11institutetext: Kyushu University, Fukuoka, Japan,
11email<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
# Meta-learning of Pooling Layers
for Character Recognition††thanks: We provide our implementation at
https://github.com/Otsuzuki/Meta-learning-of-Pooling-Layers-for-Character-
Recognition.

Takato Otsuzuki 11 Heon Song 11 Seiichi Uchida 11 0000-0001-8592-7566
Hideaki Hayashi (✉) 11 0000-0002-4800-1761

###### Abstract

In convolutional neural network-based character recognition, pooling layers
play an important role in dimensionality reduction and deformation
compensation. However, their kernel shapes and pooling operations are
empirically predetermined; typically, a fixed-size square kernel shape and max
pooling operation are used. In this paper, we propose a meta-learning
framework for pooling layers. As part of our framework, a parameterized
pooling layer is proposed in which the kernel shape and pooling operation are
trainable using two parameters, thereby allowing flexible pooling of the input
data. We also propose a meta-learning algorithm for the parameterized pooling
layer, which allows us to acquire a suitable pooling layer across multiple
tasks. In the experiment, we applied the proposed meta-learning framework to
character recognition tasks. The results demonstrate that a pooling layer that
is suitable across character recognition tasks was obtained via meta-learning,
and the obtained pooling layer improved the performance of the model in both
few-shot character recognition and noisy image recognition tasks.

###### Keywords:

Convolutional neural network Pooling layer Meta-learning Character recognition
Few-shot learning.

##  1 Introduction

In convolutional neural network (CNN)-based character recognition, pooling
layers play an important role in dimensionality reduction and deformation
compensation. In particular, the max and average pooling layers, as
illustrated in Figs. 1(a) and (b), respectively, are widely used in CNNs.
These pooling operations are effective in absorbing the deformations that
occur in character images. Even if the convolutional features undergo local
changes owing to the deformation of the character image, the dimensionally-
reduced feature map is invariant to such changes. As a result, a CNN with
pooling layers is robust to character image deformation.

Figure 1: Meta-learning of pooling and comparison with other pooling methods:
(a) max pooling, (b) average pooling, and (c) meta-learned pooling (ours). The
matrices in the top and bottom rows represent feature maps before and after
pooling processing, respectively, where the number in each cell corresponds to
the feature value in each pixel. The colored squares represent kernels, and
their colors correspond to the type of operations. In (a) and (b), the maximum
and average values are calculated for each kernel. Moreover, all kernels share
the same shape and operation. In (c), the proposed meta-learned pooling
employs different shapes and operations in individual kernels.

Nevertheless, typical pooling layers have a limitation in that the kernel
shape and pooling operation should be determined empirically and manually. In
the pooling layers, pooling operations are performed by sliding a fixed-size
kernel, and the kernel size is predetermined as a hyperparameter. In two-
dimensional pooling, the shape of the kernel is fixed to a square generally.
Although various pooling operations have been proposed [20, 12, 25, 32, 23],
the type of pooling operation is chosen manually, and the same operation is
performed in the layer.

The purpose of this study is to determine a pooling layer suitable for
character recognition tasks in a data-driven approach. Although max and
average pooling layers with fixed-size square kernels are generally used in
tasks using character images, it is unknown whether this is in fact
appropriate. If we can obtain a suitable pooling layer across multiple tasks
using character images, we can apply it to subsequent new tasks using
character images, thereby leading to improvements in recognition accuracy.

For this purpose, we utilize meta-learning. Meta-learning aims to improve the
learning performance of a machine learning model by providing datasets of
multiple tasks and then acquiring knowledge shared among the tasks.
Specifically, hyperparameters, such as a learning rate and initial weights,
that are suitable across all the tasks are obtained via meta-learning. As a
result, the meta-learned model can improve its learning efficiency when a new
task is given.

In this paper, we propose a meta-learning framework for pooling layers. As
part of the framework, a parameterized pooling layer is proposed in which the
kernel shape and pooling operation can be trained using two parameters,
thereby allowing flexible pooling of the data. We also propose a meta-learning
algorithm for the parameterized pooling layer, which allows us to acquire a
suitable pooling layer across multiple tasks.

Fig. 1 illustrates an example of a pooling layer obtained under our framework,
alongside its comparison with max and average pooling layers. In the
traditional max and average pooling layers shown in Figs. 1(a) and (b),
respectively, fixed-size square kernels are used, and the same operation is
performed in each kernel. In contrast, in the pooling layer meta-learned in
our framework, flexible kernel shapes are used. Furthermore, the operation in
each kernel is defined based on $L_{p}$ pooling [27], which means various
types of operations, including max and average pooling, can be realized
depending on the value of $p$.

In the experiments, we reveal what pooling layer is suitable for character
recognition tasks in a data-driven approach and demonstrate that the obtained
pooling layer improves the learning performance of a CNN for a subsequent new
task. Fig. 2 shows the flow of the proposed method applied to the meta-
learning of character image recognition. First, we prepare a meta-dataset
containing a large number of character recognition tasks and apply the
proposed meta-learning framework to a pooling layer in a CNN, as shown Fig.
2(a). In this step, a pooling layer suitable for multiple character
recognition tasks can be obtained. Then, the CNN with the meta-learned pooling
layer is adapted to a new task (Fig. 2(b)).

Figure 2: Meta-learning flow. The flow of the proposed framework is divided
into two steps: (a) First, the kernel shapes and pooling operations of the
parameterized pooling layer are trained via meta-learning. By learning the
pooling layer from a large number of tasks, we can acquire knowledge across
all tasks. At this time, the parameters of the other layers are updated once
for each task batch, and their gradients are used in the meta-learning of the
parameterized pooling layer, whereas the updates of the other layers
themselves are not used in the subsequent step. (b) Then, the meta-learned
pooling layer is adapted to a new task. In this step, with the meta-learned
pooling fixed, the parameters of the remaining layers are trained using the
training dataset of the new task. Next, with all parameters fixed, the trained
model is adapted to the new task by classifying the test data.

The main contributions of this study can be summarized as follows:

  * •

We propose a meta-learning framework for pooling layers. As part of the
proposed framework, a parameterized pooling layer is proposed to make the
kernel shape and pooling operation trainable. A meta-learning algorithm for
the parameterized pooling layer is also presented.

  * •

We reveal a pooling layer suitable for character recognition tasks in a data-
driven manner. We prepare a large number of character image recognition tasks
using the Omniglot dataset [17] and then apply the proposed meta-learning
framework to these tasks. After meta-learning, we analyze the obtained kernel
shapes and pooling operations by visualizing the meta-parameters of the
parameterized pooling layer. The results demonstrate that the meta-learned
pooling layer applies max pooling around the character region in an annular
shape.

  * •

We demonstrate the effectiveness of the meta-learned pooling layer for few-
shot character image recognition and noisy character image recognition tasks.
The meta-learned pooling layer is effective in improving accuracy for one-shot
character image recognition compared to max and average pooling layers. The
robustness to noisy images is also remarkably improved by the proposed method.

##  2 Related Work

###  2.1 Pooling Layers

In recent years, many pooling methods have been proposed and have demonstrated
remarkable performance in specific tasks. Gao et al. [10] proposed a global
second-order pooling (GSoP) block which can be used to calculate the second-
order information in the middle layer of a CNN. Although methods that utilize
the mean and variance of feature maps in the last layer have been proposed
[19, 9, 4, 31, 18], the GSoP block is the first method that can be plugged in
at any location of a CNN. Van et al. [22] proposed a novel scene text proposal
technique, which iteratively applies a pooling operation to the edge feature
extracted from an image, thereby improving the accuracy of scene text box
detection. Hou et al. [13] proposed a strip pooling module, which considers a
long but narrow kernel. They demonstrated that short-range and long-range
dependencies among different locations can be captured simultaneously by
combining a strip pooling module with a pyramid pooling module (PPM) [34] in
semantic segmentation tasks.

Instead of utilizing a specific pooling operation, some trials have been aimed
to generalize pooling operations for global use. A representative example is
$L_{p}$ pooling [27, 7], which calculates an $L_{p}$-norm over the input
elements. Consequently, $L_{p}$ pooling involves average pooling when $p=1$
and max pooling when $p\rightarrow\infty$, and it can perform pooling
operations between average and max pooling using $1<p<\infty$ without
completely eliminating the influence of input elements with small values.
Another example is generalized mean pooling (GeM) [5, 29, 2], which is a
generalization of average pooling and consequently has a similar operation to
$L_{p}$ pooling. A pooling method that can automatically enhance important
features by learning has also been proposed [33, 11].

The proposed method is inspired by the attempts to generalize pooling
operations. In particular, the formulation of the proposed method is an
expansion of $L_{p}$ pooling. The proposed method differs from $L_{p}$ pooling
in that the kernel shape is trainable. Another difference from conventional
trainable pooling methods is that the proposed method learns suitable pooling
across multiple tasks, instead of a single task, based on meta-learning.

###  2.2 Meta-learning

Meta-learning aims to learn knowledge shared across multiple tasks to adapt
the model to another subsequent task given with a small amount of training
data or few learning steps. According to Baik et al. [1], meta-learning
algorithms can be divided into three categories: network-based, metric-based,
and optimization-based. Network-based methods learn a fast adaptation strategy
using auxiliary networks [26, 21]. Metric-based methods learn the
relationships between inputs in the task spaces and acquire object metrics or
distance functions to facilitate problem solving [3, 30, 28]. Optimization-
based methods adjust the optimization algorithm so that the model can be
adapted to several tasks [8, 1, 24, 35].

Among existing optimization-based meta-learning algorithms, model-agnostic
meta-learning (MAML) [8] has attracted much attention because of its
simplicity and generality. MAML encodes prior knowledge so that the model can
perform well across tasks and learn new tasks rapidly. Some studies have aimed
to improve the performance of MAML while maintaining simplicity and
generality. Rusu et al. [24] proposed an approach that learns an embedding of
the model parameters into a low-dimensional space and conducts meta-learning
in that space, achieving more efficient adaptation. Baik et al. [1] improved
the MAML-based framework by attenuating the conflicts among tasks and layers,
thereby improving its performance in few-shot learning. Some researchers have
attempted to extend MAML. Zhou et al. [35] introduced a method called meta-
learning symmetries by reparameterization (MSR) for meta-learning neural
network architectures. They showed that MSR can learn symmetries shared among
tasks from data based on meta-learning and reparameterization of network
weights.

Although the algorithm of the proposed method is inspired by MAML and MSR, the
proposed algorithm differs from the above optimization-based meta-learning
algorithms in that meta-learning is applied to the pooling layer while
considering the gradients of other layers. To the best of our knowledge, this
is the first attempt in adopting meta-learning in the pooling layer. Another
unique feature of the proposed algorithm is that the kernel shape matrix is
learned with a constraint to be binary.

##  3 Meta-learning of Pooling Layers

In this section, we present our meta-learning framework for pooling layers. To
make pooling layers trainable, we first introduce a parameterized pooling
layer, wherein the shape and operation of each kernel are trained
individually. We then describe the meta-learning algorithm of the
parameterized pooling layer.

###  3.1 Parameterized Pooling Layer

The purpose of the proposed method is to acquire the knowledge shared across
multiple tasks, thereby developing a pooling layer that can be efficiently
applied to new tasks. By parameterizing the pooling layer and making it
trainable, flexible determination of kernel shapes and pooling operations can
be achieved.

