# Selective Probabilistic Classifier Based on Hypothesis Testing

###### Abstract

In this paper, we propose a simple yet effective method to deal with the
violation of the Closed-World Assumption for a classifier. Previous works tend
to apply a threshold either on the classification scores or the loss function
to reject the inputs that violate the assumption. However, these methods
cannot achieve the low False Positive Ratio (FPR) required in safety
applications. The proposed method is a rejection option based on hypothesis
testing with probabilistic networks. With probabilistic networks, it is
possible to estimate the distribution of outcomes instead of a single output.
By utilizing Z-test over the mean and standard deviation for each class, the
proposed method can estimate the statistical significance of the network
certainty and reject uncertain outputs. The proposed method was experimented
on with different configurations of the COCO and CIFAR datasets. The
performance of the proposed method is compared with the Softmax Response,
which is a known top-performing method. It is shown that the proposed method
can achieve a broader range of operation and cover a lower FPR than the
alternative.

Index Terms—  Selective Classifier, Probabilistic Neural Network, Statistical
Analysis, Uncertainty Estimation

##  1 Introduction

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Artificial Intelligence (AI) is becoming a vital part of many real-life
applications such as healthcare, logistics, surveillance, and industry.
Classification is a common concept in the AI field, and it can be considered
one of the building blocks for higher-level reasoning and decision-making
systems. With the increasing demand for robust and reliable algorithms,
especially in safety-critical systems [1], the research community has been
trying to define the robustness [2], evaluation metrics [3], and solutions to
satisfy the requirements of a robust classifier [4].

State-of-the-art classifiers have achieved high accuracy numbers when dealing
with simple datasets such as MNIST [5] or challenging ones like ImageNet [6].
However, several open questions remain on how the classifier should behave in
the circumstances not covered in the training set, for example, when unseen
classes appear (out-of-distribution samples) or when inputs are distorted in a
way not seen in the training set. In such cases, a classifier might generate
faulty results. So it becomes clear that accuracy is not enough for measuring
the performance of classifiers, and the generalization to new environments and
robustness to environmental changes should also be considered.

In their review, Zhang et al. argue that unexpected faulty result in a pattern
recognition algorithm can happen due to the violation of either of the
following assumptions[7]: (1) Closed-World Assumption where the data is
assumed to have a fixed number of classes, all covered in the training set,
(2) Independent and Identically Distributed Assumption where the classes in
the data are assumed to be independent of each other and have the same
distribution, and (3) Clean and Big Data Assumption where the data is assumed
to be well-labeled and large enough for training the network properly. While
fulfilling these assumptions is more accessible in a controlled environment,
real-world applications rarely cover them completely.

This paper deals with the violation of the Closed-World Assumption. While a
straightforward way of dealing with this issue is introducing a trash class in
the training set to cover all out-of-distribution samples, the complex
distribution of them makes it impossible to train an effective classifier in
most cases. Moreover, different distortions might make a sample not easy to
classify, even for a human. While there is ongoing research for adversarial
attacks, the phenomenon is not that common in the everyday use of AI
algorithms. In a typical case, distortions usually are from these categories:
blur, noise, occlusion, and digital alteration of the image.

Recent works try to solve this issue by formulating it to reliable rejection
of the predictions when the network is uncertain. The rejection option, also
known as selective classification, is a central concept in different
classification applications when dealing with uncertainty (e.g., optical
character recognition). Previous works either rely on using a specific type of
activation function in the classifier, such as OpenMax [8], temperature
scaling for SoftMax [9], and Sigmoid [10], modifying the loss function such as
discrepancy loss [11], using more resources such as an ensemble of multiple
classifiers [12] and Monte-Carlo dropout [13]. Moreover, some also suggest a
combination of different ideas [14].

The proposed method is a rejection option based on hypothesis testing with
probabilistic networks. By utilizing a Z-test over the distribution of
outcomes from a probabilistic network, it is possible to estimate the
statistical significance of a given output and reject insignificant results.
The main difference between the proposed method and previous state-of-the-art
methods such as ODIN [8] is the non-restricted use of different architectures.
The proposed method can be applied to any architecture and improve the
performance when dealing with violation of the Closed-World Assumption by not
limiting the network to a specific loss function or activation function.

In their work, Geifman and El-Yaniv show that Softmax Response (SR) is a
simple yet top-performing method in selective classifiers [15] that
outperforms Monte Carlo (MC) dropout. However, this paper shows that if
utilized correctly, the probabilistic network can easily outperform the SR
method, making it a viable choice.

The main contributions of this paper are as follows:

  * •

Proposing a simple yet effective method (rejection based on the statistical
significance of probabilistic network output) to deal with the violation of
the Closed-World Assumption in classifiers. This method can be utilized in any
modern network architecture by changing the structure into a probabilistic
model, which is possible with the help of existing tools.

  * •

Testing the proposed method on state-of-the-art architecture (ResNet) with a
diverse set of distortions (blur, noise, gamma correction, and occlusion) to
show the effectiveness of the proposed method over the baseline SR method.

The rest of this paper is structured as follows. The details of the proposed
method are presented in Section 2. Then Section 3 deals with the experiments
and their results. Finally, Section 4 concludes the work and suggests
potential research directions for the future.

Fig. 1: The structure of the proposed method. (1) Pass the test image through
the probabilistic classifier. (2) Repeat it $n$ times and store the class
scores for each inference. (3) Calculate the mean and standard deviation for
each class. (4) Find the maximum mean value and label it as potential output.
(5) Run two-sample Z-tests between the potential output and all other classes,
then store the Z-scores. (6) Compare Z-scores with the threshold value to
decide the acceptance or rejection of the potential output.

##  2 Methods

###  2.1 Proposed method

The proposed method requires a fully trained probabilistic classifier to work.
Due to the nature of the probabilistic classifier, each inference of it will
result in a slightly different class score. To utilize this fact, first, the
test image is passed through the network $n$ times to get the mean and
standard deviation values for each class. After that, the maximum mean value
between classes is chosen as the potential output. Next, two-sample Z-tests
[16] are deployed between the potential output and all other classes to find
the statistical significance between their difference. Finally, if the
Z-scores indicate a significant difference, then the potential output is
chosen to be correct. Algorithm 1 summarizes these steps and Figure 1 shows
the structure of the proposed method.

0: A trained probabilistic classifier.

1: run the image through the classifier $n$ times

2: find mean ($\mu$) and std. dev. ($\sigma$) for all $N$ classes

3: find the class with the highest mean value ($\mathit{c}_{M}$)

4: for $i\in 1,2,\ldots,N;\ i\neq M$ do

5: run the two-sample Z-test between $\mathit{c}_{M}$ and $\mathit{c}_{i}$

6: store the $\mathit{Z}_{i}$ score

7: end for

8: if $\mathit{Z}_{i}>z$ for $i\in 1,2,\ldots,C;i\neq M$ then

9: set output to be $\mathit{C}_{M}$

10: else

11: set output to be Reject

12: end if

return output value for the image

Algorithm 1 Selective Probabilistic Classifier

####  2.1.1 Probabilistic Neural Network

A probabilistic neural network (PNN) classifier [17] uses a stochastic
weighting system. The classifier can allocate a class to an input sample by
utilizing the posterior probability, which means each run of the network will
result in a slightly different output. The amount of difference between
several runs is the key to network certainty. A low standard deviation between
several runs indicates a higher level of certainty for the network, making
standard deviation a suitable metric for selective classification. The
convolution layers for such a network are constructed based on Flipout [18].
The code can be found in the Tensorflow probability directory [19].

