# Did I do that? Blame as a means to identify controlled effects in
reinforcement learning

Oriol Corcoll
Institute of Computer Science
University of Tartu
<EMAIL_ADDRESS>
Raul Vicente
Institute of Computer Science
University of Tartu
<EMAIL_ADDRESS>
###### Abstract

Modeling controllable aspects of the environment enable better prioritization
of interventions and has become a popular exploration strategy in
reinforcement learning methods. Despite repeatedly achieving State-of-the-Art
results, this approach has only been studied as a proxy to a reward-based task
and has not yet been evaluated on its own. We show that solutions relying on
action prediction fail to model important events. Humans, on the other hand,
assign blame to their actions to decide what they controlled. Here we propose
Controlled Effect Network (CEN), an unsupervised method based on
counterfactual measures of blame. CEN is evaluated in a wide range of
environments showing that it can identify controlled effects better than
popular models based on action prediction.

##  1 Introduction

introductionintroductionIntroductionIntroduction The recent success of
reinforcement learning (RL) methods in difficult environments such as Hide &
Seek (Baker et al., 2019), StarCraft II (Vinyals et al., 2019), or Dota2
(OpenAI et al., 2019) has shown the potential of RL to learn complex behavior.
Unfortunately, these methods also show RL’s inefficiency to learn (Espeholt et
al., 2018; Kapturowski et al., 2019; Gulcehre et al., 2020), requiring a vast
amount of interactions with the environment before meaningful learning occurs.
The inefficiency of these algorithms makes it necessary to provide a dense
reward. Environments with sparse rewards are known to be extremely difficult
to solve and a good exploration strategy is imperative.

A popular approach to exploration is to introduce behavioral biases in the
form of intrinsic motivators (Chentanez et al., 2005). This technique aims to
produce dense rewards that facilitate the learning of task-agnostic behavior.
Typical motivators drive the agent to discover novel states and by doing so
increase the chance of reaching the environment’s extrinsic reward. Numerous
motivators have been developed by taking inspiration from humans, e.g.
curiosity or control (Bellemare et al., 2012b; Pathak et al., 2017; Burda et
al., 2018; Choi et al., 2019; Badia et al., 2020b).

Recent work (Choi et al., 2019; Song et al., 2019; Badia et al., 2020a, b) has
achieved State-of-the-Art on the Atari benchmark (Bellemare et al., 2012a) by
rewarding agents for the discovery of novel ways of controlling their
environment. A common design principle among these methods is the use of an
inverse model to predict the chosen action. The hope is that the latent
representation learned by these models encloses aspects of the environment
controlled by the agent. We hypothesize that these methods are ill-suited to
model anything other than the agent, limiting their applicability. For
example, if an agent moves to the right and pushes a box by doing so, the
model only needs to represent the agent’s change of location to predict its
action. Thus, the model is free to ignore the box’s movement which the agent
controlled. Surprisingly the ability to identify controlled effects has only
been evaluated as a proxy to a reward-based task but not in isolation. In this
work, we evaluate action prediction models on the problem of controlled effect
identification and show that, indeed, they do not model controlled effects
other than those upon the agent.

A causal approach is to compare counterfactual worlds, i.e. an effect is
controllable if the agent would have had another effect by taking a different
action (Pearl, 2009). A caveat of this approach is that things become
trivially controllable. Following the previous example, a box becomes
controllable even when the agent performs the action ”do-nothing” since there
is an action, ”move-right”, that moves the box. It is believed that humans
identify controlled effects by assigning a degree of blame to their actions.
In particular, humans compare what happened to a normative imagined world
(Halpern, 2016; Morris et al., 2018; Langenhoff et al., 2019; Grinfeld et al.,
2020). If what happened is normal, humans would not assign blame to their
actions, e.g. when performing ”do-nothing”, the box’s effect would not be
controlled since normally the box would not move. However, it would be
considered controlled when performing ”move-right” since its normative state
is to not move.

To improve upon this issue we propose Controlled Effects Network (CEN), an
unsupervised method based on the human notion of blame. In contrast to models
based on action prediction, CEN is composed of a two-branch forward model that
disentangles what is normal and what is controlled to predict the next step
effects. Our experiments show that CEN can disentangle effects correctly and
that it can model both distant and agent-centric controlled effects.

##  2 Identifying controlled effects using blame

causalitycausalityIdentifying controlled effects using blameIdentifying
controlled effects using blame Our goal is to identify what was controlled by
the agent. Take Fig. 1 as an example of a skull and an agent about to collide,
leading to losing a life. In both scenarios losing a life is controllable by
the agent, i.e. there is a world (moving right) where the agent would have not
lost a life. In contrast, humans perception of causality is associated to
blame (Gerstenberg and Lagnado, 2014), e.g. in the first scenario we would say
that the skull caused the agent to lose a life, whereas in the second one the
agent would be to blame since it moved onto the skull and lost a life.

Figure 1: Is losing a life caused by the skull, the agent or both? Left: the
skull moves onto the agent. Right: the agent and skull move towards each
other. In both cases the agent loses a life.

###  2.1 Controllable effects

controllablecontrollableControllable effectsControllable effects What does it
mean to cause something? Pearl et al. (2016) provide an intuitive definition
of cause-effect relations: ”A variable X is a cause of a variable Y if Y, in
any way, relies on X for its value”. In the previous example, skulls have a
causal effect on the agent’s life since the agent’s life can decrease due to
an skull’s attack. Actual causality, proposed in Halpern (2016), studies
causal relations between individual events of $X$ and $Y$. It aims to answer
questions like, did the skull moving right cause the agent to lose a life? In
the following, we introduce the concept of (actual) causal effect to then
define controllable effect in the context of RL.

The individual causal effect (ICE) of an event $X=x$ on a variable $Y_{i}$ can
be measured by comparing counterfactual worlds

$\displaystyle\mathit{ICE}_{Y_{i}}^{x}\equiv Y^{x}_{i}\neq Y^{\tilde{x}}_{i}\
,$ (1)

where $Y^{x}_{i}$ reads as what would the value of an individual $Y_{i}$ be if
$X$ is forced to be $x$. Similarly, $Y^{\tilde{x}}_{i}$ describes the value of
$Y_{i}$ when $X$ is forced to not be $x$. Note that since we are using the
actual causality framework the sub-index $i$ refers to an individual event. In
the following, we use $Y_{i}^{x}$ and $Y^{x}$ interchangeably.

The world $Y^{\tilde{x}}$ does not exist and needs to be imagined.
Intuitively, Eq. 1 compares the world where the event $x$ happened to an
alternative world where event $x$ had not happened. Consequently, we say that
$x$ has a causal effect on $Y$ if there is an $\tilde{x}\in X$ that satisfies
Eq. 1.

Figure 2: Causal diagram of a typical RL setting. Controlled effects depend on
the agent’s choice of action.

In the realm of RL, $X$ and $Y$ take the form of actions, states and
observations. Figure 2 illustrates the causal relations present in a typical
RL setting, where a state $s$ has an effect on both the next state
$s^{\prime}$ and the produced observation $o$ which, in turn, has an effect on
the agent’s choice of action $a\in\mathcal{A}\subseteq\mathbb{N}$. Similarly,
an action has an effect on the next state. Since states are typically not
accessible by the agent, we do not use states as variables; nevertheless the
same principle can be applied if these are accessible. We define the perceived
effect $e_{p}^{a}$ as the difference between consecutive observations when
taking action $a$, i.e. $e_{p}^{a}\equiv o^{\prime}-o$. As in Eq. 1, we say
that a perceived effect was controllable by the agent’s action when

$\displaystyle\exists{\tilde{a}}\in\mathcal{A}\colon e_{p}^{a}\neq
e_{p}^{\tilde{a}}\ .$ (2)

Since we want to know what elements of the perceived effect are controllable,
the inequality is an element-wise operation.

Figure 3: CEN is a forward model that predicts effects from a given
observation and action. It disentangles controlled and normal effects in
latent space to then decode them into pixel space. Note that the weights of
the decoders are shared.

It is important to notice that the definition of controllable effect has far-
reaching consequences: a motionless agent has a causal effect on itself since
it could move right; an agent next to a button that can be pressed has a
causal effect on that button, even if it does not press it; an agent shot by a
gun has a causal effect on its life because it could have dodged it. If Eq. 2
is used as reward, the agent would be rewarded for almost every action at
every state! Note that taking $\tilde{a}$ as a special ”do-nothing” action
would not work since even doing nothing does something, e.g. if an agent’s
oxygen level decreases unless still, doing nothing has an effect on the
agent’s oxygen level. Instead, we would want a more human-like definition of
what is controlled where an agent controls a button if pressed; a box if moved
or its life if the bullet is dodged.

###  2.2 Blame

blameblameBlameBlame It has been shown that human notion of causality is
affected by what is normal (Kahneman and Miller, 1986; Cushman et al., 2008;
Knobe and Fraser, 2008; Hitchcock and Knobe, 2009). For example heat, fuel and
oxygen are necessary causes for a forest fire but people typically say that
heat (a match or lightning) and not oxygen caused the fire. Thus, heat is to
blame for the forest fire since oxygen and fuel are normally present in a
forest. Here, we resort to concepts of normality from actual causality to find
if the agent’s action is to blame for what happened.

Halpern and Hitchcock (2014) propose to compare what actually happened with
what normally would happen. Following this idea we use a normative world in
replacement to $Y^{\tilde{x}}$

$\displaystyle\mathit{ICE}_{Y}^{x}=Y^{x}-\beta_{Y}\ ,$ (3)

where $\beta_{Y}$ is the value $Y$ would normally take. Such a value is of
course contingent to the notion of normality used, which is to us to define.
Note that since we are interested in the magnitude and direction of the
effect, Eq. 3 uses the difference rather than the less specific inequality
operator used in Eq. 1.

In RL, a typical use of this formulation is to compute the causal effect of an
action on the return $G$ as

$\displaystyle\mathit{ICE}_{G}^{a}(s)$ $\displaystyle=G^{a}(s)-\beta_{G}(s)$
(4) $\displaystyle=Q(s,a)-V(s)$ $\displaystyle=A(s,a).$

$G^{a}(s)$ is the return the agent would get if action $a$ were to be taken at
state $s$ and is typically estimated using a state-action value function
$Q(s,a)$. The choice of normality for $\beta_{G}(s)$ is to estimate the
expected return with the state-value function $V(s)$, giving us the advantage
function $A(s,a)$.

As described in Richard S. and Barto (2018), Generalized Value Functions (GVF)
aim to integrate general knowledge of the world; leaving return as special
case. Following the same idea, we can reformulate Eq. 3 to compute the
controlled effect as

$\displaystyle\mathit{ICE}_{e_{p}}^{a}$
$\displaystyle=e_{p}^{a}-\beta_{e_{p}}$ (5)
$\displaystyle=e_{p}^{a}-\mathop{\mathbb{E}}_{\forall\tilde{a}\in
A}\Big{[}e_{p}^{\tilde{a}}\Big{]}.$

Intuitively, Eq. 5 builds a normal world by observing every alternative
perceived effect reached by each action and choosing the most frequent of
them. To simplify notation, the following sections use controlled effect as
$e_{c}^{a}=\mathit{ICE}_{e_{p}}^{a}$ and normal effect $e_{n}=\beta_{e_{p}}$.
Consider the example in Fig. 1, scenario on the right. Moving left would cause
the agent to die but staying or moving right would keep him alive. If the
agent chooses to move left, Eq. 5 indicates that staying put would be normal
and that losing a life is controlled by the agent. Contrarily, the scenario on
the left would take the loss of a life as normal.

