell^{1}(\mathcal{B})\rightarrow\mathcal{L}(H_{\pi})$, given by
$\langle\tilde{\pi}(f)\xi,\eta\rangle=\sum_{s\in
G}\left\langle\pi_{s}(f_{s})\xi,\eta\right\rangle$, such that

$\displaystyle|\langle\tilde{\pi}(f)\xi,\eta\rangle|$
$\displaystyle\leq\|\xi\|\|\eta\|\sum_{s\in
G}\left\|\pi_{s}\left(f_{s}\right)\right\|\leq\|\xi\|\|\eta\|\|f\|_{1}.$

For $f\in\ell^{1}(\mathcal{B})$, put
$\|f\|_{*}=\sup\left\\{\|\tilde{\pi}(f)\|:\pi\in\Sigma_{c}(\mathcal{B})\right\\}.$
For the coefficient function $u=\pi_{\xi,\eta}$,

$\big{|}\sum_{s\in G}u_{s}(f_{s})\big{|}=\big{|}\sum_{s\in
G}\langle\pi_{s}(f_{s})\xi,\eta\rangle\big{|}=|\langle\tilde{\pi}(f)\xi,\eta\rangle|\leq\|f\|_{*}\|\xi\|\|\eta\|.$

This allows us to define a Banach space structure on the set of coefficient
functions.

######  Definition 3.1.

Let $\mathcal{B}=\left\\{B_{s}\right\\}_{s\in G}$ be a Fell bundle over
discrete group $G$. The Fourier-Stieltjes space $B(\mathcal{B})$ of
$\mathcal{B}$ is the space

$B(\mathcal{B}):=\left\\{\pi_{\xi,\eta}|~{}\pi\in\Sigma_{c}(\mathcal{B}),~{}~{}\xi,\eta\in
H_{\pi}\right\\},$

with norm $\|u\|_{B(\mathcal{B})}=\sup\\{\big{|}\sum_{s\in
G}u_{s}(f_{s})\big{|}:f\in\ell^{1}(\mathcal{B}),\|f\|_{*}\leq 1\\}.$

For a representation $\pi$ of a Fell bundle $\mathcal{B}$ and a unitary
representation $u$ of $G$, we may define a representation

$u\otimes\pi:\mathcal{B}\rightarrow\mathcal{L}(H_{u}\otimes H_{\pi});\
(u\otimes\pi)_{s}(b)=u_{s}\otimes\pi_{s}(b),\ \ (s\in G,b\in B_{s}).$

Similarly, we have a representation $\pi\otimes
u:\mathcal{B}\rightarrow\mathcal{L}(H_{\pi}\otimes H_{u})$.

Let $\pi:\mathcal{B}\rightarrow\mathcal{L}(H_{\pi})$ a representation of
$\mathcal{B}$ and $u$ be a unitary representation of $G$. For $\xi,\eta\in
H_{\pi}$ and $\alpha,\beta\in H_{u}$, and functions $u_{\alpha,\beta}\in B(G)$
and $\pi_{\xi,\eta}\in B(\mathcal{B})$, put

$(u_{\alpha,\beta}\cdot\pi_{\xi,\eta})_{s}(b)=\langle(u_{s}\otimes\pi_{s}(b))(\alpha\otimes\xi),\beta\otimes\eta\rangle,\quad(s\in
G,b\in B_{s}).$

Then $B(\mathcal{B})$ is a left $B(G)$-module. Similarly, using $\pi\otimes
u$, $B(\mathcal{B})$ has a right $B(G)$-action.

A functional $\phi$ on $C^{*}(\mathcal{B})$ is positive if $\phi(u^{*}*u)\geq
0$ for each $u\in C^{*}(\mathcal{B})$. For a representation $\pi$ in
$\Sigma_{c}(\mathcal{B})$ and $\xi\in H_{\pi}$, the functional
$\tilde{\pi}_{\xi,\xi}:C^{*}(\mathcal{B})\rightarrow\mathbb{C}$ defined by
$\tilde{\pi}_{\xi,\xi}(g)=\langle\tilde{\pi}(g)\xi,\xi\rangle$ is a positive
linear functional on $C^{*}(\mathcal{B})$ and for each
$f\in\ell^{1}(\mathcal{B})$, we have $\tilde{\pi}_{\xi,\xi}(f)=\sum_{r\in
G}\langle\pi_{r}(f_{r})\xi,\xi\rangle$.

######  Definition 3.2.

An element $\varphi\in B(\mathcal{B})$ is called positive definite if there
exists a representation $\pi\in\sum_{c}(\mathcal{B})$ and a vector $\xi\in
H_{\pi}$ such that $\varphi=\pi_{\xi,\xi}$.

In [6], Fell and Doran defined the functionals of positive type on
$\mathcal{B}$ as functionals $P$ satisfying the inequality
$\sum_{i,j}P(b_{j}^{*}b_{i})\geq 0$, for any finite sequence
$b_{1},...,b_{n}\in\mathcal{B}$. By [6, 20.6], our positive definite functions
correspond to functionals of positive type in the sense of Fell and Doran. We
let $P(\mathcal{B})$ be the collection of all positive definite functions on
$\mathcal{B}$.

######  Lemma 3.3.

Let $\mathcal{B}$ be a Fell bundle over a discrete group $G$ and
$\psi=\pi_{\eta,\eta}\in P(\mathcal{B})$, for some
$\pi\in\Sigma_{c}(\mathcal{B})$ and $\eta\in H_{\pi}$. Then
$\|\psi\|=\|\eta\|^{2}$.

###### Proof.

Let us denote the unit element of $C^{*}(\mathcal{B})^{**}$ by $1$ and choose
a net $(g_{\alpha})_{\alpha}$ in $C^{*}(\mathcal{B})$ with
$g_{\alpha}\rightarrow 1$ in the weak∗ topology of $C^{*}(\mathcal{B})^{**}$.
Then, $\tilde{\pi}(g_{\alpha})\rightarrow\mathrm{id}$ in the weak∗ toplogy of
$\mathcal{L}(H_{\pi})$. Use Hahn Banach to extend
$\psi:C^{*}(\mathcal{B})\rightarrow\mathbb{C}$ to
$\tilde{\psi}:C^{*}(\mathcal{B})^{**}\rightarrow\mathbb{C}$ such that
$\|\tilde{\psi}\|=\|\psi\|$. Then,

$\tilde{\psi}(1)=\lim_{\alpha}\psi(g_{\alpha})=\lim_{\alpha}\langle\tilde{\pi}(g_{\alpha})\eta,\eta\rangle=\langle\eta,\eta\rangle=\|\eta\|^{2}.$

Therefore, $\|\tilde{\psi}\|\geq\|\eta\|^{2}\geq\|\psi\|$ and so
$\|\psi\|=\|\eta\|^{2}$.

∎

######  Proposition 3.4.

Let $\mathcal{B}$ be a saturated Fell bundle over a discrete group $G$. Then,

$(i)$ if $\varphi$ is a positive linear functional on $C^{*}(\mathcal{B})$,
then $\varphi\in P(\mathcal{B})$.

$(ii)$ every element of $B(\mathcal{B})$ is a finite linear combination of
elements of $P(\mathcal{B})$.

$(iii)$ there is an isometric isomorphism from $B(\mathcal{B})$ onto
$C^{*}(\mathcal{B})^{*}$.

$(iv)$ for $\varphi\in B(\mathcal{B})$, there exist a representation
$\pi\in\Sigma_{c}(\mathcal{B})$ and vectors $\xi,\eta\in H_{\pi}$ such that
$\varphi=\pi_{\xi,\eta}$ and $\|\varphi\|=\|\xi\|\|\eta\|.$

$(v)$ for $\varphi\in B(\mathcal{B})$,
$\|\varphi\|=\inf\\{\|\xi\|\|\eta\|~{}:~{}\varphi=\pi_{\xi,\eta}\\}.$

###### Proof.

$(i)$ Clearly $\varphi|_{\ell^{1}(\mathcal{B})}\in\ell^{1}(\mathcal{B})^{*}$
is positive and
$\ell^{1}(\mathcal{B})^{*}\cong\ell^{\infty}(\mathcal{B}^{*})$, where

$\ell^{\infty}(\mathcal{B}^{*})=\\{\psi:G\rightarrow\mathcal{B}^{*}:\
\psi(r)\in B_{r}^{*},\|\psi\|_{\infty}:=\sup_{r\in G}\|\psi_{r}\|<\infty\\}.$

For $f,g\in\ell^{1}(\mathcal{B})$, put $\langle f,g\rangle=\sum_{r\in
G}\varphi_{r}\left((g^{*}*f)_{r}\right)$ and observe that $|\langle
f,g\rangle|\leq\|\varphi\|_{\infty}\|f\|_{1}\|g\|_{1}.$ Let
$\mathcal{N}=\\{f\in\ell^{1}(\mathcal{B}):\langle f,f\rangle=0\\}$. By
Schwartz inequality, $\mathcal{N}$ is a left ideal in $\ell^{1}(\mathcal{B})$
and we get an inner product on the quotient space
$\ell^{1}(\mathcal{B})/\mathcal{N}$. Denote the Hilbert space completion of
$\ell^{1}(\mathcal{B})/\mathcal{N}$ by $H_{\varphi}$, and the image of
$f\in\ell^{1}(\mathcal{B})$ in $H_{\varphi}$ under the quotient map by
$\tilde{f}$.