The parameterized pooling layer has two (meta-)parameters that are trained via
meta-learning. One is a kernel shape matrix, $\bm{W}$, with binary elements
that determines the area in which the pooling operation is applied. By
multiplying $\bm{W}$ with the input vector/image, various types of kernel
locations and shapes can be realized, including the sliding square kernels of
ordinary pooling. The other is an operation parameter, $\bm{p}$. The
parameterized pooling layer is based on $L_{p}$ pooling, where the pooling
operation in each kernel is defined by the $L_{p}$ norm. Incidentally, $L_{p}$
pooling is a generalization of pooling operations that includes max pooling
(when $p\rightarrow\infty$) and average pooling (when $p=1$). Whereas the
value of $p$ in $L_{p}$ pooling is determined empirically, the value of
$\bm{p}$ in the proposed method is automatically determined via meta-learning.

Given an input vector $\bm{x}\in\mathbb{R}^{J}_{+}$, the parameterized pooling
layer $\bm{f}:\mathbb{R}_{+}^{J}\mapsto\mathbb{R}^{I}$ ($J\geq I$) is defined
as follows:

$f_{i}(\bm{x})=\left(\frac{1}{J}\sum_{j=1}^{J}{W_{ij}x_{j}^{p_{i}}}\right)^{\frac{1}{p_{i}}},\quad
i=1,\ldots I,$ (1)

where $\bm{W}\in\\{0,1\\}^{I\times J}$ is the kernel shape matrix,
$\bm{p}\in\mathbb{R}_{+}^{I}$ is the operation parameter, and $J$ and $I$ are
the input and output dimensions, respectively. Although the assumption that
the input elements are positive is relatively strong, in practice, it can be
satisfied by applying a rectified linear unit (ReLU) activation function to
the previous layer.

The parameterized pooling layer can also be applied to an image input, as with
conventional pooling layers. One simple method is to vectorize the image input
by raster scanning before inputting the data to the parameterized pooling
layer. Another method is to combine equation (1) and sliding windows. Taking a
small window in the input image, we vectorize the pixels in the window and
prepare independent $\bm{W}$ and $\bm{p}$ for each window. Then, we apply
equation (1) to the entire image while shifting the window across the image.
In practice, the latter method is computationally advantageous; therefore, we
employed it in our experiments using image inputs, as reported in Sections 4
and 5.

To maintain the binariness of $\bm{W}$ and positiveness of $\bm{p}$, we employ
the following variable transformation:

$\displaystyle W_{ij}$
$\displaystyle=\begin{cases}\mathrm{Sigmoid}((\tilde{W}_{ij}-0.5)/T)&\text{during
meta-learning}\\\
\mathrm{Step}(\tilde{W}_{ij}-0.5)&\text{otherwise}\end{cases},$ (2)
$\displaystyle p_{i}$ $\displaystyle=\exp{(\tilde{p}_{i})},$ (3)

where $\mathrm{Sigmoid}(\cdot)$ is the sigmoid function, $T$ is a temperature
parameter, $\mathrm{Step}(\cdot)$ is the unit step function, which returns 0
for a negative input and 1 otherwise, and $\tilde{W}_{ij}\in\mathbb{R}$ and
$\tilde{p}_{i}\in\mathbb{R}$ are auxiliary variables. The step function is
used to make $\bm{W}$ binary, but it is not differentiable; therefore, the
sigmoid function is used as an approximation during meta-learning to enable
gradient calculation. For $\bm{p}$, the exponential function is used to
satisfy $p_{i}>0$. In the meta-learning algorithm described below, the
gradients are calculated with respect to the auxiliary variables
$\tilde{W}_{ij}$ and $\tilde{p}_{i}$. However, in the following sections, we
will explain as if we deal directly with $W_{ij}$ and $p_{i}$ instead of
$\tilde{W}_{ij}$ and $\tilde{p}_{i}$ for simplicity.

###  3.2 Meta-learning Algorithm

The aim of the meta-learning of pooling is to learn and exploit a pooling
layer that is suitable for multiple tasks. Given a task distribution
$\rho\bigl{(}\tau\bigl{)}$, we assume that there is an optimal pooling layer
that is shared across $\rho\bigl{(}\tau\bigl{)}$, and we estimate it by
learning meta-parameters $\bm{W}$ and $\bm{p}$ through meta-learning.

Algorithm 1 Meta-learning of the parameterized pooling layer

Inputs: $\\{\tau_{j}\\}^{N}_{j=1}\sim\rho\bigl{(}\tau\bigl{)}$: Meta-learning
tasks

Inputs: {$\bm{W}$, $\bm{p}$}: Randomly or arbitrarily initialized meta-
parameters

Inputs: $\bm{\theta}$: Randomly initialized weights of other layers

Inputs: $\alpha$, $\eta$: Learning rates for the inner and outer loops

while $not\ done$ do

Sample minibatch
$\\{\tau^{\prime}_{i}\\}^{n}_{i=1}\sim\\{\tau_{j}\\}^{N}_{j=1}$

for $\tau^{\prime}_{i}\in\\{\tau^{\prime}_{i}\\}^{n}_{i=1}$ do

$\\{D^{\mathrm{tr}}_{i},D^{\mathrm{val}}_{i}\\}\leftarrow\ \tau^{\prime}_{i}$;
// task data

$\bm{\delta}_{i}\leftarrow\
\nabla_{\bm{\theta}}\mathcal{L}\bigl{(}\bm{W},\bm{p},\bm{\theta},D^{\mathrm{tr}}_{i}\bigl{)}$;

$\bm{\theta}^{\prime}\leftarrow\ \bm{\theta}-\alpha\bm{\delta}_{i}$; // inner
step

/* outer gradient */

$\bm{G}_{i}^{\mathrm{w}}\leftarrow\frac{\mathrm{d}}{\mathrm{d}\bm{W}}\mathcal{L}\bigl{(}\bm{W},\bm{p},\bm{\theta}^{\prime},D^{\mathrm{val}}_{i}\bigl{)}$,
$\bm{G}_{i}^{\mathrm{p}}\leftarrow\nabla_{\bm{p}}\mathcal{L}\bigl{(}\bm{W},\bm{p},\bm{\theta}^{\prime},D^{\mathrm{val}}_{i}\bigl{)}$;

end for

$\bm{W}\leftarrow\bm{W}-\eta\sum_{i}\bm{G}_{i}^{\mathrm{w}}$,
$\bm{p}\leftarrow\bm{p}-\eta\sum_{i}\bm{G}_{i}^{\mathrm{p}};$

end while

Algorithm 1 shows the meta-learning algorithm of the parameterized pooling
layer. The algorithm is a gradient-based meta-learning algorithm similar to
MSR [35] and MAML [8] algorithms. Assuming that we have a neural network with
a parameterized pooling layer as well as other layers with weights
$\bm{\theta}$, the meta-parameters $\bm{W}$ and $\bm{p}$ are updated during
the meta-learning algorithm. Note that even if there are multiple
parameterized pooling layers in a neural network, we can independently apply
this algorithm to each one.

This algorithm consists of inner and outer loops. The inner loop calculates
gradients and is nested in the outer loop, which updates the meta-parameters.
In the outer loop, a task minibatch $\\{\tau^{\prime}_{i}\\}^{n}_{i=1}$ is
first sampled from the task set $\\{\tau_{j}\\}^{N}_{j=1}$, where $n$ and $N$
are the minibatch size and number of tasks, respectively. The task minibatch
is then split into training and validation data
$\\{D^{\textrm{tr}}_{i},D^{\textrm{val}}_{i}\\}$ for any $\tau^{\prime}_{i}$.
Proceeding to the inner loop, the parameters of other layers $\bm{\theta}$ are
updated as
$\bm{\theta}^{\prime}\leftarrow\bm{\theta}-\alpha\nabla_{\bm{\theta}}\mathcal{L}\bigl{(}\bm{W},\bm{p},\bm{\theta},D^{\mathrm{tr}}_{i}\bigl{)}$,
where $\mathcal{L}(\cdot)$ is the loss function and $\alpha$ is the learning
rate for the inner loop. After calculating the gradients with respect to the
meta-parameters using the updated $\bm{\theta}^{\prime}$ and validation data,
the meta-parameters are updated in the outer loop as follows:

$\displaystyle\bm{W}$
$\displaystyle\leftarrow\bm{W}-\eta\sum_{i}\frac{\mathrm{d}}{\mathrm{d}\bm{W}}\mathcal{L}\bigl{(}\bm{W},\bm{p},\bm{\theta}^{\prime},D^{\mathrm{val}}_{i}\bigl{)},$
(4) $\displaystyle\bm{p}$
$\displaystyle\leftarrow\bm{p}-\eta\sum_{i}\nabla_{\bm{p}}\mathcal{L}\bigl{(}\bm{W},\bm{p},\bm{\theta}^{\prime},D^{\mathrm{val}}_{i}\bigl{)},$
(5)

where $\eta$ is the learning rate for the outer loop.

##  4 Experiment on Artificial Data

To verify the validity of the proposed method, we conducted an experiment
using artificially generated data. The goal of this experiment is to verify
whether the proposed method can meta-learn the kernel shapes and pooling
operations from a task set that is generated by passing random vectors or
images through a certain pooling layer.

Fig. 3 outlines the data generation process for this experiment. We prepared
datasets for one-dimensional and two-dimensional cases. For the one-
dimensional case (Fig. 3(a)), we generated sets of inputs and outputs by
passing random input vectors, whose elements were generated from a uniform
distribution over [0, 1], through a 1D pooling layer with a filter size of $2$
and a stride of $2$. For the two-dimensional case (Fig. 3(b)), we generated
sets of inputs and outputs by passing a random image, in which each pixel
value was generated from a uniform distribution over [0, 1], through a 2D
pooling layer with a filter size of $2\times 1$ (vertical rectangle) and a
stride of $2$. Here, the pooling layer consisted of max pooling for the first
half and average pooling for the second half. Each task contains 20 sets of
input and output vectors/images, and we generated $8{,}000$ training tasks. We
set the dimensions of the input and output data to $60$ and $30$ in the one-
dimensional case, and $28\times 28$ and $14\times 14$ in the two-dimensional
case, respectively.

Figure 3: Data generation process for the experiment using artificial data.
(a) In the one-dimensional case, a set of input and output data was generated
by passing a random vector through a 1D pooling layer with a kernel size of 2
and stride of 2. (b) In the two-dimensional case, a random image was passed
through a 2D pooling layer with a kernel size of $2\times 1$ and stride of 2,
i.e., pooling with a vertical rectangular kernel was performed for every other
column. In both cases, max pooling was used in the first half, and average
pooling was used in the second half.

In the meta-learning, we used a neural network with only a single
parameterized pooling layer. We set the task batch size to $32$ and the number
of epochs in the outer loop to $10{,}000$, which was sufficient to converge
the training. The data in each task were split into $1$ training sample and
$19$ validation samples. In the outer loop, we used the Adam [16] optimizer
with a learning rate of $0.001$. The temperature parameter $T$ was set to 0.2.
In the inner loop, $\bm{W}$ and $\bm{p}$ themselves were updated only once
instead of $\bm{\theta}$ because the network used in this experiment did not
have any layer other than pooling, and the stochastic gradient descent (SGD)
optimizer with a learning rate of $0.1$ was used. We used the mean squared
error as the loss function. After meta-learning, we observed the learned meta-
parameters $\bm{W}$ and $\bm{p}$.

