\section{Introduction}

The study of the beneficial role of disorder in a broad range of biological, physical, and chemical phenomena, has become a fundamental research topic in complex systems dynamics.
A seminal work in the field was the introduction of stochastic resonance (SR) in the early eighties, initially proposed to explain the occurrence of Earth ice ages~\cite{Benzi-1981a} and later on studied by numerous authors across various disciplines~\cite{Nicolis-1981a,Nicolis-1982a,Benzi-1983a,McNamara-1988a,Gammaitoni-1989a,Matteucci-1989a,Jung-1991a,Gammaitoni-1991a,Mori-2002a,Rao-2002a}. SR can happen when a nonlinear system is driven simultaneously by a periodic external forcing and noise, resulting in an amplification of the system response to the external signal~\cite{Heinsalu-2009a}.

Some of the subsequent studies showed that significant noise-driven effects, analogous to SR, can be observed also without periodicity of the external signal~\cite{Gang-1993a}, and even in the absence of any external signal, as in self-induced stochastic resonance~\cite{MURATOV2005227,Marius-2017} and coherence resonance~\cite{Pikovsky-1997a}.
Coherence resonance (CR) is an ordered response of a nonlinear excitable system to an optimal noise amplitude, resulting in regular pulses.
Beyond its effects on a single nonlinear unit, the role of noise was also extensively studied from the standpoint of its ability to improve synchronization in coupled oscillator networks, again both in the absence and in the presence of an external forcing~\cite{Jung-1995a,Liu-1999a,Busch-2003a}.
Broadly speaking, one may say that the first twenty years of research in this field focused, to a large degree, on the effects of noise in bistable or excitable systems~\cite{Gammaitoni-1998a,McDonnell-2009a}, comprising either several or just one element.

At the beginning of the new century, it was found that somewhat analogous effects to those of noise can be produced in networks of coupled oscillators through the heterogeneity of the oscillator population~\cite{Cartwright-2000a,Tessone-2006a}.
This led to the introduction of diversity-induced resonance (DIR), which denotes the amplification of a network response to an external signal, driven by the heterogeneity of network elements~\cite{Tessone-2006a,Toral-2009a,Chen-2009a,Wu-2010a,Wu-2010b,Patriarca-2012a,Tessone-2013a,Grace-2014a,Patriarca-2015a,Liang-2020a}. Just like SR, also DIR can occur both in the presence and in the absence of an external forcing.
In the latter case it has been named diversity-induced coherence~\cite{KAMAL-2015a}.

It should be clear from the above that SR and CR can occur even in systems made of a single element, therefore they are not intrinsically collective phenomena. Instead, by definition, DIR represents a collective disorder effect driven by population heterogeneity.

Most of the previous literature has emphasized either the analogies between SR and DIR~\cite{Tessone-2006a,Tessone-2007a}, considering them as two faces of the same medal, or the possibility to enhance resonance induced by noise thanks to diversity optimization (or vice versa)~\cite{degliesposti-2001a,Li-2012a,Li-2014a,Gassel-2007a}.
Relatively little work~\cite{Zhou-2001a,Glatt-2008a} has been devoted to the implications of the above-mentioned intrinsic difference between the two phenomena, which has not yet been fully analyzed. In this paper, by systematically studying the prototypical model of a heterogeneous network of pancreatic $\beta$-cells~\cite{Cartwright-2000a,Scialla-2021a} with the addition of noise, we provide insights into the different mechanisms by which diversity and noise can have markedly distinct effects on network dynamics.

\section{Model}
\label{model}

As a paradigmatic example of a system of coupled nonlinear units, we investigate an excitable network of FitzHugh-Nagumo elements.

Individual elements of such network are described by the dimensionless FitzHugh-Nagumo equations~\cite{Fitzhugh-1960a,FitzHugh-1961a,Nagumo-1962a,Cartwright-2000a,Scialla-2021a}:
\begin{eqnarray}
\dot{x} &=& a \left( x - x^3/3 + y \right) \, ,
	\label{eq_FN1a}
    \\
    \dot{y} &=& - \left( x  + by - J \right)/a .
	\label{eq_FN1b}
\end{eqnarray}
When modelling the behavior of a $\beta$-cell, the variable $x(t)$ represents the fast relaxing membrane potential, while $y(t)$ is a recovery variable mimicking the slow potassium channel gating.
Depending on the value of $J$, the unit will be in an oscillatory state if $|J|< \varepsilon$ or in an excitable state if $|J| \ge \varepsilon$, where
\begin{equation}
\label{epsilon}
\varepsilon = \frac{3 a^2 - 2 a^2 b -b^2}{3 a^3} \sqrt{a^2 - b} \, .
\end{equation}
In addition to determining the width of the oscillatory interval $ (-\varepsilon,+\varepsilon)$, parameters $a$ and $b$ define the oscillation waveform and period.

Moving from the description of a single element to that of a heterogeneous 3D network of $N$ FitzHugh-Nagumo units, we assume a cubic lattice topology.
This implies that each element is coupled to its six nearest neighbors via a coupling term $C_{ij} (x_j - x_i)$, where $i$ and $j$ are indexes that identify an element $i$ and one of its coupled neighbors $j$, and $C_{ij}$ is the interaction strength.
Notice that the choice of a cubic lattice topology is consistent with what is known about the architecture of $\beta$-cell clusters, where each cell is surrounded on average by 6-7 neighbor cells~\cite{PERSAUD-2014a,Nasteska-2018a}.
We make the simplifying assumption that the value of the coupling constants is the same for each network element, $C_{ij} \equiv C$ for any $i,j$. Since we want to study the interplay between diversity and noise, we also add a noise term $\xi_i(t)$ to the first equation.
Then the corresponding FitzHugh-Nagumo equations for the $i$th element of the network are~\cite{Cartwright-2000a,Scialla-2021a}:
\begin{eqnarray}
\dot{x_i} &=& a \!\! \left[x_i \! - \! \frac{{x_i}^3}{3} \! + \! y_i + \! C\!\!  \sum_{j \in \{ n \}_i} \!\! (x_j \! - \! x_i) \! + \! \xi_i (t)\!  \right] \!\! , \label{FNEN1} \\
\dot{y_i} &=& -(x_i + by_i - J_i)/a \, . \label{FNEN2}
\end{eqnarray}
The sum over $j$ in Eq.~(\ref{FNEN1}) is limited to the set $\{n\}_i$ of the $n=6$ neighbors coupled to the $i$th oscillator.

The $J_i$ parameters in Eq.~(\ref{FNEN2}) are different for each network element and are used to introduce diversity; the $i$th element will be in an oscillatory state if $|J_i|< \varepsilon$ or in an excitable state if $|J_i| \ge \varepsilon$.
The $J_i$ values are drawn from a Gaussian distribution with standard deviation $\sigma_d$, mean value $J_\mathrm{av}$,
and are randomly assigned to network elements.
The standard deviation $\sigma_d$ will be used in what follows as a measure of oscillator population diversity, while the mean value $J_\mathrm{av}$ expresses how far the whole population is from the oscillatory range $ (-\varepsilon,+\varepsilon)$.

The term $\xi_i(t)$ in Eq.~(\ref{FNEN1}) is a Gaussian noise with zero mean, standard deviation $\sigma_n$, and correlation function $\langle \xi_i(t) \xi_j(t') \rangle = {\sigma_n}^2 \delta_{ij} \delta (t-t')$, meaning that $\xi_i(t)$ and $\xi_j(t)$ ($i \ne j$) are statistically independent of each other. The standard deviation $\sigma_n$ will be used in this work as a measure of noise applied to each network element.

The reason why we add $\xi_i(t)$ to the first equation for the fast variable is that this maximizes the effects of noise, making it easier to study its combination with diversity.
Introducing $\xi_i(t)$ into the second equation would result in a minimal impact of noise on network synchronization~\cite{degliesposti-2001a}, due to the slower dynamics of the refractory variable.
Noise effects would be mostly averaged out to zero by time integration and coupling, as we will show below with some numerical simulations.

The above model, which to our knowledge is studied here for the first time in the version we propose, can be used to mimic various excitable biological systems, such as pancreatic $\beta$-cell clusters and some types of neurons~\cite{Poznanski-1997a,Cartwright-2000a,Andreu-2000a,degliesposti-2001a,VRAGOVIC-2006a,Scialla-2021a}.
In what follows we use the model to analyze the combined effect of diversity and noise, acting together on the same network.
In particular, we are interested in potential synergies or antagonisms, as well as in a possible hierarchy between the two sources of disorder, which in spite of some analogies have fundamentally different synchronization mechanisms.

\section{Qualitative theoretical analysis}
\label{analysis}

The white noise $\xi_i(t)$ in Eq.~(\ref{FNEN1}) can represent a randomly fluctuating external current, which is able to shift the nullcline of the $x$ variable up or down and, therefore, to instantaneously change the position of the equilibrium point of each oscillator.
Depending on the extent of the shift and on the value of $J_i$, this may result in a switch from a stable to an unstable equilibrium (or vice versa), corresponding to a transition from a resting to a spiking state of the oscillator (or vice versa).

This mechanism can be further illustrated, following Ref.~\cite{Tessone-2006a}, by introducing the global variables $X(t) = N^{-1} \sum_{i=1}^N x_i(t)$ and $Y(t) = N^{-1} \sum_{i=1}^N y_i(t)$.
We then define $\delta_i$ as the difference between $x_i$ and $X$, i.e., $x_i \equiv X + \delta_i$,
and introduce $M = N^{-1} \sum_{i=1}^N {\delta_i}^2$~\cite{Tessone-2006a,Desai-1978a}.
Therefore, $M$ will increase when diversity increases.
By averaging Eqs.~(\ref{FNEN1}) and (\ref{FNEN2}) over all $N$ network elements,
we obtain the equations for the global variables $X$, $Y$:
\begin{eqnarray}
\dot{X} &=& a  \left[X(1-M)  -  X^3 /3 + Y +  \xi_G (t)  \right] , \label{FNENG1} \\
\dot{Y} &=& -(X + bY - J_\mathrm{av})/a \, . \label{FNENG2}
\end{eqnarray}
Here noise effects are represented by a global white noise term $\xi_G(t) = N^{-1} \sum_i \xi_i(t)$ with zero mean
and correlation function $\langle \xi_G(t) \xi_G(t') \rangle = N^{-1} \sigma_n^2 \delta (t-t')$.

It is instructive to observe the different impact of diversity and noise on the nullclines of Eqs.~(\ref{FNENG1}) and (\ref{FNENG2}).
A change in diversity, i.e., in the standard deviation of the $J_i$ distribution, causes a change in $M$,
which affects the shape of the cubic nullcline by changing the coefficient of the linear term $X$ (see Fig.~\ref{nullclines}, panels (a) and (b)).
This indicates that diversity can have a significant effect on overall network dynamics, independently of whether the mean value $J_\mathrm{av}$ is inside or outside the intrinsic oscillatory range $(-\varepsilon,+\varepsilon)$.
On the other hand, the global noise term $\xi_G(t)$ can only cause rigid shifts, positive or negative, of the cubic nullcline along the vertical axis, as a consequence of its instantaneous fluctuations (compare the dashed and solid lines in Fig.~\ref{nullclines}).
This suggests that noise is unlikely to play a constructive role when diversity is optimized (i.e., in the conditions corresponding to a DIR) and $J_\mathrm{av}=0$.
In this situation, noise will likely act as a perturbation of the system, which is already in an intrinsically oscillatory and resonant state.