Special care needs to be taken when constructing the normal world
$\beta_{e_{p}}$ for continuous action spaces. Computing counterfactuals on an
infinite number of possibilities cannot be done and some approximation needs
to be implemented. Although our experiments use discrete actions, the proposed
method in the following section is equipped to handle continuous action spaces
since it does not compute counterfactuals for each possible action directly
but approximates the final normal world. It is also important to notice that
the controlled effects identifiable by Eq. 5. in a partially observable
setting ($o\neq s$), where the full state is not observed, are constrained to
those observed by the agent. Nevertheless, humans cannot perceive every change
in state but can identify relevant controlled effects for their survival and
joy.

##  3 Unsupervised learning of controlled effects

methodsmethodsUnsupervised learning of controlled effectsUnsupervised learning
of controlled effects In practice, we do not have access to every world and
cannot compute Eq. 5 directly. We propose an unsupervised method that
disentangles controlled and normal effects using perceived effects as training
signal.

Figure 4: Suite of environments used in our experiments. From left to right:
Clusters, Spiders, Lights and Montezuma’s Revenge (MZR).

Here, we introduce Controlled Effects Network (CEN), depicted in Fig. 3. CEN
is a forward model where observations are encoded into a latent representation
and used to predict the perceived effect resulting from taking a specific
action. Contrarily to conventional forward models CEN is composed of two
branches, where each branch creates a specific latent representation for
controlled and normal effects; in a similar fashion to Dueling Networks (Wang
et al., 2015).

The controlled branch has access to the action (in contrast to the normal
branch), which allows to accurately predict effects controlled by the agent,
something that the normal branch cannot do. Notice that the controlled branch
alone can predict the perceived effect resulting from an action, i.e. the
normal branch would not be needed to just act as a forward model.

Then, why do we need the normal branch? The role of the normal branch is to
force the controlled branch to predict only what is not predictable using the
observation alone and hence, modeling what is controlled by the agent. In a
way, the normal branch acts as a distillation mechanism where only what can be
controlled will be represented by the controlled branch.

Figure 5: Pixel F1 scores for CEN and the baseline in the Spiders environment
on the task of identifying the controlled effects produced by the avatar. CEN
can correctly disentangle the agent from the randomly moving spider.

In order to promote the controlled branch to model only controlled effects, we
use the following loss

$\displaystyle\mathcal{L}=\text{MSE}\left(\hat{e_{c}^{a}}+\hat{e_{n}},e_{p}^{a}\right)+\alpha\
\text{MSE}\left(\hat{e_{n}},e_{p}^{a}\right),$ (6)

where the first part of the loss promotes the modeling of perceived effects,
just as in a conventional forward model in which the predicted target
$\hat{e_{p}^{a}}=\hat{e_{c}^{a}}+\hat{e_{n}}$ is compared to the ground truth
($e_{p}^{a}$). The second part of the loss enforces the network to use the
normal branch as much as possible to model the world. Additionally, a
hyperparameter $\alpha$ regulates how much the normal branch should model the
dynamics of the environment.

Let’s take again the example in Fig. 1 (right). The normal branch is forced to
model the skull’s movement since it does not depend on the agent’s action.
Furthermore, if $\alpha$ is large enough the normal branch is forced to also
model the agent being still. Since the controlled branch can only add to what
the normal branch predicts, it needs to model the movement of the agent to
minimize its loss.

##  4 Experiments

experimentsexperimentsExperimentsExperiments In this section we
evaluate111This work’s networks, training and evaluation have been implemented
using PyTorch (Paszke et al., 2019) and NumPy (Harris et al., 2020); our
experiments are managed using Weights & Biases (Biewald, 2020). CEN’s ability
to model controlled effects and compare it to the action-prediction model
presented in Choi et al. (2019), Attentive Dynamics Model (ADM). The
experiments are divided in two main categories, pixel and attribute
prediction. Pixel prediction experiments aim to compare these two methods as
in a binary-class segmentation task where each method produces a mask that
covers as much ground truth as possible. In some applications such as for
example making use of intrinsic motivators, knowing the position/value of
controlled objects may be enough, and therefore, learning to produce a spatial
mask is unnecessarily precise. Thus, the second set of experiments aims to
measure how well these methods can model controlled effects of individual
attributes in the environment’s state instead of at pixel-level.

To achieve this, we use multiple environments each showcasing a different
aspect of what can be controlled. These environments are based on Griddly
(Bamford et al., 2020) and Atari ALE (Bellemare et al., 2012a), both using the
Gym interface (Brockman et al., 2016); see an example in Fig. 4. We give more
details of these environments in each experiment and appendix.

CEN is implemented with 2D convolutional layers and ReLU activation function
for the encoder and decoders; the normal and controlled branches are
implemented with linear layers and ReLU activation function. Throughout the
experiments we use the same neural networks and hyperparameters unless
specified otherwise. Our implementation of ADM uses the same architecture and
hyperparameters proposed in (Choi et al., 2019). See appendix for more details
on the architecture, hyperparameters, and training.

###  4.1 Controlled effects at pixel-level

controlled-pixelcontrolled-pixelControlled effects at pixel-levelControlled
effects at pixel-level Next we present the results for the pixel-level
experiments. The goal is to evaluate CEN’s predictions of controlled effects
at pixel-level. We report pixel F1 scores between ground truth and prediction
binary masks. Although Eq. 5 computes the magnitude and direction of the
effects, we thresholded the predicted controlled effect. We provide more
details in the appendix. The network is trained to minimize Eq. 6 using the
ADAM optimizer (Kingma and Ba, 2015) and 300K samples of the form
$(o,a,e_{p}^{a})$ collected with a random policy. The evaluation of the model
is done using an evaluation environment with different seed to the one used
for training during 5K steps. To generate the ground truth masks for Griddly
environments, we use the internal environment state and transform object
coordinates onto pixel coordinates. In experiments using Atari environments
the ground truth was collected by computing Eq. 5 using the ALE’s special
calls cloneSystemState and restoreSystemState.

####  4.1.1 Controlled vs uncontrolled effects

In this experiment we evaluate CEN’s ability to disentangle controlled from
uncontrolled effects. Here we use the Spiders environment which has two main
entities, the agent and an uncontrolled angry spider. If the model has learned
to identify controlled effects the masks produced should only focus on the
agent and not on the spider.

Figure 6: F1 pixel score on the Clusters environment. CEN is able to model not
just the agent but also close controlled effects like the movement of boxes.
Top) F1 score for every effect; Bottom-left) F1 score when the avatar moves;
and Bottom-right) F1 score when there is a box moving. X axis refers to the
training steps (1e5).

Fig. 5 shows the pixel F1 score for our model and the baseline. CEN is able to
correctly disentangle controlled effects and can produce accurate masks.
Although our implementation of ADM can predict the agent’s action with $88\%$
accuracy, it is not capable of modeling the agent’s controlled pixels. We
conjecture that this is due to ADM’s sparse softmax mechanism; nonetheless
this behavior persisted when increasing its entropy weight which should
produce more dense masks.

####  4.1.2 Close controlled effects

close-controlled-pixelclose-controlled-pixelClose controlled effectsClose
controlled effects Models based on action prediction are expected to work well
on aspects related to the agent. For example, if an agent moves a box due to
moving right; the box’s movement is also controlled. It is unclear why these
models would pay attention to the box since just knowing where the agent is
suffices to predict the chosen action. CEN’s controlled branch, on the other
hand, is motivated to model the box’s effect since the normal branch would
predict that the box stays where it is. We call ”close” controlled effect to
an effect that happens near the agent, like the box’s movement. To evaluate
CEN on this kind of effects we use the Clusters environment. In this
environment an agent needs to move colored boxes to their corresponding fixed
colored blocks.

Fig. 6 shows that CEN can accurately model the agent and boxes effects better
than the baseline. We breakdown individual effects to account for the class
imbalance between the agent’s movement and the boxes. CEN seems to make more
mistakes with boxes than the agent but nonetheless, it can consistently model
both.

Figure 7: F1 score on the Lights environment. CEN can model distant effects,
even when the agent is not part of the controlled effect. Top) F1 score for
every effect; Bottom-left) F1 score when the avatar moves; and Bottom-right)
F1 score when a light turns on. X axis refers to the training steps (1e5).

####  4.1.3 Distant controlled effects

distant-controlled-pixeldistant-controlled-pixelDistant controlled
effectsDistant controlled effects Similarly to the previous experiment, we
want to evaluate if CEN can model distant effects, i.e. effects that are
reasonably far away from the agent’s location. In this case, we will use the
Lights environment. Here the environment presents two buttons of different
color that, when pressed, turn on their corresponding light of the same color.
Lights are relatively far away from their corresponding buttons thus making it
difficult to model them. As show in Fig. 7, CEN is able to model this kind of
controlled effects. Although there is a clear decrease in F1 score, we believe
that this is due to the more complex shape of the lights and buttons.

####  4.1.4 Controlled effects in Montezuma’s Revenge

mzr-controlled-pixelmzr-controlled-pixelControlled effects in Montezuma’s
RevengeControlled effects in Montezuma’s Revenge This last pixel-level
experiment evaluates CEN on Montezuma’s Revenge environment. Although agents
in Atari environments have limited control over the environment, the
relatively complex dynamics of the agent makes Montezuma’s Revenge a
challenging test-bed. Results shown in Fig. 8 indicate that CEN can also model
controlled effects but that the increase in complexity affects its ability to
predict accurate masks. Additionally, Fig. 9 (last column) shows examples of
the masks generated by CEN and our baseline for this environment. Although the
F1 score is lower than in the other environments, Atari has more complex
shapes which make the model have lower F1 score even when modeling the
controlled effect.

Figure 8: Montezuma’s Revenge F1 pixel score. CEN outperforms the baseline
even in an environment with more complex dynamics.

###  4.2 Controlled effects at attribute-level

controlled-attributecontrolled-attributeControlled effects at attribute-
levelControlled effects at attribute-level In the second set of experiments,
we want to analyze if CEN can predict controlled effects of attributes present
on the environment’s state, e.g. the agent’s $(x,y)$ location or if a light
turned on. In these experiments, we employ a probing technique similar to the
one described in Anand et al. (2019). We freeze trained versions of the CEN
and baseline networks and use them to train a classifier per each attribute in
the environment’s state. We use the frozen networks to produce a binary mask
of controlled effects, as in the previous experiments, and applying the
resulting mask to occlude the perceived effects. These occluded effects are
then used to train each probing classifier to predict if there was a positive,
negative or none effect. Note that the classifier needs to predict any effect,
not just controlled. This way the classifier should only be able to predict
accurately controlled attributes such as the agent’s position but should fail
at predicting the spider’s location. Furthermore, the model does not need to
produce a perfect mask for the classifier to be able to predict these
attributes. We use a random policy to collect a dataset of 35K samples of the
form $(m*e_{p},y)$ where $m$ is the mask produced by the model and $y$ is the
ground truth class. We ensure that each class is relatively balanced, allowing
a $20\%$ class imbalance. Each dataset is split into a typical 70/20/10, we
report F1 score of each attribute on the test set.

