# MusicTraces: A collaborative music and paint activity for autistic people

Valentin Bauer<EMAIL_ADDRESS>0000-0002-3922-7507 Politecnico di
Milano, Department of Electronics, Information and BioengineeringMilanItaly ,
Tommaso Padovano  Politecnico di Milano, Department of Electronics,
Information and BioengineeringMilanItaly ,  Mattia Gianotti  Politecnico di
Milano, Department of Electronics, Information and BioengineeringMilanItaly ,
Giacomo Caslini  Politecnico di Milano, Department of Electronics, Information
and BioengineeringMilanItaly and  Franca Garzotto  Politecnico di Milano,
Department of Electronics, Information and BioengineeringMilanItaly

(2024)

###### Abstract.

Painting and music therapy approaches can help to foster social interaction
for autistic people. However, the tools sometimes lack of flexibility and fail
to keep people’s attention. Unknowns also remain about the effect of combining
these approaches. Though, very few studies have investigated how Multisensory
Environments (MSEs) could help to address these issues. This paper presents
the design of a full-body music and painting activity called “MusicTraces”
which aims to foster collaboration between people with moderate to severe
learning disabilities and complex needs, and in particular autism, within an
MSE. The co-design process with caregivers and people neurodevelopmental
conditions is detailed, including a workshop, the initial design, remote
iterations, and a design critique.

Multisensory Environments, Art Therapy, Music, Paint, Autism,
Neurodevelopmental Conditions

††copyright: acmlicensed††journalyear: 2024††doi: XXXXXXX.XXXXXXX††conference:
CHI 2024; May 11–16, 2024; Honolulu, Hawaii, USA††isbn:
978-1-4503-XXXX-X/18/06††ccs: Human-centered computing Collaborative and
social computing devices††ccs: Human-centered computing User centered
design††ccs: Human-centered computing Accessibility technologies††ccs: Applied
computing Media arts††ccs: Social and professional topics People with
disabilities

Figure 1. Main design features of the activity MusicTraces. (a.) Front User
Interface (UI). 1: Temporary Line. 2: Permanent line (with a sound). 3:
Permanent line (without sound). 4: Permanent dot (i.e., sound). 5: Animated
dot (i.e., cursor). 6: Hand of red player (b.) Floor UI: 7: player position.
8: Music blob (added after design phase 3). 9: Smart brushes or erasers being
detected as held by the users. 10: Floor area to trigger the background music.
11: Floor spots to control the music evolution. (c.) Two users playing
together in the room. 12: Line crossing blending their colors. 13. Smart
brush.

Application MusicTraces

##  1\. Introduction

Autism is a Neurodevelopmental Condition (NC) which involves social
communication and interaction difficulties and sensory issues (American
Psychiatric Association, 2013). Autistic people111This paper adopts autism
stakeholders’ language preferences (Bottema-Beutel et al., 2021), e.g.,
identity first-language (e.g., autistic people), no offending terms (e.g.,
“disorder”). can display mild learning disabilities and low support needs
(e.g., difficulty initiating conversation), or severe learning disabilities
and high support needs (e.g., minimal language), sometimes associated with
intellectual disability (ID). The latter are little represented in current
research (Russell et al., 2019). Creative practices can help to foster social
interaction and self-esteem for people with all kinds of abilities. Painting
and music improvised practices are common, and can benefit to people with
severe learning disabilities (Mayer-Benarous et al., 2021). Yet, the tools
sometimes lack of flexibility to adapt to people’s sensory issues, and
unknowns remain about the potential of combining several art practices
(Bernier et al., 2022).

Digital tools help to tackle these issues, as being flexible, predictable, and
often appealing for autistic people (Laurie et al., 2019; Mazurek et al.,
2015). More specifically, Multisensory Smart Environments (MSEs) are promising
to promote well-being and social interaction (Crowell et al., 2020; Ringland
et al., 2014; Pares et al., 2005; Garzotto et al., 2020; Brown et al., 2016).
They consist of full-body interactive spaces with multisensory stimuli (e.g.,
visuals, audio, tactile).

This paper presents the design of a full-body music and painting activity
called “MusicTraces” which aims to foster collaboration within an MSE between
young adults with moderate to severe learning disabilities and complex needs,
and in particular autism. The MSE consists of a room augmented with floor and
front visual projections, speakers, smart objects and lights, and a Microsoft
Kinect 2. Below, we report on our co-design process with caregivers and people
neurodevelopmental conditions, which includes a workshop, the initial design,
remote iterations, and a design critique.

##  2\. Related Work

Several digitally-augmented multisensory settings with full-body interactive
capabilities (e.g., smart objects, lights) have been created with clinical
teams, with positive outcomes regarding relational aspect for autistic
children (Crowell et al., 2020; Mora-Guiard et al., 2017; Ringland et al.,
2014; Pares et al., 2005; Garzotto et al., 2020; Cappelen and Andersson,
2016b). Some projects, like Mediate (Pares et al., 2005), Lands of Fogs (Mora-
Guiard et al., 2017; Crowell et al., 2020), or RHYME (Cappelen and Andersson,
2016b), promote multiplayer free exploration of the space. Others like the
Magic Room (Garzotto et al., 2020) or Sensory Paint (Ringland et al., 2014)
are task-oriented and focus on educative goals.

Creative approaches within MSEs have previously been designed for people with
NDCs (Cappelen and Andersson, 2016b; Ringland et al., 2014). Sensory Paint is
the only project combining music and paint to our knowledge. While focusing
more on sonifying the painting experience, it is promising to promote social
interaction (Ringland et al., 2014). RHYME consists of various sub projects
including musical smart objects (e.g., puppets) affording various interactions
(e.g., microphone), with benefits over social aspects and well-being (Cappelen
and Andersson, 2016b).

Other multisensory projects have been designed for autistic children outside
of MSEs with similar objectives (Cibrian et al., 2017; Ragone et al., 2022).
Bendable Sound allows to play music by touching an elastic display, with
benefits over attention and motor development. With OSMoSIS, children can play
music with their caregiver using their body, to support interactional
synchrony (Ragone et al., 2022).

Most of the above-mentioned projects consider music as a way of promoting
relational aspects within health settings, also called health musicking
(Stige, 2006; Stensæth, 2013). This concept entails multiple ways of
experiencing, rather than right or wrong way of playing, closed to the concept
of open work (Stensæth and Eide, 2017; Eco, 1989). In this context, the system
has an active role to support exploration and the co-creative experience
(e.g., with hints) (Stensæth, 2013). In particular, the interactive music
composition must adapt and evolve based on user actions, such as in the
Reactable project (Jordà et al., 2007; Villafuerte et al., 2012) or RHYME
(Cappelen and Andersson, 2016a).

At last, to cater for stakeholders’ needs, the projects must be co-designed
with them (Parsons et al., 2020). This process entails common design
principles, e.g., support understanding by structuring the information or
using video modelling techniques (Bellini and Akullian, 2007).

##  3\. Phase 1 - Co-design workshop

The design process of MusicTraces started with a one-hour workshop organized
with five caregivers and three people with neurodevelopmental conditions
(NDCs) and Intellectual Disability (ID). It aimed to brainstorm ideas about
how to best combine music and painting activities within an MSE to foster
collaboration between people with moderate to severe NDC, such as autism. It
also intended to validate the consistency of our objective. Three
investigators were present: an animator who conducted the session, a secretary
who sorted the emerging ideas, and an observer taking notes. After the
workshop, the emerging ideas were discussed with an external psychologist with
experience in autism.

###  3.1. Participants

Eight participants were recruited (5 males, 3 females): seven from a music
association for people with disabilities and one from our network. They
include: the head of the association (H), two educators (Ed1 and Ed2), two
psychologists (P1, P2), and three people with NDC and ID (PwD1, PwD2, PwD3).
PwD1 (23 year-old male) has an Adams-Oliver Disorder inducing ID and lacks of
autonomy. PwD2 (44 year-old male) and PwD3 (32 year-old female) both have
neurodevelopmental conditions inducing ID, and social and motor issues. The
external psychologist (M), who then reviewed the emerging themes, has been
working for more than ten years with autistic people at the clinical institute
called F222Anonymized for the submission. Participants participated
voluntarily without being paid after signing consent forms.