Let us now consider what happens when $J_\mathrm{av} \neq 0$ (see Fig.~\ref{nullclines}-(c)).
In this case, the constant term $J_\mathrm{av}$ determines a rigid shift, upwards or downwards, of the second nullcline.
This can significantly change the position of the equilibrium point of the system dynamics, turning the network from oscillatory into excitable.
In these conditions, noise can play a synergistic role with diversity, by causing instantaneous rigid shifts of the cubic nullcline that counterbalance the effect of $J_\mathrm{av}$, thus triggering global network oscillations.

It should be noted that we did not add a periodic driving force to our system equations, of the type $A \sin(\Omega t)$.
As mentioned in the introduction, this term is not necessary to observe either stochastic or diversity-induced resonance effects and its presence would introduce the additional constraint of matching two time scales, i.e., the driving force period and the oscillation period of the FitzHugh-Nagumo elements constituting the network.

\section{Numerical results and discussion}

In order to quantitatively study the combined effect of diversity and noise, we numerically solve the FitzHugh-Nagumo Eqs.~(\ref{FNEN1}) and (\ref{FNEN2}) for a network of $10^3$ elements with the above-mentioned topology and the following system parameters: $a=60$, $b=1.45$, $C=0.15$.
The selected values of $a$ and $b$ generate a waveform and period similar to those of bursting oscillations of pancreatic $\beta$-cells~\cite{Scialla-2021a}.
We set the coupling constant $C=0.15$, since, as we verified in an earlier work~\cite{Scialla-2021a}, the oscillatory response of the system is substantially unchanged by further increasing $C$ beyond $C=0.15$.

We run simulations for a range of diversity values $\sigma_d$ (from $\sigma_d =0$ to $\sigma_d =2.5$) and, at the same time, for a range of white noise standard deviation values $\sigma_n$ (from $\sigma_n =0$ to $\sigma_n =5.0$).
We repeat this for each of the following diversity distribution mean values: $J_\mathrm{av} = 0, \pm 0.5, \pm 1$.

For each simulation, corresponding to a set of $\sigma_d$, $\sigma_n$, and $J_\mathrm{av}$ values, we quantify the network synchronization efficiency by computing the global oscillatory activity $\rho$~\cite{Cartwright-2000a,Scialla-2021a},
\begin{equation}
\rho = N^{-1} \sqrt{\left \langle [S(t) -\bar{S}]^2 \right \rangle} \, ,
\label{FNENG7}
\end{equation}
where $N = 10^3$ is the total number of oscillators, $S(t) = \sum_i x_i(t)$, and $\bar{S} = \langle S(t) \rangle$, with $\langle \dots \rangle$ denoting a time average.
The results for the global oscillatory activity $\rho$ are plotted versus $\sigma_d$ and  $\sigma_n$, generating five three-dimensional surfaces corresponding to each of the above listed $J_\mathrm{av}$ values.

\begin{figure}[!t]
\includegraphics[width=8.8cm]{fig-6_nullclinesfin.jpg}
\caption{Nullclines of Eqs.~(\ref{FNENG1}) and (\ref{FNENG2}) for different values of $J_\mathrm{av}$ and $M$.
A comparison between panel (a) and (b) shows the effect of $M$ on the shape of the cubic nullcline.
The area delimited by the dashed curves above and below the cubic nullcline in each panel illustrates the effect of instantaneous shifts caused by noise with an amplitude of up to $\pm 1$.}
\label{nullclines}
\end{figure}

\begin{figure}[!t]
\includegraphics[width=8.0cm]{fig-1_mu0_s0.25.pdf}
\caption{Global oscillatory activity $\rho$, defined in Eq.~(\ref{FNENG7}), as a function of diversity ($\sigma_d$) and noise ($\sigma_n$), for $J_\mathrm{av}=0$. The full blue dot highlights the global surface maximum, which is coincident with the DIR maximum. The empty red dot corresponds to the noise-induced resonance maximum.}
\label{tevol-b0}
\end{figure}

In the $J_\mathrm{av}=0$ regime (Fig.~\ref{tevol-b0}), where a relatively high fraction or all of the network elements are inside the intrinsic oscillatory range, simulation results show that both diversity and noise are able to generate a resonance on their own.
If we move along the diversity axis ($\sigma_n=0$, no noise) or along the noise axis ($\sigma_d=0$, no diversity), we observe in both cases a resonance maximum that is about 20-25\% higher than the $\rho$ value corresponding to the origin.
In addition, the two sources of noise seem to act independently of one another, showing no evidence of a synergy. As a matter of fact, the global maximum of the surface coincides with the diversity-induced resonance maximum, therefore it occurs on the diversity axis, i.e., at $\sigma_d$=0.5, $\sigma_n$=0.
If, from this global maximum (shown as a full blue dot in Fig.~\ref{tevol-b0}), we move in any direction towards the middle of the surface, i.e., if we add noise, there is no gain in terms of collective oscillatory activity. This is consistent with the predictions of our qualitative theoretical analysis, indicating that noise is unlikely to play a constructive role for $J_\mathrm{av}=0$, when diversity is optimized.

\begin{figure}[!t]
\includegraphics[width=8.0cm]{fig-2_mu-1_s0.25.pdf}

\vspace{0.5cm}

\includegraphics[width=8.0cm]{fig-3_mu1_s0.25.pdf}
\caption{Global oscillatory activity $\rho$, defined in Eq.~(\ref{FNENG7}), as a function of diversity ($\sigma_d$) and noise ($\sigma_n$), for $J_\mathrm{av} = -1$ (panel (a)) and $J_\mathrm{av} = +1$ (panel (b)).
The full blue and empty green/red dots in each panel highlight the global surface maximum, the DIR maximum and the noise-induced resonance maximum, respectively.}
\label{fig-2}
\end{figure}

Moving to the opposite end of the $J_\mathrm{av}$ range, i.e., $J_\mathrm{av} = \pm1$ (Fig.~\ref{fig-2}), we observe a very different situation. In this regime most network elements are outside the intrinsic oscillatory range, either below ($J_\mathrm{av} =-1$) or above it ($J_\mathrm{av} =+1$).
Here the addition of noise to diversity always results in a significant increase of the network oscillatory activity.
In line with the theoretical analysis based on global system variables, this can be explained by considering that, in the case of $J_\mathrm{av} =-1$, most network elements are below the excitation threshold, i.e., in an excitable state, and can be pushed up into the oscillatory range by an instantaneous injection of positive external current, deriving from sufficiently large noise fluctuations with positive sign.
Vice versa, in the case of $J_\mathrm{av} =+1$, most elements are above the upper limit of the intrinsic oscillatory range, i.e., in an excitation block state, and can be pushed down into the oscillatory range by an instantaneous injection of negative external current, deriving from negative noise fluctuations with sufficiently large modulus.
In both cases, the addition of noise on top of diversity causes a synergistic effect and a remarkable network synchronization improvement: for instance, the network oscillatory activity for $J_\mathrm{av} =+1$ raises by almost $50\%$, if we compare the DIR maximum ($\rho \approx 515$, empty green dot in Fig.~\ref{fig-2}, panel (b)),
to the global maximum of the $\rho$ surface ($\rho \approx 738$, full blue dot in Fig.~\ref{fig-2}, panel (b)) resulting from the combination of diversity and noise effects.
It is also apparent from the data that, in this regime, noise is more efficient than diversity, as shown by the significantly higher noise-induced resonance maxima along the noise axis (empty red dots in Fig.~\ref{fig-2}, panel (a) and (b)), compared to their equivalents along the diversity axis (empty green dots in Fig.~\ref{fig-2}, panel (a) and (b)).

It is worth noting that the position of the DIR gets shifted towards higher values of $\sigma_d$ going from $J_\mathrm{av} = 0$ to $J_\mathrm{av} = \pm 1$.
The DIR maximum is at $\sigma_d =0.5$ for $J_\mathrm{av} =0$, versus $\sigma_d =1$ for both $J_\mathrm{av} =-1$ and $J_\mathrm{av} =+1$ (empty green dots in Fig.~\ref{fig-2}, panels (a) and (b)).
However, when we combine together noise and diversity, the position of the global maximum goes back to the same optimal diversity value found for $J_\mathrm{av} =0$
(full blue dots in Fig.~\ref{fig-2}, panels (a) and (b)).
The mechanism of this effect is that noise stochastically ''throws'' network elements towards the oscillatory range, and it does so with respect to an average position on the $J$ axis that is determined, for each element, by its $J_i$ coefficient, deriving from the diversity distribution. When this mechanism reaches the highest efficiency, i.e., at the global maximum of the surface, the optimal diversity for $J_\mathrm{av}= \pm 1$ tends to be equal to that for $J_\mathrm{av}=0$.
We may conclude that, in the $J_\mathrm{av} = \pm 1$ regime, there is a strong synergy between diversity and stochastic effects, which significantly broadens the range of resonant states of the network versus what can be observed when either source of disorder is applied individually.

Finally, in the intermediate regime corresponding to $J_\mathrm{av} = \pm 0.5$ (Fig.~\ref{fig-4}), we observe an in between situation, with various regions of the $\rho$ surface where the combination of diversity and noise produces a synergy and an extension of the resonant range of the network.
For example, at $J_\mathrm{av} =-0.5$, there are no network oscillations for diversity values $\sigma_d =0.0$ and $\sigma_d =0.25$, whereas, with the addition of noise, resonant states are observed in both cases, starting from $\sigma_n =1$ and $\sigma_n =0.5$, respectively.
We point out that, also in this regime, the global maximum of the $\rho$ surface
due the combined diversity- and noise-induced resonance occurs, for $J_\mathrm{av} = -0.5$, at $\sigma_d =0.5$ (full blue dot in Fig.~\ref{fig-4}, panel (a)) and is shifted to smaller values with respect to the DIR maximum in the absence of noise ($\sigma_d =0.75$, empty green dot in Fig.~\ref{fig-4}, panel (a)).
Therefore, the mechanism described in the previous paragraph, regarding the tendency of the optimal diversity value to be equal to that for $J_\mathrm{av} =0$, is at play here as well.

\begin{figure}[!t]
\includegraphics[width=8.0cm]{fig-4_mu-0.5_s0.25.pdf}

\vspace{0.5cm}

\includegraphics[width=8.0cm]{fig-5_mu0.5_s0.25.pdf}
\caption{Global oscillatory activity $\rho$, defined in Eq.~(\ref{FNENG7}), as a function of diversity ($\sigma_d$) and noise ($\sigma_n$), for $J_\mathrm{av} = -0.5$ (panel (a)) and $J_\mathrm{av} = +0.5$ (panel (b)). The full blue and empty green/red dots in each panel highlight the global surface maximum, the DIR maximum and the noise-induced resonance maximum, respectively.}
\label{fig-4}
\end{figure}
In order to confirm the rationale for our choice of adding the noise term $\xi_i(t)$ into the first FitzHugh-Nagumo equation, Eq.~(\ref{FNEN1}), we also performed some simulations where $\xi_i(t)$ was added instead into the second equation, Eq.~(\ref{FNEN2}).
We did this for $J_\mathrm{av} =0$ and $J_\mathrm{av} =-1$.
As expected, the results reported in Fig.~\ref{fig-7} show that in this case the effect of noise is negligible and the network dynamics is entirely determined by diversity~\cite{degliesposti-2001a}.

\begin{figure}[!t]
\includegraphics[width=8.0cm]{fig-7_mu0_s0.25_noise-sotto.pdf}

\vspace{0.5cm}

\includegraphics[width=8.0cm]{fig-7_mu-1_s0.25_noise-sotto.pdf}
\caption{Global oscillatory activity $\rho$, defined in Eq.~(\ref{FNENG7}), as a function of diversity ($\sigma_d$) and noise ($\sigma_n$), for $J_\mathrm{av} = 0$ (panel (a)) and $J_\mathrm{av} = -1$ (panel (b)), when noise is added into the second equation~(\ref{FNEN2}).}
\label{fig-7}
\end{figure}

\section{Conclusions}
Our theoretical and numerical analysis shows that, while there are some analogies between diversity- and noise-induced network synchronization, the two effects are substantially different and interact with each other differently, depending on the distance of the mean value of the diversity distribution from the intrinsic oscillatory range of the network elements.
Specifically, when the diversity distribution is centered around the intrinsic oscillatory range ($J_\mathrm{av} =0$), diversity and noise act independently of one another and there is no indication of a synergy. On the other hand, when the mean value of the diversity distribution is far away from the intrinsic oscillatory range ($J_\mathrm{av} = \pm1$), then there is a clear synergy between the two sources of disorder, which determines a major improvement of network synchronization. In addition, in this regime, noise can improve network synchronization more effectively than diversity. This provides useful indications on the relative importance of the two effects in different network configurations, and on the possibility to neglect one or the other as a consequence.