Figs. 4 and 5 show the learned meta-parameters $\bm{W}$ and $\bm{p}$ for the
one-dimensional and two-dimensional artificial data, respectively. In Fig.
4(a), higher values are concentrated on the diagonal parts of $\bm{W}$,
demonstrating that the parameterized pooling layer successfully learned the
kernel shape with a size of 2 and stride of 2. It can also be confirmed that
$\bm{p}$ takes large values in the first half and approximates $1$ in the
second half, meaning that the max and average pooling operations are also
learned approximately. In Fig. 5(a), the pooling pattern with a kernel size of
$2\times 1$ and stride of $2$ was reproduced. The pooling operations with max
pooling in the first half and average pooling in the second half were also
correctly estimated as shown in Fig. 5(b).

(a) Kernel shape matrix $\bm{W}$

(b) Operation parameter $\bm{p}$

Figure 4: Learned meta-parameters $\bm{W}$ and $\bm{p}$ in the simulation
experiment on one-dimensional artificial data. In (a), the black and white
pixels represent 0 and 1, respectively. The numbers of columns and rows
correspond to the input and output dimensions, respectively.

(a) Kernel shape matrix $\bm{W}$

(b) Operation parameter $\bm{p}$

Figure 5: Learned meta-parameters $\bm{W}$ and $\bm{p}$ in the simulation
experiment on two-dimensional artificial data. In (a), the black and white
pixels represent 0 and 1, respectively.

These results demonstrate that the proposed method can learn the kernel shapes
and pooling operations in a data-driven manner. In this experiment, we used a
network with only a parameterized pooling layer for simplicity. We have also
confirmed that the parameterized pooling layer can be meta-learned in a
network that includes other layers, such as convolutional and fully connected
layers, but these results are omitted due to page limitations. In the next
section, we describe experiments using character images with a network that
includes other layers besides a parameterized pooling layer.

##  5 Experiment on Character Datasets

To evaluate the effectiveness of the proposed method on real-world datasets,
we conducted a character image recognition experiment. The aim of this
experiment is twofold: One is to determine a pooling layer suitable for
character recognition tasks in a data-driven manner, and the other is to
evaluate the generalization capability of the meta-learned pooling for new
tasks. For the former, we first prepared a CNN that includes a parameterized
pooling layer. Then, we meta-learned the parameterized pooling layer using
multiple character recognition tasks. After that, we analyzed the learned
meta-parameters of the pooling layer, which can be regarded as the kernel
shapes and pooling operations that are suitable across multiple tasks. For the
latter, we evaluated the model using tasks that were not given during the
meta-training. Based on the assumption that the proposed meta-learning of
pooling can obtain the knowledge shared across multiple datasets, meta-learned
pooling will improve the generalization capability of the model for subsequent
new tasks. Therefore, we evaluated the model with the meta-learned pooling
layer based on few-shot image recognition and noisy image recognition tasks.
Fixing the meta-parameters, we trained the weights of the remaining layers and
evaluated the entire model.

###  5.1 Dataset

We used the Omniglot dataset [17], which is a standard benchmark dataset of
handwritten character images. Examples from the Omniglot dataset are shown in
Fig. 6. In particular, this dataset is frequently used in few-shot learning
tasks [30, 6, 15]. In this study, we used this dataset to meta-learn a
suitable pooling layer for multiple character recognition tasks and evaluate
the meta-learned pooling in few-shot and noisy image recognition tasks. The
Omniglot dataset consists of $20$ instances of $1{,}623$ characters from $50$
different alphabets, and each instance was depicted by a different person. In
this experiment, we randomly selected $1{,}200$ characters for meta-learning
regardless of the alphabet, and used the remaining $423$ characters for few-
shot and noisy image recognition tasks. We used the images after inverting
pixel colors. We applied data augmentation by rotating the images by 90, 180,
or 270 degrees.

Figure 6: Example images from the Omniglot dataset. Pixel colors were inverted
from the original images.

###  5.2 Experimental Setups

Through the experiment, we used the same network architecture. The network
consisted of a $3\times 3$ convolutional layer with $64$ filters, batch
normalization [14], ReLU activation functions, a $2\times 2$ parameterized
pooling layer, and a fully connected layer with a softmax activation function.
In the parameterized pooling layer, sliding windows with a size of $2\times 2$
and a stride of $2$, i.e., no overlap between windows, were used. The
temperature parameter $T$ was set to 0.2.

After meta-learning, we conducted few-shot recognition and noisy image
recognition tasks. In the few-shot recognition task, the model classified
images into one of five classes (five-way recognition) given either one or
five images as the training data (one-shot or five-shot, respectively). Among
the $423$ character classes that were not used in the meta-learning, five
classes were randomly sampled 100 times with duplication to generate 100 five-
way recognition tasks. In each task, one or five images were used to train the
weights of the convolutional and fully connected layers, and the remaining
images were used as test data to evaluate the recognition accuracy. In the
noisy image recognition task, we reused the network weights obtained in the
few-shot recognition tasks and evaluated the recognition accuracy by adding
salt-and-pepper noise to the test data. We varied the noise ratio in the
interval of 10–60%.

In the meta-learning, the number of task batches was set to $32$, and the
images in each batch were split into $1$ for training and $19$ for validation
in the one-shot setting and $5$ for training and $15$ for validation in the
five-shot setting. We used the Adam optimizer with a learning rate of $0.001$
for the outer loop and the SGD optimizer with a learning rate of $0.1$ for the
inner loop. The numbers of epochs for inner and outer loops were set to $1$
and $10{,}000$, respectively. The initial weights of the convolutional and
fully connected layers were determined based on the MAML algorithm [8] using
the meta-learning dataset.

For comparison, we prepared two networks with the parameterized pooling layer
replaced with either a max or average pooling layer. The results of the few-
shot recognition and noisy image recognition tasks were compared to those of
the prepared networks. The initial weights of these networks were also
determined based on the MAML algorithm.

###  5.3 Results

The learned meta-parameters $\bm{W}$ and $\bm{p}$ are shown in Fig. 7.
According to the kernel shape matrix $\bm{W}$, the proposed pooling layer
learned to extract the annular-shaped area around the region where the
character is written. Surprisingly, it did not focus on the center of the
image. From the result of $\bm{p}$, max pooling was applied around the
character region, whereas average pooling was used in other regions. The
annular shape observed in both $\bm{W}$ and $\bm{p}$ occurred because the
character was not always written in the exact center of the image in the
Omniglot dataset, and the essential information existed in somewhat peripheral
regions. Furthermore, the white annulus in $\bm{W}$ was thicker than the red
annulus in $\bm{p}$, indicating that max pooling was learned at the outer edge
of the annulus to distinguish the character from the background, and average
pooling was used in the area near the character to mitigate the effect of
noise in the same manner as a linear filter.

(a) Kernel shape matrix $\bm{W}$

(b) Operation parameter $\bm{p}$

Figure 7: Learned meta-parameters on the Omniglot dataset. (a) Kernel shape
matrix $\bm{W}$ and (b) operation parameter $\bm{p}$. In (a), the black and
white pixels represent 0 and 1, respectively. These meta-parameters were
learned using non-overlapping sliding windows with a size of $2\times 2$ and a
stride of $2$. These figures were depicted by placing the learned parameters
at the corresponding positions.

Table 1 lists the recognition accuracies for each pooling method in the one-
shot and five-shot settings. As can be seen, the proposed meta-learned pooling
showed the best performance in the one-shot setting, whereas max pooling
outperformed ours in the five-shot setting. These results indicate that the
proposed method is particularly effective when the training data are extremely
limited.

Table 1: Accuracies in the few-shot character recognition task. | Accuracy (%)
---|---
Pooling Operation | One-shot | Five-shot
Max pooling | $90.52\pm 0.68$ | $\bm{97.3\pm 0.92}$
Average pooling | $90.25\pm 0.81$ | $97.0\pm 0.93$
Meta-pooling (proposed) | $\bm{93.16\pm 0.83}$ | $96.5\pm 0.69$

The results of noisy image recognition are presented in Fig. 8. The figures
show the accuracy of each method with varying noise ratio. The proposed method
outperformed max and average pooling when the noise ratio was large. Fig. 9
exhibits the pooling features for each pooling operation when the noise ratio
was 10%. The figures display a randomly selected channel out of 64 channels
for five examples of test images. It was confirmed that the max pooling
features excessively emphasized noise, and the average pooling produced
blurred features due to noise. In contrast, the proposed method could acquire
relatively clean pooling features that preserved the strokes of the
characters. These results suggest that the proposed method can learn a pooling
layer with a high generalization capability, thus resulting in robustness to
noise.

(a) One-shot

(b) Five-shot

Figure 8: Accuracy in the noisy image recognition task. Figure 9: Pooling
features for images with noise (noise ratio: 10%): (a) input image with noise,
(b) max pooling, (c) average pooling, and (d) meta-learned pooling.

##  6 Conclusion

In this paper, we proposed a meta-learning method for pooling layers. As part
of our meta-learning framework, we proposed a parameterized pooling layer in
which the kernel shape and pooling operation are trainable using two
parameters, thereby allowing flexible pooling of the input data. We also
proposed a meta-learning algorithm for the parameterized pooling layer, which
allows us to acquire an appropriate pooling layer across the distribution of
tasks. In the experiment, we applied the proposed meta-learning framework to
character recognition tasks. The results demonstrated that a pooling layer
that is suitable across multiple character recognition tasks was obtained via
meta-learning, and the obtained pooling layer improved the performance of the
model for new tasks of few-shot character image recognition and noisy
character image recognition.

In future work, we will conduct a further comparison with other pooling
methods since we only compared the proposed method with max pooling and
average pooling. A detailed analysis of the computational complexity will also
be performed.

Acknowledgments: This work was supported by JSPS KAKENHI Grant Numbers
JP17K12752 and JP17H06100.

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# A note on Quantum Odometers

Slawomir Klimek  Department of Mathematical Sciences, Indiana University-
Purdue University Indianapolis, 402 N. Blackford St., Indianapolis, IN 46202,
U.S.A<EMAIL_ADDRESS>,  Matt McBride  Department of Mathematics and
Statistics, Mississippi State University, 175 President’s Cir., Mississippi
State, MS 39762, U.S.A<EMAIL_ADDRESS>and  J. Wilson Peoples
Department of Mathematics, Pennsylvania State University, 107 McAllister Bld.,
University Park, State College, PA 16802, U.S.A<EMAIL_ADDRESS>
###### Abstract.

We discuss various aspects of noncommutative geometry of smooth subalgebras of
Bunce-Deddens-Toeplitz Algebras.

##  1\. Introduction

In noncommutative geometry it is often necessary to consider dense
$*$-subalgebras of C∗-algebras, in particular, in connection with cyclic
cohomology or with the study of unbounded derivations on C∗-algebras [5].
Smooth subalgebras of noncommutative spaces are also naturally present in
studying spectral triples. If C∗-algebras are thought of as generalizations of
topological spaces, then dense subalgebras may be regarded as specifying
additional structures on the underlying space, like a smooth structure. In
analogy with the algebras of smooth functions on a compact manifold, such a
smooth subalgebra should be closed under holomorphic functional calculus of
all elements and under smooth-functional calculus of self-adjoint elements. It
should also be complete with respect to a locally convex algebra topology, see
[1].

The purpose of this note is to study smooth subalgebras $A_{S}^{\infty}$ of
Bunce-Deddens-Toeplitz C∗-algebras $A_{S}$ associated to a supernatural number
$S$, objects that capture their smooth structure. This work is a continuation
of, and heavily relies on, our previous papers on the subject of smooth
subalgebras, in particular [7], [8] which investigated smooth structures on
Bunce-Deddens algebras, the algebras of compact operators, and the Toeplitz
algebra.