###  3.2. Protocol

After the participants were introduced to the investigators, they tried some
non-creative activities within the MSE of our laboratory (e.g., storytelling)
during twenty minutes. Then, they went to a room cleaned from distracting
stimuli (e.g., noise), where paint material was added to foster idea
generation (i.e., brushes). The animator outlined the workshop’s rationale,
organization, and rules: (1) generate as many ideas as you can, (2) do not
judge ideas, and (3) feel free to express unusual ideas. Participants used
“concept sheets” to write or draw their ideas, thus accommodating for diverse
abilities. If possible, for each idea, they noted the best and worst aspects
to obtain more details and promote creativity.

The brainstorming session involved two 30-minute rounds, first without and
then with cards. Both started individually, to note at least one idea without
being influenced by others. The participants then shared their idea(s) with
others, while also being able to suggest other ideas. The secretary noted the
ideas on a Miro board333Miro application: https://miro.com/fr/ projected on
the wall, and sorted them based on the discussions, inspired from techniques
using sticky notes (Mackay, 2020). During the first part, the participants
could take inspiration from manipulating the paint material. Then, they could
use 20 cards that we previously designed, inspired from previous studies
(Benton et al., 2014; Scheepmaker et al., 2021). Each card bore one design
principle and one example of how to apply it. The ten green cards were design
heuristics, being picked from an preexisting set (Kramer et al., 2014). They
stated general principles such as “Allow user to rearrange”. The ten blue
cards were design principles related to the activity, inspired from previous
studies (Scheepmaker et al., 2021). For instance, they included “Think about
other music possibilities”. Examples of cards are visible on Figure 2.

The workshop ended with concluding thoughts. The participants were thanked,
asked if agreeing on participating in future testing, and given gift bags.
Then, they could share a snack with the lab members and try other
applications.

Figure 2. Examples of the material used during the co-design workshop. (a.)
Example of activity-related card, (b.) Example of design heuristic card, (c.)
Example of answers on a “concept sheet” from a participant with
neurodevelopmental condition (PwD1).

Examples of the 20 cards created for the co-design workshop.

Sessions were filmed. Two authors (secretary and observer) analyzed the data
using thematic analysis (Braun and Clarke, 2006), with deductive techniques
for themes already existing in the literature (e.g., understanding) and
inductive techniques to create new ones. Then, they discussed and precised the
themes with a third author (the animator) and (M). The identifiers of the
themes and the number of participants mentioning them are noted Tx and y/8
(where x and y are numbers).

###  3.3. Findings

Seven themes were built: Full-body multisensory interaction (T1, 5/8), Music
visualization (T2, 3/8), Collaboration (T3, 2/8), Support (T4, 2/8),
Gamification (T5, 3/8), Understanding (T6, 3/8), and Expressivity (T7, 4/8).

Full-body multisensory interaction is expressed by four caregivers and PwD2,
to better include people with motor issues. It consists of drawing with the
feet (Ed1), or using free (P1) or specific (P2) full-body gestures.
Multisensory stimuli are advised (H, P1, P2, PwD2): using tangibles as
controllers (e.g., H: a “stick glowing in the dark”), a microphone (P2), and
scents (P1). (M) agreed with this theme, but against using specific gesture,
to not confuse people with ID.

Music visualization is suggested by two caregivers, PwD1, and PwD2. It
consists of using a clear visual “grammar” for the music that would be
appealing and meaningful. For instance, PwD1 suggested to use pentagrams and
P2 emphasized to connect every music element to a visual (and conversely). The
ExP agreed with this theme.

Collaboration is expressed by two caregivers (H, P2), either with turn-taking
(H) or playing simultaneously (H, P2). Scenarios could be task-oriented, with
users competing (e.g., to learn gesture combinations) (P2, H), or open-ended
(e.g., draw how they feel) (H). Though, social anxiety could hinder
collaboration (H). (M) agreed with this theme.

Support is expressed by two caregivers (H, P2). Based on autism difficulties
with abstract thinking, the goal is to prompt creativity by seeing/hearing
music or visuals before or during the experience (e.g., painting over a
background picture or song). If users get stuck, the caregiver or system
should support them. The ExP agrees with this theme.

Gamification is evoked by two caregivers and PwD2 to promote engagement. It
first consists of having task-oriented use cases (Ed2, P2), such as the
discovery of a musical drawing (P2) or a virtual trip (P2). It also concerns
the use of a competition logic (P1, PwD2), using levels and challenges (PwD2).
ExP agreed with these possibilities.

Promoting understanding is expressed by two caregivers and PwD2, using smart
objects to accommodate for people’s abilities (H), or clear rules to lead the
music evolution (e.g., based on movements, Ed1). (M) agreed with this theme.
He also suggested to map some repetitive movements (e.g., hand flapping) to
specific sounds, to give them a meaning.

Expressivity is expressed by four caregivers. The goal is to afford
symbolization processes, by enabling users to use various movements (P2, Ed1)
or painting parameters (e.g., brush colour) based on what they want to convey
(P1, H).

At last, the participants stressed before and after the workshop to
individualize the design based on tastes (e.g., music) and the objective
(e.g., relaxation or communication). ExP advised individualizing the level of
multisensory stimuli.

##  4\. Phase 2: Initial design

MusicTraces is a music and painting activity within an MSE that aims to
promote collaboration, inspired from improvised art therapy practices and
“health musicking” (Stige, 2012; Stensæth, 2013). As such, it accommodates for
multiple ways of being and acting, rather than being task-oriented. The system
is considered as a co-creator which supports collaboration with hints
(Stensæth, 2013). The design is influenced by the themes from our workshop
(noted with [Tx], where x is a number) and previous studies.

###  4.1. Environment of the activity

This two-user activity is inspired from the contemporary music practice called
“sound painting” (Thompson, 2006). It includes two spaces: the interaction
space and the outside space. In the interaction space, delimited with foam
carpets [T6], participants can create musicographic objects on a paper-like
front user interface (UI) based on their hand position (see Figure 3.c) [T1].
Smart objects (i.e., brush and eraser) are used to draw [T1], which design was
inspired from exiting accessible controllers (Ellis and Kao, 2019). The floor
is a control space used to activate the background music [T1,T4]. This design
aims to balance stimulation while fostering the engagement of people with
severe motor issues (e.g., not moving their arms).

The experience relies on multisensory stimuli: audio (from the speakers),
visual (with lights and projections), tactile (with the smart objects), and
proprioceptive (when drawing and moving) [T1]. Lights become red when users
are outside of the interaction space, and turn green when entering it, to
prompt agency. All sounds correspond to soothing music instrument based on the
literature (i.e., marimba and handpan) to not induce over-arousal (Cibrian et
al., 2018; Bauer et al., 2023). All stimuli are simple to avoid cognitive
overload. Information is structured in terms of their roles to prompt
understanding [T6].

The activity is intended for two users, with their caregiver to provide
prompts (verbal, physical) and monitor the activity with a tablet (see section
4.5), to promote understanding and collaboration [T3, T1]. Some features aim
to foster collaboration, e.g., each user has a different color and instrument,
the players’ trails cross their colors blend [T3].

###  4.2. The MusicTraces Syntax

Each visual has a music counterpart and conversely, inspired from [T2] and the
Reactable project (Jordà et al., 2007). The experience relies on three layers
(Andersson and Cappelen, 2008): sound node, narrative structure, and
composition rules [T6]. Nodes are short music patterns (e.g., notes),
narrative structure are combinations of nodes (e.g., melodies), and rules are
ways of creating the narratives.

Three design metaphors are used [T2, T5, T6]. Short music patterns are
represented by paint spots. Melodies are displayed as open or closed lines.
They can be played (i.e., as a timeline) when hit by the players. In that
case, a cursor - represented by an animated dot - navigates over them at a
fixed speed. This metaphor stems from the projects Iannix (Jacquemin et al.,
2012) and Upic (Thiebaut et al., 2008). Open lines are only heard once when
hit, and closed lines multiple times before the cursor fades out.