Another important finding is that the optimal diversity value of the network is the same in all regimes, if noise is taken into account.
In other words, when noise effects are added, the amount of diversity that maximizes collective oscillatory efficiency seems to be an intrinsic property of the network, independent of $J_\mathrm{av}$.

The fact that diversity and noise are not equivalent sources of disorder, but have distinct effects on network dynamics, may have implications for biological systems.
Our results suggest that different network configurations can lead to a hierarchy between the two sources of disorder. This may have driven the exploitation of diversity and noise to a different degree in different biological systems during their evolution, depending on their specific nature and on the types of signals that trigger their activity.

\begin{acknowledgments}

The authors acknowledge support from the Estonian Research Council through Grant PRG1059
and the ERDF (European Development Research Fund) CoE (Center of Excellence) program through Grant TK133.
The authors would like to thank Alessandro Loppini for helpful discussions.

\end{acknowledgments}


\section{Preliminary and basic results}
\par\medskip
\par\medskip

We begin this section by recalling the definition and notation
about $C^*$-algebraic bundles.
First of all, we refer the reader to [7, VIII.2.2] for the notion
of Banach algebraic bundle $\cal B$ over a locally compact group $G$.
Let $B$ be the bundle space of $\cal B$ and $\pi$ be the bundle
projection.
Let $B_t = \pi^{-1}(t)$ be the fiber over $t\in G$.
It is clear that $B_e$ is a $C^*$-algebra if $\cal B$ is a $C^*$-algebraic
bundle (see [7, VIII.16.2] for a definition).
We will use the materials from [7, VIII] implicitly.
Following the notation of [7], we denote by $\cal L(B)$ the set of all
continuous cross-sections on $\cal B$ with compact support.
Moreover, for any $f\in \cal L(B)$, ${\rm supp}(f)$ is the closed support
of $f$.
Furthermore, let $({\cal L}_p(\mu; {\cal B}), \|\ \|_p)$ be the normed
space as defined in [7, II.15.7] (where $\mu$ is the left Haar
measure on $G$).
For simplicity, we will denote ${\cal L}_p(\mu; \cal B)$ by
${\cal L}_p(\cal B)$.
By [7, II.15.9], $\cal L(B)$ is dense in ${\cal L}_p(\cal B)$.
We also need the theory of operator valued integration from [7, II],
especially, we would like to draw the readers' attention to
[7, II.5.7] and [7, II.\S 16].
\par\medskip

Throughout this paper, $\cal B$ is a $C^*$-algebraic bundle
over a locally compact group $G$ with bundle space $B$ and bundle
projection $\pi$.
Denote by $C^*(\cal B)$ the cross-sectional $C^*$-algebra of
$\cal B$ (see [7, VIII.17.2]).
We recall from [7, VIII.5.8] that there exists a canonical map $m$
from the bundle space $B$ to the set of multipliers of ${\cal L}_1(\cal B)$
(or $C^*(\cal B)$).
\par\medskip

\begin{lemma} \label{1.4}
The map $m$ from $B$ to $M(C^*(\cal B))$ is {\it faithful} in the sense that
if $m_a = m_b$, then either $a=b$ or $a=0_r$ and $b=0_s$ ($r,s\in G$).
\end{lemma}
\noindent {\bf Proof}:
Suppose that $\pi(a) = \pi(b) = r$.
Then $m_{a-b} = 0$ will imply that $a-b = 0_r$ (since $\cal B$ has a strong
approximate unit and enough continuous cross-sections).
Suppose that $\pi(a)=r\neq s=\pi(b)$.
Then there exists a neighbourhood $V$ of $e$ such that
$rV\cap sV = \emptyset$.
For any $f\in \cal L(B)$, $a(f(r^{-1}t)) = m_a(f)(t) = m_b(f)(t) =
b(f(s^{-1}t))$.
Now let $b_i$ be a strong approximate unit of $\cal B$ and $\{ f_i \}$ be
elements in $\cal L(B)$ such that ${\rm supp}(f_i)\subseteq V$ and $f_i(e) = b_i$.
Therefore $ab_i = a(f(e)) = b(f(s^{-1}r)) =0$ and hence $a=0_r$.
Similarly, $b=0_s$.
\par\medskip

From now on, we will identify $B_r$ ($r\in G$) with its image
in $M(C^*({\cal B}))$.
\par\medskip

Let ${\cal B}\times G$ be the $C^*$-algebraic bundle over $G\times G$
with the Cartesian product $B\times G$ as its bundle space such that
the bundle projection $\pi'$ is given by $\pi'(b,t) = (\pi(b),t)$ ($b\in
B$; $t\in G$).
It is not hard to see that any non-degenerate representation $T'$ of
${\cal B}\times G$ is of the form $T'_{(b,t)} = T_bu_t$ ($b\in B$; $t\in G$)
for a non-degenerate representation $T$ of ${\cal B}$ and a unitary
representation $u$ of $G$ with commuting ranges.
This gives following lemma.
\par\medskip

\begin{lemma} \label{1.5}
$C^*({\cal B}\times G) = C^*({\cal B})\otimes_{\max}C^*(G)$.
\end{lemma}

Consider the map $\delta_{\cal B}$ from $B_r$ to
$M(C^*({\cal B})\otimes_{\max}C^*(G))$ given by $\delta_{\cal B} (b) =
b\otimes \Delta_r$ where $\Delta_r$ is the canonical image of $r$ in
$M(C^*(G))$.
Denote again by $\delta_{\cal B}$ the integral form of $\delta_{\cal
B}$.
Then we have the following equalities.

\begin{eqnarray*}
\delta_{\cal B}(f)(1\otimes k)(g\otimes l)(r,s) & = &
\int_G f(t)g(t^{-1}r)[\int_G k(u)l(u^{-1}t^{-1}s)du]dt\\
& = & \int_G\int_G f(t)g(t^{-1}r)k(t^{-1}v)l(v^{-1}s) dvdt
\end{eqnarray*}
for any $f,g\in \cal L(B)$ and $k,l\in K(G)$.
If we denote by $f\bullet k(r,s)=f(r)k(r^{-1}s)$, then
$\delta_{\cal B}(f)(1\otimes k)(g\otimes l)(r,s) =
(f\bullet k)(g\otimes l)(r,s)$.
It is not hard to see that $\delta_{\cal B}$ is a full coaction
(note that $\delta_{\cal B}$ extends to a representation of
${\cal L}_1({\cal B})$ and hence of $C^*({\cal B})$).
\par\medskip

\begin{lemma} \label{1.6}
The set $N = \{ \sum_{i=1}^n f_i\bullet k_i:f_i\in {\cal L(B)},
k_i\in K(G)\}$ is dense in $C^*({\cal B}\times G)$.
Consequently, $\delta_{\cal B}$ is a non-degenerate full coaction.
\end{lemma}
\noindent {\bf Proof}:
It is sufficient to show that for any $f\in \cal L(B)$ and $k\in K(G)$,
$f\otimes k$ can be approximated by elements in $N$ with respect
to the ${\cal L}_1$-norm.
Let $M$ and $K$ be the closed supports of $f$ and $k$ respectively.
Since $k$ is uniformly continuous, for a given $\epsilon>0$, there exists
a neighbourhood $V$ of $e$ such that $\|k(u)-k(v)\| <
\epsilon/(\mu(M)\cdot \mu(MK)\cdot \sup_{t\in M}\|f(t)\|)$ if
$u^{-1}v\in V$ (where $\mu$ is the left Haar measure of $G$).
Since $M$ is compact, there exist $r_1,..., r_n$ in $M$ such that
$\cup_{i=1}^n r_iV$ covers $M$.
Let $k_i(s) = k(r_i^{-1}s)$ and let $g_1,...,g_n$ be the partition of
unity subordinate to $\{ r_iV\}_{i=1}^n$.
Then
\begin{eqnarray*}
\| f\bullet k(r,s) - \sum_{i=1}^n (g_if\otimes k_i)(r,s)\| & \leq &
\|f(r)\| \cdot \mid \sum_{i=1}^n g_i(r)k(r^{-1}s) -
\sum_{i=1}^n g_i(r)k(r_i^{-1}s)\mid \\
& \leq & \|f(r)\|\cdot (\ \sum_{i=1}^n\mid g_i(r)\mid\cdot
\mid k(r^{-1}s) - k(r_i^{-1}s) \mid\ ).
\end{eqnarray*}
As $g_i(r)\neq 0$ if and only if $r\in r_iV$, we have $\int_{MK}\int_M
\|f\bullet k(r,s) - \sum_{i=1}^n (g_if\otimes k_i)(r,s)\| drds \leq
\epsilon$.
This proves the lemma.
\par\medskip

The following lemma is about general coactions of $C^*(G)$.
It implies, in particular that $\delta_{\cal B}$ is injective.
Note that the trivial representation of $G$ on $\mathbb{C}$
induces a $*$-homomorphism $\cal E$ from $C^*(G)$ to $\mathbb{C}$
which is a coidentity in the sense that $({\cal E}\otimes {\rm id})
\delta_G = {\rm id} = ({\rm id}\otimes {\cal E})\delta_G$ (where $\delta_G$ is
the comultiplication on $C^*(G)$)
\par\medskip

\begin{lemma} \label{1.1}
Let $\epsilon$ be a coaction on $A$ by $C^*(G)$.
Suppose that $\epsilon_{\cal E} = ({\rm id}\otimes {\cal E})\epsilon$ and
$A_{\cal E} = \epsilon_{\cal E} (A)$.
\par\noindent
(a) If $\epsilon$ is non-degenerate, then it is automatically
injective.
\par\noindent
(b) $A = A_{\cal E}\oplus \ker (\epsilon)$ (as Banach space).
\end{lemma}
\noindent {\bf Proof}:
(a) We first note that $\epsilon_{\cal E}$ is a
$*$-homomorphism from $A$ to itself and so $A_{\cal E}$ is
a $C^*$-subalgebra of $A$.
It is clear that $\epsilon$ is injective on $A_{\cal E}$ and
we want to show that $A=A_\epsilon$.
For any $a\in A$ and any $s\in C^*(G)$ such that ${\cal E}(s)=1$,
$a\otimes s$ can be approximated by elements of the form
$\sum \epsilon(b_i)(1\otimes t_i)$ (as $\epsilon$ is non-degenerate).
Therefore, $\sum \epsilon_{\cal E}(b_i){\cal E}(t_i) =
({\rm id}\otimes {\cal E})(\sum \epsilon(b_i)(1\otimes t_i))$
converges to $a$.
\par\noindent
(b) Note that for any $a\in A$, $\epsilon(a-\epsilon_{\cal E}(a)) = 0$.
Since $\epsilon_{\cal E}$ is a projection on $A$, $A=A_{\cal E}\oplus
\ker (\epsilon)$.
\par\medskip