Let $\psi_{\alpha}$ be an approximate identity in $\ell^{1}(\mathcal{B})$,
which exists as $\mathcal{B}$ is saturated [6, VIII.2.13]. Then for
$f\in\ell^{1}(\mathcal{B})$,

$\langle\tilde{f},\psi_{\alpha}\rangle=\sum_{s\in
G}\varphi_{s}\left((\psi_{\alpha}^{*}*f)_{s}\right)\rightarrow\sum_{s\in
G}\varphi_{s}(f_{s})=\varphi(f).$

Note that $\lim_{\alpha}\langle v,\psi_{\alpha}\rangle$ exists for each $v\in
H_{\varphi}$, and hence $\tilde{\psi_{\alpha}}$ converges weakly in
$H_{\varphi}$ to an element $\eta$ with
$\langle\tilde{f},\eta\rangle=\varphi(f)$, for $f\in\ell^{1}(\mathcal{B})$.
Next, we define an algebra representation

$\rho:\ell^{1}(\mathcal{B})\rightarrow\mathcal{L}(H_{\varphi});\
\rho(f)\tilde{g}=(f*g)^{\tilde{}},$

for $f,g\in\ell^{1}(\mathcal{B})$, and put $\pi_{s}(b):=\rho(J_{s}(b))$, for
$s\in G,b\in B_{s}.$ Then, $\\{\pi_{s}\\}\in\Sigma_{c}(\mathcal{B})$ and

$\langle\tilde{g},\rho(f)\eta\rangle=\langle\rho(f^{*})\tilde{g},\eta\rangle\\\
=\langle(f^{*}*g)\tilde{},\eta\rangle\\\ =\langle\tilde{g},\tilde{f}\rangle.$

It follows that, $\rho(f)\eta=\tilde{f}$, for $f\in\ell^{1}(\mathcal{B})$.
Therefore,

$\displaystyle\pi_{\eta,\eta,r}(b)$ $\displaystyle=$
$\displaystyle\langle\rho(J_{r}(b))\eta,\eta\rangle=\langle\widetilde{J_{r}(b)},\eta\rangle$
$\displaystyle=$ $\displaystyle\varphi(J_{r}(b))=\sum_{s\in
G}\varphi_{s}(J_{r}(b)(s))=\varphi_{r}(b),$

for $r\in G,b\in B_{r}.$ Thus, $\varphi=\pi_{\eta,\eta}\in P(\mathcal{B})$.

$(ii)$ To each $\varphi=\pi_{\xi,\eta}\in B(\mathcal{B})$ one could associate
$\tilde{\varphi}\in C^{*}(\mathcal{B})^{*}$, defined by
$\varphi(u)=\langle\tilde{\pi}(u)\xi,\eta\rangle$, for $u\in
C^{*}(\mathcal{B})$. Using Jordan decomposition
$\varphi=(\varphi_{1}-\varphi_{2})+i(\varphi_{3}-\varphi_{4})$ into positive
linear functionals, the assertion now follows by part $(i)$.

$(iii)$ Define $T:B(\mathcal{B})\rightarrow C^{*}(\mathcal{B})^{*}$ by
$\varphi\mapsto T_{\varphi}$, where
$T_{\varphi}:C^{*}(\mathcal{B})\rightarrow\mathbb{C}$ is defined by
$T_{\varphi}(f)=\sum_{s\in G}\varphi_{s}(f_{s})$, for each
$f\in\ell^{1}(\mathcal{B})$. By part $(ii)$,
$\|T_{\varphi}\|=\|\varphi\|_{B(\mathcal{B})}$, as required.

Next, for $u\in C^{*}(\mathcal{B})^{*}$ with Jordan decomposition

$u=(u_{1}-u_{2})+i(u_{3}-u_{4}),$

for each positive functional $u_{i}$, by part $(i)$, there is a representation
$\pi^{i}\in\Sigma_{c}(\mathcal{B})$ on a Hilbert space $H_{i}$ and $\xi_{i}\in
H_{i}$ such that $u_{i}=\pi^{i}_{\xi_{i},\xi_{i}}$. Define
$\rho:=\\{\rho_{s}\\}_{s\in G}$ on the direct sum $H_{0}$ of the Hilbert
spaces $H_{i}$ by

$\langle\rho_{s}(b)(\xi_{i}),(\eta_{i})\rangle=\sum_{i=1}^{4}\langle\pi^{i}_{s}(b)\xi_{i},\eta_{i}\rangle.$

Then, for $\xi_{0}=(\xi_{1},-\xi_{2},i\xi_{3},-i\xi_{4})$ and
$\eta_{0}=(\xi_{1},\xi_{2},\xi_{3},\xi_{4})$ in $H_{0}$,

$\langle\rho_{s}(b)\xi_{0},\eta_{0}\rangle=\langle\pi^{1}_{s}(b)\xi_{1},\xi_{1}\rangle-\langle\pi^{2}_{s}(b)\xi_{2},\xi_{2}\rangle+i\langle\pi^{3}_{s}(b)\xi_{3},\xi_{3}\rangle-i\langle\pi^{4}_{s}(b)\xi_{4},\xi_{4}\rangle=u(b),$

that is, $u=\rho_{\xi_{0},\eta_{0}}$ and $T_{\rho}=u$.

$(iv)$ Let $\varphi=R_{h}\psi$ be the polar decomposition of $\varphi$, where
$\psi\in B(\mathcal{B})$ is positive, $\|\psi\|=\|\varphi\|$, $h\in
B(\mathcal{B})^{*}=C^{*}(\mathcal{B})^{**}$ with $\|h\|\leq 1$, and
$R_{h}\psi(g)=\psi(gh)$, for $g\in B(\mathcal{B})^{*}$. By Lemma 3.4, there
exists a representation $\pi\in\Sigma_{c}(\mathcal{B})$ and a vector $\eta\in
H_{\pi}$ such that $\psi=\pi_{\eta,\eta}$. Let
${\omega}:C^{*}(\mathcal{B})\rightarrow C^{*}(\mathcal{B})^{**}$ be the
universal representation of $C^{*}(\mathcal{B})$. Then,
$\tilde{\pi}(g)=\tilde{\pi}^{**}(\omega(g))$, for $g\in C^{*}(\mathcal{B})$,
where $\tilde{\pi}^{**}$ is the second transpose of $\tilde{\pi}$ from
$C^{*}(\mathcal{B})^{**}$ into $L(H_{\pi})$. Put
$\xi:=\tilde{\pi}^{**}(h)\eta$. By Lemma 3.3,

$\|\tilde{\pi}^{**}(h)\eta\|^{2}=|\langle\tilde{\pi}^{**}(h^{*}*h)\eta,\eta\rangle|=|\langle
h^{*}*h,\psi\rangle|\leq\|\psi\|=\|\varphi\|=\|\eta\|^{2}.$

Therefore, $\|\xi\|\leq\|\eta\|$. Next, for every $g\in C^{*}(\mathcal{B})$,

$\displaystyle\langle\varphi,g\rangle$ $\displaystyle=$ $\displaystyle\langle
R_{h}\psi,g\rangle=\langle\psi,\omega(g)*h\rangle$ $\displaystyle=$
$\displaystyle\langle\tilde{\pi}^{**}(\omega(g))\tilde{\pi}^{**}(h)\eta,\eta\rangle=\langle\tilde{\pi}(g)\xi,\eta\rangle,$

that is, $\varphi=\pi_{\xi,\eta}$, thus,

$\|\varphi\|\leq\|\xi\|\|\eta\|\leq\|\eta\|^{2}=\|\psi\|=\|\varphi\|.$

$(v)$ This follows immediately from part $(iv)$. ∎

######  Example 3.5.

Certain basic examples of the Fourier-Stieltjes space are in order.