Three composition rules are used [T6]. Proximity rules help to combine
musicographic element based on their spatial proximity, as in (Jordà et al.,
2007), e.g., circles close to a line are added to the melody. Harmonization
rules guide the interactive music composition based on users’ actions. For
instance, users step together on floor circles, which appear after some
actions are done [T3], to change the music chord, computed using Markov Chains
trained on 37 popular songs444The dataset can be found at this link:
https://www.kaggle.com/datasets/taylorflandro/lyrics-and-chords-from-
ultimateguitar?select=pop_lyrics_df.csv.. Rendering rules take the node
position on the ordinate and abscissa axes to affect its pitch (relative to
the chord) and positionning.

###  4.3. Interactions

MusicTraces allows users to interact in three ways: Explore, Create, and Play
[T1,T7]. Explore consists in exploring the music space, by moving the brush
and pressing on it. This creates temporary transparent lines that quickly
vanish, and triggers sounds based on the brush position. Create is about
leaving permanent paint dots or lines (when using the brush) or erasing them
(with the eraser). Creating a spot requires to stay for more than one second
in the same position. For the lines, users have to draw a spot while drawing a
temporary line (so that every melody contains at least one sound). Users can
erase the objects by hovering over them with the eraser. Play consists of
touching the spots or lines to play the related sounds. When lines are played,
a cursor navigates them starting from the interaction point.

###  4.4. Hint system

A hint system suggests interactions when it detects idleness, isolation, or
repetitive movements, inspired from the Mediate project (Pares et al., 2005),
[T3], and [T4]. About idleness, three hint levels are used. If a user does no
action for more than 20 seconds, the brush slightly vibrates to catch their
attention. If they remain idle, it lights up. At last, a hint line appears on
the front UI from their hand and with their color. About isolation, if they
continuously draw in the same area, a notification is sent to the tablet. The
caregiver can then choose to display a hint line to redirect their attention
elsewhere. Concerning repetitive movements, if a significant similarity is
detected (e.g., in terms of shape, length, etc.) between the last line drawn
and the others, a notification is sent to the caregiver who can decide whether
to display a hint line or not.

###  4.5. Tablet interface

The caregivers use the tablet UI to monitor the activity or stop it if needed
[T4, T6]. The left panel gives control over the game mechanics, e.g., pause
the game, remove lines. The middle panel mirrors the four main areas of the
front screen, signaling overuse with red lighting or where to trigger hints
with green lighting. Four hint shapes are available (e.g., house shape). The
right panel is a notification system, alerting about repetitive behaviors or
overused areas. The UI was created during the initial design and refined after
phase 4. It is visible given on Figure 3.

Figure 3. Design of the tablet user interface (UI), smart brush, and smart
eraser. Elements added after the design phase 4 are mentioned with (D4). (a.)
Tablet UI. The Left panel contains buttons to stop the game, control the
tutorial steps, remove lines/circles, activate the background music, activate
the music evolution (D4), play all melodies, activate the blobs, swap players
or hands (security). The center panel manages the hints. The right panel
displays the notifications. It can be hidden by clicking on the column at its
left (D4). (b.) Smart brush used to draw when pushing on the rainbow button.
c. Smart eraser used to erase when holding it.

###  4.6. Apparatus

The activity is developed with Unity engine (version 2021.2.14f1). A custom
package gives access to the devices in the MSE by communicating with ad-hoc
servers, and to the tablet interface. The web service is developed with
Nuxt.js. The tablet UI is built using Vue.js. The activity runs on a windows
PC. The hardware components include two projectors, a Microsoft Kinect v2,
smart lights, a tablet, and two audio speakers. The four smart objects were
built using parts from commercial objects or being 3D printed, magnets, a
module ESP8266 Witty Cloud ESP-12F WiFi, external batteries, 2 vibration
engines, and LED strips. All sounds were created using virtual instruments on
LogicProX Digital Audio Workstation. The paint visuals were created using
unity shader graph, with textures made using Inkscape software.

##  5\. Phase 3 - Remote agile process

To be able to conduct future testing in the clinical institute called F, we
continued our design process with a psychologist (M) working there, who
participated in our previous workshop. The goal was to adapt our activity to
this context.

###  5.1. Method

As no similar projects existed to our knowledge, an agile method of working
was used with the external psychologist who participated in the first
iteration (M) (Beck et al., 2001). This approach involved doing small
iterative design cycles with (M). Email exchanges occurred every two weeks for
a total of 3 iterations. Each time, feedback was asked about new changes or
ideas stemming from the findings from our workshop and from the related work.
To clarify the changes, videos were sent for the two first iterations,
segmented by different features. Textual feedback were analyzed using a
deductive qualitative analysis process, consisting of analyzing the data
according to the activity features (Elo and Kyngäs, 2008). Throughout the
three iterations, some features were removed, validated, modified, or added,
that are reported below.

###  5.2. Findings

Two ideas were abandoned. First, using additional body movements to create
sounds and visuals was removed, to not overstimulate the users. Second,
triggering specific audiovisual effects when mimicking the other player’s was
also removed, to not confuse the user since many features were already
implemented.

Three element were validated. During the first iteration, the hint system was
deemed very useful to help the users. Then, adapting the line thickness based
on the user distance from the screen was confirmed, to make the activity more
appealing and realistic. The smart objects also received approval and made the
caregiver enthusiastic about the project.

The hint system was modified to avoid visual clutter. Indeed, (M) suggested
that, after the user draws the first line, the hint lines are no longer
automatically added, but rather manually added by the caregiver from the
tablet.

Three elements were added to promote agency: closed shapes being automatically
filled with color, blobs moving on the floor that make percussive sounds when
the users touch them, and lines without nodes. These blobs aim to allow the
users being unable to draw with their hands, to be able to make sounds by
moving in the space. The lines without nodes are thought for users wanting to
create visuals without sounds (as is already possible for sounds).

##  6\. Phase 4 - Design critique with caregivers

To further adapt our design to the clinical context of F, and validate its
acceptability among a clinical team, a design critique was conducted with four
caregivers working there: three educators who had not participated in the
initial design process and the psychologist included in the third phase (M).

###  6.1. Method

The three educators (Ed3, Ed4, Ed5) have been conducting weekly creative
activities at F, for 4 year, more than 10 years, and 5 years. Ed3 (male) and
Id4 (male) do group-based manual activities (e.g., painting, sculpting). Ed5
(female) conducts group-based painting activities using a music background,
and has a training in art therapy. Id3 specifically works with people with
severe disabilities. (M)’s profile was presented above.

With the educators, individual 45-minute sessions were organized at F’s
facility, for organizational reasons. The activity features were presented
using a video, as well as visuals of the smart object designs and tablet UI
(since not fully developed). Sessions started with an activity outline,
followed by general questions based on the video steps, and specific inquiries
about the hint system and tablet interface. One week later, (M) tested the
activity in our lab. After an activity outline, he tested it while making
comments, and being asked about his colleagues’ ideas. Interviews were audio
recorded, with caregivers’ agreement. The data were then sorted into themes
using thematic analysis (Braun and Clarke, 2006).

###  6.2. Findings

All three educators were positive about using MusicTraces at their facility
and participating in the testing. Ed5 said that it closely aligns with her
practice and could benefit to her activities. Id3 emphasized the unique
combination of music and drawing, noting its potential catch the attention and
escape from repetitive behaviors.

About the UI, all caregivers agreed with the aesthetic of the front screen.
Two suggested changing the background color (Id3 and Id5), as in Id5’s
painting activities. All of them confirmed to make it possible to change the
brush color during the game. All validated the floor UI, regarding the floor
blobs and music evolution. To adapt to the users, (M) suggested to make the
latter either “interactable” (as planned), “automatic” (without the floor
circles), or disabled.