The above lemma actually holds for a general Hopf $C^*$-algebra with
a co-identity instead of $C^*(G)$ (for a brief review of Hopf
$C^*$-algebras, please see e.g. [5] or [10]).
\par\medskip

\begin{remark}
By [5, 7.15], if $\Gamma$ is a discrete amenable group, then any
injective coaction of $C^*_r(\Gamma)$ is automatically non-degenerate.
More generally, the arguments in [5, \S7] actually show that for
any discrete group $G$, any injective coaction of $C^*(G)$ is
non-degenerate.
Hence, a coaction of $C^*(G)$ is injective if and only if it is
non-degenerate (when $G$ is discrete).
\end{remark}

We end this section with the following technical lemma.
\par\medskip

\begin{lemma} \label{1.3}
Let $A$ be a $C^*$-algebra and $E$ be a Hilbert $A$-module.
Suppose that $({\cal H},\pi)$ is a faithful representation of $A$.
Then
\par\noindent
(a) $\|x\| = \sup \{\|x\otimes_\pi \xi\|: \|\xi\|\leq 1\}$ for any $x\in E$;
\par\noindent
(b) the canonical map from ${\cal L}(E)$ to ${\cal L}(E\otimes_\pi {\cal H})$
(which sends $a$ to $a\otimes 1$) is injective.
\end{lemma}

Part (a) follows from a direct computation and the
part (b) is a consequence of part (a).
\par\medskip
\par\medskip
\par\medskip

\section{Reduced cross-sectional $C^*$-algebras}
\par\medskip
\par\medskip

In this section, we will define the reduced cross-sectional
$C^*$-algebras for $C^*$-algebraic bundles and show that they
carry canonical reduced coactions.
The intuitive idea is to consider the representation of
${\cal L}_1({\cal B})$ as bounded operators on ${\cal L}_2({\cal B})$.
However, since ${\cal L}_2({\cal B})$ is not a Hilbert $C^*$-module, it
seems unlikely that we can get a $C^*$-algebra out of this
representation.
Instead, we will consider a slightly different version of
``${\cal L}_2({\cal B})$'' which is a Hilbert $B_e$-module.
The difficulty then is to show that the representation is well
defined and bounded.
This can be proved directly by a quite heavy analytical argument but we
will use Lemma \ref{2.8} to do the trick instead.
We will also define the interesting notion of proper $C^*$-algebraic
bundles which will be needed in the next section.
\par\medskip

\begin{lemma} \label{2.1}
Consider the map $\langle \ ,\ \rangle _e$ from $\cal L(B)\times \cal L(B)$
to $B_e$ defined by
$$\langle f,g\rangle _e = \int_G f(t)^*g(t) dt$$
for all $f,g\in \cal L(B)$.
Then $\langle \ ,\ \rangle _e$ is a $B_e$-valued inner product on
$\cal L(B)$.
\end{lemma}
\noindent {\bf Proof}:
It is easily seen that $\langle \ ,\ \rangle_e$ is a well defined
$B_e$-valued pre-inner product.
Moreover, for all $f\in \cal L(B)$, $\langle f,f\rangle _e = 0$
if and only if $\int_G \varphi(f(t)^*f(t))dt = 0$
for all $\varphi\in (B_e)^*_+$ which implies that
$f(t)^*f(t) = 0$ for all $t\in G$.
\par\medskip

\begin{definition} \label{2.2}
The completion of $\cal L(B)$ with respect to the $B_e$-valued inner
product in Lemma \ref{2.1} is a Hilbert $B_e$-module and is denoted by
\mbox{$(L^2_e({\cal B}), \|\cdot\|_e)$}.
\end{definition}

It is clear that $\|\langle f,g\rangle _e\|\leq \|f\|_2\|g\|_2$
(by [7, II.5.4] and the H\" older's inequality).
Hence there is a continuous map $J$ from ${\cal L}_2(\cal B)$ to
$L^2_e({\cal B})$ with dense range.
In fact, it is not hard to see that ${\cal L}_2(\cal B)$ is a right
Banach $B_e$-module and $J$ is a module map.
\par\medskip

Throughout this paper, $T$ is a non-degenerate *-representation
of $\cal B$ on a Hilbert space ${\cal H}$ and $\phi$ is the
restriction of $T$ on $B_e$.
Moreover, $\mu_T$ is the representation of $C^*({\cal B})$ on $\cal H$
induced by $T$.
By [7, VIII.9.4], $\phi$ is a non-degenerate representation of $B_e$.
\par\medskip

\begin{lemma} \label{2.4}
There exists an isometry
$$V:L^2_e({\cal B})\otimes_\phi {\cal H}\rightarrow L^2(G; {\cal H}),$$
such that for all $f\in \cal L(B)$, $\xi\in\cal H$, and $t\in G$ one has
$$V(f\otimes \xi)(t) = T_{f(t)}\xi.$$
\end{lemma}
\noindent {\bf Proof}:
It is easy to check that the map $V$ defined as above is inner
product preserving and hence extends to the required map.
\par\medskip

One technical difficulty in the study of reduced cross-sectional
$C^*$-algebras is that $V$ is not necessarily surjective.
\par\medskip

\begin{example} \label{2.6}
(a) If $\cal B$ is saturated, then $V$ is surjective.
In fact, let $K= V(L^2_e({\cal B})\otimes_\phi {\cal H})$ and
$\Theta$ be an element in the complement of $K$.
For any $g\in \cal L(B)$ and $\eta\in {\cal H}$,
$\int_G \langle T_{g(r)}\eta, \Theta(r)\rangle \ dr = 0$ which implies
that $\int_G T_{g(r)}^*\Theta(r)\ dr = 0$.
Now for any $f\in \cal L(B)$, we have
$$(\mu_T\otimes \lambda_G)\delta_{\cal B}(f)(\Theta)(t) =
\int_G T_{f(s)}\Theta(s^{-1}t)\ ds =
\int_G T_{f(tr^{-1})}\Theta(r)\Delta(r)^{-1}\ dr.$$
Moreover, for any $b\in B_{t^{-1}}$, let
$g(r) = \Delta(r)^{-1}f(tr^{-1})^*b^*$.
Then $g\in \cal L(B)$ and
$$T_b (\mu_T\otimes \lambda_G)
\delta_{\cal B}(f)(\Theta)(t) = \int_G T_{g(r)}^*\Theta(r)\ dr = 0$$
for any $b\in B_{t^{-1}}$ (by the above equality).
Since $\cal B$ is saturated and the restriction $\phi$ of $T$ is
non-degenerate, $(\mu_T\otimes \lambda_G)\delta_{\cal B}(f)(\Theta)=0$
for any $f\in \cal L(B)$.
Thus, $\Theta=0$ (because $(\mu_T\otimes \lambda_G)\circ\delta_{\cal B}$
is non-degenerate).
\par\noindent
(b) Let $\cal B$ be the trivial bundle over a discrete group $G$
(i.e. $B_e = \mathbb{C}$ and $B_t = (0)$ if $t\neq e$).
Then $L^2_e({\cal B})\otimes_\phi {\cal H} \cong {\cal H}$ is a proper
subspace of $L^2(G; {\cal H})$.
\end{example}

For any $b\in B$, let $\hat T_b$ be the map from $L^2_e({\cal B})$
to itself defined by
$\hat T_b(f) = b\cdot f$ for any $f\in \cal L(B)$ (where $b\cdot f(t) =
bf(\pi(b)^{-1}t)$).
The argument for $\hat T$ being continuous seems not easy.
Instead, we will consider the corresponding representation of
${\cal L}_1({\cal B})$ and show that it is well defined.
\par\medskip

For any $f\in {\cal L}({\cal B})$, define a map $\lambda_{\cal B} (f)$ from ${\cal
L}({\cal B})$ to itself by $\lambda_{\cal B} (f)(g) = f\ast g$ ($g\in {\cal
L}({\cal B})$).
We would like to show that this map is bounded and induces a bounded
representation of ${\cal L}_1({\cal B})$.
In order to prove this, we will first consider a map
$\tilde\lambda_{\cal B}(f)$ from ${\cal L}({\cal B})\otimes_{\rm alg} {\cal H}$
to itself given by
$\tilde \lambda_{\cal B}(f)(g\otimes \xi) = f\ast g\otimes \xi$ ($g\in {\cal
L}({\cal B})$; $\xi\in \cal H$).
In the following, we will not distinguish ${\cal L}({\cal B})\otimes_{\rm alg}
{\cal H}$ and its image in $L_e^2({\cal B})\otimes_\phi{\cal H}$.
\par\medskip

\begin{lemma} \label{2.8}
For any $f\in \cal L(B)$, $\tilde \lambda_{\cal B}(f)$ extends to a
bounded linear operator on $L_e^2({\cal B})\otimes_\phi{\cal H}$ such that
$\mu_{\lambda,T} (f)\circ V = V\circ
(\tilde \lambda_{\cal B}(f))$
(where $\mu_{\lambda,T}$ is the composition:
$C^*({\cal B})\stackrel{\delta_{\cal B}}{\longrightarrow} C^*({\cal B})\otimes_{\rm max} C^*(G)
\stackrel{\mu_T\otimes \lambda_G}{\longrightarrow} {\cal B}({\cal H}\otimes
L^2(G))$).
\end{lemma}
\noindent {\bf Proof:}
For any $g\in \cal L(B)$, $\xi\in \cal H$ and $s\in G$, we have,
\begin{eqnarray*}
\mu_{\lambda,T}(f)V(g\otimes \xi)(s)
&=& \int_G (T_{f(t)}\otimes \lambda_t) V(g\otimes \xi)(s) dt
\ \ =\ \ \int_G T_{f(t)}T_{g(t^{-1}s)}\xi dt\\
&=& T_{f\ast g(s)}\xi
\ \ =\ \ V(f\ast g\otimes \xi)(s)
\ \ =\ \ V(\tilde \lambda_{\cal B}(f)(g\otimes\xi))(s).
\end{eqnarray*}
Since $V$ is an isometry, $\tilde \lambda_{\cal B}(f)$ extends
to a bounded linear operator on $L_e^2({\cal B})\otimes_\phi{\cal H}$ and
satisfies the required equality.
\par\medskip