####  2.1.2 Two-Sample Z-test

A Z-test [20] refers to any statistical test that can approximate the
distribution of the hypothesis by a normal distribution. The two-sample Z-test
can be used to test whether two samples are similar to each other or not. The
formula is as follows:

$Z=\frac{\mu_{1}-\mu_{2}-\Delta}{\sqrt{\frac{\sigma_{1}^{2}}{n_{1}}+\frac{\sigma_{2}^{2}}{n_{2}}}}$

Where $\mu_{1}$ and $\mu_{2}$ are the mean values for two samples, $\Delta$ is
the hypothesized difference between the means (0 if testing for equality),
$\sigma_{1}$ and $\sigma_{2}$ are the standard deviations, and $n_{1}$ and
$n_{2}$ are the sample sizes (which are equal in this paper).

By setting the null hypothesis as $H_{0}:\mu_{1}=\mu_{2}$, the alternative
hypothesis as $H_{a}:\mu_{1}\neq\mu_{2}$, and $\Delta$ to zero, the two-sample
Z-test will result in a score that indicates the likelihood of two samples
being different from each other. A higher score means more likelihood for the
samples to be different. This score can be compared to critical values to get
the percentage for the likelihood of a significant difference between samples.
These values can be found in any Z-Score table, such as [21].

###  2.2 Softmax Response

The SR method applies a threshold directly to the output of the Softmax layer
from a deep neural network (DNN) and rejects any output below the threshold.
This method was chosen as the baseline for comparison. While the method is
simple, it is a known top-performer [15].

Fig. 2: Distortions on the image. (A) Original image. (B) Motion blur. (C)
Frosted glass blur. (D) Gaussian blur. (E) Noise. (F) Gamma darkening. (G)
Gamma lightening. (H) Occlusion.

##  3 Experiments and Results

The proposed method was experimented on with the well-known ResNet-18 network
configuration [22]. The goal is to show the performance of it in case of
violating the Closed-World Assumption. A comparison with the SR was made to
evaluate the performance. This comparison was based on the area under the
Receiver Operating Characteristic curve (ROC), which is threshold-independent.
Both networks are trained from scratch with the same initial configuration to
have a fair comparison. Other state-of-the-art methods were not included in
the comparison as they either require a specific structure for the model,
limiting the use case, or were only tested on more simple datasets such as
MNIST.

Multiple experiments were conducted to represent various violations of Closed-
World Assumption in real-world applications. In these experiments, the
classifiers are trained with a limited number of classes and presented with
both in-distribution and out-of-distribution samples. Further experiments also
distort the test samples to see the effect of each distortion on the
performance. The chosen distortions were based on [23]. Before discussing the
results, the dataset and distortions are explained in detail.

###  3.1 Dataset and Distortions

COCO — COCO [24] was chosen as the first dataset. It is a complex dataset
where the objects have various sizes, qualities, and overlaps. Since the COCO
is originally an object detection dataset, all instances were extracted from
it manually based on the bounding boxes provided in the dataset. The data was
separated into four classes: Human, Vehicle (containing 4-wheeled vehicles),
Animal (containing 4-legged animals), and Background (patches of images with
no overlapping objects). 260k images were used for training, excluding the
animal class, and 40k images were used as test samples. The reason behind
using a commonly known object detection dataset for classification is to have
a more realistic dataset where an external source does not filter the samples.

CIFAR — CIFAR [25] was chosen as the second dataset. It is a more
straightforward dataset where objects are classified into ten categories. The
dataset is small yet sufficiently complex, which makes it an ideal case for
testing algorithms. 40k images were used for training, excluding the
automobile and truck classes, and 10k images were used as test samples.

Blur — Three different blurring algorithms were used to see their effect on
the performance: Motion blur, Frosted glass blur, and Gaussian blur. The
effect of each algorithm can be seen in Figure 2(B-D). Each algorithm will
simulate a situation where the object is not sharp (e.g., the camera is not
focused, the object is moving, a semi-transparent object is between the camera
and the object)

Noise — Two different noises were added to test samples to see their effect on
the performance: Gaussian noise and Salt-and-pepper noise. The effect of a
sample noise can be seen in Figure 2(E). It will simulate a situation where
the input is noisy due to internal or external sources.

Gamma Correction — The gamma correction technique was applied to each test
sample to see the illumination effect on the performance. The effect of
darkening and lightening can be seen in Figure 2(F-G). It will simulate a
situation where the amount of light in the environment changes due to
environmental factors.

Occlusion — A black patch was added to test samples to see the effect of
occlusion on the performance. The effect of occlusion can be seen in Figure
2(H). It will simulate a situation where the object is partially visible.

Dataset | Method | Out | Blur | Noise | Gamma correction | Occlusion
---|---|---|---|---|---|---
of | Motion | Frosted | Gaussian | Gaussian | S&P | Darkening | Lightening
Distribution | glass
COCO | Proposed | 0.65 | 0.34 | 0.25 | 0.38 | 0.22 | 0.21 | 0.16 | 0.17 | 0.23
SR | 0.29 | 0.20 | 0.18 | 0.22 | 0.14 | 0.09 | 0.04 | 0.05 | 0.06
CIFAR | Proposed | 0.89 | 0.50 | 0.50 | 0.59 | 0.38 | 0.39 | 0.37 | 0.42 | 0.48
SR | 0.52 | 0.44 | 0.34 | 0.47 | 0.35 | 0.25 | 0.22 | 0.26 | 0.31
Table 1: AUROC values of the tests. The values are calculated by taking the
area under the ROC where the algorithm could produce a valid response.

###  3.2 Results

After conducting the tests, ROC curves were used to examine the effectiveness
of each algorithm. These curves can be seen in Figure 3-4. In general, each
point in the ROC curve corresponds to a specific threshold value for the
rejection option. If this threshold is set to 0, the algorithm will not reject
any input, resulting in a 100% FPR. The more extreme threshold values will
result in lower FPR and True Positive Ratio (TPR) until, at some point, the
algorithm rejects all inputs (0% FPR and TPR). The SR method hits this value
when the threshold is set to 1. As the output of Softmax cannot be larger than
1, any output will be rejected. However, since a DNN typically generates high
scores for the output, this threshold ends up preventing the SR algorithm from
reaching lower FPR rates. On the other hand, the proposed method does not rely
on the limit of Softmax output, as it compares the significance of each class
to the others. Such a limit will cause a significant gap in AUROC scores, as
seen in Table 1.

Judging by the ROC curves, both algorithms start roughly on the same point.
This means that both algorithms function similarly when it comes to
classification. However, the SR method has the mentioned drawback, which is
visible in the curves.

The comparison must be threshold-independent for it to be fair. Thus, the area
under the ROC curve (AUROC) was used as a comparison method. The area
calculation must consider the limitations of both algorithms. While the SR
algorithm can reach 0% FPR, it only happens when the threshold is at one (1)
or higher, which means the output is not valid. Thus, only the area under the
valid parts of the ROC curve was used in calculating the AUROC values. These
values can be found in Table 1.