Bunce-Deddens algebras $B_{S}$ [3], [4], are crossed-product C∗-algebras
obtained from odometers and Bunce-Deddence-Toeplitz algebras $A_{S}$ are their
extensions by compact operators $\mathcal{K}$:

$0\rightarrow\mathcal{K}\rightarrow A_{S}\rightarrow B_{S}\rightarrow 0.$

Due to the topology of odometers [6], which are Cantor sets with a minimal
action of a homeomorphism, the smooth subalgebras are naturally equipped with
inductive limit Frechet (LF) topology.

Using a version of the Toeplitz map [9], we build smooth subalgebras
$A_{S}^{\infty}$ from Toeplitz operators with smooth symbols and from smooth
compact operators. Smooth compact operators, introduced in [11], were studied
in details in [8]. Smooth Bunce-Deddens algebras $B_{S}^{\infty}$, the symbols
of Toeplitz operators, were introduced in [8]. We explicitly construct
appropriate LF structures on $A_{S}^{\infty}$ and prove that those algebras
are closed under holomorphic calculus so that they have the same K-Theory as
their corresponding C∗-algebra closures, and we verify that they are closed
under smooth functional calculus of self-adjoint elements.

We also focus on describing continuous derivations [14] on smooth subalgebras
$A_{S}^{\infty}$. In particular, using results from [7], [8], we classify
derivations on $A_{S}^{\infty}$ and show that, up to inner derivations with
compact range, they are lifts of derivations on $B_{S}^{\infty}$, the factor
algebra of $A_{S}^{\infty}$ modulo the ideal $\mathcal{K}^{\infty}$ of smooth
compact operators. Since many derivations on $B_{S}^{\infty}$ are themselves
inner, the factor space of continuous inner derivations on $A_{S}^{\infty}$
modulo inner derivations turns out to be one-dimensional. Additionally we
shortly describe K-Theory and K-Homology of $A_{S}$.

The paper is organized as follows. Preliminary section 2 contains our notation
and a short review of relevant results from [9] and [7]. In section 3 we
review smooth compact operators and introduce and study smooth Bunce-Deddens-
Toeplitz. Section 4 contains a detailed discussion of stability of
$A_{S}^{\infty}$ under both the holomorphic functional calculus, and the
smooth calculus of self-adjoint elements. In sections 5 we investigate the
structure and classifications of derivations. Finally, section 6 contains
remarks on K-Theory and K-Homology.

##  2\. Preliminaries

###  2.1. Supernatural Numbers

A supernatural number $S$ is defined as the formal product:

$S=\prod_{p-\textrm{prime}}p^{\varepsilon_{p}},\;\;\;\varepsilon_{p}\in\\{0,1,\cdots,\infty\\}\,.$

We will assume $\sum\varepsilon_{p}=\infty$ so that $S$ an infinite
supernatural number. We define $S$-adic ring:

${\mathbb{Z}}/S{\mathbb{Z}}=\prod\limits_{\begin{subarray}{c}p-\textrm{prime}\end{subarray}}{\mathbb{Z}}/{p^{\varepsilon_{p}}}{\mathbb{Z}}.$

Here if $S=p^{\infty}$ for a prime p, then ${\mathbb{Z}}/S{\mathbb{Z}}$ is
equal to ${\mathbb{Z}}_{p}$, the ring of $p$-adic integers.

If the ring ${\mathbb{Z}}/S{\mathbb{Z}}$ is equipped with the Tychonoff
topology it forms a compact, Abelian topological ring with unity, though only
the group structure is relevant for this paper. In addition, if $S$ is an
infinite supernatural number then ${\mathbb{Z}}/S{\mathbb{Z}}$ is a Cantor
set.

The ring ${\mathbb{Z}}/S{\mathbb{Z}}$ contains a dense copy of ${\mathbb{Z}}$
by the following indentification:

${\mathbb{Z}}\ni k\leftrightarrow\\{k\ (\textrm{mod
}p^{\varepsilon_{p}})\\}\in\prod\limits_{\begin{subarray}{c}p-\textrm{prime}\end{subarray}}{\mathbb{Z}}/{p^{\varepsilon_{p}}}{\mathbb{Z}}.$
(2.1)

###  2.2. Hilbert Spaces

We use two concrete Hilbert spaces for this paper: $H=\ell^{2}({\mathbb{Z}})$
and $H_{+}=\ell^{2}({\mathbb{Z}}_{\geq 0})$ Let
$\\{E_{l}\\}_{l\in{\mathbb{Z}}}$ and $\\{E^{+}_{k}:k\geq 0\\}$ be the
canonical bases for $H$ and $H_{+}$ respectively. We need the following shift
operator $V:H\to H$ on $H$ and the unilateral shift operator $U:H_{+}\to
H_{+}$ on $H_{+}$:

$VE_{l}=E_{l+1}\ \textrm{ and }\ UE^{+}_{k}=E^{+}_{k+1}.$

Notice that $V$ is a unitary while $U$ is an isometry. We have:

$[U^{*},U]=P_{0},$

where $P_{0}$ is the orthogonal projection onto the one-dimensional subspace
spanned by $E^{+}_{0}$.

For a continuous function $f\in C({\mathbb{Z}}/S{\mathbb{Z}})$ we define two
operators $m_{f}:H\to H$ and $M_{f}:H_{+}\to H_{+}$ via formulas:

$m_{f}E_{l}=f(l)E_{l}\ \textrm{ and }\ M_{f}E^{+}_{k}=f(k)E^{+}_{k}.$

In those formulas we considered integers $k,l$ as elements of
${\mathbb{Z}}/S{\mathbb{Z}}$ using identification (2.1). Since ${\mathbb{Z}}$
is a dense subgroup of ${\mathbb{Z}}/S{\mathbb{Z}}$ we obtain immediately that

$\|m_{f}\|=\|M_{f}\|=\underset{l\in{\mathbb{Z}}}{\textrm{sup
}}|f(l)|=\underset{k\in{\mathbb{Z}}_{\geq 0}}{\textrm{sup
}}|f(k)|=\underset{x\in{\mathbb{Z}}/S{\mathbb{Z}}}{\textrm{sup
}}|f(x)|=\|f\|_{\infty}.$

The algebras of operators generated by the $m_{f}$’s or by the $M_{f}$’s are
thus isomorphic to $C({\mathbb{Z}}/S{\mathbb{Z}})$ and so they carry all the
information about the space ${\mathbb{Z}}/S{\mathbb{Z}}$, while operators $U$
and $V$ reflect the odometer dynamics $\varphi$ on
${\mathbb{Z}}/S{\mathbb{Z}}$ given by:

$\varphi(x)=x+1.$ (2.2)

The relation between those operators is:

$V^{-1}m_{f}V=m_{f\circ\varphi}.$ (2.3)

Similarly we have:

$M_{f}U=UM_{f\circ\varphi}.$ (2.4)

There is also another, less obvious relation between $U$ and the $M_{f}$’s,
namely:

$M_{f}P_{0}=P_{0}M_{f}=f(0)P_{0}.$ (2.5)

###  2.3. Algebras

Following [9], we define the Bunce-Deddens and Bunce-Deddens-Toeplitz
algebras, $B_{S}$ and $A_{S}$ respectively, to be the following C∗-algebras:
$B_{S}$ is generated by the operators $V$ and $m_{f}$:

$B_{S}=C^{*}\\{V,m_{f}:f\in C({\mathbb{Z}}/S{\mathbb{Z}})\\}$

while $A_{S}$ is generated by the operators $U$ and $M_{f}$:

$A_{S}=C^{*}\\{U,M_{f}:f\in C({\mathbb{Z}}/S{\mathbb{Z}})\\}.$

The algebra $A_{S}$ contains the projection $P_{0}$ and in fact all compact
operators $\mathcal{K}$ and the quotient $A_{S}/\mathcal{K}$ can be naturally
identified with $B_{S}$, see [7]. Let $\tau$ be the natural homomorphism
$\tau:A_{S}\to B_{S}$.

The algebra $B_{S}$ is isomorphic with the crossed product algebra:

$B_{S}\cong C({\mathbb{Z}}/S{\mathbb{Z}})\rtimes_{\varphi}{\mathbb{Z}}\,.$

and is simple [7]. Consequently it is isomorphic the universal C∗-algebra with
generators $v$ and $f$, where $v$ is unitary, $f\in
C({\mathbb{Z}}/S{\mathbb{Z}})$, with relations (compare with (2.3)):

$v^{-1}fv=f\circ\varphi.$

Interestingly, algebras $A_{S}$ can also be described in terms of generators
and relations as follows.

######  Proposition 2.1.

The universal C∗-algebra $A$ with generators $u$ and $f$, such that $u$ is an
isometry, $f\in C({\mathbb{Z}}/S{\mathbb{Z}})$, with relations (compare with
(2.3) and (2.5)):

$fu=u\,(f\circ\varphi)\ \textrm{ and }\ fp_{0}=f(0)p_{0},$

where $[u^{*},u]=p_{0}$, is isomorphic with $A_{S}$.

###### Proof.

We will show that any irreducible representation of $A$ either factors through
$B_{S}$ or is isomorphic to the defining representation of $A_{S}$. Since
$B_{S}\cong A_{S}/\mathcal{K}$ is a factor algebra, the defining
representation of $A_{S}$ dominates the factor representation and so, by
universality, $A$ is isomorphic to $A_{S}$.

Consider an irreducible representation of $A$ and let $U$ represents $u$ and
$M_{f}$ represent $f$. Notice that $P_{0}:=I-UU^{*}$ is the orthogonal
projection onto the kernel of $U^{*}$. If that kernel is zero then $U$ is
unitary and $U$, $M_{f}$ give a representation of $B_{S}$ by universality,
since they satisfy the crossed-product relations.

If the kernel of $U^{*}$ is not zero, pick a unit vector $E^{+}_{0}$ such that
$U^{*}E^{+}_{0}=0$. Since $U$ is an isometry, the set $\\{E^{+}_{k}\\}$,
$k=0,1,\ldots$ is orthonormal, where $E^{+}_{k}:=U^{k}E^{+}_{0}$. Moreover, we
have by using relations:

$M_{f}E^{+}_{0}=M_{f}P_{0}E^{+}_{0}=f(0)E^{+}_{0},$

and similarly:

$M_{f}E^{+}_{k}=M_{f}U^{k}E^{+}_{0}=U^{k}M_{f\circ\varphi^{k}}E^{+}_{0}=f(k)U^{k}E^{+}_{0}=f(k)E^{+}_{k}.$

It follows that vectors $\\{E^{+}_{k}\\}$ span an invariant subspace and so,
by irreducibility, $\\{E^{+}_{k}\\}$ is an orthonormal basis. Since $U$ is the
unilateral shift in this basis, we reproduced the defining representation of
$A_{S}$, finishing the proof. ∎

###  2.4. Toeplitz Map

Next we discuss the key relation between the two algebras $A_{S}$ and $B_{S}$.
Let $P_{\geq 0}:H\to H_{+}$ be the following map from $H$ onto $H_{+}$ given
by

$P_{\geq 0}E_{k}=\left\\{\begin{aligned} &E_{k}^{+}&&\textrm{ if }k\geq 0\\\
&0&&\textrm{ if }k<0.\end{aligned}\right.$

We also need another map $s:H_{+}\to H$ given by:

$sE^{+}_{k}=E_{k}.$

Define the map $T:B(H)\to B(H_{+})$, between the spaces of bounded operators
on $H$ and $H_{+}$, in the following way: given $b\in B(H)$ we set

$T(b)=P_{\geq 0}bs.$

$T$ is known as a Toeplitz map. It has the following properties [10]:

  1. (1)

$T(I_{H})=I_{H_{+}}$.