Now by considering the representation $T$ for which $\phi$ is injective
and using Lemmas \ref{1.3}(a) and \ref{2.4}, $\lambda_{\cal B}(f)$ extends
to a bounded linear map from $L_e^2({\cal B})$ to itself.
It is not hard to show that $\langle f\ast g, h\rangle_e =
\langle g,f^*\ast h\rangle_e$ (for any $g,h\in {\cal L}({\cal B})$).
Hence $\lambda_{\cal B}(f)\in {\cal L}(L_e^2({\cal B}))$.
Moreover, we have the following proposition.
\par\medskip

\begin{proposition} \label{2.10}
The map $\lambda_{\cal B}$ from ${\cal L}_1(\cal B)$ to
${\cal L}(L^2_e(\cal B))$ given by $\lambda_{\cal B} (f)(g) = f\ast g$
($f,g\in \cal L(B)$) is a well defined norm decreasing non-degenerate
*-homomorphism such that $ \mu_{\lambda,T}(f)\circ V = V\circ (\lambda_{\cal B}(f)
\otimes_\phi 1)$ ($f\in \cal L(B)$).
\end{proposition}

\begin{definition} \label{2.11}
(a) $\lambda_{\cal B}$ is called the {\it reduced representation of $C^*({\cal B})$}
and $C^*_r({\cal B}) = \lambda_{\cal B}(C^*(\cal B))$ is called the
{\it reduced cross-sectional $C^*$-algebra of $\cal B$}.
\par\noindent
(b) $\cal B$ is said to be {\it amenable} if $\lambda_{\cal B}$ is
injective.
\end{definition}

\begin{example} \label{2.12}
Suppose that $\cal B$ is the semi-direct product bundle corresponding to
an action $\alpha$ of $G$ on a $C^*$-algebra $A$.
Then $C^*({\cal B}) = A\times_\alpha G$ and $C^*_r({\cal B}) =
A\times_{\alpha, r} G$.
\end{example}

As in the case of full cross-sectional $C^*$-algebras,
we can define non-degenerated reduced coactions on
reduced cross-sectional $C^*$-algebras.
First of all, let us consider (as in the case of reduced
group $C^*$-algebras) an operator $W$ from
${\cal L(B}\times G)$ to itself defined by $W(F)(r,s) =
F(r,r^{-1}s)$ ($F\in {\cal L(B}\times G)$).
Note that for any $f\in\cal L(B)$ and $k\in K(G)$, $W(f\otimes k) =
f\bullet k$ (where $f\bullet k$ is defined in the
paragraph before Lemma \ref{1.6}) and that
$L^2_e({\cal B}\times G) = L^2_e({\cal B})\otimes L^2(G)$ as
Hilbert $B_e$-modules.
\par\medskip

\begin{lemma} \label{2.13}
$W$ is a unitary in ${\cal L}(L^2_e({\cal B})\otimes L^2(G))$.
\end{lemma}
\par
\noindent {\bf Proof}:
For any $f,g\in \cal L(B)$ and $k,l\in K(G)$, we have the following
equality:
\begin{eqnarray*}
\langle W(f\otimes k), W(g\otimes l)\rangle
& = & \int_G\int_G (f\bullet k)(r,s)^*(g\bullet l)(r,s) dsdr\\
& = & \int_G\int_G f(r)^*\overline{k(r^{-1}s)} g(r)l(r^{-1}s) drds\\
& = & \int_G\int_G f(r)^*g(r)\overline{k(t)}l(t) dtdr
\quad = \quad \langle f\otimes k, g\otimes l\rangle.
\end{eqnarray*}
Hence $W$ is continuous and extends to an operator on
$L^2_e({\cal B})\otimes L^2(G)$.
Moreover, if we define $W^*$ by $W^*(f\otimes k)(r,s) = f(r)k(rs)$,
then $W^*$ is the adjoint of $W$ and $WW^* = 1 = W^*W$.
\par\medskip

As in [12], we can define a *-homomorphism $\delta^r_{\cal B}$
from $C^*_r(\cal B)$ to ${\cal L}(L^2_e({\cal B})\otimes L^2(G))$
by $\delta^r_{\cal B}(x) = W(x\otimes 1)W^*$
($x\in C^*_r(\cal B)$).
Moreover, for any $b\in B\subseteq M(C^*(\cal B))$ (see Lemma
\ref{1.4}), $\delta^r_{\cal B}(\lambda_{\cal B}(b)) =
\lambda_{\cal B}(b)\otimes \lambda_{\pi(b)}$
(where $\lambda_t$ is the canonical image of $t$ in $M(C^*_r(G))$).
\par\medskip

\begin{proposition} \label{2.14}
The map $\delta^r_{\cal B}$ defined above is an injective non-degenerate
coaction on $C^*_r(\cal B)$ by $C^*_r(G)$.
\end{proposition}
\noindent {\bf Proof}:
It is clear that $\delta^r_{\cal B}$ is an injective *-homomorphism.
Moreover,
$(\lambda_{\cal B}\otimes\lambda_G)\circ\delta_{\cal B} =
\delta^r_{\cal B}\circ\lambda_{\cal B}$
which implies that $\delta^r_{\cal B}$ is a non-degenerate coaction
(see Lemma \ref{1.6}).
\par\medskip

There is an alternative natural way to define ``reduced''
cross-sectional $C^*$-algebra (similar to the corresponding
situation of full and reduced crossed products):
$C^*_R({\cal B}) := C^*({\cal B})/\ker (\epsilon_{\cal B})$
(where $\epsilon_{\cal B}$ is the composition: $C^*({\cal B})\stackrel
{\delta_{\cal B}}{\longrightarrow}C^*({\cal B})\otimes_{\rm max}C^*(G)
\stackrel{{\rm id}\otimes \lambda_G}{\longrightarrow}C^*({\cal B})\otimes C^*_r(G)$).
\par\medskip

\begin{remark} \label{2.b}
(a) It is clear that $\mu_{\lambda,T} = (\mu_T\otimes \lambda_G)\circ\delta_{\cal B}$
(see Lemma \ref{2.8}) induces a representation of $C^*_R({\cal B})$ on
$L^2(G; {\cal H})$.
If $\mu_T$ is faithful, then this induced representation is also
faithful and $C^*_R({\cal B})$ can be identified with the image of $\mu_{\lambda,T}$.
\par\noindent
(b) If $\mu_T$ is faithful, then so is $\phi$ and
$\lambda_{\cal B}\otimes_\phi 1$
is a faithful representation of $C^*_r({\cal B})$ (by Lemma \ref{1.3}(b)).
Therefore, part (a) and Lemma \ref{2.10} implies that $C^*_r({\cal B})$ is a
quotient of $C^*_R({\cal B})$.
\end{remark}

In [12, 3.2(1)], it was proved that these two reduced cross-sectional
$C^*$-algebras coincide in the case of semi-direct product bundles.
The corresponding result in the case of $C^*$-algebraic bundles
over discrete groups was proved implicitly in [6, 4.3].
In the following we shall see that it is true in general.
\par\medskip

The idea is to define a map $\varphi$ from $C_r^*(\cal B)$ to
${\cal L}(L^2(G; {\cal H}))$ such that $\varphi\circ\lambda_{\cal B} =
\mu_{\lambda,T}$ (see Remark \ref{2.b}(a)).
As noted above, the difficulty is that $V$ may not be surjective and,
by Lemma 2.10, $\lambda_{\cal B}\otimes_\phi 1$ may only be a proper
subrepresentation of $\mu_{\lambda,T}$ (see Example \ref{2.6}(b)).
However, we may ``move it around'' filling out the whole
representation space for $\mu_{\lambda,T}$ using the right regular
representation $\rho$ of $G$ (on $L^2(G)$):
$\rho_r(g)(s)=\Delta(r)^{1/2}g(sr)$ ($g\in L^2(G)$; $r,s\in G$) where
$\Delta$ is the modular function for $G$.
\par\medskip

\begin{lemma} \label{2.a}
For each $r\in G$,
\par\noindent
  (a) The unitary operator $\rho_r\otimes 1$ on $L_2(G)\otimes {\cal H}=
L^2(G;{\cal H})$ lies in the commutant of $\mu_{\lambda,T} (C^*(\cal B))$.
\par\noindent
  (b) Consider the isometry
  $$
V^r: L_2({\cal B})\otimes_\phi{\cal H} \to L^2(G;{\cal H}),
  $$
given by $V^r = (\rho_r\otimes 1) V$.
Then for all $a\in C^*({\cal B})$ one
has $V^r (\lambda_{\cal B}(a)\otimes 1) = \mu_{\lambda,T}(a) V^r$.
\par\noindent
  (c) Let $K_r$ be the range of\/ $V^r$.
Then $K_r$ is invariant under $\mu_{\lambda,T}$ and the restriction of $\mu_{\lambda,T}$
to $K_r$ is equivalent to $\lambda_{\cal B}\otimes 1$.
\end{lemma}
\noindent {\bf Proof:}
It is clear that $\rho_r\otimes 1$ commutes with $\mu_{\lambda,T}(b_t) =
\lambda_t\otimes T_{b_t}$ for any $b_t\in B_t$ (see Lemma \ref{1.4}).
It then follows that $\rho_r\otimes 1$ also commutes with the
range of the integrated form of $\mu_{\lambda,T}$, whence (i).
The second point follows immediately from (i) and
Proposition \ref{2.10}. Finally, (iii) follows from (ii).
\par\medskip

  Our next result is intended to show that the $K_r$'s do indeed fill
out the whole of $L^2(G;{\cal H})$.
\par\medskip

\begin{proposition} \label{2.c}
  The linear span of\/ $\bigcup_{r\in G} K_r$ is dense in $L^2(G;{\cal H})$.
\end{proposition}
\noindent {\bf Proof:}
Let
  $$
\Gamma = {\rm span}\{V^r( f\otimes \eta): r\in G,\  f\in {\cal L}({\cal B}),
\ \eta\in{\cal H}\}.
  $$
Since for any $t\in G$,
  $$
V^r( f\otimes \eta)(t) =
(\rho_r\otimes 1)V( f\otimes \eta)(t) =
\Delta(r)^{1/2}V(f\otimes \eta)(tr) =
\Delta(r)^{1/2}T_{f(tr)}\eta,
  $$
and since we are taking $f$ in ${\cal L}({\cal B})$ above, it is easy to see
that $\Gamma$ is a subset of $C_c(G,{\cal H})$.
Our strategy will be to use [7, II.15.10]
(on the Banach bundle ${\cal H}\times G$ over $G$)
for which we must prove that:
\par
\noindent (I) If $f$ is a continuous complex function on $G$ and
$\zeta\in\Gamma$, then the pointwise product $f\zeta$ is in $\Gamma$;
\par
\noindent (II) For each $t\in G$, the set $\{\zeta(t):\zeta\in \Gamma\}$
is dense in ${\cal H}$.
\par\noindent
The proof of (I) is elementary in view of the fact that ${\cal L}({\cal B})$ is
closed under pointwise multiplication by continuous scalar-valued
functions [7, II.13.14].
  In order to prove (II), let $\xi\in{\cal H}$ have the form
$\xi=T_b\eta=\phi(b)\eta$, where $b\in B_e$ and $\eta\in{\cal H}$.
  By [7, II.13.19], let $f\in {\cal L}({\cal B})$ be such that $f(e)=b$.
  It follows that
$\zeta_r:=V^r(f\otimes \eta)$ is in $\Gamma$ for all $r$.
  Also note that,
setting $r=t^{-1}$, we have
  $$
\zeta_{t^{-1}}(t) =
\Delta(t)^{-1/2}T_{ f(e)}\eta =
\Delta(t)^{-1/2}T_{b}\eta =
\Delta(t)^{-1/2}\xi.
  $$
  This shows that $\xi\in\{\zeta(t):\zeta\in \Gamma\}$.
  Since the set of such $\xi$'s is dense in ${\cal H}$ (because
$\phi$ is non-degenerate by assumption), we have that (II) is proven.
  As already indicated, it now follows from [7, II.15.10] that
$\Gamma$ is dense in $L^2(G;{\cal H})$.  Since $\Gamma$ is contained in the
linear span of $\bigcup_{r\in G} K_r$, the conclusion follows.
\par\medskip