$(i)$ Let $G$ be a finite abelian group and $(A,G,\tau)$ be a
$C^{*}$-dynamical system. For each character $\chi\in\widehat{G}$, put
$B_{\chi}=\\{a\in A:\tau_{s}(a)=\chi(s)a~{}(s\in G)\\}$. This forms a Fell
bundle $\mathcal{B}=\\{B_{\chi}\\}_{\chi\in\widehat{G}}$ over $\widehat{G}$
and $A$ can be identified with $C^{*}(\mathcal{B})$; for more details see [6,
P. 1044]. This is known to be saturated when $A$ is of compact type (i.e., a
direct sum of elementary $C^{*}$-algebras; cf., [6, VI.23.3]) and $G$ acts on
$A$ by inner automorphisms [6, X.Exercise 23]. In this case, the Fourier-
Stieltjes space $B(\mathcal{B})$ is isomorphic to $A^{*}$. As far as we know,
a necessary and sufficient condition for the bundle in this example to be
saturated is not known yet [6, page 1045].

$(ii)$ Let $\mathcal{B}$ be the Fell bundle of a $C^{*}$-dynamical system
$(A,G,\gamma)$. To every functional
$\varphi:A\rtimes_{\gamma}G\rightarrow\mathbb{C}$ associate a function
$\Phi:G\rightarrow A^{*}$ by $\Phi(t)(a)=\varphi(au_{t})$, for $t\in G$ and
$a\in A$; where $u:G\rightarrow M(A\rtimes_{\gamma}G)$ pairs with a
representation $\pi:A\rightarrow M(A\rtimes_{\gamma}G)$ in the universal
covariant representation $(\pi,u)$ of $(A,G,\gamma)$. The set of functions
$\Phi:G\rightarrow A^{*}$ is denoted by $B(A\rtimes_{\gamma}G)$ and there is a
bijection from $(A\rtimes_{\gamma}G)^{*}$ onto $B(A\rtimes_{\gamma}G)$, as
observed first by Gert Pedersen in [9]. Moreover, if
$\mathcal{B}=\\{B_{s}\\}_{s\in G}$ is the Fell bundle of $(A,G,\gamma)$, then
it is known that $C^{*}(\mathcal{B})\simeq A\rtimes_{\gamma}G$. Therefore,
$B(\mathcal{B})\simeq(A\rtimes_{\gamma}G)^{*}$, since $\mathcal{B}$ is known
to be automatically saturated in this case [6, VIII.4.3].

For this example, the Fourier-Stieltjes space $B(\mathcal{B})$ is isomorphic
to the space $B(A\rtimes_{\gamma}G)$, introduced by Pedersen [9, 7.6.6]. If
$\varphi$ is a positive functional on $A\rtimes_{\gamma}G$, then the
associated function $\Phi$ is a positive definite function. The above
isomorphism maps $P(\mathcal{B})$ on the set $B_{+}(A\rtimes_{\gamma}G)$ of
positive definite functions (cf., [9, 7.6.8]).

$(iii)$ For the Fell bundle $\mathcal{A}$ associated as in Example 2.2($iii$)
to a coaction $\delta:A\rightarrow$ $A\otimes C^{*}(G),$ we have $A\cong
C^{*}(\mathcal{A})$. Since this bundle is clearly saturated, we get
$B(\mathcal{A})\simeq A^{*}$.

$(iv)$ If $G$ is abelian, then there exists a natural action
$\widehat{\gamma}$ of the dual group $\widehat{G}$ of $G$ on
$B:=A\rtimes_{\gamma}G$ given for $f\in C_{c}(G,A)$ by

$\widehat{\gamma}_{\chi}(f)(s)=\chi(s)f(s).$

The Takesaki-Takai duality theorem asserts that
$(A\rtimes_{\gamma}G)\rtimes_{\hat{\gamma}}\widehat{G}\simeq
A\otimes\mathcal{K}\left(\ell^{2}(G)\right)$, canonically, that is, the double
crossed product is stably isomorphic to $A$. One can define the Fell bundle of
$(A\rtimes_{\gamma}G,\widehat{G},\hat{\gamma})$ by
$\widehat{\mathcal{B}}=\\{\widehat{B}_{\chi}\\}_{\chi\in\widehat{G}}$, which
gives

$C^{*}(\widehat{\mathcal{B}})\simeq(A\rtimes_{\gamma}G)\rtimes_{\hat{\gamma}}\widehat{G}\simeq
C^{*}(\mathcal{B})\rtimes_{\hat{\gamma}}\widehat{G}.$

Therefore,
$B(\widehat{\mathcal{B}})\simeq(C^{*}(\mathcal{B})\rtimes_{\hat{\gamma}}\widehat{G})^{*}$
if $\widehat{\mathcal{B}}$ is saturated (this in particular holds if
$A\rtimes_{\gamma}G$ is of compact type and $\hat{\gamma}$ is inner; see part
($i$) above).

Let
$\lambda:\mathcal{B}\rightarrow\mathcal{L}\left(\ell^{2}(\mathcal{B})\right)$
be the regular representation of a Fell bundle $\mathcal{B}=\\{B_{s}\\}_{s\in
G}$ on the Hilbert $B_{e}$-module $\ell^{2}(\mathcal{B})$. We may then
consider the map

$\lambda_{\xi,\eta,s}:B_{s}\rightarrow
B_{e};\quad\lambda_{\xi,\eta,s}(b)=\langle\lambda_{s}(b)\xi,\eta\rangle,$

for $\xi,\eta\in\ell^{2}(\mathcal{B})$, $s\in G$ and $b\in B_{s}$.

######  Definition 3.6.

The Fourier space $A(\mathcal{B})$ of a Fell bundle $\mathcal{B}$ is defined
by

$A(\mathcal{B})=\left\\{\lambda_{\xi,\eta}:\xi,\eta\in\ell^{2}(\mathcal{B})\right\\}.$

Put

$\|\lambda_{\xi,\eta}\|=\sup\\{\|\langle\lambda(f)\xi,\eta\rangle\|:f\in\ell^{1}(\mathcal{B}),~{}\|f\|_{*}\leq
1\\},$

where, $\langle\lambda(f)\xi,\eta\rangle:=\sum_{r\in
G}\langle\lambda_{r}(f_{r})\xi,\eta\rangle$.

First let us observe that $A(\mathcal{B})$ is a two-sided $A(G)$-module: given
$\lambda_{\xi,\eta}\in A(\mathcal{B})$ and $\lambda^{G}_{\alpha,\beta}\in
A(G)$, for every $s\in G$ and $b\in B_{s}$,

$(\lambda_{\xi,\eta}\cdot\lambda^{G}_{\alpha,\beta})_{s}(b)=(\lambda^{G}_{\alpha,\beta}\cdot\lambda_{\xi,\eta})_{s}(b)=\langle\lambda_{s}(b)\xi,\eta\rangle\langle\lambda_{s}^{G}\alpha,\beta\rangle\in
B_{e}.$

######  Example 3.7.

Fourier space could be calculated for typical cases of Fell bundles.

$(i)$ Let $\mathcal{B}$ be the Fell bundle of a $C^{*}$-dynamical system
$(A,G,\gamma)$. By Example 2.2$(ii)$, we have the identification

$\ell^{2}(\mathcal{B})=\\{(a_{s})\in\Pi_{s\in G}B_{s}:\|\sum_{s\in
G}\gamma_{s^{-1}}(a_{s}^{*}a_{s})\|<\infty\\}.$

In this case, $\lambda_{\xi,\eta,s}(b)=(e,\sum_{t\in
G}a_{s^{-1}t}^{*}b^{*}b_{t})\in B_{e}$. Back to Example 3.5($ii$), the
isomorphism sending $B(\mathcal{B})$ onto $B(A\rtimes_{\gamma}G)$ sends
$A(\mathcal{B})$ onto $A(A\rtimes_{\gamma}G)$, defined by Pedersen [9, page
262]. An element in $B_{+}(A\rtimes_{\gamma}G)$ belongs to
$A_{+}(A\rtimes_{\gamma}G)$ [9, 7.7.6] and $A(A\rtimes_{\gamma}G)$ is nothing
but the span closure of elements of “finite support” in
$B_{+}(A\rtimes_{\gamma}G)$ (where finite support here simply means that the
corresponding map on $G$ is of finite support). It now follows that
$A(\mathcal{B})$ coincides with the space $A(A\rtimes_{\gamma}G)$ considered
by Pedersen, when $\mathcal{B}$ is the Fell bundle of system $(A,G,\gamma)$.
In particular, it follows from (the proof of) [9, 7.7.7] that for Fell bundles
coming from actions, $A(\mathcal{B})=B(\mathcal{B})$ when $G$ is discrete
amenable (we make these vague ideas into rigorous proofs in the next section).