Fig. 3: ROC curves for proposed method in COCO test. The worst performance of
each category was chosen to present the tolerance of the algorithm to extreme
distortions.

While every distortion reduces the performance, gamma correction has the most
significant effect, and blurring has an almost negligible effect on the
proposed method. It can be justified by how a classifier works, as changing
the intensity of the image makes it harder to separate the objects from the
Background class. That being said, the proposed algorithm still outperforms
the SR method by a notable margin.

Fig. 4: ROC curves for SR method in COCO test. The worst performance of each
category was chosen to present the tolerance of the algorithm to extreme
distortions.

##  4 Conclusion

In this paper, we propose a rejection option for probabilistic classifiers
based on Z-test analysis. This method will address the violation of the
Closed-World Assumption. By utilizing a probabilistic classifier, each run
results in a slightly different class score. A Z-test analyses the mean and
standard deviation values for multiple runs to estimate network certainty and
filter out uncertain results.

We designed several experiments based on a well-known network configuration
(ResNet-18) and datasets (COCO and CIFAR). A comparison with the SR method was
made based on AUROC as a threshold-independent metric. The proposed method was
shown to have better performance than the SR method by a notable margin while
maintaining robustness in the presence of distortions. This makes the proposed
method more suitable in safety applications.

In the future, we will consider expanding the method by merging it with
existing tools such as ODIN and covering more complex systems such as object
detection.

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Properly outer and strictly outer actions of finite groups on prime
C*-algebras

Costel Peligrad

Department of Mathematical Sciences, University of Cincinnati, PO Box 210025,
Cincinnati, OH 45221-0025, USA. E-mail address<EMAIL_ADDRESS>
Key words and phrases. dynamical system, algebra of local multipliers,
strictly outer actions.

2020 Mathematics Subject Classification. Primary 46L55, 46L40; Secondary
20F29.

ABSTRACT. An action of a compact, in particular finite group on a C*-algebra
is called properly outer if no automorphism of the group that is distinct from
identity is implemented by a unitary element of the algebra of local
multipliers of the C*-algebra. In this paper I define the notion of strictly
outer action (similar to the definition for von Neumann factors in [11]) and
prove that for finite groups and prime C*-algebras, it is equivalent to the
proper outerness of the action. For finite abelian groups this is equivalent
to other relevant properties of the action.

##  1 Introduction

An action $\alpha$ of a locally compact group or a quantum group, $G,$ on a
von Neumann factor $M$ is said to be strictly outer when the relative
commutant of the factor in the crossed product is trivial. The action is
called minimal if the inclusion $M^{\alpha}\subset M$ is irreducible, that is
the relative commutant $(M^{\alpha})^{\prime}\cap M$ of the fixed point
algebra $M^{\alpha}$ in $M$ is trivial. For integrable actions of quantum
groups on factors an action is minimal if and only if it is strictly outer
[11]. It can be shown that, in particular, for finite groups, an action
$\alpha$ on a factor $M$ is strictly outer if and only if it is properly outer
(that is, no $\alpha_{g},g\neq e$ is implemented by a unitary element of $M$).
In the next Section, I define strictly outer actions of finite groups on
C*-algebras and, in Section 3.1., prove that an action of a finite group, $G,$
on a prime C*-algebra, $A,$ is strictly outer if and only if it is properly
outer (that is, no $\alpha_{g},g\neq e,$ is implemented by a unitary element
in the algebra of local multipliers of $A$). In Section 3.2. I study actions
of finite abelian groups on prime C*-algebras and state some equivalent
conditions to proper and therefore strict outerness. These results extend and
complement some known results in the litterature ([4], [5], [7]). In [2]
similar results were obtained for compact abelian group actions on separable
prime C*-algebras with an additional condition. I do not require separability
of C*-algebras.

##  2 Definitions and preliminary results

If $A$ is a C*-algebra, denote by $\mathcal{I}$ the set of all closed
essential, two sided ideals of $A$. A two sided ideal $I\subset A$ is called
essential if $\left\\{a\in A:aI=(0)\right\\}=(0).$ If $I\subset J$ are two
such ideals then according to [7, Proposition 3.12.8.] we have the inclusion
$M(J)\subset M(I)$ where $M(I),$ respectively $M(J)$ are the multiplier
algebras of $I,$ respectively $J.$ The inductive limit of the system
$\left\\{M(I):I\in\mathcal{I}\right\\}$ is called the algebra of local
multipliers of $A$ and denoted $M_{loc}(A)$ [1]. Originally, this algebra was
denoted $M^{\infty}(A)$ and called the algebra of essential multipliers of $A$
[6]. Recall that a C*-algebra, $A,$ is called prime if every (two sided) ideal
of $A$ is essential.

In the next lemma I collect some known results

###### Lemma 1

Let $A$ be a C*-algebra. The following properties are equivalent
a) $A$ is prime.
b) The center of $M_{loc}(A)$ consists of scalars.
c) If $B$ is a C*-subalgebra of $M_{loc}(A)$ such that $A\subset B\subset
M_{loc}(A),$ then the center of $B$ consists of scalars.

Proof. The equivalence of a) and b) follows from [2, Proposition 3.1.] (see
also [6, page 303, statements 6), 4) and 5)]). The implication
b)$\Rightarrow$c) follows from [1, Lemma 3.2.2.]. and the implication
c)$\Rightarrow$b) is obvious.

In the rest of this paper we will consider a C*-dynamical system
$(A,G,\alpha)$ where $A$ is a C*-algebra, $G$ a finite (or compact) group and
$\alpha$ a faithful action of $G$ on $A,$ that is for each $g\in G,$
$\alpha_{g}$ is an automorphism of $A$ such that
$\alpha_{g_{1}g_{2}}=\alpha_{g_{1}}\alpha_{g_{2}}$ for all $g_{1},g_{2}\in G.$
We wil denote by $A^{\alpha}=\left\\{a\in A:\alpha_{g}(a)=a,g\in G\right\\}$
the fixed point algebra of the action. The action $\alpha$ is called faithful
if $\alpha_{g}\neq id$ for $g\neq e$ where $e$ is the unit of $G.$ If the
group $G$ is finite, the action $\alpha$ of $G$ on $A$ has a natural extension
to $M_{loc}(A).$

###### Lemma 2

Suppose $G$ is a compact group. Then,
a) Every approximate identity of $A^{\alpha}$ is an approximate identity of
$A,$
b) $M(A)^{\alpha}=M(A^{\alpha}).$

Proof.

a) follows from the more general case of compact quantum group actions [3,
Lemma 2.7.]. To prove b), notice first that $M(A)^{\alpha}\subset
M(A^{\alpha}),$ then, using a) it follows that $M(A^{\alpha})\subset M(A),$ so
$M(A^{\alpha})\subset M(A)^{\alpha}$.

###### Lemma 3

Let $(A,G,\alpha)$ be a C*-dynamical system with $G$ finite and $A$ prime.
Then,
a) $M_{loc}(A)^{\alpha}\subset M_{loc}(A^{\alpha}).$
b) If $x\in M_{loc}(A)$ commutes with $A^{\alpha}$ then $x$ commutes with
$M_{loc}(A)^{\alpha}.$
c) If $A^{\alpha}$ is prime, then the center of $M_{loc}(A)^{\alpha}$ consists
of scalars.
d) $A^{\prime}\cap M_{loc}(A)$ consists of scalars.