  2. (2)

$T(bV^{n})=T(b)U^{n}$ and $T(V^{-n}b)=(U^{*})^{n}T(b)$ for $n\geq 0$ and all
$b\in B(H)$.

  3. (3)

$T(bm_{f})=T(b)M_{f}$ and $T(m_{f}\,b)=M_{f}T(b)$ for all $f\in
C({\mathbb{Z}}/S{\mathbb{Z}})$ and all $b\in B(H)$

  4. (4)

$T(b^{*})=T(b)^{*}$ for all $b\in B(H)$.

Consequently, it follows that $T$ is a $*$-preserving map from $B_{S}$ to
$A_{S}$. If $\tau$ is the natural homomorphism from $A_{S}$ to $B_{S}$ then we
have

$\tau T(b)=b$

for all $b\in B_{S}$. It follows that for any $a$ in $A_{S}$ there is a
compact operator $c$ such that we have a decomposition:

$a=T(b)+c,$ (2.6)

where $b=\tau(a)\in B_{S}$. One can verify that if $b$ is an element in
$B_{S}$ then $T(b)$ is compact if and only if $b=0$. This implies the
uniqueness of the above decomposition (2.6).

###  2.5. Fourier Series

There are natural one-parameter groups of automorphisms of $B_{S}$ and $A_{S}$
respectively. They are given by the formulas:

$\rho^{\mathbb{L}}_{\theta}(b)=e^{2\pi i\theta\mathbb{L}}be^{-2\pi
i\theta\mathbb{L}}\ \textrm{ for }b\in B_{S}\ \textrm{ and }\
\rho^{\mathbb{K}}_{\theta}(a)=e^{2\pi i\theta{\mathbb{K}}}ae^{-2\pi
i\theta{\mathbb{K}}}\ \textrm{ for }a\in A_{S},$

where $\theta\in{\mathbb{R}}/{\mathbb{Z}}$. Here we using the following
diagonal label operators on $H$ and $H_{+}$ respectively:

$\mathbb{L}E_{l}=lE_{l}\ \textrm{ and }\ {\mathbb{K}}E^{+}_{k}=kE^{+}_{k}.$

We have the following relations:

$\rho^{\mathbb{L}}_{\theta}(V)=e^{2\pi i\theta}V\textrm{ and
}\rho^{\mathbb{L}}_{\theta}(m_{f})=m_{f}\,.$

Automorphisms $\rho^{\mathbb{K}}_{\theta}$ satisfy analogous relations and the
extra relation on $U^{*}$, namely

$\rho^{\mathbb{K}}_{\theta}(U^{*})=e^{-2\pi i\theta}U^{*}\,.$

Define $E:B_{S}\to C^{*}\\{m_{f}:f\in C({\mathbb{Z}}/S{\mathbb{Z}})\\}\cong
C({\mathbb{Z}}/S{\mathbb{Z}})$ via

$E(b)=\int_{0}^{1}\rho^{\mathbb{L}}_{\theta}(b)\,d\theta\,.$

It’s easily checked that $E$ is an expectation on $B_{S}$. For a $b\in B_{S}$
we define the $n$-th Fourier coefficient $b_{n}$ by the following:

$b_{n}=e(V^{-n}b)=\int_{0}^{1}\rho_{\theta}(V^{-n}b)\,d\theta=\int_{0}^{1}e^{-2\pi
in\theta}V^{-n}\rho_{\theta}(b)\,d\theta.$

From this definition, it’s clear that $b_{n}\in C^{*}\\{M_{f}:f\in
C({\mathbb{Z}}/S{\mathbb{Z}})\\}$ so we can write $b_{n}=m_{f_{n}}$ for some
$f_{n}\in C({\mathbb{Z}}/S{\mathbb{Z}})$. We define an expectation, $E$ on
$A_{S}$, in a similar fashion:

$E:A_{S}\to C^{*}\\{M_{f}:f\in C({\mathbb{Z}}/S{\mathbb{Z}})\\}\cong
C({\mathbb{Z}}/S{\mathbb{Z}})\,.$

For an $a\in A_{S}$, its $n$-th Fourier coefficient $a_{n}$ is also defined
similarly and also $a_{n}=M_{f_{n}}$ for some $f_{n}\in
C({\mathbb{Z}}/S{\mathbb{Z}})$. Additionally, notice that we have the
following relation with the Toeplitz map:

$(T(b))_{n}=T(b_{n})\ \textrm{ for all }n\,.$

##  3\. Smooth Subalgebras

###  3.1. Smooth Compact Operators

We begin by reviewing properties of smooth compact operators from [8]. Let
$\mathcal{K}$ be the algebra of compact operators on $H_{+}$. The orthonormal
basis $\\{E^{+}_{k}\\}_{k\geq 0}$ of $H_{+}$ determines a system of units
$\\{P_{ks}\\}_{k,s\geq 0}$ in $\mathcal{K}$ that satisfy the following
relations:

$P_{ks}^{*}=P_{sk}\quad\textrm{and}\quad P_{ks}P_{rt}=\delta_{sr}P_{kt}\,,$

where $\delta_{sr}=1$ for $s=r$ and is equal to zero otherwise. The set of
smooth compact operators with respect to $\\{E^{+}_{k}\\}$ is the set of
operators of the form

$c=\sum_{k,s\geq 0}c_{ks}P_{ks}\,,$

so that the coefficients $\\{c_{ks}\\}_{k,s\geq 0}$ are rapidly decaying (RD).
We denote the set of smooth compact operators by $\mathcal{K}^{\infty}$.

We now introduce norms on $\mathcal{K}^{\infty}$. They are constructed using
the following useful derivation on $\mathcal{K}^{\infty}$:

$d_{\mathbb{K}}(c)=[{\mathbb{K}},c]\,.$

Clearly $d_{\mathbb{K}}$ is linear and satisfies the the Leibniz rule as
$d_{\mathbb{K}}$ is a commutator. We define $\|\cdot\|_{M,N}$ norms on
$\mathcal{K}^{\infty}$ by the following formulas:

$\|c\|_{M,N}=\sum_{j=0}^{M}\begin{pmatrix}M\\\
j\end{pmatrix}\|d_{\mathbb{K}}^{j}(c)(I+{\mathbb{K}})^{N}\|\,,$

with $\delta_{\mathbb{K}}^{0}(c):=c$. The following proposition from [8]
summarizes the basic properties of $\|\cdot\|_{M,N}$ norms.

######  Proposition 3.1.

Let $a$ and $b$ be bounded operators in $H$, then

  1. (1)

$a\in\mathcal{K}^{\infty}$ if and only if $\|a\|_{M,N}<\infty$ for all
nonnegative integers $M$ and $N$.

  2. (2)

$\|a\|_{M+1,N}=\|a\|_{M,N}+\|d_{\mathbb{K}}(a)\|_{M,N}$.

  3. (3)

$\|a\|_{M,N}\leq\|a\|_{M,N+1}$.

  4. (4)

$\|ab\|_{M,N}\leq\|a\|_{M,0}\|b\|_{M,N}\leq\|a\|_{M,N}\|b\|_{M,N}$.

  5. (5)

$\|d_{\mathbb{K}}(a)\|_{M,N}\leq\|a\|_{M+1,N}$.

  6. (6)

$\|a^{*}\|_{M,N}\leq\|a\|_{M+N,N}$.

  7. (7)

$\mathcal{K}^{\infty}$ is a complete topological vector space.

This proposition implies that $\mathcal{K}^{\infty}$ is a Fréchet $*$-algebra
with respect to the norms, $\|\cdot\|_{M,N}$.

###  3.2. Smooth Bunce-Deddens Algebras

Next we review smooth Bunce-Deddens algebras $B_{S}^{\infty}$ from [7]. We
need the following terminology. We say a family of locally constant functions
on ${\mathbb{Z}}/S{\mathbb{Z}}$ is Uniformly Locally Constant, ULC, if there
exists a divisor $l$ of $S$ such that for every $f$ in the family we have

$f(x+l)=f(x)$

for all $x\in{\mathbb{Z}}/S{\mathbb{Z}}$.

We define the space of smooth elements of the Bunce-Deddens algebra,
$B_{S}^{\infty}$, to be the space of elements in $B_{S}$ whose Fourier
coefficients are ULC and whose norms are RD. Using Fourier series those
conditions can be written as:

$B_{S}^{\infty}=\left\\{b=\sum_{n\in{\mathbb{Z}}}V^{n}m_{f_{n}}:\\{\|f_{n}\|\\}\textrm{
is }RD,\textrm{ there is an }l|S,\,\,V^{l}bV^{-l}=b\right\\}\,.$

It’s immediate that $B_{S}^{\infty}$ is indeed a nonempty subset of $B_{S}$
and it was proved in [7] that $B_{S}^{\infty}$ is a $*$-subalgebra of $B_{S}$.

Let $\delta_{\mathbb{L}}:B^{\infty}_{S}\to B^{\infty}_{S}$ be given by

$\delta_{\mathbb{L}}(b)=[\mathbb{L},b]$

This derivation is very fundamental below. We have the following simple
relations:

$\delta_{\mathbb{L}}(v^{n})=nV^{n}\ \textrm{ and }\
\delta_{\mathbb{L}}(m_{f})=0.$

This derivative is in particular used to define the following norms on
$B_{S}^{\infty}$ that capture the RD property of the Fourier coefficients of
elements of $B_{S}^{\infty}$. They are defined by:

$\|b\|_{P}=\sum_{j=0}^{P}\ \begin{pmatrix}P\\\
j\end{pmatrix}\|\delta_{\mathbb{L}}^{j}(b)\|\,.$

The following proposition from [7] states the basic properties the $P$-norms.

######  Proposition 3.2.

Let $b_{1}$ and $b_{2}$ be in $B_{S}^{\infty}$, then

  1. (1)

$\|b_{1}\|_{P+1}=\|b_{1}\|_{P}+\|\delta_{\mathbb{L}}(b_{1})\|_{P}$ with
$\|b_{1}\|_{0}:=\|b_{1}\|$.

  2. (2)

$\|b_{1}b_{2}\|_{P}\leq\|b_{1}\|_{P}\|b_{2}\|_{P}$.

  3. (3)

$\|\delta_{\mathbb{L}}(b_{1})\|_{P}\leq\|b_{1}\|_{P+1}$.

It follows that we have the following useful way to describe elements in
$B_{S}^{\infty}$:

$B_{S}^{\infty}=\\{b\in B_{S}:\|b\|_{M}<\infty\,,\textrm{ for every
}M,\textrm{ there is an }l|S,\,\,V^{l}bV^{-l}=b\\}\,.$

###  3.3. Smooth Bunce-Deddens-Toeplitz Algebras

Finally, following similar considerations for the Toeplitz algebra in [8], we
define the smooth Bunce-Deddens-Toeplitz algebra $A_{S}^{\infty}$ by

$A_{S}^{\infty}=\\{a=T(b)+c:b\in B_{S}^{\infty},\
c\in\mathcal{K}^{\infty}\\}\subseteq A_{S}\,.$

Much like with the short exact sequence for $A_{S}$ and $B_{S}$, these smooth
subalgebras have the following related short exact sequence:

$0\longrightarrow\mathcal{K}^{\infty}\longrightarrow
A_{S}^{\infty}\longrightarrow B_{S}^{\infty}\longrightarrow 0\,.$

Thus, we can view the topology on $A_{S}^{\infty}$, as a vector space, in the
usual way:

$A_{S}^{\infty}\cong B_{S}^{\infty}\oplus\mathcal{K}^{\infty}\,.$

This gives $A_{S}$ its LF topology.