$(ii)$ Let $\mathcal{B}=\\{A_{s}\\}_{s\in G}$ be the Fell bundle whicof a
coaction $\delta:A\rightarrow$ $A\otimes C^{*}(G)$ as in Example 2.2$(iii)$.
Then

$\ell^{2}(\mathcal{B})=\\{(a_{s})\in\Pi_{s\in G}A_{s}:\|\sum_{s\in
G}a_{s}^{*}a_{s}\|<\infty\\}.$

In this case, $\lambda_{\xi,\eta,s}(b)=\sum_{t\in
G}a_{s^{-1}t}^{*}b^{*}b_{t}$, for $\xi=(a_{t})_{t\in G},\eta=(b_{t})_{t\in
G}\in\ell^{2}(\mathcal{B})$ and $b\in A_{s}$. Back to Example 3.5($iii$), we
say that a positive functional $\phi\in A^{*}_{+}$ is of “compact support” if
$\phi(\sum_{s\in G}a_{s}^{*}a_{s})<\infty$. One could now identify the Fourier
space $A(\mathcal{A})$ of the Fell bundle $\mathcal{A}$ of a coaction with the
span closure of positive functionals.

Let $c_{0}(G,\mathcal{L}(\mathcal{B},B_{e}))$ be the Banach space of all maps
$\varphi$ from $G$ into disjoint union of spaces $\mathcal{L}({B_{s}},B_{e})$,
vanishing at infinity with $\varphi(s)\in\mathcal{L}(B_{s},B_{e})$, for every
$s\in G$, with norm $\|\varphi\|_{\infty}=\sup_{s\in G}\|\varphi(s)\|$. We
denote by $c_{c}(G,\mathcal{L}(\mathcal{B},B_{e}))$ the set of all $\varphi\in
c_{0}(G,\mathcal{L}(\mathcal{B},B_{e}))$ with finite support.

It is known that for a (locally compact) group $G$, the Fourier algebra $A(G)$
is a subspace of $C_{0}(G)$. This also holds for Fell bundles in the following
sense.

######  Proposition 3.8.

$A(\mathcal{B})$ is a subspace of $c_{0}(G,\mathcal{L}(\mathcal{B},B_{e}))$.

###### Proof.

Given $\varphi,\psi\in C_{c}(\mathcal{B})$ with
$K:=\operatorname{supp}(\varphi)$ and $L:=\operatorname{supp}(\psi)$,
$KL^{-1}$ is compact, and given $s\notin KL^{-1}$, it is easy to see that
$\lambda_{\varphi,\psi,s}=0$. Therefore, $\lambda_{\varphi,\psi}\in
c_{c}(G,\mathcal{L}(\mathcal{B},B_{e}))$. If
$\xi,\eta\in\ell^{2}(\mathcal{B})$, there exist sequences $(\varphi_{n})$ and
$(\psi_{n})$ in $c_{c}(\mathcal{B})$ such that
$\|\varphi_{n}-\xi\|_{2}\rightarrow 0$ and $\|\psi_{n}-\eta\|_{2}\rightarrow
0$. We have,

$\displaystyle\|\lambda_{\varphi_{n},\psi_{n}}-\lambda_{\xi,\eta}\|_{\infty}$
$\displaystyle=$ $\displaystyle\sup_{s\in
G}\|\lambda_{\varphi_{n},\psi_{n},s}-\lambda_{\psi,\eta,s}\|$
$\displaystyle\leq$
$\displaystyle\|\varphi_{n}\|_{2}\|\psi_{n}-\eta\|_{2}+\|\varphi_{n}-\xi\|_{2}\|\eta\|_{2}\rightarrow
0.$

Thus, $\lambda_{\xi,\eta}\in c_{0}(G,\mathcal{L}(\mathcal{B},B_{e}))$. ∎

##  4\. multiplier algebra and bounded approximate identities

In this section we find sufficient conditions on Fell bundles for the Fourier
and Fourier-Stieltjes spaces to be Banach algebras.

Let $\mathcal{A}=\\{A_{s}\\}_{s\in G}$ and $\mathcal{B}=\\{B_{t}\\}_{t\in H}$
be Fell bundles over discrete groups $G$ and $H$. For $s\in G$ and $t\in H$,
consider the algebraic tensor product $A_{s}\odot B_{t}$ and put
$\mathcal{A}\odot\mathcal{B}=\\{A_{s}\odot B_{t}\\}_{(s,t)\in G\times H}$.
This is called the algebraic tensor product of $\mathcal{A}$ and
$\mathcal{B}$. Let $p$ be a $C^{*}$-norm on $A_{e}\odot B_{e}$ satisfying
$p(x^{*}x)=p(xx^{*})$, for all $x\in\mathcal{A}\odot\mathcal{B}$. Put
$\|x\|_{p}=\sqrt{p(x^{*}x)}$ and let $A_{s}\otimes_{p}B_{t}$ be the
corresponding completion of $A_{s}\odot B_{t}$. Then,
$\mathcal{A}\otimes_{p}\mathcal{B}:=\\{A_{s}\otimes_{p}B_{t}\\}_{(s,t)\in
G\times H}$ is a Fell bundle over $G\times H$. We refer the reader for more
details to [1].

The notion of Hopf coalgebra is introduced by Turaevin in [12] and studied by
Virelizier in [13]. Roughly speaking, a Hopf $G$-coalgebra is a family
$D=\\{D_{s}\\}_{s\in G}$ of algebras endowed with a co-multiplication
$\Delta=\\{\Delta_{s,t}:D_{st}\rightarrow D_{s}\otimes D_{t}\\}_{s,t\in G}$, a
counit $\epsilon:D_{e}\rightarrow\mathbb{C}$ and an antipode
$S=\\{S_{t}:H_{t}\rightarrow H_{t^{-1}}\\}$, satisfying natural compatibility
conditions.

######  Proposition 4.1.

Let $\mathcal{B}=\\{B_{s}\\}_{s\in G}$ be a Fell bundle over $G$ endowed with
a family $\Delta=\\{\Delta_{s,t}:B_{st}\rightarrow
B_{s}\otimes_{p}B_{t}\\}_{s,t\in G}$ of bounded linear maps satisfying

$(\Delta_{r,s}\otimes\mathrm{id}_{B_{t}})\Delta_{rs,t}=(\mathrm{id}_{B_{r}}\otimes\Delta_{s,t})\Delta_{r,st},$

for $r,s,t\in G$. Then $B(\mathcal{B})$ and $A(\mathcal{B})$ are Banach
algebras.

###### Proof.

Given representations $\pi$ and $\rho$ of the bundle
$\mathcal{B}=\\{B_{s}\\}_{s\in G}$, we define
$(\pi\otimes\rho)_{(s,t)}:B_{s}\otimes_{p}B_{t}\rightarrow\mathcal{L}(H_{\pi}\otimes
H_{\rho})$ by

$(\pi\otimes\rho)_{(s,t)}(b\otimes
c):=(\pi_{s}\otimes\rho_{t})\Delta_{s,t}(bc),\quad(s,t\in G,b\in B_{s},c\in
B_{t}).$

Then, $\pi\otimes\rho$ is a representation for Fell bundle
$\mathcal{B}\otimes_{p}\mathcal{B}=\\{B_{s}{\otimes}_{p}B_{t}\\}_{(s,t)\in
G\times G}$ over $G\times G$. For $\xi,\eta\in H_{\pi}$, $\alpha,\beta\in
H_{\rho}$, and $s\in G$, and a fixed element $b^{s}_{0}\in B_{s}$, put

$(\pi_{\xi,\eta}\rho_{\alpha,\beta})_{s}(b):=(\pi\otimes\rho)_{\xi\otimes\alpha,\eta\otimes\beta,(s,s)}(b\otimes
b^{s}_{0}),$

and observe that

$(\pi_{\xi,\eta}\rho_{\alpha,\beta})_{s}(b)=\langle(\pi_{s}\otimes\rho_{s})\Delta_{s,s}(bb^{s}_{0})(\xi\otimes\alpha),\eta\otimes\beta\rangle.$

In particular, $\pi_{\xi,\eta}\rho_{\alpha,\beta}\in B(\mathcal{B})$ and it is
not difficult to see that
$\|\pi_{\xi,\eta}\rho_{\alpha,\beta}\|\leq\|\pi_{\xi,\eta}\|\|\rho_{\alpha,\beta}\|$.
Thus, $B(\mathcal{B})$ is a Banach algebra.