The interactions seemed easy understandable (all), despite some imprecisions
in the body tracking (M), that we fixed through coding and with the smart
objects. The latter seemed to ease the gameplay (all). (M) noted the colors of
environmental lights were too close from the brush colors and prevented from
clearly seeing the screen. Thus, we used a different color for the brushes
(yellow), and made them brighter outside of the interaction space and darker
within.

All caregivers validated the use of hints, especially for isolation and
repetitiveness. Three aspects were changed. First, hint lines became dashed,
to differ from the players’ lines. Second, wavy lines were added to guide
users’ attention toward specific areas, after Ed3’s comment. Finally, while
all educators enjoyed the use of hints on the smart objects (vibrations,
lights), vibrations became an and/off feature, as (M) said that they could
confuse or startle some users.

About the tutorial, all caregivers agreed on the different steps, the use of
video modelling techniques, and the addition of audio recorded instructions.
Id3 advised including all steps, contrary to Id4 who said that these “users
cannot keep in mind more than three things.” Thus, we decided to choose the
steps to include before to start with the tablet. (M) agreed with this change.
At last, Id3 suggested adding dashed lines that users could follow with their
hands.

All caregivers were positive about the tablet features. The notifications were
helpful (Id5). However, the UI contained too much information (M), leading to
move some buttons to the left and to allow to hide the notification panel.
Indeed, when playing with a user the caregiver would not need to see this
panel (M).

One additional insight was suggested by Id3, which (M) agreed on. Since “some
individuals may only stare at the floor”, he suggested to activate only one
screen at first, and then complement it with the second screen (e.g., fading
in). Though, it was kept for perspectives to not include too many features and
potentially confuse the users.

##  7\. Conclusion and Future Works

This paper has introduced the design of “MusicTraces”: an open-ended music and
paint activity within an MSE, inspired from improvised art therapy practices,
intended to foster collaboration between young adults with moderate to severe
neurodevelopmental conditions, and more particularly autism. The co-design
process was conducted with caregivers and people with NDC, through a design
workshop, the initial design, a remote design process, and a design critique.
It also took inspiration from existing studies (Jordà et al., 2007; Cibrian et
al., 2017).

MusicTraces has three main contributions to our knowledge. First, it is the
only project that equally combines features about music and paint. Indeed,
similar projects leaned more toward paint (Ringland et al., 2014) or music
(Cappelen and Andersson, 2016b). Secondly, it is the only full-body music
activity designed for two individuals with NDC, and not a child and their
caregiver, as in Osmosis (Ragone et al., 2022). Thirdly, it is one a the few
creative activities intended for people severe conditions, such as Mediate
(Pares et al., 2005). Yet, the project is currently bounded to a MSE, inducing
issues in terms of portability. Thus, future plans involve porting it to
Virtual or Augmented Reality headsets. Including other insights is also
considered to support agency, e.g., a microphone control as in RHYME
(Andersson and Cappelen, 2014; Cappelen and Andersson, 2016b).

Next research steps include acceptability and usability testing with people
with disabilities, followed by an empirical study with around ten people with
moderate to severe NDC at F. A within-group experimental design will be used,
where our activity will be compared with a group-based painting activity using
background music.

###### Acknowledgements.

The authors would like to thank all the participants who participated in this
research. Without them, designing this application would not have been
possible.

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# Idempotents of large norm and homomorphisms of Fourier algebras

M. Anoussis, G. K. Eleftherakis, A. Katavolos  M. Anoussis
Department of Mathematics
University of the Aegean
832 00 Karlovassi, Greece<EMAIL_ADDRESS>G. K. Eleftherakis
Department of Mathematics
Faculty of Sciences
University of Patras
265 00 Patra, Greece<EMAIL_ADDRESS>A. Katavolos
Department of Mathematics
National and Kapodistrian University of Athens
1578 84 Athens, Greece<EMAIL_ADDRESS>
###### Abstract.

We provide necessary and sufficient conditions for the existence of
idempotents of arbitrarily large norms in the Fourier algebra $A(G)$ and the
Fourier-Stieltjes algebra $B(G)$ of a locally compact group $G$. We prove that
the existence of idempotents of arbitrarily large norm in $B(G)$ implies the
existence of homomorphisms of arbitrarily large norm from $A(H)$ into $B(G)$
for every locally compact group $H$. A partial converse is also obtained: the
existence of homomorphisms of arbitrarily large norm from $A(H)$ into $B(G)$
for some amenable locally compact group $H$ implies the existence of
idempotents of arbitrarily large norm in $B(G)$.

###### Key words and phrases:

Group homomorphism, idempotents, Fourier algebras

2010 Mathematics Subject Classification. primary: 43A22, secondary: 43A30

##  1\. introduction

Let $G$ be a locally compact group. The Fourier algebra $A(G)$ and the
Fourier-Stieltjes algebra $B(G)$ of the group $G$ were introduced by Eymard
[7]. The Fourier-Stieltjes algebra of $G$ consists of the matrix coefficients
$\left(\pi(\cdot)\xi,\eta\right)$ of all continuous unitary representations
$\pi$ of $G$, while the Fourier algebra of $G$ consists of the matrix
coefficients of the left regular representation of $G$. If $G$ is abelian,
$A(G)$ and $B(G)$ can be identified, via the Fourier transform, with
$L^{1}(\widehat{G})$ and the measure algebra $M(\widehat{G})$ of the dual
group $\widehat{G}$, respectively. In [2] Cohen characterized the
homomorphisms from $A(H)$ into $B(G)$ in terms of piecewise affine maps when
$H,G$ are locally compact abelian groups. To obtain his result he proved a
characterization of idempotents in $B(G)$ [3]. Host in [10] extended the
characterization of idempotents in $B(G)$ to general locally compact groups.

Homomorphisms of Fourier algebras for locally compact groups were studied by
Ilie [11] and Ilie and Spronk [12]. They characterized completely bounded
homomorphisms from $A(H)$ into $B(G)$ for locally compact groups $H,G$ with
$H$ amenable in terms of piecewise affine maps [12] (see also [5]).

Let $G,H$ be locally compact groups. Let $K$ be a subgroup of $G$ and $C$ a
left coset of $K$ in $G$. A map $\alpha:C\rightarrow H$ is called affine if
there exists a continuous homomorphism $\theta:K\rightarrow H$ and elements
$s_{0}\in H,t_{0}\in G$ such that $C=t_{0}^{-1}K$ and

$\alpha(t)=s_{0}\theta(t_{0}t),$

for all $t\in C$.

A map $\alpha:Y\rightarrow H$ is called piecewise affine if $Y$ can be written
as a disjoint union $Y=\bigcup\limits_{i=1}^{m}Y_{i}$, where each $Y_{i}$
belongs to the open coset ring $\Omega_{0}(G)$, such that each restriction
$\alpha|_{Y_{i}}$ extends to an affine map $\alpha_{i}:C_{i}\rightarrow H$
defined on an open coset $C_{i}\supseteq Y_{i}$.

Recall that the open coset ring $\Omega_{0}(G)$ is the ring generated by the
open cosets of the group $G$.

Let $Y$ be an open and closed subset of $G$ and $\alpha$ a piecewise affine
map $Y\rightarrow H$. Define $\rho:A(H)\rightarrow B(G)$ by

(1.1) $\rho(u)(t)=\left\\{\begin{array}[]{ll}u\\!\circ\\!\alpha(t),&t\in Y\\\
0,&t\in G\setminus Y\end{array}\right.$

Ilie and Spronk proved in [12, Proposition 3.1 and Theorem 3.7] that $\rho$ is
a completely bounded homomorphism and that, if the group $H$ is amenable,
every completely bounded homomorphism $\rho:A(H)\rightarrow B(G)$ is of this
form.

Notation The symbol $\chi_{F}$ denotes the characteristic function of a set
$F$.