Proof. a) Since $G$ is finite, every essential ideal $J\subset A$ contains the
$\alpha$-invariant essential ideal $I=\cap_{g\in G}\alpha_{g}(J).$ Therefore,
$M_{loc}(A)$ is the inductive limit of $\left\\{M(I):I\subset A\text{
}\alpha\text{-invariant essential ideal}\right\\}.$If $I\subset A$ is a closed
two sided $\alpha$-invariant essential ideal of $A,$ it is obvious that
$I^{\alpha}$ is an essential ideal of $A^{\alpha}$. From Lemma 2 b) applied to
$I$ instead of $A$ it follows that $M(I)^{\alpha}=M(I^{\alpha})\subset
M_{loc}(A^{\alpha})$ and, therefore, $M_{loc}(A)^{\alpha}\subset
M_{loc}(A^{\alpha}).$
To prove b), let $I\subset A$ be an $\alpha$-invariant essential ideal. Then
$I^{\alpha}$ is ideal of $A^{\alpha}$ so $x$ commutes with $I^{\alpha}.$ If
$m\in M(I^{\alpha})$ and $a\in I^{\alpha}$ we have

$xam=amx.$

and

$xam=axm.$

Therefore

$a(xm-mx)=0.$

so $I^{\alpha}(xm-mx)=\left\\{0\right\\}.$ Applying Lemma 2 a) to $I$ instead
of $A$, it follows that $I(xm-mx)=\left\\{0\right\\}.$ From [1, Lemma
2.3.3.ii)] we get $xm-mx=0.$ Applying Lemma 2 b) to $I,$ we have
$M(I^{\alpha})=M(I)^{\alpha},$ so $x$ commutes with $M(I)^{\alpha}$ for every
$\alpha$-invariant essential ideal of $A.$ Therefore, $x$ commutes with
$\underline{lim}_{I}M(I)^{\alpha}=M_{loc}(A)^{\alpha}.$
c) If $A^{\alpha}$ is prime, from Lemma 1 b) applied to $A^{\alpha}$ instead
of $A$ it follows that the center of $M_{loc}(A^{\alpha})$ consists of
scalars. Applying a) it follows that $M_{loc}(A)^{\alpha}\subset
M_{loc}(A^{\alpha}).$ From Lemma 1 c) applied to $A^{\alpha}$ instead of $A$
it follows that the center of $M_{loc}(A)^{\alpha}$ consists of scalars.
d) Follows from part b) for trivial $\alpha$.

The cross product $A\times_{\alpha}G$ of $A$ by a finite group, $G,$ is
defined as the C*-completion of the algebra of all functions $f:G\rightarrow
A$ with the following operations:

$(f\cdot h)(g)=\sum_{p\in G}f(p)\alpha_{p}(h(p^{-1}g).$
$f^{\ast}(g)=\alpha_{g}(f(g^{-1})^{\ast}.$

As $G$ is a finite group, each $f\in A\times_{\alpha}G$ can be written $f=\sum
a_{g}\delta_{g}$ where $a_{g}\in A$ and $\delta_{g}$ is the function on $G$
with values in the multiplier algebra $M(A)$ of $A$ such that
$\delta_{g}(g)=1$ and $\delta_{g}(p)=0$ for $p\neq g.$

If $A$ is a C*-algebra then, $A\subset A^{\prime\prime}\subset B(H_{u})$ where
$A^{{}^{\prime\prime}}$ is enveloping von Neumann algebra of $A$ in $B(H_{u})$
and $H_{u}$ is the Hilbert space of the universal representation of $A$ [7,
Section 3.7.].Denote by $id$ the identity representation of $A\ $in
$B(H_{u}).$ If $G$ is a finite group, denote by $\lambda$ the left regular
representation of $G$ on $l^{2}(G)$. Denote by
($\widetilde{id},\lambda,l^{2}(G,H_{u})),$the covariant representation of
$(A,G,\alpha)$ on $l^{2}(G,H_{u}),$ where

$\widetilde{id}(a)(h)=\alpha_{h^{-1}}(a),\lambda_{g}(\xi)(h)=\xi(g^{-1}h).$

Consider the regular representation, $\widetilde{id}\times\lambda,$ of
$A\times_{\alpha}G$ induced by ($\widetilde{id},\lambda,l^{2}(G,H_{u})):$

$(\widetilde{id}\times\lambda)(f)(p)={\displaystyle\sum\limits_{g\in
G}}\alpha_{p^{-1}}(f(g))\xi(g^{-1}p).$

Then, $\widetilde{id}\times\lambda$ is a faithful non degenerate
representation of $A\times_{\alpha}G$ on $l^{2}(G,H_{u}$) [7, Section 7.7.].

###### Remark 4

$A\times_{\alpha}G\subset M(A)\times_{\alpha}G\subset
A^{{}^{\prime\prime}}\times_{\alpha}G$ where
$A^{{}^{\prime\prime}}\times_{\alpha}G$ is the W*-cross product of
$A^{{}^{\prime\prime}}$ by the double dual action $\alpha^{{}^{\prime\prime}}$
of $\alpha.$

I will also use the following

###### Lemma 5

Suppose $G\ $is finite. Then, $M_{loc}(A)\subset M_{loc}(A\times_{\alpha}G).$

Proof. According to the proof of Lemma 3. a), $M_{loc}(A)$ is the inductive
limit of $\left\\{M(I):I\subset A\text{ }\alpha\text{-invariant essential
ideal}\right\\}$. A standard argument, using that $G$ is finite, shows that
for every $\alpha$-invariant essential ideal $I\subset A,\ I\times_{\alpha}G$
is an essential ideal of $A\times_{\alpha}G.$ Since $M(I)\subset
M(I\times_{\alpha}G)\subset M_{loc}(A\times_{\alpha}G)$, it follows that
$M_{loc}(A)\subset M_{loc}(A\times_{\alpha}G).$

In the following definitions we assume that the group $G$ is finite.

###### Definition 6

a) If $(A,G,\alpha)$ is a C*-dynamical system with $A$ prime, the action
$\alpha$ is called properly outer if no $\alpha_{g},g\neq e$ is implemented by
a unitary element in $M_{loc}(A).$
b) The action is called strictly outer if $A^{\prime}\cap
M_{loc}(A\times_{\alpha}G)=\mathbb{C}I,$ where $A^{\prime}$ is the commutant
of $A.$

In [2, Proposition 3.2.] the authors prove that two other statements are
equivalent with the above Definition 6. part a) for separable C*-algebras. For
not necessarily separable C*-algebras, I use the statement given in the above
Definition 6. a). Actually, as proven in [2], the other two statements imply
the Definition 6. a) for not necessarily separable C*-algebras.