Next, we may view $\ell^{2}(\mathcal{B})\otimes\ell^{2}(\mathcal{B})$ as a
right Hilbert $B_{e}$-module via $(\xi_{1}\otimes\xi_{2})\cdot
a=\xi_{1}\otimes\xi_{2}\cdot a$ and $B_{e}$-valued inner product

$\langle\xi_{1}\otimes\xi_{2},\xi_{3}\otimes\xi_{4}\rangle=\langle\xi_{2},\lambda_{e}(\langle\xi_{1},\xi_{3}\rangle)\xi_{4}\rangle.$

Define
$(\lambda\otimes\lambda)_{s,t}:B_{s}\otimes_{\mathrm{min}}B_{t}\rightarrow\mathcal{L}(\ell^{2}(\mathcal{B})\otimes\ell^{2}(\mathcal{B}))$
by

$(\lambda\otimes\lambda)_{s,t}(b\otimes
c)=(\lambda_{s}\otimes\lambda_{t})\Delta_{s,t}(bc),$

and observe that,

$\displaystyle(\lambda_{\xi_{1},\xi_{2}}\lambda_{\xi_{3},\xi_{4}})_{s}(b)$
$\displaystyle:=$
$\displaystyle(\lambda\otimes\lambda)_{\xi_{1}\otimes\xi_{3},\xi_{2}\otimes\xi_{4},(s,s)}(b\otimes
b_{0}^{s})$ $\displaystyle=$
$\displaystyle\langle(\lambda_{s}\otimes\lambda_{s})\Delta_{s,s}(bb^{s}_{0})(\xi_{1}\otimes\xi_{3}),\xi_{2}\otimes\xi_{4}\rangle\in
B_{e}.$

A similar norm inequality holds in this case as well, and $A(\mathcal{B})$ is
also a Banach algebra. ∎

######  Example 4.2.

We provide concrete examples of cases in which Fourier-Stieltjes space is a
Banach algebra.

$(i)$ Let $(A,G,\gamma)$ be a $C^{*}$-dynamical system such that $A$ is a Hopf
$C^{*}$-algebra; i.e., there is a comultiplication $\Delta:A\rightarrow
M(A\otimes_{\mathrm{min}}A)$. It is known that $A^{*}$ is a Banach algebra
under $fg=(f\otimes g)\circ\Delta$, $f,g\in A^{*}$. Suppose that
$\mathcal{B}=\\{A_{s}\\}_{s\in G}$ is the Fell bundle of $(A,G,\gamma)$. Let
$\pi$ and $\rho$ be representations of $\mathcal{A}$ on Hilbert spaces
$H_{\pi}$ and $H_{\rho}$. Then each $A_{s}$ is isomorphic as a Banach space to
$A$ and for $\xi,\eta\in H_{\pi}$, $\alpha,\beta\in H_{\rho}$, and $s\in G$,
one may consider $\pi_{\xi,\eta,s}$ and $\rho_{\alpha,\beta,s}$ as bounded
linear functionals on $A$. This allows ud to consider
$\pi_{\xi,\eta,s}\otimes\rho_{\alpha,\beta,s}:M(A\otimes_{\mathrm{min}}A)\rightarrow\mathbb{C}$,
defined by

$\pi_{\xi,\eta,s}\otimes\rho_{\alpha,\beta,s}(a\otimes
b)=\langle\pi_{s}((s,a))\xi,\eta\rangle\langle\rho_{s}((s,b))\alpha,\beta\rangle.$

Thus, $\pi\otimes\rho$ is a representation of the Fell bundle
$\mathcal{A}\otimes_{\mathrm{min}}\mathcal{A}=\\{A_{s}\otimes_{\mathrm{min}}A_{t}\\}_{(s,t)\in
G\times G}$ on Hilbert space $H_{\pi}\otimes H_{\rho}$, with

$(\pi\otimes\rho)_{(s,t)}(a\otimes b)=\pi_{s}((s,a))\otimes\rho_{t}((t,b)).$

Finally, given $\pi_{\xi,\eta},\rho_{\alpha,\beta}\in B(\mathcal{A})$,

$(\pi_{\xi,\eta}\rho_{\alpha,\beta})_{s}((s,a))=\langle(\pi_{\xi,\eta,s}\otimes\rho_{\alpha,\beta,s}),\Delta(a)\rangle,$

showing that the Fourier-Stieltjes space $B(\mathcal{A})$ is closed under
multiplication. The other conditions are now straightforward.

$(ii)$ Let $(M,\Delta)$ be a Hopf-von Neumann algebra and $(M,G,\gamma)$ be a
$W^{*}$-dynamical system. Let $\mathcal{M}=\\{M_{s}\\}_{s\in G}$ be the
corresponding Fell bundle. For every $s\in G$, define $f_{s}:M\rightarrow
M_{s}$ by $f_{s}(a)=(s,a)$. It is straightforward to see that $f_{r}\otimes
f_{s}:M\bar{\otimes}M\rightarrow M_{r}{\otimes}_{p}M_{s}$ is a bounded linear
map, where $(f_{r}\otimes f_{s})(a\otimes b)=f_{r}(a)\otimes f_{s}(b)$, for
$r,s\in G$ and $a,b\in M$, where the domain is the von Neumann algebra tensor
product. The coproduct family

$\Delta_{r,s}:M_{rs}\rightarrow
M_{r}{\otimes}_{p}M_{s},\quad\Delta_{r,s}((rs,a))=(f_{r}\otimes
f_{s})(\Delta(a))$

satisfies
$(\Delta_{r,s}\otimes\mathrm{id}_{B_{t}})\Delta_{rs,t}=(\mathrm{id}_{B_{r}}\otimes\Delta_{s,t})\Delta_{r,st}$.
In particular, the Fourier-Stieltjes space $B(\mathcal{B})$ and Fourier space
$A(\mathcal{B})$ are Banach algebras. In fact, given
$\pi_{\xi,\eta},\rho_{\alpha,\beta}\in B(\mathcal{B})$,

$\displaystyle(\pi_{\xi,\eta}\rho_{\alpha,\beta})_{s}(s,b)$ $\displaystyle=$
$\displaystyle\langle(\pi_{s}\otimes\rho_{s})\Delta_{s,s}(s^{2},b)\xi\otimes\alpha,\eta\otimes\beta\rangle$
$\displaystyle=$ $\displaystyle\langle(\pi_{s}\otimes\rho_{s})(f_{s}\otimes
f_{s})(\Delta(b))\xi\otimes\alpha,\eta\otimes\beta\rangle.$

$(iii)$ For the trivial Fell bundle
$\mathcal{B}=G\times\mathbb{C}=\\{B_{s}\\}_{s\in G}$, let us observe that
$A(\mathcal{B})$ is canonically isomorphic to the classical Fourier algebra
$A(G)$. Define

$\Delta_{s,t}:B_{st}\rightarrow
B_{s}{\otimes}_{p}B_{t},\quad(st,z)\mapsto(s,z){\otimes}(t,1).$

This a compatible coproduct family. Define,
$\theta:\mathcal{B}\rightarrow\mathbb{C}$ by $\theta(b)=z_{b}$, for $s\in G$
and $b=(s,z_{b})\in B_{s}$. Then, $\theta(b^{*})=\overline{z_{b}}$ and

$\langle\lambda_{s}(b)\xi,\eta\rangle=\left(e,\overline{z_{b}}\langle\lambda^{G}_{s}\theta\xi,\theta\eta\rangle_{\ell^{2}(G)}\right),$

where $\lambda^{G}$ is the left regular representation of $G$. Define
$T_{b}:A(\mathcal{B})\rightarrow A(G)$ by

$T_{b}(\lambda_{\xi,\eta})(s)=\frac{1}{\overline{z_{b}}}\theta\left(\langle\lambda_{s}(b)\xi,\eta\rangle\right)=\langle\lambda^{G}_{s}{\theta\xi,\theta\eta}\rangle_{\ell^{2}(G)},$

where $b=(s,z_{b})$ is a nonzero element of $B_{s}$. Consider the map
$T_{b}:A(\mathcal{B})\rightarrow A(G)$. It is easy to see that $T_{b}$ is a
bounded linear map. Moreover,