To motivate our work, consider the following simple example of completely
bounded homomorphisms from $A(\mathbb{Z})$ to $A(\mathbb{Z})$ of arbitrarily
large norm:

For $F=\\{-k,\dots,k\\}\subseteq\mathbb{Z}$ we define the map
$\rho_{F}:A(\mathbb{Z})\to A(\mathbb{Z})$ by

$\rho_{F}(u)(j)=\begin{cases}u(0)&\text{if }j\in F\\\ 0&\text{if }j\notin
F\end{cases}$

Then it follows from [11] that $\rho_{F}$ is a completely bounded
homomorphism. Consider the function $u_{0}:\mathbb{Z}\to\mathbb{Z}$ given by
$u_{0}(i)=\delta_{i,0}$. Clearly $u_{0}\in A(\mathbb{Z})$ and
$\|u_{0}\|_{A(\mathbb{Z})}\leq 1$.

Since $\rho_{F}(u_{0})=\chi_{F}$, its Fourier transform is
$\widehat{\rho_{F}(u_{0})}(z)=\widehat{\chi_{F}}(z)=\sum_{i\in F}z^{-i}$ and
so

$\|\rho_{F}\|\geq\|\rho_{F}(u_{0})\|_{A(\mathbb{Z})}=\|\chi_{F}\|_{A(\mathbb{Z})}=\|\widehat{\chi_{F}}\|_{L^{1}(\mathbb{T})}$
$=\int_{\mathbb{T}}\left|\sum_{i\in
F}z^{-i}\right|dz=\int_{0}^{2\pi}|D_{k}(x)|\frac{dx}{2\pi}$

where $D_{k}$ is the Dirichlet kernel, and it is known that the $L^{1}$ norm
of $D_{k}$ grows like $\log k$.

In the above example we used the existence of idempotents in $A(\mathbb{Z})$
of large norm to construct homomorphisms of $A(\mathbb{Z})$ with large norm.

In this work we study the following questions: a) for which locally compact
groups $G$ there exist idempotents of arbitrarily large norms in the Fourier
algebra $A(G)$ (resp. in the Fourier-Stieltjes algebra $B(G)$) and b) how is
the existence of idempotents of arbitrarily large norm related to the
existence of homomorphisms of arbitrarily large norm between Fourier algebras.

We provide necessary and sufficient conditions for the existence of
idempotents of arbitrarily large norms in the Fourier algebra $A(G)$ (resp. in
the Fourier-Stieltjes algebra $B(G)$) of a locally compact group $G$. To prove
our results we reduce the problem to the case where $G$ is totally
disconnected. Then we first consider the case where $G$ is a discrete group in
Proposition 2.2, and for the general case we use a result of Leiderman, Morris
and Tkachenko for totally disconnected groups [14]. We also prove that the
existence of idempotents of arbitrarily large norm in $B(G)$ implies the
existence of homomorphisms of arbitrarily large norm from $A(H)$ into $B(G)$
for every locally compact group $H$. Finally, we obtain a partial converse:
the existence of homomorphisms of arbitrarily large norm from $A(H)$ into
$B(G)$ for some amenable locally compact group $H$ implies the existence of
idempotents of arbitrarily large norm in $B(G)$.

##  2\. norms of idempotents

Let $G$ be a locally compact group. A function $u:G\rightarrow\mathbb{C}$ is
called a multiplier of $A(G)$ if $uA(G)\subseteq A(G).$ In this case the map
$m_{u}:A(G)\to A(G):v\mapsto uv$ is bounded. In case it is completely bounded
we call $u$ a completely bounded multiplier. We denote by $M_{cb}A(G)$ the
algebra of completely bounded multipliers of $A(G)$.

The space $M_{cb}A(G)$ inherits the operator space structure from the space
$CB(A(G))$ of completely bounded maps $A(G)\rightarrow A(G)$. We write
$\|u\|_{CB(A(G))}$, and simply $\|u\|_{cb}$ when there is no risk of
confusion, for the completely bounded norm $\|m_{u}\|_{CB(A(G))}$ of an
element $u\in M_{cb}A(G)$. Note that $B(G)$ consists of completely bounded
multipliers on $A(G)$ [6, Corollary 1.8]; thus $B(G)$ (and also $A(G)$)
inherits the operator space structure from $M_{cb}A(G)$.

It is shown in [6] that the space $M_{cb}A(G)$ is the dual of the normed space
$(L^{1}(G),\|\cdot\|_{Q(G)})$ where the norm $\|\cdot\|_{Q(G)}$ is given by

$\|f\|_{Q(G)}=\sup\left\\{\left|\int_{G}f(s)\phi(s)ds\right|:\phi\in
M_{cb}A(G),\|\phi\|_{cb}\leq 1\right\\},\quad f\in L^{1}(G)\,.$

We shall use the following Theorem, combining [18, Corollary 6.3 (iv)] and [7,
2.26, Corollaire 3 and 3.25, Proposition]:

######  Theorem 2.1 (Eymard, Spronk).

Let $G$ be a locally compact group and $H$ a closed, normal subgroup of $G$.
Let $\pi:G\rightarrow G/H$ be the quotient map. The map

$j_{\pi}:M_{cb}(A(G/H))\rightarrow M_{cb}(A(G)):u\mapsto u\circ\pi$

is a complete isometry. Moreover, $j_{\pi}(B(G/H))\subseteq B(G)$; if $H$ is
compact, then $j_{\pi}(A(G/H))\subseteq A(G)$ .

######  Proposition 2.2.

Let $G$ be a discrete infinite group. Then

$\sup\\{\|\chi_{F}\|_{cb}:F\subseteq G\text{ finite}\\}=+\infty.$

###### Proof.

Assuming that

$\sup\\{\|\chi_{F}\|_{cb}:F\subseteq G\text{ finite}\\}=M<+\infty\,,$

we shall prove that $\ell^{\infty}(G)\subseteq M_{cb}(A(G))$. This means that
$G$ is a strong Leinert set, which, by a result of Pisier [16, Theorem 3.3],
implies that $G$ must be finite.

If $v$ is an extreme point of the positive part $\Omega$ of the unit ball of
$\ell^{\infty}(G)$, then $v(t)\in\\{0,1\\}$ for all $t\in G.$ Thus for any
finite $F\subseteq G$, the function $\chi_{F}v$ takes values in $\\{0,1\\}$
and so $\chi_{F}v=\chi_{F^{\prime}}$ for some finite subset $F^{\prime}$ of
$G$. Thus

$\|\chi_{F}v\|_{cb}\leq M$

by our assumption.

Now fix an arbitrary $u\in\Omega$ and a finite subset $F\subseteq G$. By the
Krein-Milman theorem, $u$ is a weak-* limit of a net $(u_{i})$ of convex
combinations of extreme points of $\Omega$. By the previous paragraph, each
$u_{i}$ will satisfy $\|\chi_{F}u_{i}\|_{cb}\leq M$.

Since $M_{cb}A(G)$ is the dual of $(\ell^{1}(G),\|\cdot\|_{Q(G)})$, given
$\varepsilon>0$ there exists $f\in\ell^{1}(G)$ with $\|f\|_{Q(G)}\leq 1$ such
that

$\|\chi_{F}u\|_{cb}-\varepsilon<\left|\sum_{t\in G}(\chi_{F}uf)(t)\right|\,.$

Now

$\sum_{t\in G}(\chi_{F}uf)(t)=\lim_{i}\sum_{t}(\chi_{F}u_{i}f)(t),$

since $u$ is the weak-* limit of the net $(u_{i})$ and so

$\sum_{t\in
G}(\chi_{F}uf)(t)=\lim_{i}\sum_{t}(\chi_{F}u_{i}f)(t)=\lim_{i}\langle
f,\chi_{F}u_{i}\rangle,$

where $\langle\cdot,\cdot\rangle$ denotes the duality between $\ell^{1}(G)$
and $M_{cb}A(G)$. But

$|\langle f,\chi_{F}u_{i}\rangle|\leq\|f\|_{Q(G)}\|\chi_{F}u_{i}\|_{cb}\leq M$

for each $i$, and therefore

$\displaystyle\|\chi_{F}u\|_{cb}-\varepsilon<\left|\sum_{t\in
G}(\chi_{F}uf)(t)\right|$ $\displaystyle=\lim_{i}|\langle
f,\chi_{F}u_{i}\rangle|\leq M.$