The results in Table 1 indicate that CEN significantly outperforms the
baseline when predicting controlled effects for state’s attributes, and thus
modeling controlled effects accurately. Furthermore, for both Spiders and
Montezuma’s Revenge environments the model cannot predict the uncontrolled
effects, as expected. Even though ADM’s action prediction accuracy was high
($\sim 88\%$) on every environment, it is not able to consistently predict
controlled effects at the attribute-level.

|  | F1
---|---|---
Environment | Attribute | CEN (ours) | ADM
Spiders | Agent | 1.0$\pm$0.00 |  0.47$\pm$0.23
Spider $\downarrow$ |  0.35$\pm$0.03 | 0.25$\pm$0.03
Clusters | Agent | 0.76$\pm$0.41 |  0.28$\pm$0.08
Box | 0.78$\pm$0.37 |  0.32$\pm$0.19
Lights | Agent | 0.97$\pm$0.01 |  0.33$\pm$0.15
Button | 0.93$\pm$0.05 |  0.33$\pm$0.01
Light | 0.93$\pm$0.04 |  0.41$\pm$0.14
MZR | Agent | 0.66$\pm$0.08 |  0.42$\pm$0.23
Skull $\downarrow$ | 0.19$\pm$0.03 |  0.20$\pm$0.08
Table 1: F1 score for the predicted state attributes. CEN outperforms the
baseline and is able to model effects distant from the agent. Note that the
locations of the spider and skull are not controlled by the agent thus, the
worse it can be predicted the better.

##  5 Related work

related-workrelated-workRelated workRelated work Intrinsic motivators: A
popular way of introducing behavioral biases in RL agents is the use of
intrinsic motivators (Singh et al., 2005). These motivators can promote
different types of exploration, from observational surprise (Burda et al.,
2018) to control seeking agents (Pathak et al., 2017; Choi et al., 2019).
Methods in the latter category have shown extremely good results achieving
State-of-the-Art in important benchmarks. Choi et al. (2019) proposed
Attentive Dynamics Model (ADM), an attention based method to discover
controlled elements in the environment and used it in combination to a count-
based exploration to reward the agent for discovering these elements. This
method showed State-of-the-Art in Montezuma’s Revenge. This work was extended
in Song et al. (2019) to incorporate multi-step controlled effects as part of
the intrinsic reward, again achieving human-level performance just with
intrinsic rewards. Badia et al. (2020b) combined control and observational
surprise motivators to create a so called ”reward soup” where multiple rewards
were combined into one. Their method uses an episodic memory in combination to
an inverse model to promote the discovery of controlled effects in a single
episode and Random Network Distillation (Burda et al., 2018) to promote long
term progress through the game. This method, again, achieves State-of-the-Art
in Atari’s hard exploration environments. One thing in common among these
methods is the use of an inverse model to model controlled aspects of the
environment. Our work, instead, uses a two-branch forward model showing a
better coverage of these effects. These methods show the importance of
identifying what an agent can control, an avenue that deserves to be explored
in depth.

Figure 9: Samples of the masks generated by CEN and the baseline on each
environment. Samples were picked for both algorithms based on their high F1
score and diversity. The original perceived effect is marked in black and
white; the generated masks are marked in red. F1 score is overlaid in white.

Causality in deep reinforcement learning: Causality is central to humans; we
think in terms of cause-effect. A similar method was proposed in Chattopadhyay
et al. (2019), where they use causal attribution methods to analyzed the
effect of inputs on a neural network’s outputs via causal attribution. Recent
work has introduced causality into deep reinforcement learning (Foerster et
al., 2018; Buesing et al., 2018; Jaques et al., 2018; Dasgupta et al., 2019;
Goyal et al., 2019; Nair et al., 2019; Madumal et al., 2020) showing that this
is a promising avenue for the training of agents. Corcoll and Vicente (2020)
proposed an attribution method to learn temporal abstractions for object-
centric hierarchical RL. Bellemare et al. (2012b) compute controllable aspects
of the environment. Their method generates a mask with all possible
controllable areas of an image and uses it as part of the policy’s input. In
this work, we instead identify controlled effects using causal concepts of
normality and blame.

##  6 Conclusions

conclusionsconclusionsConclusionsConclusions The identification of controlled
effects is a central task in RL. This work proposes an unsupervised approach
to this problem named Controlled Effect Network (CEN). CEN computes
counterfactual worlds to assign blame to the agent’s actions for what
happened. The presented experiments show that, despite of being unsupervised,
this method learns to identify controlled effects successfully. Furthermore,
we show that popular methods based on action prediction do not model
controlled effects besides the agent.

Understanding the world in terms of controlled effects, instead of actions, is
a promising avenue. This approach narrows down the large amount of
possibilities an agent faces and can be used to prioritize the next action to
take based on how its consequences align with the agent’s goals. The
expressive power of effects ought to be explored to achieve better
prioritization of effects. In this direction, more sophisticated modeling of
normality, e.g. social norms or time-based norms; can provide a measure of
relevance or importance useful to prioritize the learning of effects. This
study relies on the notion that effects can only be controlled at the next
time step and that the environment controls follow up effects. For example, a
ball pushed by the agent will be identified as controlled the moment the agent
pushes it but it will be governed by the environment’s dynamics thereafter. A
useful extension to this work would be to identify the consequences of an
action taken N steps ago. We believe that the fast pace of the graph neural
network ecosystem allows for this extension. These improvements would
represent a major leap towards more efficient RL agents.

## Acknowledgments

The authors would like to thank Jaan Aru and Meelis Kull for insightful
comments on the manuscript. This work was supported by the University of Tartu
ASTRA Project PER ASPERA, financed by the European Regional Development Fund
and the University of Tartu HPC.

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##  Appendix A Experiments

###  A.1 Qualitative results

Here we provide additional qualitative results for our first set of
experiments. CEN constantly find the agent and other controlled entities.
Errors in Griddly environments are typically due to not recognizing the
movement of the agent or predicting it will move when it did not. Montezuma’s
Revenge shows more complex patterns which the model fails to find a mask that
perfectly suits the ground truth, even then the pixels outside the ground
truth are near the agent.

Figure 10: Additional examples of masks generated by CEN (left) and ADM
(right).

###  A.2 Inverse model and latent probing

In this experiment, we evaluate the inverse model used in (Badia et al.,
2020b). More precisely, we evaluate CEN and this new baseline on the task of
predicting controlled attributes. The main difference with experiment 4.2 is
that the latent representation is used to predict these attributes instead of
the masked effects. Note that ADM does not have a clear latent representation
of what is controlled. Similarly to the previous experiment, we train a linear
probe to predict controlled attributes from the latent space of these models.
In the case of CEN, we use the output of the controlled branch. For the
inverse model, we concatenate the features of current and next observations.
The probe predicts changes on the x, y and direction (if applies) for agent,
spider, skull and boxes; and on/off for lights and buttons.

|  | F1
---|---|---
Environment | Attribute | CEN (ours) | Inverse
Spiders | Agent | 0.97$\pm$0.01 |  0.67$\pm$0.05
Spider $\downarrow$ | 0.41$\pm$0.01 |  0.44$\pm$0.02
Clusters | Agent | 0.97$\pm$0.01 |  0.56$\pm$0.09
Box | 0.95$\pm$0.02 |  0.77$\pm$0.00
Lights | Agent | 1.0$\pm$0.01 |  0.84$\pm$0.08
Button | 0.99$\pm$0.0 | 0.99$\pm$0.0
Light | 1.0$\pm$0.0 |  0.99$\pm$0.0
MZR | Agent | 0.91$\pm$0.02 |  0.88$\pm$0.02
Skull $\downarrow$ | 0.61$\pm$0.03 |  0.61$\pm$0.04
Table 2: F1 score for the predicted state attributes. CEN outperforms the
baseline and is able to model effects distant from the agent. Note that the
locations of the spider and skull are not controlled by the agent thus, the
worse it can be predicted the better.

Results are presented in Table 2 showing that CEN improves on the baseline’s
performance. Although the inverse model is closer to CEN’s performance than
ADM, it still has difficulties predicting the agent and box changes in
location. We hypothesize that when probing from latent space instead of masked
effects, correlations between attributes can be more easily exploited. For
example, in the lights environment the representation of pushing a button may
be enough to predict the button and light turning on. In the case of pixels,
predicting the light turning on from the pixels representing the agent becomes
a harder task.

##  Appendix B Environments

###  B.1 Griddly

Griddly (Bamford et al., 2020) is a highly optimized grid-world based suite of
environments. Environments used in this work based on Griddly generate
$64\times 64$ pixel observations, although the size of the grid-world may
vary. Griddly supports multiple rendering formats, this work uses the 2D
rendering of sprites.

Spiders: is a 6x6 arena where a Gnome (the agent) has to grab a Gem without
being killed by a Spider. The agent dies if it collides with the spider. In
this environment, the agent can move left, right, up, down or stay. The spider
takes an action randomly from the following: rotate left, rotate right or move
forward. This environment’s controlled entities are: Agent.

Clusters: is a 13x10 arena where a Knight has to move boxes of the same color
to their corresponding colored-block without touching present spikes. There
are two different colors, blue and red. The agent dies if it collides with
spikes or if a box is destroyed by spikes. The agent can move left, right, up,
down or stay. This environment’s controlled entities are: Agent and Boxes.

Lights: is a 11x8 arena where a Ghost (the agent) has to turn all the lights
on by pressing each button. Buttons and lights are colored either blue or
green. Pressing a button of one color turns the light of the same color on.
The agent can move left, right, up, down or stay. This environment’s
controlled entities are: Agent, Buttons and Lights.

Ground truth: Griddly provides access to each entity’s state (x, y, light is
on/off, etc); a binary mask is produced for each controlled entity with any of
its attributes changed when transitioning between two time steps. x and y
coordinates are projected into pixel space and a bounding box is generated
using the size of that entity. Note that the coordinates may be different
between steps thus the generated mask may enclose multiple locations. The
resulting masks for each entity are combined into a single mask $m$ by taking
the maximum value among them. Since we want to know what pixels were actually
controlled, the final ground truth mask is produced as: $m\cdot e^{p}_{a}$.

###  B.2 Atari Montezuma’s Revenge

The ALE (Bellemare et al., 2012a) provides access to Atari 2600 games to
learning methods like RL. As it has been a popular choice by methods using
inverse models, in this work we use the game Montezuma’s Revenge to evaluate
CEN. This environment provides uncontrolled as well as controlled effects with
more complex entities. The environment typically generates observations of
210x160 pixels which we downscale to 64x64 pixels. Additionally the action
space is of size 10.

Ground truth: in contrast to Griddly, we can actually compute counterfactual
worlds by saving and loading the state of the game (RAM) multiple times when
taking different actions. For this, we directly compute Eq. 5 using ALE’s
special calls cloneSystemState and restoreSystemState. More precisely, we
compute every possible perceived effect reachable from the current state and
build a normal effect using the mode over all possible effects. Then, we
compare the perceived effect for the agent’s chosen action against the normal
effect. The ground truth mask will have 1s where these two effects are
different.

##  Appendix C Training

###  C.1 Architecture

Encoder: is composed of two 2D convolutional layers with 4x4 kernels, stride 2
and padding of 1. Additionally, we have 2 residual blocks each with two 2D
convolutional layers with stride 1 and padding of 1. The first layer has a
kernel of 3x3 and the second layer of 1x1. ReLU is used as activation function
throughout the network; BatchNorm is used between each layer; and 64 channels
on every convolutional layer. We project the resulting maps into a flatten
vector of size 32 using a linear layer with ReLU activation function.