$\displaystyle T_{b}(\lambda_{\xi,\eta}\lambda_{\alpha,\beta})(s)$
$\displaystyle=$ $\displaystyle
T_{b}((\lambda\otimes\lambda)_{\xi\otimes\alpha,\eta\otimes\beta})(s)$
$\displaystyle=$
$\displaystyle\frac{1}{\overline{z_{b}}}\theta(\langle(\lambda_{s}\otimes\lambda_{s})\Delta_{s,s}(b(s,1))(\xi\otimes\alpha),\eta\otimes\beta\rangle)$
$\displaystyle=$
$\displaystyle\frac{1}{\overline{z_{b}}}\theta(\langle(\lambda_{s}\otimes\lambda_{s})(b\otimes(s,1))(\xi\otimes\alpha),\eta\otimes\beta\rangle)$
$\displaystyle=$
$\displaystyle\frac{1}{\overline{z_{b}}}\theta\left(\langle\lambda_{s}(b)\xi,\eta\rangle\otimes\langle\lambda_{s}((s,1))\alpha,\beta\rangle\right)$
$\displaystyle=$
$\displaystyle\frac{1}{\overline{z_{b}}}\theta\left((e,\overline{z_{b}}\langle\lambda^{G}_{s}{\theta\xi,\theta\eta}\rangle_{\ell^{2}(G)})\otimes(e,\langle\lambda^{G}_{s}{\theta\alpha,\theta\beta}\rangle_{\ell^{2}(G)})\right)$
$\displaystyle=$
$\displaystyle\langle\lambda^{G}_{s}{\theta\xi,\theta\eta}\rangle_{\ell^{2}(G)}\langle\lambda^{G}_{s}{\theta\alpha,\theta\beta}\rangle_{\ell^{2}(G)}$
$\displaystyle=$ $\displaystyle
T_{b}(\lambda_{\xi,\eta})(s)T_{b}(\lambda_{\alpha,\beta})(s).$

Thus, $T_{b}$ is homomorphism. In fact,
$T_{b}(\lambda_{\xi,\eta})=\lambda^{G}_{\theta\xi,\theta\eta}$, and since
$\|\lambda_{\xi,\eta}\|=\|\lambda^{G}_{\theta\xi,\theta\eta}\|$, it follows
that $T_{b}$ is an isometric isomorphism.

Back to general Fell bundles, since $B_{s}B_{e}\subseteq B_{s}$, it follows
that $\ell^{2}(\mathcal{B})$ is a Hilbert $B_{e}$-module if we put $(\xi\cdot
a)(s)=\xi(s)a$, for $\xi\in\ell^{2}(\mathcal{B})$ and $a\in B_{e}$. Then for
any Hilbert space $H$, $\ell^{2}(\mathcal{B})\otimes H$ is a Hilbert
$B_{e}$-module via $(\xi\otimes h)\cdot a=\xi\cdot a\otimes h$ and

$\langle\xi\otimes h,\eta\otimes k\rangle=\langle
h,k\rangle\langle\xi,\eta\rangle\in
B_{e},\quad(\xi,\eta\in\ell^{2}(\mathcal{B})\;\mathrm{and}\;h,k\in H).$

For a representation $\pi:\mathcal{B}\rightarrow\mathcal{L}(H_{\pi})$,
consider
$\lambda\otimes\pi:\mathcal{B}\rightarrow\mathcal{L}(\ell^{2}(\mathcal{B})\otimes
H_{\pi})$, defined by
$(\lambda\otimes\pi)_{s}(b):=(\lambda_{s}\otimes\pi_{s})\Delta_{s,s}(bb^{s}_{0})$,
for $s\in G$ and $b\in B_{s}$.

For a Banach algebra $A$, let $M(A)$ be the Banach algebra of double
centralisers, that is, pairs $(L,R)$ of linear maps on $A$ with

$\left\\{\begin{array}[]{c}L(ab)=L(a)b,\quad R(ab)=aR(b)\\\
aL(b)=R(a)b\end{array}\quad(a,b\in\mathcal{A})\right..$

We denote by $M(A(\mathcal{B}))$ the multiplier algebra of $A(\mathcal{B})$.
In the group case, $A(G)$ has a bounded approximate identity if and only if
$M(A(G))=B(G)$. We show that a similar statement holds for Fell bundles.

######  Lemma 4.3.

Let $\mathcal{B}$ be a Fell bundle over a discrete group $G$. If $\pi$ is a
representation of $\mathcal{B}$, then $\lambda_{\xi,\eta}\pi_{\alpha,\beta}\in
A(\mathcal{B})$.

###### Proof.

Let $\lambda_{\xi,\eta}\in A(\mathcal{B})$ and $\pi_{\alpha,\beta}\in
B(\mathcal{B})$. Then

$\displaystyle(\lambda_{\xi,\eta}\pi_{\alpha,\beta})_{s}(b)$ $\displaystyle=$
$\displaystyle\langle(\lambda_{s}\otimes\pi_{s})\Delta_{s,s}(bb^{s}_{0})(\xi\otimes\alpha),\eta\otimes\beta\rangle$
$\displaystyle=$
$\displaystyle\langle(\lambda_{s}\otimes\pi_{s})\sum_{i=1}^{\infty}(b_{i}\otimes
d_{i})(\xi\otimes\alpha),\eta\otimes\beta\rangle$ $\displaystyle=$
$\displaystyle\sum_{i=1}^{\infty}\langle\lambda_{s}(b_{i})\xi\otimes\pi_{s}(d_{i})\alpha,\eta\otimes\beta\rangle$
$\displaystyle=$
$\displaystyle\sum_{i=1}^{\infty}\langle\lambda_{s}(b_{i})\xi,\eta\rangle\langle\pi_{s}(d_{i})\alpha,\beta\rangle\in
B_{e}.$

Thus, $\lambda_{\xi,\eta}\pi_{\alpha,\beta}\in A(\mathcal{B})$. ∎

######  Theorem 4.4.

Let $\mathcal{B}$ be a Fell bundle over a discrete group $G$. If
$A(\mathcal{B})$ is a Banach algebra with a bounded approximate identity, then
$M(A(\mathcal{B}))=B(\mathcal{B})$.

###### Proof.

Let $(e_{i})_{i}=(\lambda_{\xi_{i},\eta_{i}})_{i}$ be a bounded approximate
identity for $A(\mathcal{B})$ with $\|e_{i}\|=1$. By Lemma 4.3, for any $w\in
B(\mathcal{B})$, the bounded linear maps $L_{w}:A(\mathcal{B})\rightarrow
A(\mathcal{B})$ and $R_{w}:A(\mathcal{B})\rightarrow A(\mathcal{B})$, given by
$L_{w}(v)=wv$ and $R_{w}(v)=vw$, are well defined.

Let $(L,R)\in M(A(\mathcal{B}))$. Each element $R(e_{i})$ may be regarded as
bounded linear map from $C^{*}(\mathcal{B})$ into $B_{e}$ via

$R(e_{i})(f)=\sum_{s\in
G}\langle\lambda_{s}(f_{s})\xi_{i},\eta_{i}\rangle,\quad(f\in\ell^{1}(\mathcal{B})).$

Since, $\sup_{i\in I}\|R(e_{i})(f)\|<\infty$, by the uniform boundedness
principle, there exists $w\in\mathcal{L}(C^{*}(\mathcal{B}),B_{e})$ such that
$w(f)=\lim_{i}R(e_{i})(f)$, for $f\in\ell^{1}(\mathcal{B})$.

Fix $u\in A(\mathcal{B}))$. Then,

$uw(f)=\lim_{i}(uR(e_{i}))(f)=\lim_{i}R(ue_{i})(f)=R(u)(f),\quad(f\in\ell^{1}(\mathcal{B})).$

Therefore, $R=R_{w}$, for some $w\in B(\mathcal{B})$, where $R_{w}(v)=vw$.
Similarly, $L(v)=L_{w}(v)$, where $L_{w}(v)=wv$. Thus,
$M(A(\mathcal{B}))=B(\mathcal{B})$, as required. ∎

Let $(A,G,\gamma)$ be a $C^{*}$-dynamical system such that $A$ is unital with
unit element $1$ and let $\mathcal{B}=\\{B_{s}\\}_{s\in G}$ be the
corresponding Fell bundle. For any subset $V\subseteq G$, define
$1_{V}:G\rightarrow\mathcal{B}$ by $1_{V}(s)=(s,1)$, for $s\in V$, and
$1_{V}(s)=(s,0)$, otherwise.

The next lemma extends Eymard’s trick [4].