The Toeplitz map $T:B_{S}\to A_{S}$ can naturally be restricted to
$B_{S}^{\infty}$ and considered as a map $T:B_{S}^{\infty}\to A_{S}^{\infty}$.
In addition, the homomorphism $\tau$ can be restricted to $A_{S}^{\infty}$ and
we have a homomorphism $\tau:A_{S}^{\infty}\to B_{S}^{\infty}$.

It is easy to verify on generators that we have

$d_{\mathbb{K}}(T(b))=T(\delta_{\mathbb{L}}(b)).$

As a consequence of continuity of $T$ this formula is true for all $b\in
B_{S}^{\infty}$.

It remains to verify that $A_{S}^{\infty}$ is indeed a subalgebra of $A_{S}$.
This follows from the following two propositions.

######  Proposition 3.3.

Let $b$ be in $B_{S}^{\infty}$ and $c$ be in $\mathcal{K}^{\infty}$. Then
$T(b)c$ and $cT(b)$ are in $\mathcal{K}^{\infty}$.

###### Proof.

Because $T(b^{*})=T(b)^{*}$, we only need to prove $T(b)c$ is in
$\mathcal{K}^{\infty}$. Proceeding as in [8] we prove by induction on $M$ that
we have the following estimate:

$\|T(b)c\|_{M,N}\leq\|b\|_{M}\|c\|_{M,N}\,.$ (3.1)

The $M=0$ case is immediate from the definition of the norms. The inductive
step is:

$\displaystyle\|T(b)c\|_{M+1,N}=\|T(b)c\|_{M,N}+\|d_{\mathbb{K}}(T(b))c+T(b)d_{\mathbb{K}}(c)\|_{M,N}\leq$
$\displaystyle\leq\left(\|b\|_{M}+\|\delta_{\mathbb{L}}(b)\|_{M}\right)(\|c\|_{M,N}+\|d_{\mathbb{K}}(c)\|_{M,N})=\|b\|_{M+1}\|c\|_{M+1,N}\,.$

Notice also that, again proceeding as in [8], we can obtain the following
inequality:

$\|cT(b)\|_{M,N}\leq\|b\|_{M+N}\|c\|_{M,N}\,.$ (3.2)

∎

######  Proposition 3.4.

Let $b_{1}$ and $b_{2}$ be smooth Bunce-Deddens elements, then the following
expression is a smooth compact element:

$T(b_{1})T(b_{2})-T(b_{1}b_{2})\,.$

###### Proof.

We follow [8]. Let $b_{1}$ and $b_{2}$ be in $B_{S}^{\infty}$ with the
following decompositions:

$b_{1}=b_{1}^{+}+b_{1}^{-}=\sum_{n\geq
0}V^{n}m_{f_{n}}+\sum_{n<0}m_{f_{n}}V^{n}\quad\textrm{and}\quad
b_{2}=b_{2}^{+}+b_{2}^{-}=\sum_{n\geq
0}V^{n}m_{g_{n}}+\sum_{n<0}m_{g_{n}}V^{n}$

where $\\{\|f_{n}\|\\}$ and $\\{\|g_{n}\|\\}$ are RD sequences and
$\\{f_{n}\\}$ and $\\{g_{n}\\}$ are ULC. Since $T$ is linear we only need to
study the following differences:

$\displaystyle T(b_{1}^{+})T(b_{2}^{+})-T(b_{1}^{+}b_{2}^{+}),$ $\displaystyle
T(b_{1}^{-})T(b_{2}^{-})-T(b_{1}^{-}b_{2}^{-})$ $\displaystyle
T(b_{1}^{-})T(b_{2}^{+})-T(b_{1}^{-}b_{2}^{+}),$ $\displaystyle
T(b_{1}^{+})T(b_{2}^{-})-T(b_{1}^{-}b_{2}^{+})\,.$

First consider the following:

$\displaystyle T(b_{1}^{+})T(b_{2}^{+})-T(b_{1}^{+}b_{2}^{+})$
$\displaystyle=\sum_{m,n\geq 0}U^{n}M_{f_{n}}U^{m}M_{g_{m}}-\sum_{m,n\geq
0}T\left(V^{n}m{f_{n}}V^{m}m_{g_{m}}\right)$ $\displaystyle=\sum_{m,n\geq
0}U^{n+m}M_{f_{n}\circ\varphi^{m}}M_{g_{m}}-\sum_{m,n\geq
0}T\left(V^{n+m}m_{f_{n}\circ\varphi^{m}}m_{g_{m}}\right)$
$\displaystyle=\sum_{m,n\geq
0}U^{n+m}M_{f_{n}\circ\varphi^{m}}M_{g_{m}}-\sum_{m,n\geq
0}T\left(V^{n+m}\right)M_{f_{n}\circ\varphi^{m}}M_{g_{m}}.$

Since $T(V^{n+m})=U^{n+m}$, so the above is zero. A similar argument can be
made for $T(b_{1}^{-})T(b_{2}^{-})-T(b_{1}^{-}b_{2}^{-})$. For the next
difference we have

$T(b_{1}^{-})T(b_{2}^{+})-T(b_{1}^{-}b_{2}^{+})=\sum_{m\geq
0,n<0}M_{f_{n}}(U^{*})^{-n}U^{m}M_{g_{m}}-\sum_{m\geq
0,n<0}M_{f_{n}}T(V^{n}V^{m})M_{g_{m}}.$

However, since $T(V^{n+m})=(U^{*})^{-n}U^{m}$ since $n<0$, this difference is
also zero. Finally, for the last difference, we have

$\displaystyle C:=T(b_{1}^{+})T(b_{2}^{-})-T(b_{1}^{+}b_{2}^{-})$
$\displaystyle=T(b_{1}^{+})\sum_{m<0}M_{g_{m}}(U^{*})^{-m}-\sum_{m<0}T(b_{1}^{+}m_{g_{m}}V^{m})$
$\displaystyle=\sum_{m<0}\left(T(b_{1}^{+}m_{g_{m}})(U^{*})^{-m}-T(b_{1}^{+}m_{g_{m}}V^{m})\right)$
$\displaystyle=-\sum_{m<0}T(b_{1}^{+}m_{g_{m}}V^{m})P_{<-m}$

where we used the following formula for $m<0$:

$U^{-m}(U^{*})^{-m}-I=-P_{<-m}\,.$

Clearly, $C$ is compact but we still need to prove it’s smooth compact. To
this end, we prove the $M,N$-norms of $C$ are finite. A straightforward
calculation gives:

$d_{\mathbb{K}}^{j}(C)=-\sum_{m<0}d_{\mathbb{K}}^{j}\left(T(b_{1}^{+}m_{g_{m}}V^{m})P_{<-m}\right)=-\sum_{m<0}T\left(d_{\mathbb{K}}^{j}(b_{1}^{+}V^{m})\right)P_{<-m}$

Next we estimate norms of $C$ using $\|P_{<-m}\|_{0,N}=|m|^{N}$ to obtain:

$\displaystyle\|d_{\mathbb{K}}^{j}(C)\|_{0,N}$
$\displaystyle\leq\sum_{m<0}\sum_{l=0}^{j}\begin{pmatrix}j\\\
l\end{pmatrix}|m|^{j-l+N}\|d_{\mathbb{K}}^{l}(b_{1}^{+})\|\|g_{m}\|$
$\displaystyle\leq\sum_{m<0}(1+|m|)^{N+j}\left(\sum_{l=0}^{j}\begin{pmatrix}j\\\
l\end{pmatrix}\|d_{\mathbb{K}}^{l}(b_{1}^{+})\|\right)\|g_{m}\|$
$\displaystyle=\sum_{m<0}\|b_{1}^{+}\|_{j}(1+|m|)^{N+j}\|g_{m}\|\leq\textrm{const}\|b_{1}^{+}\|_{j}\|b_{2}^{-}\|_{N+j+2}\,.$

Consequently, since $b_{1}$ and $b_{2}$ are in $B_{S}^{\infty}$ we get
$||C||_{M,N}<\infty$. This shows $T(b_{1})T(b_{2})-T(b_{1}b_{2})$ is smooth
compact. A more careful analysis following [8] yields the following estimate:

$\|T(b_{1})T(b_{2})-T(b_{1}b_{2})\|_{M,N}\leq\textrm{const}\|b_{1}\|_{j}\|b_{2}\|_{N+j+2}\\\
,.$ (3.3)

∎

##  4\. Stability of Smooth Bunce-Deddens-Toeplitz Algebra

The purpose of this section is to establish stability of $A_{S}^{\infty}$
under both the holomorphic functional calculus, and the smooth calculus of
self-adjoint elements. It is well known that showing the former automatically
implies that the $K$-Theories of $A_{S}^{\infty}$ and $A_{S}$ coincide [2].

######  Proposition 4.1.

The smooth Bunce-Deddens-Toeplitz algebra $A_{S}^{\infty}$ is closed under the
holomorphic functional calculus.

###### Proof.

Since $A_{S}^{\infty}$ is a complete locally convex topological vector space,
it is enough to check that if $a\in A_{S}^{\infty}$ and invertible in $A_{S}$,
then $a^{-1}\in A_{S}^{\infty}$. Consequently, the Cauchy integral
representation finishes the proof. To this end, let $a\in A_{S}^{\infty}$ and
thus $a=T(b)+c$ with $b\in B_{S}^{\infty}$ and $c\in\mathcal{K}^{\infty}$ and
suppose $a$ is invertible in $A_{S}$. Since $\tau$ is a homomorphism,
$\tau(a)=b$ is invertible in $B_{S}^{\infty}$. It is proved in [7] that if
$b\in B_{S}^{\infty}$ and invertible, then $b^{-1}\in B_{S}^{\infty}$. Since
$\mathcal{K}$ is an ideal of $A_{S}$ and $\tau T$ is the identity map, it
follows that

$a^{-1}=T(b^{-1})+c^{\prime}$

for some $c^{\prime}\in\mathcal{K}$. The proof will be complete if we can show
that $c^{\prime}\in\mathcal{K}^{\infty}$. Notice that

$c^{\prime}=a^{-1}-T(b^{-1})=a^{-1}(I-aT(b^{-1})=a^{-1}(I-T(b)T(b^{-1})+cT(b^{-1}))\,.$

From Propositions 3.3 and 3.4, we have that both $I-T(b)T(b^{-1})$ and
$cT(b^{-1})$ are in $\mathcal{K}^{\infty}$. Consequently, there is a
$\tilde{c}\in\mathcal{K}^{\infty}$ such that $c^{\prime}=a^{-1}\tilde{c}$. It
follows from the properties of norms on $\mathcal{K}^{\infty}$ that

$\|c^{\prime}\|_{0,N}\leq\|a^{-1}\|\|\tilde{c}\|_{0,N}<\infty\,.$ (4.1)

Computing $\delta_{\mathbb{K}}$ on $c$ we have

$\delta_{\mathbb{K}}(c^{\prime})=\delta_{\mathbb{K}}(a^{-1})\tilde{c})=-a^{-1}\delta_{\mathbb{K}}(a)a^{-1}\tilde{c}+a^{-1}\delta_{\mathbb{K}}(\tilde{c})\,.$

Similarly to the proof of Proposition 3.3, we have, inductively, for any $j$
that

$\delta_{\mathbb{K}}^{j}(b)=\sum_{i}a_{i}b_{i}\quad\textrm{finite sum,}$

with $a_{i}$ bounded and $b_{i}$ are smooth compact. Using this and the
estimate in equation (4.1), we see that $\|c^{\prime}\|_{M,N}$ is finite for
all $M$ and $N$. Thus $c^{\prime}\in\mathcal{K}^{\infty}$, completing the
proof. ∎

To prove closure under the calculus of self-adjoint elements, the approach
used in [7] works in this setting as well. Hence, we need results regarding
the growth of exponentials of elements of $B_{S}^{\infty}$ and
$\mathcal{K}^{\infty}$. For $\mathcal{K}^{\infty}$, the exact result needed
was proved in [7]. We state it here for convenience.