Now, let $(A,G,\alpha)$ be a C*-dynamical system with $G$ compact abelian.
Denote by $\widehat{G}$ the dual of $G.$ If $\gamma\in\widehat{G}$ let
$A_{\gamma}=\left\\{a\in A:\alpha_{g}(a)=\left\langle g,\gamma\right\rangle
a,g\in G\right\\}$ be the corresponding spectral subspace of $A.$ Denote by
$sp(\alpha)$ the Arveson spectrum of $\alpha$ and by $\Gamma(\alpha)$ the
Connes spectrum of $\alpha$:

$sp(\alpha)=\left\\{\gamma\in\widehat{G}:A_{\gamma}\neq\left\\{0\right\\}\right\\}.$

and, if $\mathcal{H}$ is the set of all non zero $\alpha$-invariant hereditary
subalgebras of $A$

$\Gamma(\alpha)=\cap\left\\{sp(\alpha|_{B}:B\in\mathcal{H}\right\\}.$

I will list some standard facts about the Arveson spectrum of actions of
compact abelian groups (in particular finite abelian groups) that will be used
in Section 3.2.:

1) The linear span of $\cup\left\\{A_{\gamma}:\gamma\in Sp(\alpha)\right\\}$
is dense in $A.$ Therefore, if $\left\langle t,\gamma\right\rangle=1$ for
every $\gamma\in Sp(\alpha),$ then $\alpha_{t}$ is the identity automorphism.

2) If $\gamma_{1},\gamma_{2}\in Sp(\alpha),$ then
$A_{\gamma_{1}}A_{\gamma_{2}}\subset A_{\gamma_{1}+\gamma_{2}}.$ Here,
$A_{\gamma_{1}}A_{\gamma_{2}}$ denotes the linear span of $\left\\{ab:a\in
A_{\gamma_{1}},b\in A_{\gamma_{2}}\right\\}.$

3) If $\gamma\in Sp(\alpha),$ $A_{\gamma}A_{\gamma}^{\ast}$ and
$A_{\gamma}^{\ast}A_{\gamma}$ are ideals of $A^{\alpha}$. Here
$A_{\gamma}A_{\gamma}^{\ast}$ denotes the linear span of
$\left\\{ab^{\ast}:a,b\in A_{\gamma}\right\\}.$

4) If $A^{\alpha}$ is prime, $Sp(\alpha)$ is a subgroup of $\widehat{G}.$This
follows from 2) and 3).

5) If $A^{\alpha}$ is prime and the action $\alpha$ is faithful (that is
$\alpha_{t}\neq id$ for $t\neq 0$), then $Sp(\alpha)=\widehat{G}$. This
follows from 4) and 1).

The following result was proven by Olesen for simple C*-algebras, $A$ and
actions for which $sp(\alpha)/\Gamma(\alpha)$ is compact in
$\widehat{G}/\Gamma(\alpha)$ in [4, Theorem 4.2.] (see also [7, Theorem
8.9.7.]). If ($A,G,\alpha)$ is a C*-dynamical system, $A$ is called $G$-prime
if every non zero $\alpha$-invariant ideal of $A$ is an essential ideal.

###### Proposition 7

Let ($A,G,\alpha)$ be a C*-dynamical system with $A$ $G$-prime and $G$ compact
abelian such that $\widehat{G}/\Gamma(\alpha)$ is finite (in particular $G$
finite group). If $t_{0}\in G$ is such that $\left\langle
t_{0},\gamma\right\rangle=1$ for every $\gamma\in\Gamma(\alpha)$ then there
exists an $\alpha$-invariant (essential) ideal $J\subset A$ and a unitary
operator $u\in M(J)^{\alpha}$ such that $\alpha_{t_{0}}=Adu.$

Proof.

Applying [7. Proposition 8.8.7.] for $A$ $G$-prime and $G$ compact with
$\widehat{G}/\Gamma(\alpha)$ finite, it follows that there exists a non zero
$\alpha-$invariant hereditary C*-subalgebra, $C\subset A$ such that
$sp(\alpha|_{C})=\Gamma(\alpha).$Therefore, $\alpha_{t_{0}}|_{C}=id.$ Let
$J=\overline{ACA}$ be the closed two sided ideal of $A$ generated by $C.$
Since $C$ is $\alpha$-invariant and $A$ is $G$-prime, it follows that $J$ is
an esseential $\alpha$-invariant ideal of $A.$ Using an approximate identity
of $A$ it follows that $C\subset J\subset J^{\prime\prime}$.

Let $p$ be the open projection corresponding to $C$ in
$A^{{}^{\prime\prime}}$. Then $p\in J^{\prime\prime}$. Denote by $1_{J}$ the
unit of $J^{{}^{\prime\prime}}$ ($1_{J}$ is the open projection in
$A^{{}^{\prime\prime}}\ $corresponding.to $J).$ Notice that the central
support $c(p)$ of $p$ in $J^{{}^{\prime\prime}}$ equals $1_{J}.$ Indeed if
$q\geqslant p$ is a central projection in $J^{{}^{\prime\prime}},$ then
$qJ^{{}^{\prime\prime}}$ is an ideal of $J"$ that contains $C,$ so $q=1_{J}$.

Applying [7, Lemma 8.9.1.] to $M=J^{{}^{\prime\prime}}$ and $w=p,$ it follows
that there exists a unique unitary $u\in J^{{}^{\prime\prime}}\ $such that
$up=p$ and
$\alpha_{t_{0}}^{\prime\prime}|_{J^{{}^{\prime\prime}}}=Adu|_{J^{\prime\prime}}.$
Since $\alpha_{t_{0}}^{{}^{\prime\prime}}(p)=p=upu^{\ast}$ we have $up=pu=p.$
Since
$\alpha_{t_{0}}^{{}^{\prime\prime}}|_{J^{{}^{{}^{\prime\prime}}\prime}}=Adu|_{J^{{}^{\prime\prime}}}$
and $M(J)\subset J^{{}^{\prime\prime}},$ it follows that
$\alpha_{t_{0}}|_{M(J)}=Adu|_{M(J)}.$ Also, since $G$ is abelian,
$\alpha_{t}(p)=p,$ for every $t\in G,$ and $u$ is unique with the listed
properties, it follows that, $\alpha_{t}(u)=u$ for every $t\in G.$

We will prove next that $u\in M(J).$ Let $J_{0}=\left\\{x\in J:ux\in
J\right\\}.$ Clearly $J_{0}$ is a closed right ideal of $J.$ Now, if $y\in J,$
and $x\in J_{0}$ we have $u\operatorname{y}x=u\operatorname{y}u^{\ast}ux$
=$\alpha_{t_{0}}(y)ux\in J,$ so $J_{0}$ is also a left ideal. Notice that
$C\subset J_{0}.$ Indeed, if $x\in C,$ we have $ux=upx=px=x\in C\subset J,$ so
$x\in J_{0}.$ Therefore, $C\subset J_{0}$ $\subset J$ . Since $J$ is the (two
sided) ideal generated by $C$ it follows that $J_{0}=J,$ so $u$ is a left
multiplier of $J.$ To prove that $u$ is also a right multiplier of $J$, notice
that if $x\in J$ we have $xu=uu^{\ast}xu=u\alpha_{t_{0}}^{-1}(x)\in J.$ Since
$M(A)\subset M(J)$ (in $M_{loc}(A)),$ it follows that $\alpha_{t_{0}}=Adu.$

###### Corollary 8

Let ($A,G,\alpha)$ be a C*-dynamical system with $A$ $G$-prime and $G$ finite
abelian. If $t_{0}\in G$ is such that $\left\langle
t_{0},\gamma\right\rangle=1$ for every $\gamma\in\Gamma(\alpha)$ then
$\alpha_{t_{0}}$ is inner in $M_{loc}(A)^{\alpha}.$

Proof.