Thus, for all nonnegative $u$ in the unit ball of $\ell^{\infty}(G)$ we have

$\sup_{u\in\Omega}\|\chi_{F}u\|_{cb}\leq M$

for every finite subset $F\subseteq G$. In particular, if $u\in c_{oo}(G)$ is
nonnegative then $\frac{u}{\|u\|_{\infty}}\in\Omega$ and thus

$\|u\|_{cb}\leq M\|u\|_{\infty}.$

Therefore for all $u\in c_{oo}(G)$, we have that

$\displaystyle\|u\|_{cb}$ $\displaystyle\leq\|(\mathop{\rm
Re}u)^{+}\|_{cb}+\|(\mathop{\rm Re}u)^{-}\|_{cb}+\|(\mathop{\rm
Im}u)^{+}\|_{cb}+\|(\mathop{\rm Im}u)^{-}\|_{cb}$ $\displaystyle\leq
M(\|(\mathop{\rm Re}u)^{+}\|_{\infty}+\|(\mathop{\rm
Re}u)^{-}\|_{\infty}+\|(\mathop{\rm Im}u)^{+}\|_{\infty}+\|(\mathop{\rm
Im}u)^{-}\|_{\infty})$ $\displaystyle\leq 4M\|u\|_{\infty}.$

We conclude that the norms $\|\cdot\|_{cb}$ and $\|\cdot\|_{\infty}$ are
equivalent on $c_{00}(G).$ Thus the identity map ${\rm
id}:(c_{00}(G),\|\cdot\|_{\infty})\rightarrow(M_{cb}(A(G)),\|\cdot\|_{cb})$ is
continuous. Since $M_{cb}(A(G))$ is a dual Banach space we can consider the
unique weak-* continuous extension of ${\rm id}$ to its double dual
$\ell^{\infty}(G)$ (see e.g. [1, Lemma A.2.2.]) which we denote by $T:$

$T:(\ell^{\infty}(G),\|\cdot\|_{\infty})=(c_{00}(G),\|\cdot\|_{\infty})^{**}\to(M_{cb}(A(G)),\|\cdot\|_{cb}).$

We claim that $T$ is the identity. If $u\in\ell^{\infty}(G)$; we will show
that $u=Tu$. Indeed, if $(u_{i})$ is a net in $c_{00}(G)$ such that
$u=\lim_{i}u_{i}$ in the weak* topology
$\sigma(\ell^{\infty}(G),\ell^{1}(G)),$ then $(Tu_{i})$ converges to $Tu$ is
the weak-* topology of $M_{cb}(A(G))$, and hence pointwise (since
$(Tu)(t)=\langle Tu,\delta_{t}\rangle$ for $t\in G$). Thus, for all $t\in G$,

$(Tu)(t)=\lim_{i}(Tu_{i})(t)=\lim_{i}u_{i}(t)=u(t)$

since $u=\lim_{i}u_{i}$ in the weak* topology
$\sigma(\ell^{\infty}(G),\ell^{1}(G))$ and hence pointwise. This proves our
claim.

We have shown that $\ell^{\infty}(G)\subseteq M_{cb}(A(G))$ and thus $G$ must
be finite, as observed above. ∎

Since $\left\|u\right\|_{cb}\leq\left\|u\right\|_{B(G)}$ when $u\in B(G)$ [6,
Corollary 1.8], we obtain the following

######  Corollary 2.3.

If $G$ is a discrete infinite group then

$\sup_{F}\\{\|\chi_{F}\|_{A(G)}:F\subseteq G\text{ finite}\\}=+\infty.$

_Note_ We thank the referee for providing the following alternative argument
for Corollary 2.3:

Using the analogous arguments as in the proof of Proposition 2.2, we can show
that if $\sup_{F}\\{\|\chi_{F}\|_{A(G)}:F\subseteq G\text{ finite}\\}<+\infty$
then $\ell^{\infty}(G)=B(G)$.

It follows from this equality that $G$ must be finite. Indeed if
$\ell^{\infty}(G)=B(G)$, then $\ell^{1}(G)=C^{*}(G)$ with equivalent norms.
However, this would imply that $\ell^{1}(G)$ is Arens regular, which by [19]
shows that $G$ is finite.

######  Theorem 2.4.

Let $G$ be an infinite totally disconnected group. Then

$\sup\\{\|\chi_{F}\|_{cb}:F\subseteq G,\chi_{F}\in A(G)\\}=+\infty.$

###### Proof.

From the theorem of van Dantzig [4], [9, Theorem II.7.7] there exists a
compact open subgroup $H\subseteq G.$

If $H$ is finite then $\\{e\\}$ is an open subgroup of $G$ and thus $G$ is
discrete. In this case the conclusion follows from Proposition 2.2.

If $H$ is infinite, by Theorem 2.6 in [14] there exists a closed normal
subgroup $N$ of $H$ such that the quotient $H/N$ is homeomorphic to a
countably infinite product of finite groups. We write $K=H/N.$ Clearly $K$ is
compact and separable. We also denote by $K_{d}$ the group $K$ with the
discrete topology. The inclusion

$\iota:K_{d}\rightarrow K$

is a continuous homomorphism; thus it induces a contractive homomorphism

$\rho:A(K)\rightarrow B(K_{d}):u\rightarrow u\circ\iota.$

Let $\epsilon>0$. By Proposition 2.2 there exists a finite $F\subseteq K$ such
that $\chi_{F}\in A(K_{d})$ and

$\|\chi_{F}\|_{A(K_{d})}>\epsilon\,.$

Since $K$ is a totally disconnected and separable group, there exists a
decreasing sequence of compact-open subgroups such that

$\cap_{n=1}^{\infty}K_{n}=\\{e\\}\quad\text{and hence
}\;\cap_{n=1}^{\infty}FK_{n}=F.$

Now $FK_{n}$ is a finite disjoint union of sets of the form $x_{i}K_{n}$ where
$x_{i}\in F$ and since each $K_{n}$ is a compact open subgroup,
$\chi_{x_{i}K_{n}}$ is in $A(K)$ and has norm 1. Indeed, the constant function
1 on the compact group $K_{n}$ belongs to $A(K_{n})$ and has norm 1. It
follows from [7, Proposition 3.21 (1)] that $\chi_{K_{n}}$ belongs to $A(K)$
and has norm 1 and hence the same holds for its translate $\chi_{x_{i}K_{n}}$.
Thus $\chi_{FK_{n}}$ is in $A(K)$ and the sequence
$(\|\chi_{FK_{n}}\|_{A(K)})_{n}$ is bounded by the cardinality of $F$. Since
$\rho$ is bounded, the sequence $(\|\chi_{FK_{n}}\|_{B(K_{d})})_{n}$ is also
bounded. For all $f\in\ell^{1}(K_{d})$, since $(\chi_{FK_{n}})_{n}$ converges
pointwise to $\chi_{F}$, by dominated convergence we have

$\lim_{n}\sum_{t\in K_{d}}f(t)\chi_{FK_{n}}(t)=\sum_{t\in
K_{d}}f(t)\chi_{F}(t)\,.$

Since $\ell^{1}(K_{d})$ is dense in the predual $C^{*}(K_{d})$ of $B(K_{d})$,
we obtain

$\text{w*-}\lim_{n}\chi_{FK_{n}}=\chi_{F}$

in the weak* topology of $B(K_{d}).$ Therefore
$\sup_{n}\|\chi_{FK_{n}}\|_{B(K_{d})}>\epsilon$ and hence

$\sup_{n}\|\chi_{FK_{n}}\|_{A(K)}>\epsilon$

which implies that there exists $\chi_{FK_{n}}\in A(K)$ such that

$\|\chi_{FK_{n}}\|_{A(K)}>\epsilon\,.$

This shows that

$\sup\\{\|\chi_{V}\|_{A(H/N)}:V\subseteq H/N,\chi_{V}\in A(H/N)\\}=+\infty$

and since $H/N$ is compact, it follows that

$\sup\\{\|\chi_{V}\|_{cb}:V\subseteq H/N,\chi_{V}\in A(H/N)\\}=+\infty.$

Let $\pi:H\rightarrow H/N$ be the quotient map. It follows from Theorem 2.1
that if $F\subseteq H/N$ is such that $\chi_{F}\in A(H/N)$, then
$\chi_{F}\circ\pi=\chi_{\pi^{-1}(F)}\in A(H)$ and

$\|\chi_{F}\|_{CB(A(H/N))}=\|\chi_{F}\circ\pi\|_{CB(A(H))}=\|\chi_{\pi^{-1}(F)}\|_{CB(A(H))}.$

We conclude that

$\sup\\{\|\chi_{F}\|_{cb}:F\subseteq H,\chi_{F}\in A(H)\\}=+\infty.$

Let $F\subseteq H$ be such that $\chi_{F}\in A(H)$. There exists $v\in A(G)$
such that $v|_{H}=\chi_{F}$ [7, 3.21(1)]. By [7, 3.21(2)], $\chi_{H}\in A(G)$
and hence $\chi_{F}=v\chi_{H}\in A(G)$.