######  Proposition 4.2.

Suppose that $c\in\mathcal{K}^{\infty}$ is a self-adjoint smooth compact
operator. Then we have an estimate:

$\|e^{ic}\|_{M,0}\leq\prod_{j=1}^{M}(1+\|c\|_{j,0})^{2^{M-j}}\,.$

The second result needed is a minor adaptation of Proposition 3.4 in [7].

######  Proposition 4.3.

If $b\in B^{\infty}_{S}$ is self-adjoint, then we have an estimate:

$\|e^{ib}\|_{M}\leq\prod_{j=1}^{M}(1+\|b\|_{j})^{2^{M-j}}\,.$

###### Proof.

For $M=0$, notice that $\|e^{ib}\|_{0}=1$. We continue by induction, utilizing
part (1) of Proposition 3.2:

$\|e^{ib}\|_{M+1}=\|e^{ib}\|_{M}+\|\delta_{\mathbb{L}}(e^{ib})\|_{M}.$

Using that

$\delta_{\mathbb{L}}(e^{ib})=i\int_{0}^{1}e^{i(1-t)b}\delta_{\mathbb{L}}(b)e^{itb}\,dt\,,$

we have the following estimate for the inductive step:

$\displaystyle\|e^{ib}\|_{M+1}\leq\|e^{ib}\|_{M}+i\int_{0}^{1}\|e^{i(1-t)b}\|_{M}\|\delta_{\mathbb{L}}(b)\|_{M}\|e^{itb}\|_{M}\,dt\leq$
$\displaystyle\leq\prod_{j=1}^{M}(1+\|b\|_{j})^{2^{M-j}}+\left[\prod_{j=1}^{M}(1+\|b\|_{j})^{2^{M-j}}\right]^{2}\|\delta_{\mathbb{L}}(b)\|_{M}\,.$

Since $\|\delta_{\mathbb{L}}(b)\|_{M}\leq\|b\|_{M+1}$, we have:

$\displaystyle\|e^{ib}\|_{M+1}\leq\prod_{j=1}^{M}(1+\|b\|_{j})^{2^{M-j}}(1+\prod_{j=1}^{M}(1+\|b\|_{j})^{2^{M-j}}\|b\|_{M+1})\leq$
$\displaystyle\leq\prod_{j=1}^{M}(1+\|b\|_{j})^{2^{M-j}}\prod_{j=1}^{M}(1+\|b\|_{j})^{2^{M-j}}(1+\|b\|_{M+1})=\prod_{j=1}^{M+1}(1+\|b\|_{j})^{2^{M+1-j}}\,.$

This establishes the inductive step and finishes the proof. ∎

######  Theorem 4.4.

The smooth Bunce-Deddens-Toeplitz algebra $A_{S}^{\infty}$ is closed under the
smooth functional calculus of self-adjoint elements.

###### Proof.

We need to prove that, given a self-adjoint element $a$ of $A_{S}^{\infty}$
and a smooth function $f(x)$ defined on an open neighborhood of the spectrum
$\sigma(a)$ of $a$ we have $f(a)$ is in $A_{S}^{\infty}$. It is without loss
of generality to assume that $f(x)$ is smooth on ${\mathbb{R}}$ and is
$L$-periodic: $f(x+L)=f(x)$ for some $L$.. Then $f(x)$ admits a Fourier series
representation with rapid decay coefficients $\\{f_{n}\\}$, and hence

$f(a)=\sum_{n\in{\mathbb{Z}}}f_{n}e^{2\pi ina/L}$

for a self-adjoint $a=T(b)+c\in A_{S}^{\infty}$. Thus, it remains to establish
at most polynomial growth in $n$ of norms $\|e^{2\pi ina/L}\|_{M,N}$.

Notice that ${\tau}\left(e^{2\pi ina/L}\right)$ in $B_{S}^{\infty}$ is
$e^{2\pi inb/L}$, which indeed grows at most polynomially in $n$, by
Proposition 4.3. Thus, we only need to show that the $\|\cdot\|_{M,N}$ of the
difference

$e^{2\pi in(T(b)+c)/L}-T\left(e^{2\pi inb/L}\right)\in\mathcal{K}^{\infty}$

are at most polynomially growing in $n$.

To analyze the above, we use a version of the Duhamel’s formula:

$\displaystyle
e^{i(T(b)+c)}-T\left(e^{ib}\right)=\int_{0}^{1}\frac{d}{dt}\left(e^{it(T(b)+c)}T\left(e^{i(1-t)b}\right)\right)dt=$
$\displaystyle=\int_{0}^{1}e^{it(T(b)+c)}c\,T\left(e^{i(1-t)b}\right)dt+\int_{0}^{1}e^{it(T(b)+c)}\left[T(b)T\left(e^{i(1-t)b}\right)-T\left(be^{i(1-t)b}\right)\right]dt\,.$

Employing Proposition 3.1 we can estimate the norms as follows:

$\displaystyle\|e^{i(T(b)+c)}-T\left(e^{ib}\right)\|_{M,N}\leq\int_{0}^{1}\|e^{it(T(b)+c)}\|_{M,0}\|c\,T\left(e^{i(1-t)b}\right)\|_{M,N}\,dt+$
$\displaystyle+\int_{0}^{1}\|e^{it(T(b)+c)}\|_{M,0}\|T(b)T\left(e^{i(1-t)b}\right)-T\left(be^{i(1-t)b}\right)\|_{M,N}\,dt\,.$

All terms above can now be estimated using (3.2), as well as Propositions 4.2
and 4.3. We obtain the following bounds:

$\displaystyle\|e^{i(T(b)+c)}-T\left(e^{ib}\right)\|_{M,N}\leq\prod_{j=1}^{M}(1+\|b\|_{j}+\|c\|_{j,0})^{2^{M-j}}\,\|c\|_{M,N}\prod_{j=1}^{M+N}(1+\|b\|_{j})^{2^{M+N-j}}+$
$\displaystyle+\textrm{const}\prod_{j=1}^{M}(1+\|b\|_{j}+\|c\|_{j,0})^{2^{M-j}}\,\|b\|_{M}\prod_{j=1}^{M+N+2}(1+\|b\|_{j})^{2^{M+N+2-j}}\,.$

Clearly those estimates establish the desired at most polynomial growth,
finishing the proof. ∎

##  5\. Classification of Derivations

We begin with recalling the basic concepts from [9]. Let $A$ be a complete
locally compact topological algebra and let $d:A\to A$ be continuous
derivation on $A$. Suppose that there is a continuous one-parameter family of
automorphisms $\rho_{\theta}:A\to A$ of $A$,
$\theta\in{\mathbb{R}}/{\mathbb{Z}}$.

Given $n\in{\mathbb{Z}}$, a continuous derivation $d:A\to A$ is said to be a
$n$-covariant derivation if the relation

$\rho_{\theta}^{-1}d\rho_{\theta}(a)=e^{-2\pi in\theta}d(a)$

holds for all $\theta$. When $n=0$ we say the derivation is invariant. In this
definition $A$ could be any of the following algebras: $A_{S}^{\infty}$,
$B_{S}^{\infty}$, or $\mathcal{K}^{\infty}$ and with the appropriate one-
parameter family of automorphisms $\rho_{\theta}^{\mathbb{K}}$ or
$\rho_{\theta}^{\mathbb{L}}$. With this definition, we point out that
$\delta_{\mathbb{L}}:B_{S}^{\infty}\to B_{S}^{\infty}$ is an invariant
continuous derivation as is $d_{\mathbb{K}}:A_{S}^{\infty}\to A_{S}^{\infty}$
and $d_{\mathbb{K}}:\mathcal{K}^{\infty}\to\mathcal{K}^{\infty}$.

If $d$ is a continuous derivation on $A$, the $n$-th Fourier component of $d$
is defined as:

$d_{n}(a)=\int_{0}^{1}e^{2\pi
in\theta}\rho_{\theta}^{-1}d\rho_{\theta}(a)\,d\theta\,.$

We have the following simple observation [9].

######  Proposition 5.1.

With the above notation the $n$-th Fourier component $d_{n}:A_{S}^{\infty}\to
A_{S}^{\infty}$ is a continuous $n$-covariant derivation.

To classify continuous derivations on $A_{S}^{\infty}$ we follow the strategy
from [9]. We use the classification of derivations on $B_{S}^{\infty}$ from
[7] and show how to lift derivations from $B_{S}$ to $A_{S}$. We handle the
remaining derivations, those with range in $\mathcal{K}^{\infty}$, by using
the Fourier decomposition components. This is the heart of the argument and
will be described next.

Let $\mathcal{A}_{S}\subseteq A_{S}^{\infty}$ be the subspace of
$A_{S}^{\infty}$ consisting of elements $a=T(b)+c$ such that $b$ has only
finitely many non-zero Fourier components and $c$ has only finitely many non-
zero matrix coefficients (in the standard basis). It was observed in [9] that
$\mathcal{A}_{S}$ is a dense subalgebra of $A_{S}$. In turn, we note that it
is also a dense subalgebra of $A_{S}^{\infty}$.

######  Theorem 5.2.

If $d:A_{S}^{\infty}\to\mathcal{K}^{\infty}$ is a continuous derivation, then
there is $c\in\mathcal{K}^{\infty}$ such that $d(a)=[c,a]$ for every $a\in
A_{S}^{\infty}$. In particular, $d$ is an inner derivation.

###### Proof.

Let $d:A_{S}^{\infty}\to\mathcal{K}^{\infty}$ be a continuous derivation. Let
$d_{n}$ be the nth-Fourier component of $d$. From Proposition 5.1, $d_{n}$ are
$n$-covariant derivations and $d_{n}:A_{S}^{\infty}\to\mathcal{K}^{\infty}$.
We only consider the case $n\geq 0$ as $n<0$ can be treated similarly. All
$n$-covariant derivations $d_{n}:\mathcal{A}_{S}\to A_{S}$ were classified in
[9] . Thus, we know there exists a sequence, $\\{\beta_{n}(k)\\}$, possibly
unbounded in $k$, such that

$d_{n}(a)=[U^{n}\beta_{n}({\mathbb{K}}),a]$ (5.1)

for any $a\in\mathcal{A}_{S}$. We are requiring here the range of $d$ to
belong to $\mathcal{K}^{\infty}$, which places restrictions on
$\\{\beta_{n}(k)\\}$.

Let $\chi$ be a character on ${\mathbb{Z}}/S{\mathbb{Z}}$ and since
$d_{n}(a)\in\mathcal{K}^{\infty}$ for any $a\in A_{S}^{\infty}$ we have

$\left\\{\begin{aligned}
d_{n}(U)&=U^{n+1}(\beta_{n}({\mathbb{K}}+I)-\beta_{n}({\mathbb{K}})):=U^{n+1}\alpha_{n}({\mathbb{K}})\in\mathcal{K}^{\infty}&&\textrm{for
}n\geq 0\\\
d_{n}(M_{\chi})&=U^{n}\beta_{n}({\mathbb{K}})(1-\chi(n))\in\mathcal{K}^{\infty}&&\textrm{for
}n\geq 0\,.\end{aligned}\right.$

Since for each $n>0$ we can choose $\chi$ such that $\chi(n)\neq 1$, and thus
we have $\\{\alpha_{n}(k)\\}$ and $\\{\beta_{n}(k)\\}$ are RD in $k$ for every
$n>0$.