  We can now obtain the desired result.
\par\medskip

\begin{theorem} \label{2.d}
For all $a\in C^*({\cal B})$ one has that $\|\mu_{\lambda,T}(a)\| \leq
\|\lambda_{\cal B}(a)\|$.
Consequently, $C^*_R({\cal B})=C^*_r({\cal B})$.
\end{theorem}
\noindent {\bf Proof:}
  We first claim that for all $a\in C^*({\cal B})$ one has
that
  $$
  \lambda_{\cal B}(a) = 0 \quad =\!\!\Rightarrow \quad \mu_{\lambda,T}(a) = 0.
  $$
  Suppose that $\lambda_{\cal B}(a) = 0$.
  Then for each $r\in G$ we have by Lemma \ref{2.a}(b) that
  $$
  \mu_{\lambda,T}(a) V^r = V^r (\lambda_{\cal B}(a)\otimes 1) =0.
  $$
  Therefore
  $\mu_{\lambda,T}(a)=0$ in the range $K_r$ of $V^r$.
  By Proposition \ref{2.c} it follows that $\mu_{\lambda,T}(a)=0$, thus proving our
claim.
  Now define a map
  $$
  \varphi : C^*_r({\cal B}) \longrightarrow {\cal B}(L^2(G;{\cal H}))
  $$
by $\varphi (\lambda_{\cal B}(a)) := \mu_{\lambda,T}(a)$, for all $a$ in $C^*({\cal B})$.
  By the claim above we have that $\varphi$ is well defined.
  Also, it is easy to see that $\varphi$ is a *-homomorphism.
  It follows that $\varphi$ is contractive and hence that for all
$a$ in $C^*({\cal B})$
  $$
\|\mu_{\lambda,T}(a)\| =
\|\varphi (\lambda_{\cal B}(a))\| \leq
\|\lambda_{\cal B}(a)\|.
  $$
  For the final statement, we note that if $\mu_T$ is faithful, then
the map $\varphi$ defined above is the inverse of the quotient map
from $C^*_R({\cal B})$ to $C^*_r({\cal B})$ given in Remark \ref{2.b}(ii).
\par\medskip

  The following generalises [11, 7.7.5] to the context of $C^*$-
algebraic bundles:
\par\medskip

\begin{corollary} \label{2.e}
  Let $T:{\cal B}\rightarrow {\cal L}({\cal H})$ be a non-degenerate
$*$-representation of the $C^*$-algebraic bundle ${\cal B}$
and let $\mu_{\lambda,T}$ be the representation of ${\cal B}$ on $L^2(G;{\cal H})$ given by
$\mu_{\lambda,T}(b_t) = \lambda_t\otimes T_{b_t}$, for $t\in G$, and $b_t\in B_t$.
  Then $\mu_{\lambda,T}$ is a well defined representation and induces a
representation of $C^*({\cal B}) ($again denoted by $\mu_{\lambda,T}$).
  In this case, $\mu_{\lambda,T}$ factors through $C^*_r({\cal B})$.
  Moreover, if $\phi = T\!\mid_{B_e}$ is faithful, the
representation of $C^*_r({\cal B})$ arising from this factorisation
is also faithful.
\end{corollary}
\noindent {\bf Proof:}
By Remark \ref{2.b}(a), $\mu_{\lambda,T}$ factors through a representation of
$C^*_R({\cal B})=C^*_r({\cal B})$ (Theorem \ref{2.d}).
Now if $\phi$ is faithful, then by Theorem \ref{2.d}, Lemmas \ref{2.10}
and \ref{1.3}(a), $\|\mu_{\lambda,T}(a)\| = \|\lambda_{\cal B}(a)\|$.
This proves the second statement.
\par\medskip
\par\medskip
\par\medskip

\section{The approximation property of $C^*$-algebraic bundles}
\par\medskip
\par\medskip

From now on, we assume that $\mu_T$ (see the paragraph
before Lemma \ref{2.4}) is faithful.
Moreover, we will not distinguish ${\cal L}({\cal B})$ and its image in $C^*({\cal B})$.
\par\medskip

The materials in this section is similar to the discrete case in [6].
Let ${\cal B}_e$ be the $C^*$-algebraic bundle $B_e\times G$ over $G$.
We will first define a map from $L^2_e({\cal B}_e)\times
C^*_r({\cal B})\times L^2_e({\cal B}_e)$ to $C^*(\cal B)$.
For any $\alpha\in {\cal L(B}_e)$, let $V_\alpha$ be a map
from ${\cal H}$ to $L^2(G; {\cal H})$ given by
$$V_\alpha(\xi)(s) = \phi(\alpha(s))\xi$$
($\xi\in {\cal H}$; $s\in G$).
It is clear that $V_\alpha$ is continuous and $\|V_\alpha\|\leq \|\alpha\|$.
Moreover, we have $V_\alpha^*(\Theta) =
\int_G \phi(\alpha(r)^*)\Theta(r)\ dr$
($\alpha\in {\cal L(B}_e)$; $\Theta\in L^2(G)\otimes {\cal H} =
L^2(G;{\cal H})$) and $\|V_\alpha^*\|\leq \|\alpha\|$.
Thus, for any $\alpha, \beta\in L^2_e({\cal B}_e)$, we obtain a continuous
linear map $\Psi_{\alpha,\beta}$ from ${\cal L}(L^2(G)\otimes {\cal H})$
to ${\cal L}({\cal H})$ defined by
$$\Psi_{\alpha,\beta}(x) = V_\alpha^*xV_\beta$$
with $\|\Psi_{\alpha,\beta}\|\leq \|\alpha\|\|\beta\|$.
Recall from Remark \ref{2.b}(a) and Theorem \ref{2.d} that $C^*_r(\cal B)$
is isomorphic to the image of $C^*(\cal B)$ in ${\cal L}(L^2(G)\otimes
{\cal H})$ under $\mu_{\lambda,T} = (\mu_T\otimes \lambda_G)\circ \delta_{\cal B}$.
\par\medskip

\begin{lemma} \label{3.1}
Let $\alpha, \beta\in {\cal L(B}_e)$ and $f\in \cal L(B)$.
Then $\Psi_{\alpha,\beta}(\mu_{\lambda,T}(f)) =
\alpha\cdot f\cdot \beta$ where $\alpha\cdot f\cdot \beta\in {\cal L(B)}$
is defined by $\alpha\cdot f\cdot \beta(s) =
\int_G \alpha(t)^*f(s)\beta(s^{-1}t)\ dt$.
\end{lemma}
\noindent {\bf Proof}:
For any $\xi \in {\cal H}$, we have
\begin{eqnarray*}
\Psi_{\alpha,\beta}(\mu_{\lambda,T}(f))\xi & = &
\int_G \phi(\alpha(t)^*)(\mu_{\lambda,T}(f)V_\beta\xi)(t)\ dt\\
& = & \int_G\int_G \phi(\alpha^*(t))T_{f(s)}\phi(\beta(s^{-1}t))
\xi\ ds dt\\
& = & \int_G T_{(\alpha\cdot f\cdot \beta)(s)}\xi\ ds.
\end{eqnarray*}
\par\medskip

Hence we have a map from $L^2_e({\cal B}_e)\times
C^*_r({\cal B})\times L^2_e({\cal B}_e)$ to $C^*(\cal B)$ such
that $\|\alpha\cdot x\cdot \beta\|\leq \|\alpha\|\|x\|\|\beta\|$.
Next, we will show that this map sends
$L^2_e({\cal B}_e)\times \mu_{\lambda,T}({\cal L(B}))\times
L^2_e({\cal B}_e)$ to $\cal L(B)$.
\par\medskip

\begin{lemma} \label{3.2}
For any $\alpha, \beta\in L^2_e({\cal B}_e)$ and $f\in \cal L(B)$,
$\alpha\cdot f\cdot \beta\in \cal L(B)$.
\end{lemma}
\noindent {\bf Proof}:
If $\alpha', \beta' \in {\cal L(B}_e)$
\begin{eqnarray*}
\|(\alpha'\cdot f\cdot \beta')(s)\| & = &
\sup \{\mid \langle \eta,\int_G T_{\alpha'(t)^*f(s)\beta'(s^{-1}t)}
\xi\ dt\rangle \mid: \|\eta\|\leq 1; \|\xi\|\leq 1\}\\
& \leq & \sup \{ \|f(s)\| (\int_G \|\phi(\alpha'(t))\eta\|^2\ dt)^{1/2}
(\int_G \|\phi(\beta'(t))\xi\|^2\ dt)^{1/2}: \|\eta\|\leq 1; \|\xi\|\leq
1\}\\
& = & \|f(s)\|\|\alpha'\|\|\beta'\|.
\end{eqnarray*}
Let $\alpha_n$ and $\beta_n$ be two sequences of elements in ${\cal L(B}_e)$
that converge to $\alpha$ and $\beta$ respectively.
Then $(\alpha_n\cdot f\cdot \beta_n)(s)$ converges to an element
$g(s)\in B_s$.
Moreover, since $f$ is of compact support and the convergence is uniform,
$g\in \cal L(B)$ and ${\rm supp}(g)\subseteq {\rm supp}(f)$.
In fact, this convergence actually takes place in
${\cal L}_1({\cal B})$ and hence in $C^*(\cal B)$.
Therefore $\Psi_{\alpha,\beta}(\mu_{\lambda,T}(f))=\mu_T(\alpha\cdot f\cdot \beta)$.
\par\medskip

\begin{remark} \label{3.3}
The proof of the above lemma also shows that $\Psi_{\alpha,\beta}$
sends the image of $B_t$ in $M(C^*_r(\cal B))$ to the image
of $B_t$ in $M(C^*(\cal B))$.
Hence $\Psi_{\alpha,\beta}$ induces a map $\Phi_{\alpha,\beta}$ from $B$
to $B$ which preserves fibers.
\end{remark}

\begin{definition} \label{3.4}
Let $\{\Phi_i\}$ be a net of maps from $B$ to itself such that
they preserve fibers and are linear on each fiber.
\par\noindent
(a) $\{\Phi_i\}$ is said to be {\it converging to $1$ on compact slices
of $B$} if for any $f\in {\cal L(B})$ and any $\epsilon > 0$, there
exists $i_0$ such that for any $i\geq i_0$, $\|\Phi_i(b)-b\|<\epsilon$
for any $b\in f(G)$ ($f(G)$ is called a {\it compact slice} of B).
\par\noindent
(b) Then $\{\Phi_i\}$ is said to be {\it converging to
$1$ uniformly on compact-bounded subsets of $B$} if for any compact
subset $K$ of $G$ and any $\epsilon > 0$, there exists $i_0$ such that
for any $i\geq i_0$, $\|\Phi_i(b)-b\|<\epsilon$ if
$\pi(b)\in K$ and $\|b\|\leq 1$.
\end{definition}

\begin{lemma} \label{3.5}
Let $\{\Phi_i\}$ be a net as in Definition \ref{3.4}.
Then each of the following conditions is stronger than the next one.
\begin{enumerate}
\item[i.] $\{\Phi_i\}$ converges to $1$ uniformly on
compact-bounded subsets of $B$.
\item[ii.] $\{\Phi_i\}$ converges to $1$ uniformly on
compact slices of $B$.
\item[iii.] For any $f\in \cal L(B)$, the net $\Phi_i\circ f$ converges
to $f$ in ${\cal L}_1(\cal B)$.
\end{enumerate}
\end{lemma}
\par\noindent {\bf Proof:}
Since every element in ${\cal L(B})$ has compact support and is bounded,
it is clear that (i) implies (ii).
On the other hand, (ii) implies (iii) is obvious.
\par\medskip