Proposition 7 shows that $\alpha_{t_{0}}=Adu,u\in M(J)^{\alpha}\subset
M_{loc}(J)$. By [6, page 303, statement 3)], since $J$ is an essential ideal
of $A,$ $M_{loc}(J)=M_{loc}(A),$ so $\alpha_{t_{0}}$ is inner in
$M_{loc}(A)^{\alpha}.$

I will close this section with the following

###### Theorem 9

Let $(A,G,\alpha)$ be a C*-dynamical system with $G$ finite abelian, $\alpha$
faithful and $A$ $G$-prime. Then the following statements are equivalent:
a) $Sp$($\alpha)=\Gamma(\alpha).$
b) $A^{\alpha}$ is prime.
c) The center of $M_{loc}(A)^{\alpha}$ consists of scalars.
d) No $\alpha_{t}\neq id$ is implemented by a unitary in
$M_{loc}(A)^{\alpha}.$

Proof. $a)\Leftrightarrow b)$ This is [7 Theorem 8.10.4.].
$b)\Rightarrow c)$ Since $A^{\alpha}$ is prime, from Lemma 1,
a)$\Leftrightarrow$b$),$ with $A^{\alpha}$ instead of $A$ it follows that the
center of $M_{loc}(A^{\alpha})$ is $\mathbb{C}I.$ Applying Lemma 1,
b)$\Leftrightarrow$c) to $A^{\alpha}$ instead of $A$, it follows that the
center of $M_{loc}(A)^{\alpha}\subset M_{loc}(A^{\alpha})$ is $\mathbb{C}I.$
$c)\Rightarrow d)$ If for some $t_{0}\in G,$ $\alpha_{t_{0}}$ is implemented
by a unitary $u\in M_{loc}(A)^{\alpha},$ then $u$ commutes with $A^{\alpha}$.
From Lemma 3 b) it follows that $u\ $commutes with $M_{loc}(A)^{\alpha},$ so
$u$ belongs to the center of $M_{loc}(A)^{\alpha},$ hence $u$ is a scalar and
$\alpha_{t_{0}}=id.$
$d)\Rightarrow a)$ Suppose $Sp$($\alpha)\neq\Gamma(\alpha).$ Since
$\Gamma(\alpha)$ is a subgroup of $\widehat{G}$, there exists $t_{0}\in G$
such that $\left\langle t_{0},\gamma\right\rangle=1$ for every
$\gamma\in\Gamma(\alpha)$ and $\left\langle t_{0},\gamma\right\rangle\neq 1$
for some $\gamma\in Sp(\alpha).$ From Corollary 8 it follows that
$\alpha_{t_{0}}\neq id$ and $\alpha_{t_{0}}$ is implemented by a unitary in
$M_{loc}(A)^{\alpha},$ contradiction.

##  3 Outer and strictly outer actions of finite groups

###  3.1 Finite non abelian groups

###### Lemma 10

Let $(A,G,\alpha)$ be a C*-dynamical system with $G$ finite.
a) Then $M(A\times_{\alpha}G)=M(A)\times_{\alpha}G.$
b) If A is prime and $\alpha$ is properly outer, then
$M_{loc}(A\times_{\alpha}G)$ is isomorphic to $M_{loc}(A)\times_{\alpha}G.$

Proof. a) It is obvious that
($\widetilde{id}\times\lambda)(A\times_{\alpha}G)$ is included in the
W*-crossed product ($\widetilde{id}\times\lambda)(A^{"}\times_{\alpha^{"}}G).$
Since $\widetilde{id}\times\lambda$ is a faithful and non degenerate
representation of $A\times_{\alpha}G,$ the multiplier algebra
$M(A\times_{\alpha}G)$ is isometrically isomorphic with the algebra of
multipliers of ($\widetilde{id}\times\lambda)(A\times_{\alpha}G)$ in
(($\widetilde{id}\times\lambda)(A^{"}\times_{\alpha^{"}}G))\subset
B(l^{2}(G,H_{u}))$ [7, Prop. 3.12.3]. Let $b={\displaystyle\sum\limits_{g\in
G}}b_{g}\delta_{g}\in A^{"}\times_{\alpha^{"}}G$ be a multiplier of
$A\times_{\alpha}G.$ If $a\in A$ is arbitrary, $\widetilde{a}=a\delta_{e}\in
A\times_{\alpha}G,$ where $e$ is the neutral elementof $G$ and $\cdot$ is the
multiplication of $A^{"}\times_{\alpha^{"}}G,$ we have

$(b\cdot\widetilde{a})_{g}={\displaystyle\sum\limits_{p\in
G}}b_{p}\alpha_{p}(\widetilde{a}_{p^{-1}g})=b_{g}\alpha_{g}(a)\in A.$
$(\widetilde{a}\cdot b)_{g}={\displaystyle\sum\limits_{p\in
G}}\widetilde{a}_{p}\alpha_{p}(b_{p^{-1}g})=ab_{g}\in A.$

Therefore, $b_{g}\in M(A)$ for every $g\in G,$ so $b\in M(A)\times_{\alpha}G$.

b) Let $I$ be an $\alpha$-invariant essential ideal of $A.$ Then, a simple
computation shows that $I\times_{\alpha}G$ is an essential ideal of
$A\times_{\alpha}G$ [10, Proposition 2.2.]. By a),
$M(I\times_{\alpha}G)=M(I)\times_{\alpha}G.$ If $J$ is an essential ideal of
$A\times_{\alpha}G$ then there exists an $\alpha$-invariant essential ideal
$I$ of $A$ such that $I\times_{\alpha}G\subset J,$ so $M(J)\subset
M(I\times_{\alpha}G)=M(I)\times_{\alpha}G.$ To prove that such an ideal $I$
exists, notice that, since $A$ is prime and $\alpha$ properly outer, the
action $\alpha$ is, in particular, ”purely outer” in the sense used by Rieffel
in [10]. From [10 Theorem 1.1.] it follows that $I_{0}=J\cap
A\neq\left\\{0\right\\}.$ Since $J\subset A\times_{\alpha}G$ is essential it
follows that $I_{0}$ is essential in $A.$ Since $G$ is finite, it follows that
$I=\cap_{g\in G}\alpha_{g}(I_{0})\subset I_{0}$ is an essential
$\alpha$-invariant ideal of $A$ and $I\times_{\alpha}G\subset J.$ Therefore,
$M(J)\subset M(I\times_{\alpha}G)$. According to a),
$M(I\times_{\alpha}G)=M(I)\times_{\alpha}G.$ Therefore

$M(J)\subset M(I)\times_{\alpha}G\subset M_{loc}(A)\times_{\alpha}G.$

Hence

$M_{loc}(A\times_{\alpha}G)\subset M_{loc}(A)\times_{\alpha}G.$

On the other hand, by Lemma 5, $M_{loc}(A)\subset M_{loc}(A\times_{\alpha}G),$
so

$M_{loc}(A)\times_{\alpha}G\subset M_{loc}(A\times_{\alpha}G).$

###### Theorem 11

Let $(A,G,\alpha)$ be a C*-dynamical system with $G$ finite, $A$ prime and
$\alpha$ faithful. Then $\alpha$ is properly outer if and only if it is
strictly outer.