For $n=0$, the above equation only implies that $\\{\alpha_{0}(k)\\}$ is RD in
$k$. We have the following difference equation:

$\alpha_{n}(k)=\beta_{n}(k+1)-\beta_{n}(k)\,.$

This equation has a solution of the form

$\beta_{n}(k)=-\sum_{r=k}^{\infty}\alpha_{n}(r)\,.$ (5.2)

It follows, since $\\{\alpha_{0}(k)\\}$ is RD in $k$, so is
$\\{\beta_{0}(k)\\}$. Thus $\\{\beta_{n}(k)\\}$ is RD for any $n$ and the
formula (5.1) extends by continuity to any $a\in A_{S}^{\infty}$.

We want to establish that $\\{\beta_{n}(k)\\}$ is RD in both $n$ and $k$.
Since $d_{n}(U)\in\mathcal{K}^{\infty}$ we have that $\|d_{n}(U)\|_{M,N}$ are
finite for all $M$ and $N$. So, for any $N$ and $j$ there exists a constant
$C_{j,N}$ such that

$\|d_{\mathbb{K}}^{j}(d_{n}(U))(I+{\mathbb{K}})^{N}\|\leq C_{j,N}$

On the other hand, consider the following calculation for $n\geq 0$:

$d_{\mathbb{K}}^{j}(d_{n}(U))=d_{\mathbb{K}}^{j}(U^{n+1}\alpha_{n}({\mathbb{K}}))=(n+1)^{j}U^{n+1}\alpha_{n}({\mathbb{K}})$

since $\alpha_{n}({\mathbb{K}})$ is diagonal. Therefore, we have that

$(n+1)^{j}\|\alpha_{n}({\mathbb{K}})(I+{\mathbb{K}})^{N}\|\leq C_{j,N}\,.$

However,

$(n+1)^{j}\|\alpha_{n}({\mathbb{K}})(I+{\mathbb{K}})^{N}\|=(n+1)^{j}\underset{k}{\textrm{sup}}\left\\{(1+k)^{N}|\alpha_{n}(k)|\right\\}\,.$

It follows that

$(1+n)^{j}(1+k)^{N}|\alpha_{n}(k)|\leq C_{j,N}$

and thus $\\{\alpha_{n}(k)\\}$ is RD in both $n$ and $k$. Consequently, by
(5.2), $\\{\beta_{n}(k)\\}$ is RD in both $n$ and $k$. Therefore

$\displaystyle d(a)$
$\displaystyle=\sum_{n\in{\mathbb{Z}}}d_{n}(a)=\sum_{n\geq
0}[U^{n}\beta_{n}({\mathbb{K}}),a]+\sum_{n<0}[\beta_{n}({\mathbb{K}})(U^{*})^{-n},a]$
$\displaystyle=\left[\sum_{n\geq
0}U^{n}\beta_{n}({\mathbb{K}})+\sum_{n<0}\beta_{n}({\mathbb{K}})(U^{*})^{-n},a\right]=[c,a]$

where all the sums converge and $c\in\mathcal{K}^{\infty}$. Thus $d$ is inner,
completing the proof. ∎

To analyze general derivations $d:A_{S}^{\infty}\to A_{S}^{\infty}$ we first
notice the following.

######  Proposition 5.3.

Let $d:A_{S}^{\infty}\to A_{S}^{\infty}$ be a continuous derivation, then
$d(\mathcal{K}^{\infty})\subseteq\mathcal{K}^{\infty}$.

###### Proof.

Since $\mathcal{K}^{\infty}$ is generated by the system of units
$\\{P_{ks}\\}$ and since $d$ is continuous we only need to verify that
$d(P_{ks})$ is in $\mathcal{K}^{\infty}$. Since $P_{ks}=P_{kr}P_{rs}$, by the
Leibniz rule we have that

$d(P_{ks})=P_{kr}d(P_{rs})+d(P_{kr})P_{rs}\,.$

Since the right-hand side is clearly in $\mathcal{K}^{\infty}$, the claim
follows. ∎

It follows from this proposition that any continuous derivation
$d:A_{S}^{\infty}\to A_{S}^{\infty}$ defines a continuous derivation on
$B_{S}^{\infty}$, which is isomorphic to the factor algebra
$A_{S}^{\infty}/\mathcal{K}^{\infty}$. We use this observation in the proof of
the following main result of this section.

######  Theorem 5.4.

Let $d:A_{S}^{\infty}\to A_{S}^{\infty}$ be any continuous derivation. Then
there exist: a constant $\gamma$, $b\in B_{S}^{\infty}$ and
$c\in\mathcal{K}^{\infty}$ such that:

$d=\gamma d_{\mathbb{K}}+[T(b)+c,\cdot].$

###### Proof.

Let $d:A_{S}^{\infty}\to A_{S}^{\infty}$ be a continuous derivation and define
a derivation $\delta:B_{S}^{\infty}\to B_{S}^{\infty}$ by

$\delta(a+\mathcal{K}^{\infty})=d(a)+\mathcal{K}^{\infty}\,.$

In other words, $\delta$ is the class of $d$ in the factor algebra
$A_{S}^{\infty}/\mathcal{K}^{\infty}\cong B_{S}^{\infty}$. The continuity of
$d$ implies the continuity of $\delta$. But all continuous derivations
$\delta:B_{S}^{\infty}\to B_{S}^{\infty}$ were classified in [7]. Therefore,
by that paper, there exists a constant $\gamma$ such that

$\delta=\gamma\delta_{\mathbb{L}}+\tilde{\delta}$

where $\tilde{\delta}$ is inner. Thus there exists a $b\in B_{S}^{\infty}$
such that $\tilde{\delta}=[b,\cdot]$.

Next notice that $[T(b),\cdot]$ is an inner derivation on $A_{S}^{\infty}$
whose class in $B_{S}^{\infty}$ is precisely $[b,\cdot]$. Define a derivation
$\tilde{d}:A_{S}^{\infty}\to A_{S}^{\infty}$ by

$\tilde{d}=d-cd_{\mathbb{K}}-[T(b),\cdot]\,.$

Since the class of $d_{\mathbb{K}}$ in is $\delta_{\mathbb{L}}$, we have that
$\tilde{d}:A_{S}^{\infty}\to\mathcal{K}^{\infty}$ and hence by Theorem 5.2,
$\tilde{d}=[c,\cdot]$ for some $c\in\mathcal{K}^{\infty}$. This concludes the
proof. ∎

##  6\. K-Theory and $K$-Homology

Since $\mathcal{K}^{\infty},A_{S}^{\infty},B_{S}^{\infty}$ are closed under
the holomorphic functional calculus, each inclusion induces an isomorphism in
$K$-Theory. Using this fact, along with the $6$-term exact sequence [12]
induced by the short exact sequence of smooth subalgebras, we compute the
$K$-Theory of $A_{S}^{\infty}$. We then make use of the Universal Coefficient
Theorem [13] to compute the $K$-Homology of $A_{S}$.

###  6.1. K Theory

Recall the short exact sequence

$0\longrightarrow\mathcal{K}^{\infty}\longrightarrow
A^{\infty}_{S}\longrightarrow B^{\infty}_{S}\longrightarrow 0$

of smooth subalgebras. This induces the following $6$-term exact sequence in
$K$-Theory:

${K_{0}(\mathcal{K}^{\infty})}$${K_{0}(A^{\infty}_{S})}$${K_{0}(B^{\infty}_{S})}$${K_{1}(B^{\infty}_{S})}$${K_{1}(A_{S}^{\infty})}$${K_{1}(\mathcal{K}^{\infty})}$$\scriptstyle{K_{0}(\tau)}$$\scriptstyle{\exp}$ind$\scriptstyle{K_{1}(\tau)}$

For details regarding the $K$-Theory of $B^{\infty}_{S}$, see [7]. Since the
generating unitary $V$ in $B^{\infty}_{S}$ lifts to the partial isometry $U$,
it follows that

$\textrm{ind}([V]_{1})=[I-U^{*}U]_{0}-[I-UU^{*}]_{0}=-[P_{00}],$

which generates $K_{0}(\mathcal{K}^{\infty})$. Hence, the index map is an
isomorphism. By exactness, it follows that $K_{1}(\tau)$ is the trivial map.
Since $K_{1}(\mathcal{K}^{\infty})=0$, by exactness $K_{1}(\tau)$ is also
injective, and hence $K_{1}(A^{\infty}_{S})=0$. Since $\exp$ is trivial, by
exactness $K_{0}(\tau)$ is surjective. But again, since ind is an isomorphism,
it follows that the map $K_{0}(\mathcal{K}^{\infty})\to K_{0}(A^{\infty}_{S})$
is trivial. Hence, $K_{0}(\tau)$ is injective as well. Using the computation
done in [7], it follows that we have:

$K_{0}(A^{\infty}_{S})\cong G_{S}\ \textrm{ where }\
G_{S}=\\{k/l\in\mathbb{Q}:k\in\mathbb{Z},l|S\\}.$

Let us summarize the results in the following proposition.

######  Proposition 6.1.

The $K$-Theory of $A_{S}$ is given by

$K_{0}(A_{S})\cong G_{S}\ \textrm{ and }\ K_{1}(A_{S})\cong 0.$

###  6.2. $K$-Homology

The Universal Coefficient Theorem of Rosenberg and Schochet [13] states that
we have two exact sequences:

${0}$${\operatorname{Ext}^{1}_{\mathbb{Z}}(K_{1}(A_{S}),{\mathbb{Z}})}$${K^{0}(A_{S})}$${\operatorname{Hom}(K_{0}(A_{S}),{\mathbb{Z}})}$${0,}$

and

${0}$${\operatorname{Ext}^{1}_{\mathbb{Z}}(K_{0}(A_{S}),{\mathbb{Z}})}$${K^{1}(A_{S})}$${\operatorname{Hom}(K_{1}(A_{S}),{\mathbb{Z}})}$${0,}$

where in the above, we have used the identification
$KK^{i}(A_{S},\mathbb{C})=K^{i}(A_{S})$. From the first sequence, it is clear
that

$\operatorname{Ext}^{1}_{\mathbb{Z}}(K_{1}(A_{S}),{\mathbb{Z}})\cong 0.$

In [7] it was shown that

$\operatorname{Hom}(K_{0}(A_{S}),{\mathbb{Z}})\cong 0.$

Hence, we have $K^{0}(A_{S})=0$. From the second sequence, it is immediate
that

$K^{1}(A_{S})\cong\operatorname{Ext}^{1}_{\mathbb{Z}}(K_{0}(A_{S}),{\mathbb{Z}})\cong
K^{1}(B_{S}),$

where the last isomorphism is derived in [7]. This group was computed in [7]
to be isomorphic to $({\mathbb{Z}}/S{\mathbb{Z}})/{\mathbb{Z}}.$ This
reference also contains an explicit description of the precise subgroup being
modded out. In fact, this subgroup turns out to be the natural dense copy of
${\mathbb{Z}}\subseteq{\mathbb{Z}}/S{\mathbb{Z}}$. We summarize the above
computations in the following proposition.

######  Proposition 6.2.

The $K$-Homology of $A_{S}$ is given by

$K^{0}(A_{S})\cong 0\ \textrm{ and }\
K^{1}(A_{S})\cong({\mathbb{Z}}/S{\mathbb{Z}})/{\mathbb{Z}}.$

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