Following the idea of [6], we define the approximation property
of $\cal B$.
\par\medskip

\begin{definition} \label{3.6}
(a) Let $\cal B$ be a $C^*$-algebraic bundle.
For $M>0$, $\cal B$ is said to have the {\it $M$-approximation property}
(respectively, {\it strong $M$-approximation property}) if
there exist nets $(\alpha_i)$ and $(\beta_i)$ in ${\cal L(B}_e)$ such that
\begin{enumerate}
\item[i.] $\sup_i \|\alpha_i\|\|\beta_i\| \leq M$;
\item[ii.] the map $\Phi_i = \Phi_{\alpha_i,\beta_i}$ (see Remark \ref{3.3})
converges to $1$ uniformly on compact slices of $B$ (respectively,
uniformly on compact-bounded subsets of $B$).
\end{enumerate}
$\cal B$ is said to have the (respectively, {\it strong})
{\it approximation property} if it has
the (respectively, strong) $M$-approximation property
for some $M > 0$.
\par\noindent
(b) We will use the terms ({\it strong}) {\it positive
$M$-approximation property} and ({\it strong}) {\it positive
approximation property} if we can choose $\beta_i = \alpha_i$
in part (a).
\end{definition}

Because of Remark 3.7(b) below as well as [11, 7.3.8], we believe that
the above is the weakest condition one can think of to ensure the
amenability of the $C^*$-algebraic bundle.
\par\medskip

\begin{remark} \label{3.7}
(a) Since any compact subset of a discrete group is finite and any
$C^*$-algebraic bundle has enough cross-sections,
the approximation property defined in [6] is the
same as positive 1-approximation property defined above.
\par\noindent
(b) It is easy to see that the amenability of $G$ implies
the positive 1-approximation property of $\cal B$ (note that the
positive 1-approximation property is similar to the condition in
[11, 7.3.8]).
In fact, let $\xi_i$ be the net given by [11, 7.3.8] and let
$\eta_i(t) = \overline{\xi_i(t)}$.
If $u_j$ is an approximate unit of $B_e$ (which is also a strong
approximate unit of $\cal B$ by [7, VIII.16.3]),
then the net $\alpha_{i,j} = \beta_{i,j} = \eta_i u_j$
will satisfy the required property.
\par\noindent
(c) We can also formulate the approximation property as follows:
there exists $M>0$ such that for any compact slice $S$ of $B$ and any
$\epsilon > 0$, there exist $\alpha,\beta\in L^2_e({\cal B}_e)$ with
$$\|\alpha\|\|\beta\|\leq M\qquad {\rm and} \qquad
\|\alpha\cdot b\cdot \beta - b\| < \epsilon$$
if $b\in S$.
In fact, we can replace $L^2_e({\cal B}_e)$ by ${\cal L(B}_e)$ and
consider the directed set $D= \{ (K,\epsilon): K$ is a compact subset
of $G$ and $\epsilon > 0 \}$.
For any $d=(K,\epsilon)\in D$, we take $\alpha_d$ and $\beta_d$
that satisfying the above condition.
These are the required nets.
\end{remark}

We can now prove the main results of this section.
\par\medskip

\begin{proposition} \label{3.8}
If $\cal B$ has the approximation property, then the coaction
$\epsilon_{\cal B} = ({\rm id}\otimes \lambda_G)\circ\delta_{\cal B}$ is injective.
\end{proposition}
\noindent {\bf Proof}:
Let $\Phi_i = \Phi_{\alpha_i,\beta_i}$ be the map from $B$ to itself
as given by Definition \ref{3.6}(a)(ii) and $\Psi_i = \Psi_{\alpha_i,\beta_i}$.
Let $J_i=\Psi_i\circ\mu_{\lambda,T}$.
By Lemma \ref{3.2}, for any $f\in\cal L(B)$, $J_i(f)\in {\cal L}({\cal B})$
(note that we regard ${\cal L}({\cal B})\subseteq C^*({\cal B})$) and $J_i(f)(s) =
\Phi_i(f(s))$ ($s\in G$).
Since $\Phi_i\circ f$ converges to $f$ in ${\cal L}_1(\cal B)$
(by Lemma \ref{3.5}), $J_i(f)$ converges to $f$ in $C^*(\cal B)$.
Now because $\|J_i\|\leq \|\Psi_i\|\leq \sup_i\|\alpha_i\|
\|\beta_i\|\leq M$, we know that $J_i(x)$ converges to $x$ for all
$x\in C^*(\cal B)$ and $\epsilon_{\cal B}$ is injective.
\par\medskip

Note that if $G$ is amenable, we can also obtain directly from
the Lemma \ref{1.1}(a) that $\epsilon_{\cal B}$ is injective.
\par\medskip

\begin{theorem} \label{3.9}
Let $\cal B$ be a $C^*$-algebraic bundle having the
approximation property (in particular, if $G$ is amenable).
Then $\cal B$ is amenable.
\end{theorem}
\noindent {\bf Proof:}
Proposition \ref{3.8} implies that $C_R^*({\cal B}) = C^*({\cal B})$
(see the paragraph before Remark \ref{2.b}).
Now the amenability of ${\cal B}$ clearly follows from Theorem \ref{2.d}.
\par\medskip
\par\medskip
\par\medskip

\section{Two special cases}
\par\medskip
\par\medskip

\noindent {\em I. Semi-direct product bundles and nuclearity of
crossed products.}
\par\medskip

Let $A$ be a $C^*$-algebra with action $\alpha$ by a locally compact
group $G$.
Let ${\cal B}$ be the semi-direct product bundle of $\alpha$.
\par\medskip

\begin{remark} \label{4.1}
$\cal B$ has the (respectively, strong) $M$-approximation property
if there exist nets $\{\gamma_i\}_{i\in I}$ and
$\{\theta_i\}_{i\in I}$ in $K(G;A)$ such that
$$\|\int_G \gamma_i(r)^*\gamma_i(r)\
dr\|\cdot\|\int_G \theta_i(r)^*\theta_i(r)\ dr\| \leq M^2$$
and for any $f\in K(G;A)$ (respectively, for any compact subset
$K$ of $G$),
$\int_G \gamma_i(r)^*a\alpha_t (\theta_i(t^{-1}r))\ dr$ converges to
$a\in A$ uniformly for $(t,a)$ in the graph of $f$ (respectively,
uniformly for $t\in K$ and $\|a\|\leq 1$).
\end{remark}

\begin{definition} \label{4.2}
An action $\alpha$ is said to have the (respectively, strong)
{\it ($M$-)approximation property}
(respectively, $\alpha$ is said to be {\it weakly amenable}) if
the $C^*$-algebraic bundle $\cal B$ associated with $\alpha$ has the
(respectively, strong) ($M$-)approximation property
(respectively, ${\cal B}$ is amenable).
\end{definition}

Let $G$ and $H$ be two locally compact groups.
Let $A$ and $B$ be $C^*$-algebras with actions $\alpha$ and $\beta$ by
$G$ and $H$ respectively.
Suppose that $\tau = \alpha\otimes \beta$ is the product action on
$A\otimes B$ by $G\times H$.
\par\medskip

\begin{lemma} \label{4.3}
With the notation as above, if $A$ is nuclear and both $\alpha$ and
$\beta$ have the approximation property, then $(A\otimes B)
\times_\tau(G\times H)=(A\otimes B)\times_{\tau,r}(G\times H)$.
\end{lemma}
\noindent {\bf Proof}:
Let $\cal B$, $\cal D$ and $\cal F$ be the semi-direct product bundles of
$\alpha$, $\beta$ and $\tau$ respectively.
Then $C^*_r({\cal F})=C^*_r({\cal B})\otimes C^*_r({\cal D})$
(by Example \ref{2.12}).
Moreover, since $A$ is nuclear, $C^*({\cal F}) =
C^*({\cal B})\otimes_{\max} C^*({\cal D})$ (by Example \ref{2.12} and
[10, 3.2]).
It is not hard to see that the coaction, $\delta_{\cal F}$, on
$C^*(\cal F)$ is the tensor product of the coactions on
$C^*(\cal B)$ and $C^*(\cal D)$.
Suppose that $C^*(\cal F)$ is a $C^*$-subalgebra of $\cal L(H)$.
Consider as in Section 2, the composition:
$$\mu_{\cal F}: C^*({\cal F})\stackrel{\delta_{\cal F}}
{\longrightarrow} C^*({\cal F})\otimes_{\rm max} C^*(G\times H)
\stackrel{{\rm id}\otimes \lambda_{G\times H}}{\longrightarrow}
{\cal L} ({\cal H}\otimes L^2(G\times H))$$
and identify its image with $C^*_R({\cal F})=C^*_r({\cal F})$
(see Remark \ref{2.b}(a)).
We also consider similarly the maps $\mu_{\cal B}$ and $\mu_{\cal D}$
from $C^*(\cal B)$ and $C^*(\cal D)$ to ${\cal L}({\cal H}\otimes
L^2(G))$ and ${\cal L}({\cal H}\otimes L^2(H))$ respectively.
Now for any $f\in \cal L(B)$ and $g\in \cal L(D)$,
we have $\mu_{\cal F}(f\otimes g) = (\mu_{\cal B}(f))_{12}
(\mu_{\cal D}(g))_{13}\in {\cal L}({\cal H}\otimes L^2(G)\otimes L^2(H))$.
As in Section 3, we define, for any $k\in {\cal L(B}_e)$ and
$l\in {\cal L(D}_e)$, an operator $V_{k\otimes l}$ from $\cal H$ to
${\cal H}\otimes L^2(G\times H)$ by
$V_{k\otimes l}\zeta (r,s) = k(r)(l(s)\zeta)$
($r\in G$; $s\in H$; $\zeta\in {\cal H}$).
It is not hard to see that $V_{k\otimes l}(\zeta) =
(V_k\otimes 1)V_l(\zeta)$ and
$$V_{k\otimes l}^*\mu_{\cal F}
(f\otimes g)V_{k'\otimes l'} = (V_k^*\mu_{\cal B}(f)V_{k'})
(V_l^*\mu_{\cal D}(g)V_{l'})\in \cal L(H)$$
(note that $B_r$
commutes with $D_s$ in ${\cal L}(\cal H)$).
Now let $k_i, k'_i\in \cal L(B)$ and $l_j, l'_j\in \cal L(D)$ be
the nets that give the corresponding approximation property on
$\cal B$ and $\cal D$ respectively.
Then $V_{k_i}^*\mu_{\cal B}(f)V_{k'_i}$ converges to $f$ in
$C^*(\cal B)$ and $V_{l_j}^*\mu_{\cal D}(g)V_{l'_j}$
converges to $g$ in $C^*(\cal D)$.
Hence $J_{i,j}(z) =
V_{k_i\otimes l_j}^*\mu_{\cal F}(z)V_{k'_i\otimes l'_j}$
converges to $z$ in $C^*(\cal F)$
for all $z\in {\cal L(B)}\otimes_{alg} {\cal L(D)}$.
Since $\|J_{i,j}\|$ is uniformly bounded, $\mu_{\cal F}$ is
injective and $C^*({\cal F}) = C^*_r({\cal F})$.
\par\medskip