Decoder: this module is composed of six 2D transposed convolutional layers all
having 4x4 kernels, stride 2 and padding of 1. Each layer uses ReLU as
activation function but the output layer which uses Tanh activation. Every
layer uses 64 channels with the exception of the last layer which outputs a 1
channel prediction of the perceived effects. Parameters are shared among the
controlled and normal branch decoders.

Controlled and normal modules: both modules are composed of three linear
layers with 32 hidden units, each with a ReLU as activation function. The
input to the controlled branch are the encoded observation and an embedding of
size 8 of the chosen action.

###  C.2 Mask generation

CEN’s controlled masks are generated using the encoder, controlled branch and
decoder. The predicted controlled effect is binarized using a threshold as
$m_{\text{CEN}}=(-T<\hat{e^{c}})|(\hat{e^{c}}>T)$. In the case of ADM, its
attention mask is thresholded in the same way and resized to the size of the
effect.

###  C.3 Hyperparameters

Name | Value | Sweep
---|---|---
hidden size | 32 | [16, 32, 64, 128]
latent size | 128 | [16, 32, 64, 128, 256]
channels | 64 | [16, 32, 64, 128]
learning rate | 0.0001 | [0.0001, 0.0005, 0.001, 0.005]
$\alpha$ | 0.01 | [0.001, 0.01, 0.1, 1, 5, 10, 20, 30, 50]
$T$ | 0.01 | -
Table 3: CEN hyperparameter sweeps and final values used. Name | Value | Sweep
---|---|---
entropy | 0.05 | [0.01, 0.05, 0.1, 0.5, 1, 5]
hidden size | 64 | [16, 32, 64, 128]
attention size | 128 | [32, 64, 128, 256]
learning rate | 0.0001 | [0.0001, 0.0005, 0.001, 0.005]
$T$ | 0.01 | -
Table 4: ADM hyperparameter sweeps and final values used.


# The Mazur-Ulam property for uniform algebras

Osamu Hatori  Institute of Science and Technology, Niigata University, Niigata
950-2181, Japan<EMAIL_ADDRESS>
###### Abstract.

We give a sufficient condition for a Banach space with which the homogeneous
extension of a surjective isometry from the unit sphere of it onto another one
is real-linear. The condition is satisfied by a uniform algebra and a certain
extremely $C$-regular space of real-valued continuous functions.

###### Key words and phrases:

Tingley’s problem, the Mazur-Ulam property, surjective isometries, uniform
algebras, maximal convex sets, analytic functions

###### 2020 Mathematics Subject Classification:

46B04, 46B20, 46J10, 46J15

##  1\. Introduction

In 1987 Tingley [25] proposed a problem if a surjective isometry between the
unit spheres of Banach spaces is extended to a surjective isometry between
whole spaces. Wang [26] seems to be the first to solve Tingley’s problem
between specific spaces. He dealt with $C_{0}(Y)$, the space of all real
(resp. complex) valued continuous functions which vanish at infinity on a
locally compact Hausdorff space $Y$. Although we do not exhibit each of the
literatures, a considerable number of interesting results have shown that
Tingley’s problem has an affirmative answer. No counterexample is known. Due
to [29, p.730] Ding was the first to consider Tingley’s problem between
different type of spaces [9]. Ding [10, Corollary 2] in fact proved that the
real Banach space of all null sequences of real numbers satisfies now we call
the Mazur-Ulam property. Later Cheng and Dong [5] introduced the concept of
the Mazur-Ulam property. Following Cheng and Dong we say that a real Banach
space $E$ satisfies the Mazur-Ulam property if a surjective isometry between
the unit sphere of $E$ and that of any real Banach space is extended to a
surjective real-linear isometry between the whole spaces. Tan [19, 20, 21]
showed that the space $L^{p}(\mathbb{R})$ for $\sigma$-finite positive measure
space satisfies the Mazur-Ulam property. In [22] Tan, Huang and Liu introduced
the notion of generalized lush spaces and local GL spaces and proved that
every local GL space satisfies the Mazur-Ulam property. New achievements by
Mori and Ozawa [17] prove that the Mazur-Ulam property is satisfied by unital
$C^{*}$-algebras and real von Neumann algebras. Cueto-Avellaneda and Peralta
[7] proved that a complex (resp. real) Banach space of all continuous maps
with the value in a complex (resp. real) Hilbert space satisfies the Mazur-
Ulam property (cf. [8]). The result proving that all general JBW*-triples
satisfy the Mazur-Ulam property is established by Becerra-Guerrero, Cueto-
Avellaneda, Fernández-Polo and Peralta [2] and Kalenda and Peralta [14]. The
study of the Mazur-Ulam property is nowadays a challenging subject of study
(cf. [1, 4, 8, 28]).

In this paper we say that a complex Banach space $B$ satisfies the complex
Mazur-Ulam property, emphasizing the term ‘complex’, if a surjective isometry
between the unit spheres of $B$ and any complex Banach space is extended to a
surjective real-linear isometry between the whole spaces. Jiménez-Vargas,
Morales-Compoy, Peralta and Ramírez [13, Theorems 3.8, 3.9] probably provides
the first examples of complex Banach spaces satisfying the complex Mazur-Ulam
property (cf. [18]). Note that a complex Banach space satisfies the complex
Mazur-Ulam property provided that it satisfies the Mazur-Ulam property as a
real Banach space (a complex Banach space is a real Banach space
simultaneously).

In [12] we proved that a surjective isometry between the unit spheres of
uniform algebras is extended to a surjective real-linear isometry between
whole of the uniform algebras. In this paper we show the complex Mazur-Ulam
property for uniform algebras. Typical examples of a uniform algebra consist
of analytic functions of one and several complex-variables such as the disk
algebra, the polydisk algebra and the ball algebra. Through the Gelfand
transform, the algebra of all bounded analytic functions on a certain domain
is considered as a uniform algebra on the maximal ideal space. Hence the main
result in this paper provides the first example of a Banach space of analytic
functions which satisfies the complex Mazur-Ulam property. For further
information about uniform algebras, see [3].

##  2\. Is a homogeneous extension linear?

For a real or complex Banach space $B$, we denote $S(B)$ the unit sphere
$\\{a\in B:\|a\|=1\\}$ of $B$. A maximal convex subset of $S(B)$ is denoted by
$\mathfrak{F}_{B}$. Throughout the paper the map $T:S(B_{1})\to S(B_{2})$
always denotes a surjective isometry with respect to the metric induced by the
norm, where $B_{1}$ and $B_{2}$ are both real Banach spaces or both complex
Banach spaces. We define the homogeneous extension $\widetilde{T}:B_{1}\to
B_{2}$ of $T$ by

$\widetilde{T}(a)=\begin{cases}\|a\|T\left(\frac{a}{\|a\|}\right),&0\neq a\in
B_{1}\\\ 0,&a=0.\end{cases}$

By the definition $\widetilde{T}$ is a bijection which satisfies that
$\|\widetilde{T}(a)\|=\|a\|$ for every $a\in B_{1}$. The Tingley’s problem
asks if $\widetilde{T}$ is real-linear or not. In this paper we prove that
$\widetilde{T}$ is real-linear for certain Banach spaces $B_{1}$ including
uniform algebras.

It is well known that for every $F\in\mathfrak{F}_{B}$ of a real or complex
Banach space $B$, there exists an extreme point $p$ in the closed unit ball
$B(B^{*})$ of the dual space $B^{*}$ of $B$ such that $F=p^{-1}(1)\cap S(B)$
(cf. [24, Lemma 3.3], [12, Lemma 3.1]). Let $Q$ be the set of all extreme
points $p$ in $B(B^{*})$ such that $p^{-1}(1)\cap S(B)\in\mathfrak{F}_{B}$. We
define an equivalence relation $\sim$ in $Q$. We write
$\mathbb{T}=\\{z\in{\mathbb{C}}:|z|=1\\}$ if $B$ is a complex Banach space,
where $\mathbb{C}$ denotes the space of all complex numbers, and
$\mathbb{T}=\\{\pm 1\\}$ if $B$ is a real Banach space.

From Definition 2.1 through Definition 2.6 $B$ is a real or complex Banach
space.

######  Definition 2.1.

Let $p_{1},p_{2}\in Q$. We denote $p_{1}\sim p_{2}$ if there exits
$\gamma\in\mathbb{T}$ such that $p_{1}^{-1}(1)\cap S(B)=(\gamma
p_{2})^{-1}(1)\cap S(B)$.

Note that $\gamma p\in Q$ provided that $\gamma\in\mathbb{T}$ and $p\in Q$.

######  Lemma 2.2.

The binary relation $\sim$ is an equivalence relation in $Q$.

A proof is by a routine argument and is omitted.

######  Definition 2.3.

A set of all representatives with respect to the equivalence relation $\sim$
is simply called a set of representatives for $\mathfrak{F}_{B}$.

Note that a set of representatives exists due to the choice axiom. Note also
that a set of representatives for $\mathfrak{F}_{B}$ is a norming family for
$B$, hence it is a uniqueness set for $B$.

######  Example 2.4.

Let $A$ be a uniform algebra. We assume a uniform algebra as a complex Banach
space here and after. We denote the choquet boundary for $A$ by
$\operatorname{Ch}(A)$. By the Arens-Kelley theorem (cf. [11, p.29]) we have
$Q=\\{\gamma\delta_{x}:x\in\operatorname{Ch}(A),\,\,\gamma\in\mathbb{T}\\}$,
where $\delta_{x}$ denotes the point evaluation at $x$. In this case
$\\{\delta_{x}:x\in\operatorname{Ch}(A)\\}$ is a set of representatives for
$\mathfrak{F}_{A}$.

######  Lemma 2.5.

Let $P$ be a set of representatives. For $F\in\mathfrak{F}_{B}$ there exists a
unique $(p,\lambda)\in P\times\mathbb{T}$ such that $F=\\{a\in
S(B):p(a)=\lambda\\}$. Conversely, for $(p,\lambda)\in P\times\mathbb{T}$ we
have $\\{a\in S(B):p(a)=\lambda\\}$ is in $\mathfrak{F}_{B}$.

###### Proof.

Let $F\in\mathfrak{F}_{B}$. We first prove the existence of $(p,\lambda)\in
P\times\mathbb{T}$ satisfying the condition. There exists $p\in Q$ such that
$F=p^{-1}(1)\cap S(B)$. By the definition, there exists $q\in P$ such that
$p\sim q$. Hence there exists $\gamma\in\mathbb{T}$ such that $F=(\gamma
q)^{-1}(1)\cap S(B)$. Letting $\lambda=\overline{\gamma}$ we have $F=\\{a\in
S(B):q(a)=\lambda\\}$. We prove the uniqueness of $(q,\lambda)$. Suppose that
$\\{a\in S(B):q(a)=\lambda\\}=\\{a\in S(B):q^{\prime}(a)=\lambda^{\prime}\\}$
for $(q^{\prime},\lambda^{\prime})\in P\times\mathbb{T}$. Then we have
$q^{-1}(1)\cap S(B)=(\lambda\overline{\lambda^{\prime}}q^{\prime})^{-1}\cap
S(B)$, that is $q\sim q^{\prime}$. As $P$ is the set of all representatives
with respect to the equivalence relation $\sim$ and $q,q^{\prime}\in P$ we
have $q=q^{\prime}$. It follows that $\lambda=\lambda^{\prime}$.