Proof. Suppose $\alpha$ is properly outer and let $v=\sum_{h\in
G}v(h)\delta_{h}\in A^{\prime}\cap
M_{loc}(A\times_{\alpha}G)(=M_{loc}(A)\times_{\alpha}G$ by Lemma 10) be a
unitary element. Then, for $a\in A$ denote $\widetilde{a}=a\delta_{e}\in
A\times_{\alpha}G\subset M_{loc}(A)\times_{\alpha}G$
($=M_{loc}(A\times_{\alpha}G)$ by Lemma 10.). We have

$v\widetilde{a}=\widetilde{a}v\text{ {in} }M_{loc}(A)\times_{\alpha}G\text{
{for all} }a\in A.$

Therefore

$\sum_{h}v(h)\alpha_{h}(\widetilde{a}(h^{-1}g))=\sum\widetilde{a}(h)\alpha_{h}(v(h^{-1}g)),g\in
G.$

So

$v(g)\alpha_{g}(a)=av(g)\text{ {for all} }g\in G,a\in A.\text{ \ }(\ast)$

We will prove next that if $v(g)\neq 0$ then $v(g)$ is a scalar multiple of a
unitary element in $M_{loc}(A).$ If we denote

$\widetilde{v}_{g_{0}}=v(g_{0})\delta_{g_{0}}\in M_{loc}(A)\times_{\alpha}G.$

then, since $v(g)\alpha_{g}(a)=av(g),g\in G$, we have
$\widetilde{v}_{g_{0}}\widetilde{a}=\widetilde{a}\widetilde{v}_{g_{0}}$ and
therefore
$\widetilde{v}_{g_{0}}^{\ast}\widetilde{a}=\widetilde{a}\widetilde{v}_{g_{0}}^{\ast}$
for all $g_{0}\in G,a\in A.$ Then,

$\widetilde{v}_{g_{0}}\widetilde{v}_{g_{0}}^{\ast},\widetilde{v}_{g_{0}}^{\ast}\widetilde{v}_{g_{0}}\in
A^{\prime}\cap\left(M_{loc}(A)\times_{\alpha}G\right).$

Standard calculations in $M_{loc}(A)\times_{\alpha}G$ show that
($\widetilde{v}_{g_{0}}\widetilde{v}_{g_{0}}^{\ast})=v(g_{0})^{\ast}v(g_{0})\delta_{e},$
and
($\widetilde{v}_{g_{0}}^{\ast}\widetilde{v}_{g_{0}})=\alpha_{g_{0}^{-1}}(v^{\ast}(g_{0})v(g_{0}))\delta_{e}$
so,

$\widetilde{v}_{g_{0}}\widetilde{v}_{g_{0}}^{\ast},\widetilde{v}_{g_{0}}^{\ast}\widetilde{v}_{g_{0}}\in
A^{\prime}\cap M_{loc}(A).$

Since A is prime, by Lemma 3 d) we have that
$v(g_{0})v(g_{0})^{\ast}\in\mathbb{C}1$ and
$v(g_{0})^{\ast}v(g_{0})\in\mathbb{C}1.$Therefore, there exists
$\omega\in\mathbb{C},$ $\omega\neq 0$ such that $\omega v(g_{0})$ is unitary.
We can assume that $\omega=1.$ By relation $(\ast$) above we hav

$\alpha_{g_{0}}(a)=v^{\ast}(g_{0})av(g_{0}),a\in A.$

Hence $\alpha_{g_{0}}$ is inner in $M_{loc}(A),$ so $g_{0}=e.$ Therefore
$v\in\mathbb{C}1$ in $M_{loc}(A)\times_{\alpha}G=M_{loc}(A\times_{\alpha}G)$
and we are done.

Now, suppose that $\alpha$ is strictly outer, i.e.
$\left(A\right)^{\prime}\cap M_{loc}(A\times_{\alpha}G)=\mathbb{C}I.$ Let
$g_{0}\in G$ with $\alpha_{g_{0}}(a)=v_{0}av_{0}^{\ast}$ for a unitary element
$v_{0}\in M_{loc}(A).$ I will show that $g_{0}=e$ where $e$ is the unit of
$G.$ If we denote $\widehat{v}_{0}=v_{0}\delta_{e}\in
M_{loc}(A)\times_{\alpha}G$ $(=M_{loc}(A\times_{\alpha}G)$ according to Lemma
10 b)) and $\widetilde{b}=b\delta_{e}\in A\times_{\alpha}G\subset
M_{loc}(A)\times_{\alpha}G$ for $b\in A,$ then

$\widetilde{\alpha_{g_{0}}(a)}=\widehat{v}_{0}\widetilde{a}\widehat{v}_{0}^{\ast},a\in
A.$

On the other hand, a standard calculation in $M_{loc}(A)\times_{\alpha}G$
gives

$\widetilde{\alpha_{g_{0}}(a)}=\delta_{g_{0}}\widetilde{a}\delta_{g_{0}^{-1}}.$

Therefore $\delta_{g_{0}^{-1}}\widehat{v}_{0}\in$ $\left(A\right)^{\prime}\cap
M_{loc}(A\times_{\alpha}G)=\mathbb{C}1,$ so
$\widehat{v}_{0}=\mu\delta_{g_{0}}$ for some scalar $\mu.$ Since the support
of $\widehat{v}_{0}\ $as a function on $G$ is $\left\\{e\right\\}$ and the
support of $\delta_{g_{0}}$ is $\left\\{g_{0}\right\\}$ it follows that
$g_{0}=e.$ Hence every $\alpha_{g_{0}}$ with g${}_{0}\neq e$ is properly
outer.

###### Remark 12

a) If $(A^{\alpha})^{\prime}\cap M_{loc}(A)=\mathbb{C}I,$ then every
automorphism $\alpha_{t},t\neq e$ is properly outer and therefore, the action
is strictly outer by the above theorem. Indeed if some $\alpha_{g_{0}}$ is
implemented by a unitary $u\in M_{loc}(A)$ then, clearly
$u\in(A^{\alpha})^{\prime}\cap M_{loc}(A),$ so $g_{0}=e.$
b) Suppose $A$ and $A\times_{\alpha}G$ are prime, $G$ compact (in particular
finite) and $\alpha$ faithful. If for every automorphism, $\beta$ of $A$ whose
restriction to $A^{\alpha}$ is the identity, there exists $g\in G$ such that
$\beta=\alpha_{g},$ then $(A^{\alpha})^{\prime}\cap M_{loc}(A)=\mathbb{C}I.$
Indeed, if $A$ and $A\times_{\alpha}G$ are prime and $\alpha$ is faithful, by
[8, Corollary 3.12] and [9, Lemma 2.2.] the hypotheses of [9, Theorem 2.3.]
are satisfied and the conclusion follows from that result.

In the next Section I will prove that, for finite abelian groups, the
condition $(A^{\alpha})^{\prime}\cap M_{loc}(A)=\mathbb{C}I$ is equivalent
with the action being properly outer, and therefore strictly outer.

###  3.2 Finite abelian groups

The proof of the following result is inspired by the proof of [5, Theorem 4]
for simple C*-algebras (see also [7, Proposition 8.10.13.]). The hypothesis
that $A$ is simple is essential in their result.