An interesting consequence of this proposition is the nuclearity
of the crossed products of group actions with the approximation
property (which is a generalisation of the case of actions
of amenable groups).
Note that in the case of discrete group, this was also proved by
Abadie in [1].
\par\medskip

\begin{theorem} \label{4.4}
Let $A$ be a $C^*$-algebra and $\alpha$ be an action on $A$ by a
locally compact group $G$.
If $A$ is nuclear and $\alpha$ has the approximation property,
then $A\times_{\alpha} G = A\times_{\alpha , r} G$
is also nuclear.
\end{theorem}
\noindent {\bf Proof}:
By Lemma \ref{4.3} (or Theorem \ref{3.9}),
$A\times_{\alpha} G = A\times_{\alpha , r} G$.
For any $C^*$-algebra $B$, let $\beta$ be the trivial action
on $B$ by the trivial group $\{e\}$.
Then $(A\times_\alpha G)\otimes_{\max} B =
(A\otimes_{\max}B)\times_{\alpha\otimes\beta} G =
(A\otimes B)\times_{\alpha\otimes\beta, r} G =
(A\times_{\alpha , r} G)\otimes B$ (by Lemma \ref{4.3} again).
\par\medskip

One application of Theorem \ref{4.4} is to relate the amenability of
Anantharaman-Delaroche (see [4, 4.1]) to the approximation property
in the case when $A$ is nuclear and $G$ is discrete.
The following corollary clearly follows from this theorem and [4, 4.5].
\par\medskip

\begin{corollary} \label{4.5}
Let $A$ be a nuclear $C^*$-algebra with an action $\alpha$ by
a discrete group $G$.
If $\alpha$ has the approximation property,
then $\alpha$ is amenable in the sense of Anantharaman-Delaroche.
\end{corollary}

We don't know if the two properties coincide in general.
However, we can have a more direct and transparent comparison of
them in the case of commutative $C^*$-algebras and show that they
are the same.
Furthermore, they also coincide in the case of finite
dimensional $C^*$-algebras.
\par\medskip

\begin{corollary} \label{4.a}
Let $A$ be $C^*$-algebra with action $\alpha$ by a discrete group $G$.
\par\noindent
(a) If $A$ is commutative, the followings are equivalent:
\begin{enumerate}
\item[i.] $\alpha$ is amenable in the sense of Anantharaman-Delaroche;
\item[ii.] $\alpha$ has the positive 1-approximation property;
\item[iii.] $\alpha$ has the approximation property.
\end{enumerate}
\par\noindent
(b) If $A$ is unital and commutative or if $A$ is finite
dimensional, then (i)-(iii) are also equivalent to the
following conditions:
\begin{enumerate}
\item[iv.] $\alpha$ has the strong positive 1-approximation property;
\item[v.] $\alpha$ has the strong approximation property.
\end{enumerate}
\end{corollary}
\noindent {\bf Proof:}
(a) By [4, 4.9(h')], $\alpha$ is amenable in the sense of
Anantharaman-Delaroche if and only if
there exists a net $\{\gamma_i\}$ in $K(G;A)$ such that
$\|\sum_{r\in G} \gamma_i(r)^*\gamma_i(r)\|\leq 1$
and $\sum_{r\in G} \gamma_i(r)^*\alpha_t(\gamma_i(t^{-1}r))$ converges
to $1$ strictly for any $t\in G$.
It is exactly the original definition of the approximation property
given in [1].
Hence conditions (i) is equivalent to conditions (ii) (see Remark
\ref{3.7}(a)).
Now part (a) follows from Corollary \ref{4.5}.
\par\noindent
(b) Suppose that $A$ is both unital and commutative.
Let $\alpha$ satisfy condition (i) and $\{\gamma_i\}$ be the net as
given in the proof of part (a) above.
As $A$ is unital, the strict convergence and the norm convergence are
equivalent.
Moreover, as $G$ is discrete, any compact subset $K$ of $G$ is finite.
These, together with the commutativity of $A$, imply that
$\sum_{r\in G} \gamma_i(r)^*\alpha_t(\gamma_i(t^{-1}r))$ converges to 1
strictly for any $t\in G$ if and only if
$\sum_{r\in G} \gamma_i(r)^*a\alpha_t(\gamma_i(t^{-1}r))$ converges to
$a\in A$ uniformly for $t\in K$ and $\|a\|\leq 1$.
Thus, by Remark \ref{4.1}, we have the equivalence of (i) and (iv)
in the case of commutative unital $C^*$-algebras and the equivalence
of (i)-(v) follows from Lemma \ref{3.5} and Corollary \ref{4.5}.
Now suppose that $A$ is a finite dimensional $C^*$-algebra (but not
necessary commutative).
By [4, 4.1] and [4, 3.3(b)], $\alpha$ satisfies condition (i) if and
only if there exists a net $\{\gamma_i\}$ in $K(G; Z(A))$
(where $Z(A)$ is the centre of $A=A^{**}$) such that
$\|\sum_{r\in G} \gamma_i(r)^*\gamma_i(r)\|\leq 1$
and for any $t\in G$, $\sum_{r\in G} \gamma_i(r)^*\alpha_t
(\gamma_i(t^{-1}r))$ converges to $1$ weakly (and hence converges
to $1$ in norm as $A$ is finite dimensional).
Let $K$ be any compact (and hence finite) subset of $G$.
Since $\alpha_t(\gamma_i(t^{-1}r))\in Z(A)$,
$\sum_{r\in G} \gamma_i(r)^*a \alpha_t(\gamma_i(t^{-1}r))$
converges to $a\in A$ uniformly for $t\in K$ and $\|a\|\leq 1$.
This shows that $\alpha$ satisfies condition (iv)
(see Remark \ref{4.1}).
The equivalence follows again from Lemma \ref{3.5} and
Corollary \ref{4.5}.
\par\medskip

Because of the above results, we believe that approximation property
is a good candidate for the notion of amenability
of actions of locally compact groups on general $C^*$-algebras.
\par\medskip
\par\medskip

\noindent {\em II. Discrete groups: $G$-gradings and coactions.}
\par\medskip

Let $G$ be a discrete group and let $D$ be a $C^*$-algebra with a
$G$-grading (i.e. $D=\overline{\oplus_{r\in G} D_r}$ such that
$D_r\cdot D_s \subseteq D_{rs}$ and $D_r^*\subseteq D_{r^{-1}}$).
Then there exists a canonical $C^*$-algebraic bundle structure
(over $G$) on $D$.
We denote this bundle by $\cal D$.
Now by [6, \S 3], $D$ is a quotient of $C^*(\cal D)$.
Moreover, if the grading is topological in the sense that there exists a
continuous conditional expectation from $D$ to $D_e$ (see [6, 3.4]),
then $C^*_r({\cal D})$ is a quotient of $D$ (see [6, 3.3]).
Hence by [12, 3.2(1)] (or [10, 2.17]), there is an induced
non-degenerate coaction on $D$ by $C^*_r(G)$
which define the given grading.
Now the proof of [9, 2.6], [6, 3.3] and the above
observation imply the following equivalence.
\par\medskip

\begin{proposition} \label{4.6}
Let $G$ be a discrete group and $D$ be a $C^*$-algebra.
Then a $G$-grading $D=\overline{\oplus_{r\in G} D_r}$ is topological
if and only if it is induced by a non-degenerate coaction
of $C^*_r(G)$ on $D$.
\end{proposition}

\begin{corollary} \label{4.7}
Let $D$ be a $C^*$-algebra with a non-degenerate coaction $\epsilon$
by $C^*_r(G)$.
Then it can be ``lifted'' to a full coaction i.e. there exist a
$C^*$-algebra $A$ with a full coaction $\epsilon_A$ by
$G$ and a quotient map $q$ from $A$ to $D$ such that
$\epsilon\circ q = (q\otimes \lambda_G)\circ\epsilon_A$.
\end{corollary}

In fact, if $\cal D$ is the bundle as defined above, then we can take
$A=C^*({\cal D})$ and $\epsilon_A = \delta_{\cal D}$.
\par\medskip
\par\medskip
\par\medskip

\noindent {\bf References}
\par\medskip
\noindent
[1] F. Abadie, Tensor products of Fell bundles over discrete groups,
preprint (funct-an/9712006), Universidade de S\~{a}o Paulo, 1997.
\par\noindent
[2] C. Anantharaman-Delaroche, Action moyennable d'un groupe
localement compact sur une alg\`ebre de von Neumann,
Math. Scand. 45 (1979), 289-304.
\par\noindent
[3] C. Anantharaman-Delaroche, Action moyennable d'un groupe
localement compact sur une alg\`ebre de von Neumann II,
Math. Scand. 50 (1982), 251-268.
\par\noindent
[4] C. Anantharaman-Delaroche, Sys\`emes dynamiques non
commutatifs et moyennabilit\`e, Math. Ann. 279 (1987), 297-315.
\par\noindent
[5] S. Baaj and G. Skandalis, $C^{*}$-alg\`ebres de Hopf et
th\'eorie de Kasparov \'equivariante, $K$-theory 2 (1989), 683-721.
\par\noindent
[6] R. Exel, Amenability for Fell bundles, J. Reine Angew. Math.,
492 (1997), 41--73.
\par\noindent
[7] J. M. G. Fell and R. S. Doran, {\it Representations of *-algebras,
locally compact groups, and Banach *-algebraic bundles vol. 1 and 2},
Academic Press, 1988.
\par\noindent
[8] K. Jensen and K. Thomsen, {\it Elements of $KK$-Theory},
Birkh\"auser, 1991.
\par\noindent
[9] C. K. Ng, Discrete coactions on $C^*$-algebras,
J. Austral. Math. Soc. (Series A), 60 (1996), 118-127.
\par\noindent
[10] C. K. Ng, Coactions and crossed products of Hopf
$C^{*}$-algebras, Proc. London. Math. Soc. (3), 72 (1996), 638-656.
\par\noindent
[11] G. K. Pedersen, {\it $C^*$-algebras and their automorphism groups},
Academic Press, 1979.
\par\noindent
[12] I. Raeburn, On crossed products by coactions and their
representation theory, Proc. London Math. Soc. (3), 64 (1992), 625-652.
\par\noindent
[13] M. A. Rieffel, Induced representations of $C^*$-algebras,
Adv. Math. 13 (1974), 176--257.
\par\noindent
\par
\medskip
\noindent Departamento de Matem\'{a}tica, Universidade Federal de Santa
Catarina, 88010-970 Florian\'{o}polis SC, Brazil.
\par
\noindent $E$-mail address: exel@mtm.ufsc.br
\par\medskip

\noindent Mathematical Institute, Oxford University, 24-29 St. Giles, Oxford
OX1 3LB, United Kingdom.
\par
\noindent $E$-mail address: ng@maths.ox.ac.uk
\par
\end{document}