Conversely, let $(p,\lambda)\in P\times\mathbb{T}$. Then
$\overline{\lambda}p\in Q$ and $\\{a\in
S(B):p(a)=\lambda\\}=(\overline{\lambda}p)^{-1}(1)\cap
S(B)\in\mathfrak{F}_{B}$. ∎

######  Definition 2.6.

For $(q,\lambda)\in Q\times\mathbb{T}$, we denote $F_{q,\lambda}=\\{a\in
S(B):q(a)=\lambda\\}$. A map

$I_{B}:\mathfrak{F}_{B}\to P\times\mathbb{T}$

is defined by $I_{B}(F)=(p,\lambda)$ for $F=F_{p,\lambda}\in\mathfrak{F}_{B}$.

By Lemma 2.5 the map $I_{B}$ is well defined and bijective. An important
theorem of Cheng, Dong and Tanaka states that a surjective isometry between
the unit spheres of Banach spaces preserves maximal convex subsets of the unit
spheres. This theorem was first exhibited by Cheng and Dong in [5, Lemma 5.1]
and a crystal proof was given by Tanaka [23, Lemma 3.5].

In the following $P_{j}$ is a set of representatives for
$\mathfrak{F}_{B_{j}}$ for $j=1,2$. Due to the theorem of Cheng, Dong and
Tanaka a bijection $\mathfrak{T}:\mathfrak{F}_{B_{1}}\to\mathfrak{F}_{B_{2}}$
is induced.

######  Definition 2.7.

The map $\mathfrak{T}:\mathfrak{F}_{B_{1}}\to\mathfrak{F}_{B_{2}}$ is defined
by $\mathfrak{T}(F)=T(F)$ for $F\in\mathfrak{F}_{B_{1}}$. The map
$\mathfrak{T}$ is well defined and bijective. Put

$\Psi=I_{B_{2}}\circ\mathfrak{T}\circ I_{B_{1}}^{-1}:P_{1}\times\mathbb{T}\to
P_{2}\times\mathbb{T}.$

Define two maps

$\phi:P_{1}\times\mathbb{T}\to P_{2}$

and

$\tau:P_{1}\times\mathbb{T}\to\mathbb{T}$

by

(1) $\Psi(p,\lambda)=(\phi(p,\lambda),\tau(p,\lambda)),\,\,(p,\lambda)\in
P_{1}\times\mathbb{T}.$

If $\phi(p,\lambda)=\phi(p,\lambda^{\prime})$ for every $p\in P_{1}$ and
$\lambda,\lambda^{\prime}\in\mathbb{T}$ we simply write $\phi(p)$ instead of
$\phi(p,\lambda)$ by discarding the second term $\lambda$.

Rewriting the equation (1) we have

(2) $T(F_{p,\lambda})=F_{\phi(p,\lambda),\tau(p,\lambda)},\quad(p,\lambda)\in
P_{1}\times\mathbb{T}.$

We point out that

(3) $\phi(p,-\lambda)=\phi(p,\lambda),\,\,\tau(p,-\lambda)=-\tau(p,\lambda)$

for every $(p,\lambda)\in P_{1}\times\mathbb{T}$. The reason is as follows. It
is well known that $T(-F)=-T(F)$ for every $F\in\mathfrak{F}_{B_{1}}$ (cf.
[16, Proposition 2.3]). Hence

$F_{\phi(p,-\lambda),\tau(p,-\lambda)}=T(F_{p,-\lambda})=T(-F_{p,\lambda})=-T(F_{p,\lambda})\\\
=-F_{\phi(p,\lambda),\tau(p,\lambda)}=F_{\phi(p,\lambda),-\tau(p,\lambda)}$

for every $p\in P_{1}$. Since the map $I_{B_{2}}$ is a bijection we have (3).

Rewriting (2) we get a basic equation in our argument :

(4) $\phi(p,\lambda)(T(a))=\tau(p,\lambda),\quad a\in F_{p,\lambda}.$

We will prove that under some condition on $B_{1}$, which we will exhibit
explicitly later, we have that

$\phi(p,\lambda)=\phi(p,\lambda^{\prime}),\quad p\in P_{1}$

for every $\lambda$ and $\lambda^{\prime}$ in $\mathbb{T}$, and

$\tau(p,\lambda)=\tau(p,1)\times\begin{cases}\text{$\lambda$, \quad for some
$p\in P_{1}$}\\\ \text{$\overline{\lambda}$, \quad for other
$p$'s}\end{cases}$

for $\lambda\in\mathbb{T}$. We get, under some condition on $B_{1}$, that

(5) $\phi(p)(T(a))=\tau(p,1)\times\begin{cases}\text{$p(a)$, \quad for some
$p\in P_{1}$}\\\ \text{$\overline{p(a)}$, \quad for other $p$'s}\end{cases}$

for $a\in F_{p,p(a)}$. If the equation (5) holds for any $a\in S(B_{1})$,
without the restriction that $a\in F_{p,p(a)}$, then applying the definition
of $\widetilde{T}$ we get

(6) $\phi(p)(\widetilde{T}(a))=\tau(p,1)\times\begin{cases}\text{$p(a)$, \quad
for some $p\in P_{1}$}\\\ \text{$\overline{p(a)}$, \quad for other
$p$'s}\end{cases}$

for every $a\in B_{1}$, with which we infer that

$\phi(p)(\widetilde{T}(a+rb))=\phi(p)(\widetilde{T}(a))+\phi(p)(r\widetilde{T}(b))$

for every pair $a,b\in B_{1}$ and every real number $r$. As
$\phi(P_{1})=P_{2}$ is a norming family, we conclude that $\widetilde{T}$ is
real-linear. It means that we arrive at the final positive solution for
Tingley’s problem under some conditions on $B_{1}$.

##  3\. Hausdorff distance between the maximal convex subsets

Recall that the Hausdorff distance $d_{H}(K,L)$ between non-empty closed
subsets $K$ and $L$ of a metric space with metric $d(\cdot,\cdot)$ is defined
by

$d_{H}(K,L)=\max\\{\sup_{a\in K}d(a,L),\sup_{b\in L}d(b,K)\\}.$

######  Lemma 3.1.

Let $B$ be a complex Banach space and $P$ a set of representatives for
$\mathfrak{F}_{B}$. We consider $B$ as a metric space induced by the norm. For
every $p\in P$ we have

$d_{H}(F_{p,\lambda},F_{p,\lambda^{\prime}})=|\lambda-\lambda^{\prime}|$

for every $\lambda,\lambda^{\prime}\in\mathbb{T}$. Suppose further that
$p^{\prime}\in P$ is different from $p$. If
$\mathfrak{F}_{p,\gamma}\cap\mathfrak{F}_{p^{\prime},-\gamma^{\prime}}\neq\emptyset$
for some $\gamma,\gamma^{\prime}\in\mathbb{T}$, then we have

$d_{H}(F_{p,\gamma},F_{p^{\prime},\gamma^{\prime}})=2.$

###### Proof.

Let $a\in F_{p,\lambda}$. Then $\lambda^{\prime}\overline{\lambda}a\in
F_{p,\lambda^{\prime}}$. Thus
$d(a,F_{p,\lambda^{\prime}})\leq\|a-\lambda^{\prime}\overline{\lambda}a\|=|\lambda-\lambda^{\prime}|$.
On the other hand, for every $b\in F_{p,\lambda^{\prime}}$ we have
$|\lambda-\lambda^{\prime}|=|p(a)-p(b)|\leq\|a-b\|$. Hence
$|\lambda-\lambda^{\prime}|\leq d(a,F_{p,\lambda^{\prime}})$. Therefore
$d(a,F_{p,\lambda^{\prime}})=|\lambda-\lambda^{\prime}|$ holds for every $a\in
F_{p,\lambda}$. In the similar way we obtain that
$d(b,F_{p,\lambda})=|\lambda-\lambda^{\prime}|$ for every $b\in
F_{p,\lambda^{\prime}}$. We conclude that
$d_{H}(F_{p,\lambda},F_{p,\lambda^{\prime}})=|\lambda-\lambda^{\prime}|$ for
every $\lambda,\lambda^{\prime}\in\mathbb{T}$.

Suppose that $P\ni p^{\prime}\neq p$ and $\mathfrak{F}_{p,\gamma}\cap
F_{p^{\prime}-\gamma^{\prime}}\neq\emptyset$ for some
$\gamma,\gamma^{\prime}\in\mathbb{T}$. Let $a\in\mathfrak{F}_{p,\gamma}\cap
F_{p^{\prime}-\gamma^{\prime}}$. Then for every $b\in
F_{p^{\prime},\gamma^{\prime}}$ we have

$2=|-\gamma^{\prime}-\gamma^{\prime}|=|p^{\prime}(a)-p^{\prime}(b)|\leq\|a-b\|\leq
2.$

Hence $d(a,F_{p^{\prime},\gamma^{\prime}})=2$. It follows that
$d_{H}(F_{p,\gamma},F_{p^{\prime},\gamma^{\prime}})=2$. ∎

Note that the notion of the condition of the Hausdorff distance does not
depend on the choice of $P$. In fact, we can describe the condition applying
the terms of $Q$; the condition of the Hausdorff distance is satisfied by $B$
if and only if

$\displaystyle
d_{H}(F_{q,\lambda},F_{q^{\prime},\lambda^{\prime}})=\begin{cases}|\lambda-\gamma\lambda^{\prime}|,\quad&q^{-1}(1)\cap
S(B)=(\gamma q^{\prime})^{-1}(1)\cap S(B)\\\ 2,&q\not\sim
q^{\prime}\end{cases}$

for $q,q^{\prime}\in Q$.

######  Example 3.2.

Let $A$ be a uniform algebra and
$P=\\{\delta_{x}:x\in\operatorname{Ch}(A)\\}$, where $\delta_{x}$ denotes the
point evaluation at $x$. Then
$\mathfrak{F}_{\delta_{x},\lambda}\cap\mathfrak{F}_{\delta_{x^{\prime}},\lambda^{\prime}}\neq\emptyset$
for any pair of different poins $x$ and $x^{\prime}$ in $\operatorname{Ch}(A)$
and any $\lambda,\lambda^{\prime}\in\mathbb{T}$ [12, Lemma 4.1]. Thus a
uniform algebra satisfies the condition of the Hausdorff distance.