Since by [18, Corollary 6.3(iii)] the map

$M_{cb}A(G)\rightarrow M_{cb}A(H):u\rightarrow u|_{H}$

is completely contractive, we obtain

$\|\chi_{F}\|_{CB(A(G))}\geq\|\chi_{F}|_{H}\|_{CB(A(H))}=\|\chi_{F}\|_{CB(A(H))}\,.$

We conclude that

$\sup\\{\|\chi_{F}\|_{cb}:F\subseteq G,\chi_{F}\in A(G)\\}=+\infty.$

∎

Note the crucial use of [14] in obtaining a countable family $(K_{n})$ of
compact open subgroups with $\cap_{n=1}^{\infty}K_{n}=\\{e\\}$.

The proof of the above theorem is not constructive. Below we provide a
different proof for the case where G is an infinite direct product of finite
groups. Ilie and Spronk [12, Theorem 2.1] proved that if $\chi_{F}$ is an
idempotent in $B(G)$, then $\|\chi_{F}\|_{B(G)}=1$ if and only if $F$ is a
coset of an open subgroup of $G$. Forrest and Runde in [8] and Stan in [17]
proved that if the cb-norm of an idempotent $\chi_{F}\in B(G)$ satisfies
$\|\chi_{F}\|_{cb}<\frac{2}{\sqrt{3}}$ then $F$ is a coset of an open subgroup
of $G$. The ‘gap’ $[1,\frac{2}{\sqrt{3}})$ was improved by Mudge and Pham in
[15] to $[1,\frac{1+\sqrt{2}}{2})$.

######  Proposition 2.5.

Let $G$ be an infinite direct product of finite groups. Then

$\sup\\{\|\chi_{F}\|_{cb}:F\subseteq G\;,\;\chi_{F}\in A(G)\\}=+\infty.$

###### Proof.

Since $G$ is compact, $A(G)=B(G)$ and $\|u\|_{cb}=\|u\|_{A(G)}$ for all $u\in
A(G)$ [13, Corollary 5.4.11] and hence it is sufficient to prove the
proposition for $\|\cdot\|_{A(G)}$.

Let $G_{0}$ be a finite group with $|G_{0}|\geq 3$. Then there exists
$A\subseteq G_{0}$ such that $A$ is not a coset of a subgroup of $G_{0}$ (for
example, take $A$ such that $|A|$ does not divide $|G_{0}|$). Then, since
$A\in\Omega_{0}(G_{0})$, it follows from [15] that
$\|\chi_{A}\|_{A(G_{0})}\geq\frac{1+\sqrt{2}}{2}$.

Now consider finite groups $G_{1},G_{2},\dots,G_{n}$ with $|G_{i}|\geq 3$ for
all $i=1,2,\dots,n$ and set $G=\prod_{i=1}^{n}G_{i}$. Choose $A_{i}\in G_{i}$
with $\|\chi_{A_{i}}\|_{A(G_{i})}\geq\frac{1+\sqrt{2}}{2}$ and set
$A={A_{1}}\times{A_{2}}\times\dots\times{A_{n}}$. Since $A(G)$ is
isometrically isomorphic to the operator space projective tensor product
${A(G_{1})}\hat{\otimes}{A(G_{2})}\hat{\otimes}\dots\hat{\otimes}{A(G_{n})}$
[13, Lemma 4.1.2] we obtain

$\|\chi_{A}\|_{A(G)}=\|\chi_{A_{1}}\otimes\chi_{A_{2}}\otimes...\chi_{A_{n}}\|_{{A(G_{1})}\hat{\otimes}{A(G_{2})}\hat{\otimes}\dots\hat{\otimes}{A(G_{n})}}\geq\left(\frac{1+\sqrt{2}}{2}\right)^{n}.$

Now let $G=\prod_{i=1}^{\infty}G_{i}$ be an infinite product of finite groups
$G_{i}$. Without loss of generality we may assume that $|G_{i}|\geq 3$ for all
$i\in\mathbb{N}$ (lumping together some of the $G_{i}$’s if necessary). Set
$H_{n}=\prod_{i=n+1}^{\infty}G_{i}$ and let $\pi_{n}$ be the quotient map
$G\rightarrow G/H_{n}$. It follows from Theorem 2.1 that the map
$A(G/H_{n})\rightarrow A(G):u\mapsto u\circ\pi_{n}$ is isometric. Since
$G/H_{n}\simeq\prod_{i=1}^{n}G_{i}$, we can choose $A\subseteq G/H_{n}$ such
that $\|\chi_{A}\|_{{A(G/H_{n})}}\geq(\frac{1+\sqrt{2}}{2})^{n}.$ Setting
$F=\pi_{n}^{-1}(A)$, we obtain
$\|\chi_{F}\|_{cb}=\|\chi_{F}\|_{A(G)}\geq(\frac{1+\sqrt{2}}{2})^{n}$ and the
conclusion follows. ∎

######  Corollary 2.6.

Let $G$ be a locally compact group and $G_{0}$ be the connected component of
$e\in G.$ If the quotient $G/G_{0}$ is infinite then

$\sup\\{\|\chi_{F}\|_{cb}:F\subseteq G,\chi_{F}\in B(G)\\}=+\infty.$

###### Proof.

Since $G/G_{0}$ is infinite and totally disconnected, it follows from Theorem
2.4 that

$\sup\\{\|\chi_{F}\|_{cb}:F\subseteq G/G_{0},\chi_{F}\in
A(G/G_{0})\\}=+\infty.$

Let $\pi:H\rightarrow G/G_{0}$ be the quotient map. Since
$\chi_{\pi^{-1}(F)}=\chi_{F}\circ\pi$, it follows from Theorem 2.1 that

$\sup\\{\|\chi_{F}\|_{cb}:F\subseteq G,\chi_{F}\in B(G)\\}=+\infty.$

∎

######  Remark 2.7.

Let $G$ be a locally compact group and $N$ a closed normal subgroup of $G$. It
follows from Theorem 2.1 that if

$\sup\\{\|\chi_{F}\|_{cb}:F\subseteq G/N,\chi_{F}\in B(G/N)\\}=+\infty$

then

$\sup\\{\|\chi_{F}\|_{cb}:F\subseteq G,\chi_{F}\in B(G)\\}=+\infty.$

##  3\. groups with idempotents with large norms

Let $G$ be a locally compact group and $G_{0}$ the connected component of the
identity of $G$. In this section we show that $B(G)$ contains idempotents of
arbitrarily large norms if and only if $G/G_{0}$ is infinite. We also prove a
related result for $A(G)$. It follows from [10] that an idempotent is in
$B(G)$ if and only if it is of the form $\chi_{F}$ with $F$ in the open coset
ring of $G$.