######  Lemma 4.5.

Let $\mathcal{B}=\\{B_{s}\\}_{s\in G}$ be the Fell bundle of a C*-dynamical
system $(A,G,\gamma)$, $U$ be a subset of $G$ and $K\subseteq U$ be a compact
set. There exists $u\in A(\mathcal{B})$ such that $u_{s}(s,a)=(e,a^{*})$, for
$s\in K$ and $u_{s}(s,a)=(s,0)$, otherwise.

###### Proof.

Let $V$ be a finite neighborhood of $e\in G$. Define
$u=\frac{1}{|V|}\lambda_{1_{V},1_{KV}}$, where $|V|$ is the cardinal of $V$.
Then

$\displaystyle u_{s}(s,a)$ $\displaystyle=$
$\displaystyle\frac{1}{|V|}\langle\lambda_{s}(s,a)1_{V},1_{KV}\rangle$
$\displaystyle=$ $\displaystyle\frac{1}{|V|}\sum_{r\in
G}(\lambda_{s}(s,a)1_{V}(r))^{*}1_{KV}(r)$ $\displaystyle=$
$\displaystyle\frac{1}{|V|}\sum_{r\in
G}(1_{V}(s^{-1}r))^{*}(s^{-1},a^{*})1_{KV}(r)$ $\displaystyle=$
$\displaystyle\frac{1}{|V|}\sum_{r\in sV\cap
KV}(r^{-1}s,1)(s^{-1},a^{*})(r,1)$ $\displaystyle=$
$\displaystyle\frac{1}{|V|}\sum_{r\in sV\cap KV}(e,a^{*})=\frac{|sV\cap
KV|}{|V|}(e,a^{*}).$

Now, if $s\in K$, then $u_{s}(s,a)=(e,a^{*})$, and if $s\notin U$, then
$sV\cap KV=\emptyset$, and so $u_{s}(s,a)=(e,0)$. ∎

Let $\mathcal{B}=\\{B_{s}\\}_{s\in G}$ be the Fell bundle of a system
$(A,G,\gamma)$, where $A$ is unital with unit element $1$. Define
$\Delta_{s,t}:B_{st}\rightarrow B_{s}\otimes_{min}B_{t}$ by
$\Delta_{s,t}(st,a)=(s,1)\otimes(t,a)$. It is easy to see that,

$(\Delta_{r,s}\otimes\mathrm{id}_{B_{t}})\Delta_{rs,t}=(\mathrm{id}_{B_{r}}\otimes\Delta_{s,t})\Delta_{r,st}.$

Next, if $\pi_{\alpha_{1},\beta_{1}},\rho_{\alpha_{2},\beta_{2}}\in
B(\mathcal{B})$, then

$\displaystyle(\pi_{\alpha_{1},\beta_{1}}\rho_{\alpha_{2},\beta_{2}})_{s}(s,a)$
$\displaystyle=$
$\displaystyle\langle(\pi_{s}\otimes\rho_{s})(\Delta_{s,s}(s^{2},a))\alpha_{1}\otimes\alpha_{2},\beta_{1}\otimes\beta_{2}\rangle$
$\displaystyle=$
$\displaystyle\langle(\pi_{s}\otimes\rho_{s})((s,1)\otimes(s,a))\alpha_{1}\otimes\alpha_{2},\beta_{1}\otimes\beta_{2}\rangle$
$\displaystyle=$
$\displaystyle\langle\pi_{s}(s,1)\alpha_{1},\beta_{1}\rangle\langle\rho_{s}(s,a)\beta_{1},\beta_{2}\rangle.$

Therefore, $B(\mathcal{B})$ is a Banach algebra. Similarly, one can show that
$A(\mathcal{B})$ is a Banach algebra. Moreover, for any $\lambda_{\xi,\eta}\in
A(\mathcal{B})$ and $\pi_{\alpha,\beta}\in B(\mathcal{B})$,

$\lambda_{\xi,\eta}\pi_{\alpha,\beta}\in A(\mathcal{B}).$

Now we have $\ell^{1}(\mathcal{B})\cong\ell^{1}(G,A)$, and
$B(\mathcal{B})=C^{*}(\mathcal{B})^{*}=(A\rtimes G)^{*}$. Also, under the
canonical correspondence discussed before, $P(\mathcal{B})$ is associated to

$\\{\phi:G\rightarrow
A^{*}:\phi(s)(a)=\varphi(au_{s}),~{}\mathrm{for~{}some}~{}\mathrm{positive~{}functional~{}}\varphi~{}\mathrm{on}~{}C^{*}(\mathcal{B})\\}$

In this case, for every $\lambda_{\xi,\eta}\in A(\mathcal{B})$,

$\lambda^{*}_{\xi,\eta,s}(s,a):=\langle\lambda_{s}(s,a)\xi,\eta\rangle^{*}=\langle\lambda_{s^{-1}}(s^{-1},a^{*})\eta,\xi\rangle=\lambda_{\eta,\xi,s^{-1}}(s^{-1},a^{*}).$

Therefore, $A(\mathcal{B})$ is a Banach $*$-algebra. Similarly
$B(\mathcal{B})$ is a Banach $*$-algebra.

######  Theorem 4.6.

Let $\mathcal{B}$ be the Fell bundle of C*-dynamical system $(A,G,\gamma)$
with $A$ unital. Then,

${(i)}$ the span closure of $P(\mathcal{B})\cap c_{c}(G,A^{*})$ is isomorphic
to $A(\mathcal{B})$, as a Banach space.

${(ii)}$ for a positive linear functional $\psi:B_{e}\rightarrow\mathbb{C}$
with $\psi(e,1)=1$, there is a surjective bounded linear map
$T_{\psi}:A(\mathcal{B})\rightarrow\overline{\mathrm{span}}(P(\mathcal{B})\cap
c_{c}(G,A^{*}))$.

###### Proof.

${(i)}$ Given $\phi\in P(\mathcal{B})\cap c_{c}(G,A^{*})$ with
$K=\mathrm{supp}(\phi)$, by Lemma 4.5, there exists an element $u_{\phi}\in
A(\mathcal{B})$ with $(u_{\phi})_{s}(s,a)=(e,a^{*})$, for $s\in K$ and $a\in
A$. Define $T:\overline{\mathrm{span}}~{}P(\mathcal{B})\cap
c_{c}(G,A^{*})\rightarrow A(\mathcal{B})$ by $T(\phi)=u_{\phi}\phi$.

For $\phi\in P(\mathcal{B})\cap c_{c}(G,A^{*})$, there are
$\xi,\eta\in\ell^{2}(\mathcal{B})$ and $\pi_{\alpha,\alpha}\in B(\mathcal{B})$
such that $u_{\phi}=\lambda_{\xi,\eta}$ and $\phi=\pi_{\alpha,\alpha}$. Thus,
for every $s\in K$,

$\displaystyle(u_{\phi}\phi)_{s}(s,a)$ $\displaystyle=$
$\displaystyle(\lambda_{\xi,\eta}\pi_{\alpha,\alpha})_{s}(s,a)$
$\displaystyle=$
$\displaystyle\langle\lambda_{s}(s,1)\xi,\eta\rangle\langle\pi_{s}(s,a)\alpha,\alpha\rangle$
$\displaystyle=$ $\displaystyle(e,1)\langle\pi_{s}(s,a)\alpha,\alpha\rangle$
$\displaystyle=$ $\displaystyle(e,1)\phi_{s}(s,a).$

Therefore, $T(\phi)=(e,1)\phi$, and it $T$ is linear with
$\|T(\phi)\|=\|\phi\|$. Moreover, for $\pi_{\alpha,\beta}$ and
$\rho_{\alpha^{\prime},\beta^{\prime}}$ in $B(\mathcal{B})$,

$\displaystyle(T(\pi_{\alpha,\alpha}\rho_{\beta,\beta}))_{s}(s,a)$
$\displaystyle=$
$\displaystyle(e,1)(\pi_{\alpha,\alpha}\rho_{\beta,\beta})_{s}(s,a)$
$\displaystyle=$
$\displaystyle(e,1)\langle\pi_{s}(s,1)\alpha,\alpha\rangle(e,1)\langle\rho_{s}(s,a)\beta,\beta\rangle$
$\displaystyle=$
$\displaystyle(T(\pi_{\alpha,\alpha})T(\rho_{\beta,\beta}))_{s}(s,a).$

Thus, $T$ is homomorphism.