######  Lemma 3.3.

Suppose that $B_{1}$ satisfies the condition of the Hausdorff distance. Let
$P_{1}$ be a set of representatives for $\mathfrak{F}_{B_{1}}$. Then we have
$\phi(p,\lambda)=\phi(p,\lambda^{\prime})$ for every $p\in P_{1}$ and
$\lambda,\lambda^{\prime}\in\mathbb{T}$. Put

$P_{1}^{+}=\\{p\in P_{1}:\tau(p,i)=i\tau(p,1)\\}$

and

$P_{1}^{-}=\\{p\in P_{1}:\tau(p,i)=\bar{i}\tau(p,1)\\}.$

Then $P_{1}^{+}$ and $P_{1}^{-}$ are possibly empty disjoint subsets of
$P_{1}$ such that $P_{1}^{+}\cup P_{1}^{-}=P_{1}$. Furthermore we have

$\tau(p,\lambda)=\lambda\tau(p,1),\quad p\in P_{1}^{+},\lambda\in\mathbb{T}$

and

$\tau(p,\lambda)=\bar{\lambda}\tau(p,1),\quad p\in
P_{1}^{-},\lambda\in\mathbb{T}.$

###### Proof.

First we prove that $\phi(p,\lambda)=\phi(p,\lambda^{\prime})$ for every $p\in
P_{1}$ and $\lambda,\lambda^{\prime}\in\mathbb{T}$. Suppose not: there exist
$p\in P_{1}$ and $\lambda,\lambda^{\prime}\in\mathbb{T}$ such that
$\phi(p,\lambda)\neq\phi(p,\lambda^{\prime})$. We may assume that
$|\lambda-\lambda^{\prime}|<2$. (If
$\phi(p,\lambda)\neq\phi(p,\lambda^{\prime})$ with
$\|\lambda-\lambda^{\prime}|=2$, then $\phi(p,\lambda)\neq\phi(p,i\lambda)$ or
$\phi(p,\lambda^{\prime})\neq\phi(p,i\lambda)$. Replacing $i\lambda$ by
$\lambda^{\prime}$ in the first case, and $i\lambda$ by $\lambda$ in the later
case we have $\phi(p,\lambda)\neq\phi(p,\lambda^{\prime})$
with$|\lambda-\lambda^{\prime}|=\sqrt{2}<2$.) Letting
$\lambda^{\prime\prime}=\lambda^{2}\overline{\lambda^{\prime}}$ we have
$|\lambda-\lambda^{\prime}|=|\lambda-\lambda^{\prime\prime}|$ and
$\lambda^{\prime}\neq\lambda^{\prime\prime}$. Since $B_{1}$ satisfies the
condition of the Hausdorff distance, an element $(q,\alpha)\in
P_{1}\times\mathbb{T}$ which satisfies

(7) $d_{H}(F_{p,\lambda},F_{q,\alpha})=|\lambda-\lambda^{\prime}|$

is only two elements $(p,\lambda^{\prime})$ and $(p,\lambda^{\prime\prime})$.
As $\mathfrak{T}$ preserves the Hausdorff distance we have

$d_{H}(F_{\phi(p,\lambda),\tau(p,\lambda)},F_{\phi(p,\lambda^{\prime}),\tau(p,\lambda^{\prime})})=d_{H}(F_{p,\lambda},F_{p,\lambda^{\prime}})=|\lambda-\lambda^{\prime}|.$

Applying Lemma 3.1 we have

(8)
$d_{H}(F_{\phi(p,\lambda),\tau(p,\lambda)},F_{t,\beta})=|\lambda-\lambda^{\prime}|$

if
$(t,\beta)=(\phi(p,\lambda),\lambda\overline{\lambda^{\prime}}\tau(p,\lambda)),(\phi(p,\lambda),\lambda\overline{\lambda^{\prime\prime}}\tau(p,\lambda)),(\phi(p,\lambda^{\prime}),\tau(p,\lambda^{\prime}))$.
As we suppose that $\phi(p,\lambda)\neq\phi(p,\lambda^{\prime})$, the number
of points $(t,\beta)\in P_{2}\times\mathbb{T}$ which satisfy (8) is at least
three, while the number of points $(q,\alpha)\in P_{1}\times\mathbb{T}$ which
satisfies (7) is two since $B_{1}$ satisfies the condition of the Hausdorff
distance. On the other hand the numbers of $(q,\alpha)$ and $(t,\beta)$ which
satisfy (7) and (8) respectively must coincide each other because
$\mathfrak{T}$ preserves the Hausdorff distance between the maximal convex
subset. We arrive at a contradiction proving that
$\phi(p,\lambda)=\phi(p,\lambda^{\prime})$ for every $p\in P_{1}$ and
$\lambda,\lambda^{\prime}\in\mathbb{T}$. In the following we simply write
$\phi(p)$ instead of $\phi(p,\lambda)$ by discarding the second term.

We prove that $\tau(p,\lambda)=\lambda\tau(p,1)$ or $\bar{\lambda}\tau(p,1)$
for $p\in P_{1}$ and $\lambda\in\mathbb{T}$. Since $\mathfrak{T}$ preserves
the Hausdorff distance we have by Lemma 3.1 that

$|\lambda-1|=d_{H}(F_{p,\lambda},F_{p,1})=d_{H}(F_{\phi(p),\tau(p,\lambda)},F_{\phi(p),\tau(p,1)})\\\
=|\tau(p,\lambda)-\tau(p,1)|=|\tau(p,\lambda)\overline{\tau(p,1)}-1|,$

hence

(9) $\tau(p,\lambda)\overline{\tau(p,1)}=\lambda$ or $\bar{\lambda}$

Thus we have $\tau(p,\lambda)=\lambda\tau(p,1)$ or $\bar{\lambda}\tau(p,1)$.
Letting $\lambda=i$ we infer that $P_{1}^{+}\cup P_{1}^{-}=P_{1}$ and
$P_{1}^{+}\cap P_{1}^{-}=\emptyset$. We have

$2=d_{H}(F_{p,i},F_{p,-i})=d_{H}(F_{\phi(p),\tau(p,i)},F_{\phi(p),\tau(p,-i)})=|\tau(p,i)-\tau(p,-i)|,$

hence $\tau(p,-i)=-\tau(p,i)$ for every $p\in P_{1}$. We show that
$\tau(p,\lambda)=\lambda\tau(p,1)$ for $p\in P_{1}^{+}$ and
$\lambda\in\mathbb{T}$. By (9) we may suppose that
$\operatorname{Im}\lambda\neq 0$. In the case of $\operatorname{Im}\lambda>0$
we have

$|i-\lambda|=d_{H}(F_{p,i},F_{p,\lambda})=d_{H}(F_{\phi(p),\tau(p,i)},F_{\phi(p),\tau(p,\lambda)})\\\
=|\tau(p,i)-\tau(p,\lambda)|=|i\tau(p,1)-\tau(p,\lambda)|=|i-\tau(p,\lambda)\overline{\tau(p,1)}|.$

Applying (9) we infer that $\tau(p,\lambda)\overline{\tau(p,1)}=\lambda$ as
$\operatorname{Im}\lambda>0$, hence $\tau(p,\lambda)=\lambda\tau(p,1)$. Next
we consider the case of $\operatorname{Im}\lambda<0$. We have

$|-i-\lambda|=d_{H}(F_{p,-i},F_{p,\lambda})\\\
=d_{H}(F_{\phi(p),\tau(p,-i)},F_{\phi(p),\tau(p,\lambda)})=|\tau(p,-i)-\tau(p,\lambda)|,$

as $\tau(p,-i)=-\tau(p,i)$ we have

$=|-\tau(p,i)-\tau(p,\lambda)|=|-i\tau(p,1)-\tau(p,\lambda)|=|-i-\tau(p,\lambda)\overline{\tau(p,1)}|.$

As $\operatorname{Im}\lambda<0$, we infer that
$\tau(p,\lambda)=\lambda\tau(p,1)$ by (9).

Similarly, it can be proved that $\tau(p,\lambda)=\bar{\lambda}\tau(p,1)$ if
$p\in P_{1}^{-}$ and $\lambda\in\mathbb{T}$. ∎

##  4\. A sufficient condition for the complex Mazur-Ulam property

We define a set $M_{p,\alpha}$ with which the map $\Psi$ plays a crucial role
to work out the complex Mazur-Ulam property for a uniform algebra. We exhibit
the definition of $M_{p,\alpha}$. We denote
$\bar{\mathbb{D}}=\\{z\in{\mathbb{F}}:|z|\leq 1\\}$, where
$\mathbb{F}=\mathbb{R}$ if the corresponding Banach space is a real one and
$\mathbb{F}=\mathbb{C}$ if the corresponding Banach space is a complex one.

######  Definition 4.1.

Let $B$ be a real or complex Banach space and $P$ a set of representatives for
$\mathfrak{F}_{B}$. For $p\in P$ and $\alpha\in\bar{\mathbb{D}}$ we denote

$M_{p,\alpha}=\\{a\in S(B):d(a,F_{p,\alpha/|\alpha|})\leq
1-|\alpha|,d(a,F_{p,-\alpha/|\alpha|})\leq 1+|\alpha|\\},$

where we read $\alpha/|\alpha|=1$ if $\alpha=0$.

######  Lemma 4.2.

If $B_{j}$ is a real Banach space for $j=1,2$, then we have

$T(M_{p,\alpha})=M_{\phi(p),\alpha\tau(p,1)}$

for every $(p,\alpha)\in P_{1}\times\mathbb{T}$. If $B_{j}$ is a complex
Banach space which satisfy the condition of the Hausdorff distance for
$j=1,2$, then we have

$T(M_{p,\alpha})=\begin{cases}M_{\phi(p),\alpha\tau(p,1)},\quad p\in
P_{1}^{+}\\\ M_{\phi(p),\overline{\alpha}\tau(p,1)},\quad p\in
P_{1}^{-}\end{cases}$

for every $(p,\alpha)\in P_{1}\times\mathbb{T}$.

###### Proof.

Due to the definition of the map $\Psi$ we have

$T(F_{p,\frac{\alpha}{|\alpha|}})=F_{\phi(p,\frac{\alpha}{|\alpha|}),\tau(p,\frac{\alpha}{|\alpha|})}$

and

$T(F_{p,-\frac{\alpha}{|\alpha|}})=F_{\phi(p,-\frac{\alpha}{|\alpha|}),\tau(p,-\frac{\alpha}{|\alpha|})}$

Suppose that $B_{j}$ is a real Banach space first. Then by the definition
$\mathbb{T}=\\{\pm 1\\}$. By (3) we have $\phi(p,1)=\phi(p,-1)$ for every
$p\in P_{1}$. Hence $\phi(p,\lambda)$ does not depend on the second term for a
real Banach space. We also have $\tau(p,-1)=-\tau(p,1)$ for every $p\in P_{1}$
by (3). It follows that

$T(F_{p,\frac{\alpha}{|\alpha|}})=F_{\phi(p),\frac{\alpha}{|\alpha|}\tau(p,1)}$

and

$T(F_{p,-\frac{\alpha}{|\alpha|}})=F_{\phi(p),-\frac{\alpha}{|\alpha|}\tau(p,1)}$

As $T$ is a surjective isometry we have

$d(a,F_{p,\frac{\alpha}{|\alpha|}})=d(T(a),F_{\phi(p),\frac{\alpha}{|\alpha|}\tau(p,1)})$

and

$d(a,F_{p,-\frac{\alpha}{|\alpha|}})=d(T(a),F_{\phi(p),-\frac{\alpha}{|\alpha|}\tau(p,1)})$

As $T$ is a bijection we conclude that

$T(M_{p,\alpha})=M_{\phi(p),\alpha\tau(p,1)}$

for every $p\in P_{1}$ and $\alpha\in\bar{\mathbb{D}}$.