Let $H$ be an open subgroup of $G$. Then $H\cap G_{0}$ is open and closed in
$G_{0}$, and hence equal to $G_{0}$; thus $H\supseteq G_{0}$. Since $G_{0}$ is
contained in every open subgroup of $G$, we have that that if $E$ is an left
coset of an open subgroup in $G$, then $E=EG_{0}$. It is easy to see that if
$E$ is a left coset of an open subgroup $G$, we also have $E^{c}=E^{c}G_{0}$
(here $E^{c}$ is the complement of $E$).

######  Lemma 3.1.

Let $X$ be in the open coset ring $\Omega_{0}(G)$. Then $X=XG_{0}$.

###### Proof.

We show that if $X=XG_{0}$ and $Y=YG_{0}$, then $X\cap Y=(X\cap Y)G_{0}$.
Indeed, let $z\in X\cap Y$ and $g\in G_{0}$. Then
$z=xg^{\prime}=yg^{\prime\prime}$ for some $x\in X$, $y\in Y$ and
$g^{\prime},g^{\prime\prime}\in G_{0}$. Then
$zg=xg^{\prime}g=yg^{\prime\prime}g$ and since $xg^{\prime}g\in X$ and
$yg^{\prime\prime}g\in Y$ we obtain $zg\in X\cap Y$. In view of the above
remark on complements, the assertion follows. ∎

######  Corollary 3.2.

Let $\phi$ be the map defined on $\Omega_{0}(G)$ by $X\mapsto q(X)$ (where
$q:G\to G/G_{0}$ is the quotient map). Then $\phi$ is a ring isomorphism from
$\Omega_{0}(G)$ onto $\Omega_{0}(G/G_{0})$.

###### Proof.

Clearly $\phi(X\cap Y)=\phi(X)\cap\phi(Y)$ and $\phi(X^{c})=\phi(X)^{c}$. Let
$aK$ be a coset of an open subgroup in $G/G_{0}$. Then $\phi^{-1}(aK)$ is a
coset of an open subgroup in $G$ and $\phi(\phi^{-1}(aK))=aK$. Finally,
$XG_{0}=YG_{0}$ for $X,Y\in\Omega_{0}(G)$ implies $X=Y$, hence $\phi$ is
injective. ∎

######  Theorem 3.3.

Let $G$ be a locally compact group and $G_{0}$ be the connected component of
$e\in G.$ The following are equivalent

  1. (1)

The quotient $G/G_{0}$ is infinite and $G_{0}$ is compact

  2. (2)

$\;\sup\\{\|\chi_{F}\|_{cb}:\chi_{F}\in A(G)\\}=+\infty.$

  3. (3)

$\;\sup\\{\|\chi_{F}\|_{A(G)}:\chi_{F}\in A(G)\\}=+\infty.$

###### Proof.

That (1) implies (2) follows from Theorems 2.1 and 2.4.

That (2) implies (3) follows since $\|\chi_{F}\|_{cb}\leq\|\chi_{F}\|_{A(G)}$.

We show that (3) implies (1): If $G_{0}$ is not compact, it follows from Lemma
3.1 that there are no idempotents in $A(G)$. If $G/G_{0}$ is finite, the open
coset ring is finite from Corollary 3.2, and hence the set of idempotents is
finite. ∎

The proof of the following theorem is similar.

######  Theorem 3.4.

Let $G$ be a locally compact group and $G_{0}$ be the connected component of
$e\in G.$ The following are equivalent

  1. (1)

The quotient $G/G_{0}$ is infinite.

  2. (2)

$\;\sup\\{\|\chi_{F}\|_{cb}:\chi_{F}\in B(G)\\}=+\infty.$

  3. (3)

$\;\sup\\{\|\chi_{F}\|_{B(G)}:\chi_{F}\in B(G)\\}=+\infty.$

##  4\. norms of homomorphisms

In this section we show that if $G$ is a locally compact group with connected
component of the identity $G_{0}$ such that $G/G_{0}$ is infinite, and $H$ is
a locally compact group, then there exist homomorphisms of arbitrarily large
norm from $A(H)$ into $B(G)$. We also prove that if there exists an amenable
group $H$ such that homomorphisms of arbitrarily large norm from $A(H)$ into
$B(G)$ exist, then $G/G_{0}$ is infinite.

######  Proposition 4.1.

Let $G,H$ be locally compact groups and $F\in\Omega_{0}(G)$. For $u\in A(H)$
we define

$\rho_{F}(u)(t)=\left\\{\begin{array}[]{ll}u(e),&t\in F\\\ 0,&t\notin
F\end{array}\right.$

Then the map $\rho_{F}$ is a completely bounded homomorphism $A(H)\rightarrow
B(G)$ and

$\|\rho_{F}\|_{cb}=\|\rho_{F}\|=\|\chi_{F}\|_{B(G)}.$

###### Proof.

It follows from [12, Theorem 3.1] that the map $\rho_{F}$ is a completely
bounded homomorphism. Choose $u\in A(H)$ such that $u(e)=1$ and
$\|u\|_{A(H)}\leq 1$. Then

$\|\rho_{F}\|\geq\|\rho_{F}(u)\|_{B(G)}=\|u(e)\chi_{F}\|_{B(G)}=\|\chi_{F}\|_{B(G)}.$

We also have, for $u\in A(H)$,

$\|\rho_{F}(u)\|_{B(G)}=\|u(e)\chi_{F}\|_{B(G)}=|u(e)|\|\chi_{F}\|_{B(G)}\leq\|u\|_{A(H)}\|\chi_{F}\|_{B(G)}\,,$

and hence $\|\rho_{F}\|=\|\chi_{F}\|_{B(G)}.$

Since the image of $\rho_{F}$ is one-dimensional, it follows that

$\|\rho_{F}\|_{cb}=\|\rho_{F}\|=\|\chi_{F}\|_{B(G)}.$

∎

Applying Theorem 3.4 to Proposition 4.1, we obtain the following

######  Corollary 4.2.

Let $G,H$ be locally compact groups and assume that

$\sup\\{\|\chi_{F}\|_{cb}:F\in\Omega_{0}(G)\\}=+\infty\,.$

Then

$\sup\\{\|\rho:A(H)\rightarrow B(G)\|:\rho\text{ is a cb
homomorphism}\\}=+\infty\,.$

######  Theorem 4.3.

Let $G$ be a locally compact group and $G_{0}$ be the connected component of
$e\in G.$ The following are equivalent:

$(i)\quad\sup\\{\|\rho:A(H)\rightarrow B(G)\|:\rho\text{ is a cb
homomorphism}\\}=+\infty.$

for every locally compact group $H$.

(ii) There exists an amenable locally compact group $H$ such that

$\sup\\{\|\rho:A(H)\rightarrow B(G)\|:\rho\text{ is a cb
homomorphism}\\}=+\infty.$

(iii) The group $G/G_{0}$ is infinite.

(iv) $\sup\\{\|\chi_{F}\|_{B(G)}:F\in\Omega_{0}(G)\\}=+\infty.$

###### Proof.

Clearly $(i)$ implies $(ii)$. The equivalence $(iii)\Leftrightarrow(iv)$
follows from Theorem 3.4. Also the implication $(iv)\Rightarrow(i)$ follows
from Corollary 4.2.

It remains to show that $(ii)$ implies $(iii)$. Suppose not, i.e. that
$|G/G_{0}|<+\infty.$ By Corollary 3.2, there exists $m\in\mathbb{N}$ such that
$|\Omega_{0}(G)|\leq m.$ Let $\rho:A(H)\rightarrow B(G)$ be a completely
bounded homomorphism. By [12, Theorem 3.7], $\rho$ is of the form 1.1 for some
$Y\in\Omega_{0}(G)$ and a piecewise affine map $\alpha:Y\rightarrow H$. By
Proposition 3.1 in [12] we have that

$\|\rho\|_{cb}\leq m\cdot\\!\\!\sum_{F\in\Omega_{0}(G)}\|\chi_{F}\|_{B(G)}\leq
m^{2}\max\\{\|\chi_{F}\|_{B(G)}:F\in\Omega_{0}(G)\\}$

which is a contradiction. ∎

Acknowledgment  Warm thanks are due to the referee for his useful comments and
in particular for indicating a gap in the original proof of Theorem 2.4.

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