###### Proposition 13

Let $(A,G,\alpha)$ be a C*-dynamical system with $G$ finite abelian, $\alpha$
faithful and $A$ prime. Then $\left(A^{\alpha}\right)^{\prime}\cap
M_{loc}(A)=\mathbb{C}I$ if and only if for every $t\neq 0,$ $\alpha_{t}$ is a
properly outer automorphism.

Proof. Suppose $\left(A^{\alpha}\right)^{\prime}\cap M_{loc}(A)=\mathbb{C}I$.
If $\alpha_{t_{0}}$ is implemented by a unitary $u_{t_{0}}$ in $M_{loc}(A)$,
then, clearly, $u_{t_{9}}\in$ $\left(A^{\alpha}\right)^{\prime}\cap
M_{loc}(A)=\mathbb{C}I.$ Conversely suppose that $\alpha_{t}$ is properly
outer for every $t\in G,t\neq 0.$ According to Theorem 9, the center of
$M_{loc}(A)^{\alpha}$ is trivial.

Let $B=\left(A^{\alpha}\right)^{\prime}\cap M_{loc}(A).$ Since the center of
$M_{loc}(A)^{\alpha}$ is trivial, from Lemma 3 b) it follows that the fixed
point algebra of the action $\alpha$ restricted to the $\alpha$-invariant
subalgebra $B$ is trivial. Suppose that $B\neq\mathbb{C}I.$ Then, there exists
$\gamma_{0}\in\widehat{G}$ such that $B_{\gamma_{0}}\neq\left\\{0\right\\}.$
If $x,y\in B_{\gamma_{0}},$ then $x^{\ast}y\in\mathbb{C}I$ (the fixed point
algebra of the restriction of $\alpha$ to $B$). Therefore there exists a
unitary, $u\in B_{\gamma_{0}}$, such that $B_{\gamma_{0}}=\mathbb{C}u.$ Since
$u$ is unitary, it follows that $n\gamma_{0}\in Sp(\alpha|_{B})$ for every
$n\in\mathbb{Z},$ so $B_{n\gamma_{0}}=\mathbb{C}u^{n}$ for every
$n\in\mathbb{Z}.$

Let $H=\left\\{t\in G:\left\langle t,\gamma_{0}\right\rangle=1\right\\}.$
Then, since $\alpha_{t}(u)=\left\langle t,\gamma_{0}\right\rangle u$ for every
$t\in G,$ it follows that $\alpha_{t}(u)=u$ for all $t\in H.$ Notice that the
annihilator, $H^{\perp},$ of $H$ in $\widehat{G}$ equals
$\mathbb{Z}\gamma_{0}.$ Now, consider the C*-subalgebra

$M_{loc}(A)^{H}=\left\\{x\in M_{loc}(A):\alpha_{t}(x)=x\text{ for all }t\in
H\right\\}.$

By [7, 8.10.3.], if $\widetilde{\alpha}$ is the action of $G/H$ on
$M_{loc}(A)^{H},$ we have $Sp(\widetilde{\alpha})\subset\mathbb{Z}\gamma_{0}.$
Therefore, the spectral subspaces $(M_{loc}(A)^{H})_{n\gamma_{0}}$ have dense
span in $M_{loc}(A)^{H}.$ Next, we will prove that $u$ commutes with
$(M_{loc}(A)^{H})_{n\gamma_{0}},n\in\mathbb{Z}$ and thus $u$ belongs to the
center of $M_{loc}(A)^{H}.$ Let
$x\in(M_{loc}(A)^{H})_{n\gamma_{0}}\subset(M_{loc}(A))_{n\gamma_{0}}$ Then

$x=x(u^{\ast})^{n}u^{n}\in(M_{loc}(A))^{\alpha}u^{n}.$

On the other hand, since $u\in B=(A^{\alpha})^{\prime}\cap M_{loc}(A),$ in
particular $u\in(A^{\alpha})^{\prime}.$ applying Lemma 3 b) again, it follows
that $u$ commutes with $M_{loc}(A)^{\alpha}$ , so $u$ commutes with $x.$ Hence
$u$ belongs to the center of $M_{loc}(A)^{H}.$ Thus the center of
$M_{loc}(A)^{H}$ contains a non scalar element. Since, by hypothesis, $A$ is
prime, in particular $H$-prime, applying Theorem 9 for $G=H$, it follows that
there exists $t_{0}\in H$ such that $\alpha_{t_{0}}$ is implemented by a
unitary in $M_{loc}(A)^{H},$ thus $\alpha_{t_{0}}$ is not properly outer,
which is a contradiction.

As stated before, all the results obtained in this paper, hold for not
necessarily separable C*-algebras. The next theorem was considered in [2] for
compact groups and separable C*-algebras. In addition to separability, in the
equivalence of a) and b) below the authors make the assumption that
$A^{\alpha}$ is prime.

###### Theorem 14

Let $(A,G,\alpha)$ be a C*-dynamical system with $A$ prime, $G$ finite abelian
and $\alpha$ faithful. The following conditions are equivalent
a) $\alpha_{g}$ is properly outer for every $g\in G,g\neq 0.$
b) $(A^{\alpha})^{\prime}\cap M_{loc}(A)=\mathbb{C}I.$
c) $(A)^{\prime}\cap M_{loc}(A\times_{\alpha}G)=\mathbb{C}I.$
d) $A^{\alpha}$ is prime and $\widehat{\alpha}_{\gamma}$ is properly outer (in
$M_{loc}(A\times_{\alpha}G))$ for every $\gamma\in\widehat{G},\gamma\neq 0.$

Proof.

The equivalence of a) and b) follows from Proposition 13. The equivalence of
a) and c) follows from Theorem 11 in the particular case of abelian groups.

c)$\Rightarrow$d), Notice first that from c) it follows that the center of
$M_{loc}(A\times_{\alpha}G)$ consists of scalars. Therefore, by Lemma 1,
$A\times_{\alpha}G$ is prime, so, by [8, Corollkary 3.13], $A^{\alpha}$ is
prime. If for some $\gamma\in\widehat{G},\widehat{\alpha}_{\gamma}$ is
implemented by a unitary $u$ in $M_{loc}(A\times_{\alpha}G),$ then, as
$(A\times_{\alpha}G)^{\widehat{\alpha}}=A,$ it follows that
$u\in(A)^{\prime}\cap M_{loc}(A\times_{\alpha}G)=\mathbb{C}I.$ Hence
$\gamma=0.$

d)$\Rightarrow$c) If $A^{\alpha}$ is prime and $\alpha$ is faithful, from
property 5) in the list of properties of $Sp(\alpha)$ discussed after
Definition 6, Section 2, it follows that $Sp(\alpha)=\widehat{G}.$ Applying [8
Cor 3.13.] it follows that $A\times_{\alpha}G$ is prime. Therefore
$A\times_{\alpha}G$ is prime and $\widehat{\alpha}_{\gamma}$ is properly outer
(in $M_{loc}(A\times_{\alpha}G))$ for every $\gamma\in\widehat{G},\gamma\neq
0.$ Since $(A\times_{\alpha}G)^{\widehat{\alpha}}=A,$ from Proposition 13
applied to the C*-dynamical system
$(A\times_{\alpha}G,\widehat{G},\widehat{\alpha})$ (instead of $(A,G,\alpha)$)
it follows that $(A)^{\prime}\cap M_{loc}(A\times_{\alpha}G)=\mathbb{C}I.$

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