Suppose next that $B_{j}$ is a complex Banach space which satisfies that the
condition of the Hausdorff distance. Let $p\in P_{1}^{+}$. Then by Lemma 3.3
we have

$T(F_{p,\frac{\alpha}{|\alpha|}})=F_{\phi(p),\frac{\alpha}{|\alpha|}\tau(p,1)}$

and

$T(F_{p,-\frac{\alpha}{|\alpha|}})=F_{\phi(p),-\frac{\alpha}{|\alpha|}\tau(p,1)}.$

The left of the proof is similar to the case that $B_{j}$ is a real Banach
space, hence we see that

$T(M_{p},\alpha)=M_{\phi(p),\alpha\tau(p,1)}$

for every $p\in P_{1}^{+}$ and $\alpha\in\bar{\mathbb{D}}$. The proof for the
case of $p\in P_{1}^{-}$ is similar and is omitted. ∎

######  Lemma 4.3.

Suppose that $B$ is a real or complex Banach space and $P$ is a set of
representatives for $\mathfrak{F}_{B}$. For every $p\in P$ and
$\alpha\in\mathbb{D}$ we have

$M_{p,\alpha}\subset\\{a\in S(B):p(a)=\alpha\\}.$

###### Proof.

Let $a\in M_{p,\alpha}$. Then for arbitrary $b\in
F_{p,\frac{\alpha}{|\alpha|}}$

$\left|p(a)-\frac{\alpha}{|\alpha|}\right|=|p(a)-p(b)|\leq\|a-b\|.$

As $d(a,F_{p,\frac{\alpha}{|\alpha|}})\leq 1-|\alpha|$

$\left|p(a)-\frac{\alpha}{|\alpha|}\right|\leq
d(a,F_{p,\frac{\alpha}{|\alpha|}})\leq 1-|\alpha|.$

In the same way we have

$\left|p(a)-\left(-\frac{\alpha}{|\alpha|}\right)\right|\leq
d(a,F_{p,-\frac{\alpha}{|\alpha|}})\leq 1+|\alpha|.$

Then by the two inequalities $p(a)$ have to be $\alpha$. ∎

The following is an auxiliary result.

######  Proposition 4.4.

Let $B_{1}$ be a complex Banach space and $P_{1}$ a set of representatives for
$\mathfrak{F}_{B_{1}}$. Assume the following two conditions:

  * i)

$B_{1}$ satisfies the condition of the Hausdorff distance,

  * ii)

$M_{p,\alpha}=\\{a\in S(B_{1}):p(a)=\alpha\\}$ for every $p\in P_{1}$ and
$\alpha\in\mathbb{D}$.

Then $B_{1}$ satisfies the complex Mazur-Ulam property.

###### Proof.

Suppose that $B_{2}$ is a complex Banach space and $T:B_{1}\to B_{2}$ a
surjective isometry. Applying Lemma 3.3 for the equation (4) we get

$\phi(p)(T(a))=\tau(p,1)\times\begin{cases}p(a),\quad&p\in P_{1}^{+}\\\
\overline{p(a)},\quad&p\in P_{1}^{-},\end{cases}$

for $a\in S(B_{1})$ such that $|p(a)|=1$. We prove that

$\phi(p)(T(a))=\tau(p,1)\times\begin{cases}p(a),\quad&p\in P_{1}^{+}\\\
\overline{p(a)},\quad&p\in P_{1}^{-}\end{cases}$

for any $a\in S(B_{1})$. Let $a\in S(B_{1})$ and $\alpha=p(a)$. Then
$|\alpha|\leq 1$. By condition ii), $a\in M_{p,\alpha}$ and $T(a)\in
T(M_{p,\alpha})$. As i) is assumed, we have by Lemma 4.2 that

$T(M_{p,\alpha})=\begin{cases}M_{\phi(p),\alpha\tau(p,1)},\quad p\in
P_{1}^{+}\\\ M_{\phi(p),\overline{\alpha}\tau(p,1)},\quad p\in
P_{1}^{-}.\end{cases}$

Therefore

$\phi(p)(T(a))=\alpha\tau(p,1)=p(a)\tau(p,1)$

if $p\in P_{1}^{+}$. In a similar way we have

$\phi(p)(T(a))=\overline{p(a)}\tau(p,1)$

if $p\in P_{1}^{-}$. We conclude that

$\phi(p)(T(a))=\tau(p,1)\times\begin{cases}p(a),\quad&p\in P_{1}^{+}\\\
\overline{p(a)},\quad&p\in P_{1}^{-}\end{cases}$

for every $a\in S(B_{1})$. Let $c\in B_{1}$ and $c\neq 0$. Since
$\frac{c}{\|c\|}\in S(B_{1})$, and $\phi(p)$ and $p$ are real-linear we have

$\displaystyle\phi(p)(\widetilde{T}(c))$
$\displaystyle=\phi(p)\left(\|c\|T\left(\frac{c}{\|c\|}\right)\right)=\|c\|\tau(p,1)\times\begin{cases}p\left(\frac{c}{\|c\|}\right),\quad
p\in P_{1}^{+}\\\ \overline{p\left(\frac{c}{\|c\|}\right)},\quad p\in
P_{1}^{-}\end{cases}$
$\displaystyle=\tau(p,1)\times\begin{cases}p(c),\quad&p\in P_{1}^{+}\\\
\overline{p(c)},\quad&p\in P_{1}^{-}\end{cases}$

We infer that

$\phi(p)(\widetilde{T}(a+rb))=\tau(p,1)\times\begin{cases}p(a+rb),\quad p\in
P_{1}^{+}\\\ \overline{p(a+rb)},\quad p\in P_{1}^{-}\end{cases}\\\
=\tau(p,1)\times\begin{cases}p(a),\quad p\in P_{1}^{+}\\\
\overline{p(a)},\quad p\in
P_{1}^{-}\end{cases}+\tau(p,1)\times\begin{cases}rp(b),\quad p\in P_{1}^{+}\\\
r\overline{p(b)},\quad p\in P_{1}^{-}\end{cases}\\\
=\phi(p)(\widetilde{T}(a)+r\widetilde{T}(b))$

for every pair $a,b\in B_{1}$ and every real number $r$. As
$\phi(P_{1})=P_{2}$ is a norming family we conclude that

$\widetilde{T}(a+rb)=\widetilde{T}(a)+r\widetilde{T}(b)$

for every pair $a,b\in$ and every real number $r$. By the definition of
$\widetilde{T}$ it is a bijection from $B_{1}$ onto $B_{2}$ and it satisfies
the equality $\|\widetilde{T}(a)\|=\|a\|$ for every $a\in B_{1}$. Thus
$\widetilde{T}$ is a surjective real-linear isometry from $B_{1}$ onto $B_{2}$
which extend $T$. ∎

The following is the main result in this paper.

######  Theorem 4.5.

A uniform algebra satisfies the complex Mazur-Ulam property.

###### Proof.

Let $A$ be a uniform algebra. Put $P=\\{\delta_{x}:x\operatorname{Ch}(A)\\}$,
where $\delta_{x}$ is the point evaluation at $x\in\operatorname{Ch}(A)$, the
Choquet boundary. Then $P$ is a set of representatives for $\mathfrak{F}_{A}$.
It is known that a uniform algebra satisfies the condition of the Hausdorff
distance [12, Lemma 4.1]. By [12, Lemma 6.3] we have
$M_{\delta_{x},\alpha}=\\{f\in S(A):f(x)=\alpha\\}$ for every
$x\in\operatorname{Ch}(A)$ and $\alpha\in\bar{\mathbb{D}}$. Thus the
conditions i) and ii) of Proposition 4.4 holds for $A$. Thus $A$ satisfies the
complex Mazur-Ulam property by Proposition 4.4. ∎

##  5\. The case of a real Banach space

Throughout the section we denote $B_{j}$ a real Banach space, $P_{j}$ a set of
representatives for $\mathfrak{F}_{B_{j}}$ and $T:S(B_{1})\to S(B_{2})$ is a
surjective isometry. We have by (3) that $\phi(p,1)=\phi(p,-1)$ for every
$p\in P_{1}$. As $\mathbb{T}=\\{\pm 1\\}$ for real Banach spaces we have that
$\phi(p,\lambda)$ does not depend the second term for real Banach spaces. We
also have $\tau(p,-1)=-\tau(p,1)$ for every $p\in P_{1}$ by (3). The situation
is rather simple than the case of complex Banach spaces, and by (4) we have
the following equation (10) without further assumption on $B_{1}$, i.e.,

(10) $\phi(p)(T(a))=\tau(p,1)p(a)$

for every $p\in P_{1}$ and $a\in S(B_{1})$ with $|p(a)|=1$.

######  Proposition 5.1.

Suppose that

(11) $M_{p,\alpha}=\\{a\in S(B_{1}):p(a)=\alpha\\}$

for every $p\in P$ and $-1\leq\alpha\leq 1$. Then $T$ is extended to a
surjective real-linear isometry form $B_{1}$ onto $B_{2}$. Hence $B_{1}$
satisfies the Mazur-Ulam property.

###### Proof.

We first prove the equation (10) for every $p\in P_{1}$ and $a\in S(B_{1})$
without assuming that $|p(a)|=1$. Let $p\in P_{1}$ and $a\in S(B_{1})$. Put
$\alpha=p(a)$. Then by (11) $a\in M_{p,\alpha}$. We have by Lemma 4.2 that

$\phi(p)(T(a))=\alpha\tau(p,1)=\tau(p,1)p(a).$

It follows that for the homogeneous extension $\widetilde{T}$ of $T$ we have

$\phi(p)(\widetilde{T}(c))=\phi(p)\left(\|c\|T\left(\frac{c}{\|c\|}\right)\right)\\\
=\|c\|\tau(p,1)p\left(\frac{c}{\|c\|}\right)=\tau(p,1)p(c)$

for every $0\neq c\in B_{1}$. As the equality
$\phi(p)(\widetilde{T}(0))=\tau(p,1)p(0)$ holds, we obtain for $a,b\in B_{1}$
and a real number $r$ that

$\phi(p)(\widetilde{T}(a+rb)=\tau(p,1)p(a+rb)=\tau(p,1)p(a)+r\tau(p,1)p(b)$

and

$\phi(p)(\widetilde{T}(a)+r\widetilde{T}(b))=\phi(p)(\widetilde{T}(a))+r\phi(p)(\widetilde{T}(b))\\\
=\tau(p,1)p(a)+r\tau(p,1)p(b).$

It follows that

$\phi(p)(\widetilde{T}(a+rb))=\phi(p)(\widetilde{T}(a)+r\widetilde{T}(b))$

for every $p\in P_{1}$, $a,b\in B_{1}$, and every real number $r$. As
$\phi(P_{1})=P_{2}$ is a norming family we see that $\widetilde{T}$ is real-
linear on $B_{1}$. As the homogeneous extension is a norm-preserving bijection
as is described in the first parat of section 2 we complete the proof. ∎

######  Definition 5.2.

For a locally compact Hausdorff space, we denote by $C_{0}(Y,{\mathbb{R}})$ a
real Banach space of all real-valued continuous functions which vanish at
infinity on $Y$. Let $E$ be a closed subspace of $C_{0}(Y,\mathbb{R})$ which
separates the points of $Y$, that is, for any pair $y_{1}$ and $y_{2}$ of
different points in $Y$ there exists a function $e\in E$ such that
$e(y_{1})\neq e(y_{2})$. In this paper we say that $E$ satisfies the condition
$(r)$ if for any triple $y\in\operatorname{Ch}(E)$, a neighborhood $V$, and
$\varepsilon>0$ there exists $u\in E$ such that $0\leq u\leq 1=u(y)$ on $Y$
and $0\leq u\leq\varepsilon$ on $Y\setminus V$.