$(ii)$ Given $\xi\in c_{c}(\mathcal{B})$, consider

$\varphi_{\xi}:\ell^{1}(\mathcal{B})\rightarrow B_{e};\
\varphi_{\xi}(f)=\overline{\langle\lambda(f)\xi,\xi\rangle}=\overline{\sum_{r\in
G}\langle\lambda_{r}(f_{r})\xi,\xi\rangle}.$

One may extend $\varphi_{\xi}$ to a positive linear map on
$C^{*}(\mathcal{B})$. Put
$T_{\psi}(\lambda_{\xi,\xi})(f)=\psi\circ\phi_{\xi}(f)$. Then,
$T_{\psi}(\lambda_{\xi,\xi})$ is a positive linear functional on
$C^{*}(\mathcal{B})$ and
$T_{\psi}(\lambda_{\xi,\xi})_{s}(s,a)=\psi(\overline{\langle\lambda_{s}(s,a)\xi,\xi\rangle})$.
Moreover, by Lemma 3.4, we have $T_{\psi}(\lambda_{\xi,\xi})\in
P(\mathcal{B})\cap c_{c}(G,A^{*}).$

For $\lambda_{\xi,\eta}\in A(\mathcal{B})$ let
$\varphi_{\xi,\eta}:C^{*}(\mathcal{B})\rightarrow B_{e}$ be defined by
$\varphi_{\xi,\eta}(f)=\overline{\langle\lambda(f)\xi,\eta\rangle}$ and put
$T_{\psi}(\lambda_{\xi,\eta})=\psi\circ\phi_{\xi,\eta}$. There are sequences
$(\xi_{n})_{n}$ and $(\eta_{n})_{n}$ in $c_{c}(\mathcal{B})$ such that
$\|\lambda_{\xi_{n},\eta_{n}}-\lambda_{\xi,\eta}\|_{A(\mathcal{B})}\rightarrow
0$. Moreover,

$\lambda_{\xi_{n},\eta_{n}}=\frac{1}{4}\sum_{k=0}^{3}i^{k}\lambda_{i^{k}\xi_{n}+\eta_{n},i^{k}\xi_{n}+\eta_{n}}.$

Therefore, for each $n$,
$T(\lambda_{\xi_{n},\eta_{n}})\in\mathrm{span}~{}P(\mathcal{B})\cap
c_{c}(G,A^{*}).$

A straightforward calculation gives
$\|T_{\psi}(\lambda_{\xi_{n},\eta_{n}})-T_{\psi}(\lambda_{\xi,\eta})\|_{B(\mathcal{B})}\rightarrow
0$, that is,
$T_{\psi}(\lambda_{\xi,\eta})\in\overline{\mathrm{span}}~{}P(\mathcal{B})\cap
c_{c}(G,A^{*})$.

Finally, if $\phi\in P(\mathcal{B})\cap c_{c}(G,A^{*})$, by Lemma 4.5, there
exists $u_{\phi}\in A(\mathcal{B})$ such that $u_{\phi}\phi=(e,1)\phi\in
A(\mathcal{B})$. Thus,

$T_{\psi}(u_{\phi}\phi)_{s}(s,a)=\psi((e,1)\phi_{s}(s,a))=\phi_{s}(s,a),$

therefore, $T_{\psi}$ is onto. ∎

We close this section by proving one side of the celebrated Leptin’s theorem
(cf., [4]) for Fell bundles coming from actions. We leave the inverse
implication as an open problem.

######  Theorem 4.7.

Let $G$ be an amenable discrete group and $\mathcal{B}$ be the Fell bundle of
a $C^{*}$-dynamical system $(A,G,\gamma)$ with $A$ unital. Then
$A(\mathcal{B})$ has a bounded approximate identity.

###### Proof.

Given $K\subseteq G$ compact and $\epsilon>0$, since $G$ is amenable, one may
find a finite set $C$ such that $|KC|<(1+\epsilon)|C|$ [10, Theorem 7.9].
Next, consider the collection of maps $u_{K,\epsilon}\in A(\mathcal{B})\cap
c_{c}(G,\mathcal{L}(\mathcal{B},B_{e}))$, defined by

$u_{K,\epsilon}=\frac{1}{(1+\epsilon)|C|}\lambda_{1_{C},1_{KC}}.$

If $\mathcal{K}(G)$ denote the set of all finite subsets of $G$, with preorder
on the set $\mathcal{K}(G)\times\mathbb{R}^{+}$ defined by
$(K,\epsilon)<(K^{\prime},\epsilon^{\prime})$, whenever $K\subseteq
K^{\prime}$ and $\epsilon>\epsilon^{\prime}$, then

$\|u_{K,\epsilon}\|_{A(\mathcal{B})}\leq\frac{1}{(1+\epsilon)|C|}\|1_{C}\|_{2}\|1_{KC}\|_{2}=\frac{|KC|^{1/2}}{(1+\epsilon)|C|^{1/2}}\leq\frac{\sqrt{1+\epsilon}}{1+\epsilon}\leq
1.$

Now let $v=\lambda_{\xi,\eta}\in A(\mathcal{B})\cap
c_{c}(G,\mathcal{L}(\mathcal{B},B_{e}))$, and set $K=\mathrm{supp}(v)$. Then
for $s\in K$,

$\displaystyle(u_{K,\epsilon})_{s}(s,a)$ $\displaystyle=$
$\displaystyle\frac{1}{(1+\epsilon)|C|}\langle\lambda_{s}(s,a)1_{C},1_{KC}\rangle$
$\displaystyle=$ $\displaystyle\frac{1}{(1+\epsilon)|C|}\sum_{r\in
G}(1_{C}(s^{-1}r))^{*}(s^{-1},a^{*})1_{KC}(r)$ $\displaystyle=$
$\displaystyle\frac{(e,a^{*})}{(1+\epsilon)|C|}\sum_{r\in C}1_{KC}(sr).$

It follows that

$\displaystyle((u_{K,\epsilon})v)_{s}(s,a)$ $\displaystyle=$
$\displaystyle\langle\lambda_{s}(s,a)\xi,\lambda_{e}((u_{K,\epsilon})_{s}(s,1))\eta\rangle$
$\displaystyle=$
$\displaystyle\langle\lambda_{s}(s,a)\xi,\lambda_{e}(\frac{(e,1)}{1+\epsilon})\eta\rangle$
$\displaystyle=$ $\displaystyle\frac{1}{1+\epsilon}v_{s}(s,a),$

that is,
$\|u_{K,\epsilon}v-v\|_{A(\mathcal{B})}=\frac{\epsilon}{\epsilon+1}\|v\|_{A(\mathcal{B})}$.
Thus, $\lim_{(K,\epsilon)}\|u_{K,\epsilon}v-v\|_{A(\mathcal{B})}=0$, for $v\in
A(\mathcal{B})\cap c_{c}(G,\mathcal{L}(\mathcal{B},B_{e}))$.

Next, given $w=\lambda_{\xi,\eta}\in A(\mathcal{B})$, there are sequences
$(\xi_{n})$ and $(\eta_{n})$ in $c_{c}(\mathcal{B})$ such that
$\|\lambda_{\xi_{n},\eta_{n}}-\lambda_{\xi,\eta}\|_{A(\mathcal{B})}\rightarrow
0$. Thus, there is a $v\in A(\mathcal{B})\cap
c_{c}(G,\mathcal{L}(\mathcal{B},B_{e}))$ such that $\|w-v\|<\epsilon$.
Finally,

$\displaystyle\left\|u_{K,\epsilon}w-w\right\|_{A(\mathcal{B})}$
$\displaystyle=$
$\displaystyle\left\|u_{K,\epsilon}w-u_{K,\epsilon}v+u_{K,\epsilon}v-v+v-w\right\|_{A(\mathcal{B})}$
$\displaystyle\leq$
$\displaystyle\left\|u_{K,\epsilon}w-u_{K,\epsilon}v\right\|_{A(\mathcal{B})}+\left\|u_{K,\epsilon}v-v\right\|_{A(\mathcal{B})}+\|v-w\|_{A(\mathcal{B})}$
$\displaystyle\leq$
$\displaystyle\left\|u_{K,\epsilon}\right\|_{A(\mathcal{B})}\|w-v\|_{A(\mathcal{B})}+\left\|u_{K,\epsilon}v-v\right\|_{A(\mathcal{B})}+\|v-w\|_{A(\mathcal{B})}$
$\displaystyle\leq$
$\displaystyle\left\|u_{K,\epsilon}v-v\right\|_{A(\mathcal{B})}+2\epsilon,$

namely, $(u_{K,\epsilon})$ is a left bounded approximate identity for
$A(\mathcal{B})$. Since $A(\mathcal{B})$ is a Banach $*$-algebra, it follows
that $A(\mathcal{B})$ has a two-sided bounded approximate identity. ∎

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