Note that if $E$ satisfies the condition $(r)$, then it is extremely
$C$-regular (cf. [11, Definition 2.3.9]).

######  Example 5.3.

The space $C_{0}(Y,\mathbb{R})$ satisfies the condition $(r)$ for any locally
compact Hausdorff space $Y$. Let

$E=\\{f\in C_{0}((0,2],\mathbb{R}):\text{$f(t)=at$ on $(0,1]$ for some
$a\in\mathbb{R}$}\\}.$

Then $\operatorname{Ch}(E)=[1,2]$ and $E$ satisfies the condition $(r)$.

Liu [15, Corollary 6] established the Mazur-Ulam property for
$C_{0}(Y,\mathbb{R})$ when $Y$ is compact. The following generalizes the
result of Liu.

######  Corollary 5.4.

Let $Y$ be a locally compact Hausdorff space. Suppose that $E$ is a closed
subspace of $C_{0}(Y,\mathbb{R})$ which separates the points of $Y$. Suppose
that $E$ satisfies the condition $(r)$. Then $E$ satisfies the Mazur-Ulam
property. In particular, $C_{0}(Y,\mathbb{R})$ satisfies the Mazur-Ulam
property.

###### Proof.

Let $\operatorname{Ch}(E)$ be the Choquet boundary for $E$. Then by the Arens-
Kelley theorem $P=\\{\delta_{x}:x\in\operatorname{Ch}(E)\\}$ is a set of
representatives for $\mathfrak{F}_{E}$. Suppose that $p\in P$ and
$-1\leq\alpha\leq 1$. Proving the inclusion

$M_{p,\alpha}\supset\\{a\in S(E):p(a)=\alpha\\},$

we get (11) by Lemma 4.3. It will follw by Proposition 5.1 that $E$ satisfies
the Mazur-Ulam property. Let $a\in S(B_{1})$ such that $p(a)=\alpha$. First we
consider the case of $\alpha=1$. By definition $a\in F_{p,1}$, hence
$d(a,F_{p,1})=0=1-|\alpha|$. For every $b\in F_{p,-1}$ we have
$2=|p(a)-p(b)|\leq\|a-b\|\leq 2$. Hence $d(a,F_{p,-1})=2=1+|\alpha|$. Thus
$a\in M_{p,\alpha}$ if $\alpha=1$. A proof for $\alpha=-1$ is similar and is
omitted.

We consider the case that $-1<\alpha<1$. Let $a\in S(B_{1})$ such that
$p(a)=\alpha$. Let $\varepsilon$ be $0<\varepsilon<1-|\alpha|$. Put $K=\\{q\in
P:q(a)=\alpha\\}$ and $F_{0}=\\{q\in P:|q(a)-\alpha|\geq\varepsilon/4\\}$. For
each positive integer $n$, put $F_{n}=\\{q\in
P:\varepsilon/2^{n+2}\leq|q(a)-\alpha|\leq\varepsilon/2^{n+1}\\}$. Then
$P=K\cup\left(\cup_{m=0}^{\infty}F_{m}\right)$. For each positive integer $n$,
choose $u_{n}\in S(B_{1})$ such that $0\leq u_{n}\leq 1=p(u_{n})$ on $Y$ and
$u_{n}=0$ on $F_{0}\cup F_{n}$. Since
$P=\\{\delta_{x}:x\in\operatorname{Ch}(E)\\}$ and $E$ satisfies the condition
$(r)$ such $u_{n}\in E$ exists. Put $u=\sum_{n=1}^{\infty}u_{n}/2^{n}$. As the
supremum norm of $u_{n}$ is dominated by $1$ for every $n$, $u$ converges
uniformly on $Y$ in $E$. Then put

$g_{+}=\left(\frac{\alpha}{|\alpha|}-\alpha\right)u+a,$

and

$g_{-}=\left(-\frac{\alpha}{|\alpha|}-\alpha\right)u+a.$

We see that $g_{+}\in F_{p,\frac{\alpha}{|\alpha|}}$,
$\|g_{+}-a\|=1-|\alpha|$, and $d(a,F_{p,\frac{\alpha}{|\alpha|}})\leq
1-|\alpha|$. A proof is as follows. All what is really needed is to prove that
$|q(g_{-})|\leq 1$ for every $q\in P$ as it is evident that $g_{+}\in E$ and
$p(g_{+})=\frac{\alpha}{|\alpha|}$ by the definition of $g_{+}$. If $q\in
F_{0}$, then $q(u)=0$ asserts that $|q(g_{+})|=|q(a)|\leq 1$. Suppose that
$q\in F_{n}$ for some positive integer $n$. Then $q(u_{n})=0$ and $0\leq
u_{k}\leq 1$ on $Y$ we have $|q(u)|\leq\sum_{k\neq n}1/2^{k}=1-1/2^{n}$. Thus

$\displaystyle|q(g_{+})|$
$\displaystyle\leq\left|\frac{\alpha}{|\alpha|}-\alpha\right|(1-1/2^{n})+|q(a)-\alpha|+|\alpha|$
$\displaystyle\leq(1-|\alpha|)(1-1/2^{n})+\varepsilon/2^{n+1}+|\alpha|$
$\displaystyle\leq(1-|\alpha|)(1-1/2^{n})+(1-|\alpha|)/2^{n+1}+|\alpha|\leq 1$

Suppose that $q\in K$. Then

$|q(g_{+})|\leq 1-|\alpha|+|\alpha|=1.$

We conclude that $g_{+}\in F_{p,\frac{\alpha}{|\alpha|}}$. By the definition
of $g_{+}$, we infere that $\|g_{+}-a\|=(1-|\alpha|)\|u\|=1-|\alpha|$. It
follows that

(12) $d(a,F_{p,\frac{\alpha}{|\alpha|}})\leq 1-|\alpha|.$

We also see that $g_{-}\in F_{p,-\frac{\alpha}{|\alpha|}}$,
$\|g_{-}-a\|=1+|\alpha|$ and $d(a,F_{p,-\frac{\alpha}{|\alpha|}})\leq
1+|\alpha|$. A proof is as follows. All what is really needed is to prove that
$|q(g_{-})|\leq 1$ for every $q\in P$ as it is evident that $g_{-}\in E$ and
$p(g_{-})=-\frac{\alpha}{|\alpha|}$. If $q\in F_{0}$, then $q(u)=0$ asserts
that $|q(g_{+})|=|q(a)|\leq 1$. Suppose that $q\in F_{n}$ for some positive
integer $n$. We have

$q(g_{-})=\left(-\frac{\alpha}{|\alpha|}+\alpha\right)q(u)-2\alpha
q(u)+\alpha+q(a)-\alpha,$

hence

$|q(g_{-})|\leq(1-|\alpha|)|q(u)|+|\alpha||1-2q(u)|+|q(a)-\alpha|.$

As $0\leq u\leq 1$ on $Y$ by the definition of $u$, we have $|1-2q(u)|\leq 1$
for every $q\in P=\\{\delta_{x}:x\in\operatorname{Ch}(E)\\}$. As we have shown
that $|q(u)|\leq 1-1/2^{n}$ for $q\in F_{n}$

$\displaystyle|q(g_{-})$
$\displaystyle|\leq(1-|\alpha|)|q(u)|+|\alpha|+|q(a)-\alpha|$
$\displaystyle\leq(1-|\alpha|)(1-1/2^{n})+|\alpha|+\varepsilon/2^{n+1}$
$\displaystyle\leq(1-|\alpha|)(1-1/2^{n})+|\alpha|+(1-|\alpha|)/2^{n+1}\leq
1.$

Suppose that $q\in K$. Then we have

$q(g_{-})=\left(-\frac{\alpha}{|\alpha|}+\alpha\right)q(u)-2\alpha
q(u)+\alpha.$

As $0\leq u\leq 1$ on $Y$ we have

$|q(g_{-})|\leq 1-|\alpha|+|\alpha||1-2q(u)|\leq 1.$

We conclude that $g_{-}\in F_{p,-\frac{\alpha}{|\alpha|}}$. By the definition
of $g_{-}$, we infer that $\|g_{-}-a\|=(1+|\alpha|)\|u\|=1+|\alpha|$. It
follows that

(13) $d(a,F_{p,-\frac{\alpha}{|\alpha|}})\leq 1+|\alpha|.$

By (12) and (13) we conclude that $a\in M_{p,\alpha}$.

Finally, as $C_{0}(Y,\mathbb{R})$ satisfies the condition $(r)$ by Urysohn’s
lemma, we see that $C_{0}(Y,\mathbb{R})$ satisfies the Mazur-Ulam property. ∎

######  Example 5.5.

Let $\ell^{\infty}(\Gamma,\mathbb{R})$ be the real Banach space of all real-
valued bounded functions on a discrete space $\Gamma$. Then there is a compact
Hausdorff space $X$ such that $\ell^{\infty}(\Gamma,\mathbb{R})$ is
isometrically isomorphic to $C_{0}(X,\mathbb{R})$ as a real Banach space.
Therefore $\ell^{\infty}(\Gamma,\mathbb{R})$ satisfies the Mazur-Ulam property
(cf. [10, 15]). The Mazur-Ulam property of the space
$c_{0}(\Gamma,\mathbb{R})$ is established by Ding[10, Corollary 2]. As
$c_{0}(\Gamma,\mathbb{R})$ is isometrically isomorphic to
$C_{0}(Y,\mathbb{R})$ for a locally compact Hausdorff space $Y$, Corollary 5.4
gives an alternative proof of the result of Ding.

######  Example 5.6.

Let $\mu$ be a positive measure on a $\sigma$-algebra $\Sigma$ of a subsets of
a set $\Omega$. Let $L^{\infty}(\Omega,\Sigma,\mu,\mathbb{R})$ be the usual
real Banach space of all real-valued bounded measurable functions on
$(\Omega,\Sigma,\mu)$, Then there exists a compact Hausdorff space $X$ such
that $L^{\infty}(\Omega,\Sigma,\mu,\mathbb{R})$ is isometrically isomorphic to
$C_{0}(X,\mathbb{R})$ as a real Banach space. Hence the space
$L^{\infty}(\Omega,\Sigma,\mu,\mathbb{R})$ satisfies the Mazur-Ulam property
(cf. [15, Corollary 6]). Note that Tan [19, Theorem 2.5] exhibits the result
for the case of the measure being $\sigma$-finite by an alternative proof.

##  6\. Remarks

We close the paper with a few remarks. In this paper we merely prove the
complex Mazur-Ulam property for a uniform algebra. We conjecture that a
uniform algebra, generally a closed subalgebra of $C_{0}(Y,\mathbb{C})$,
satisfies the Mazur-Ulam property.

The second remark concerns Tingley’s problem on a Banach space of analytic
functions. The main result of this paper concerns with the complex Mazur-Ulam
property. We expect that several Banach space of analytic functions including
the Hardy spaces satisfy the complex Mazur-Ulam property and the Mazur-Ulam
property.

As a final remark we encourage researches on Tingley’s problem on Banach
algebras of continuous functions. Comparing with the theorem of Wang [27] on
the Banach algebra of $C^{(n)}(X)$, it is interesting to study a surjective
isometry on the unit sphere of a Banach space or algebra of Lipschitz
functions, it has already been pointed out by Cueto-Avellaneda [6, Problem
4.0.8].

### Acknowledgments

The author was supported by JSPS KAKENHI Grant Numbers JP19K03536.

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