# Spectral Shaping in a Multimode Fiber by All-fiber Modulation

Zohar Finkelstein  Racah Institute of Physics, The Hebrew University of
Jerusalem, Jerusalem 91904, Israel  Kfir Sulimany  Racah Institute of Physics,
The Hebrew University of Jerusalem, Jerusalem 91904, Israel  Shachar Resisi
Racah Institute of Physics, The Hebrew University of Jerusalem, Jerusalem
91904, Israel  Yaron Bromberg  Corresponding author:
<EMAIL_ADDRESS>
###### Abstract

In the past few years, there is a renewed interest in using multimode fibers
for a wide range of technologies such as communication, imaging and
spectroscopy. Practical implementations of multimode fiber in such
applications, however, are held back, due to the challenges in dealing with
modal dispersion, mode coupling and the fiber’s sensitivity to mechanical
perturbations. Here, we utilize these features of multimode fibers to generate
all-fiber reconfigurable spectral filters. By applying computer-controlled
mechanical deformations to the fiber, along with an optimization algorithm, we
manipulate the light propagation in the fiber and control its output field.
Using this approach we demonstrate tunable bandpass filters and dual-band
filters, with spectral resolutions as low as 5pm.

## Introduction

In the past decade there is ongoing effort to utilize multimode optical fibers
(MMF) in a wide range of photonic applications such as optical fiber
communication [1, 2, 3], high energy fiber lasers [4, 5, 6], quantum optics[7,
8, 9, 10, 11], nonlinear optics [12, 13, 14, 15, 16, 17, 18] and imaging [19,
20, 21, 22, 23, 24, 25, 26]. The challenge of MMF-based technologies is that
transverse modes of the fiber are interrupted by inter-modal interference,
mode coupling, and modal dispersion. Therefore, the information delivered by a
MMF is scrambled, forming at its output a spatiotemporal speckle pattern. To
fully utilize the potential of MMFs, it is desired to have coherent control
over the spectral and spatial degrees of freedom of the light that is
transferred by the fiber.

The traditional way to unscramble information transferred by MMFs is using
spatial light modulators (SLM) that shape the wavefront at the input and/or
output of the fiber [19]. However, due to the wavelength-dependent
transmission through multimode fibers, SLMs do not allow efficient control
over a wide spectral bandwidth. Surprisingly, however, modal interference and
modal dispersion in multimode fibers can in fact be harnessed for a wide range
of applications [27], such as fiber spectrometers [28, 29], cryptography [30,
31], and optical implementations of neural networks [32]. Another challenge of
multimode fibers is their extreme sensitivity to mechanical perturbations. We
have recently utilized this sensitivity to make an all-fiber SLM, by applying
controlled deformations to the fiber [33]. Controlling the medium that the
light propagates through, rather than the wavefront of the incident field,
opens new opportunities for wavefront shaping, as was recently demonstrated in
the microwave regime [34, 35]. These works highlight the great potential of
controlling the medium instead of the incident field for extending wavefront
shaping to the spectral domain, thanks to the strong coupling between the
spectral and spatial degrees of freedom in complex media [36, 37].

In this work, we apply computer-controlled deformations to MMFs to realize
all-fiber, reconfigurable spectral filters. To this end, we apply local bends
at multiple positions along the fiber. The bends change the local propagation
constants of the fiber’s guided modes and induce mode mixing, thus changing
the fiber’s wavelength-dependent transmission matrix. The curvatures of the
bends are determined by an optimization algorithm that finds the optimal
configuration that maximizes the overlap between the output spectrum and a
target spectrum. Using this configuration, which we call fiber piano, we
realize reconfigurable spectral filters with a 5pm resolution.

## Experimental results

The experimental setup is depicted in Fig.1a. A broadband fiber-coupled LED
source is coupled to a single mode fiber (SMF), to obtain spatially coherent
broadband light. The light is then coupled to a 1m-long step-index MMF, with a
numerical aperture (NA) of 0.22 and a core diameter of 50$\mu$m. The output
facet of the MMF is spliced to another SMF that is coupled to a spectrometer.
The measured spectrum is determined by the complex wavelength-dependent
interference of the fiber modes at the splicing point. We apply computer-
controlled deformations to the fiber by placing 28 piezoelectric actuators
along the fiber, with a 1.5cm separation between adjacent actuators
(Fig.1b,c). We limit the minimal radius of curvature the actuators apply to
1cm, to avoid significant bending loss. As different deformations of the fiber
yield different spectra at the output of the SMF, we search for the bend
configuration that yields the required spectrum, using a simulated annealing
optimization algorithm [38].

Figure 1: Experimental setup for all-fiber spectral shaping. (a) Light from a
broadband LED source (Thorlabs MWWHF2) that is coupled to a single mode fiber
(SMF), is injected into a 1m-long multimode fiber (MMF, Thorlabs FG050LGA).
The output light is projected onto a single transverse mode by splicing the
distal end of the MMF to a SMF (Thorlabs P1-630A-FC-1) that delivers the
collected light to a spectrometer (Teledyne IsoPlane-320 and ProEm-HS:1024).
(b) To control the transmission through the fiber we place an array of 28
computer-controlled piezoelectric actuators (both CTS ceramic NAC2225-A01 and
Thorlabs PB4NB2W) along the fiber. Each actuator can travel $\sim$0.5mm in the
vertical direction, pressing the fiber from above (c). The actuators create
local deformations that change the propagation constants of the fiber’s guided
modes. An optimization algorithm (simulated annealing) searches for the
optimal travel of each actuator, that maximizes the overlap of the output
spectrum with the target spectrum. (d) To demonstrate spectral shaping with
resolution beyond the spectral resolution of the spectrometer, we replaced the
LED source with a fiber-coupled tunable laser at the telecom band (Keysight
81490A). The output facet for the MMF is imaged by an InGaAs camera (Xenics
cheetah CL-7207) which records the wavelength-dependent speckle patterns.

To demonstrate spectral shaping, we realize a tunable bandpass filter, by
maximizing the output intensity at different spectral bands (Fig.2a). The
full-width-half maximum (FWHM) bandwidth of each band is 245pm, determined by
the spectral bandwidth of the MMF [28]. Defining the enhancement factor as the
peak-to-background ratio in the obtained spectra, we achieve enhancements in
the range of 13 to 30. The efficiency of the system, defined by the ratio of
output and input intensities at the chosen spectral band, is 0.06. To
demonstrate the versatility of the spectral shaping, we also implement a dual
bandpass filter, with two peaks separated by 1.5 nm and a FWHM of 241pm for
each peak (see Fig.2b).

Figure 2: Experimental realization of a tunable bandpass filter and a dual-
band filter. (a) Measured transmission spectra of the all-fiber tunable
bandpass filter. Each curve represents an average of 6 spectra that were
optimized to maximize the enhancement at the desired wavelength. We achieve a
full-width-half-maximum (FHWM) of 245pm and enhancements in the range of 13 to
30. (b) Optimized transmission for a dual filter, with a FWHM of 241pm and
separation of 1.5nm, obtained by maximizing the sum of square roots of the
intensities at the chosen wavelengths. For the dual-band filters, we obtain
enhancements in the range of 7.5 to 10.5.

Since the spectral bandwidth of the fiber is inversely proportional to the
fiber’s length [28, 29], it is in principle possible to realize bandpass
filters with extremely narrow linewidths by using long multimode fibers. In
practice, however, the linewidths we could achieve were limited by the 60 pm
resolution of the spectrometer used in the optimization process. We therefore
replaced the spectrometer and the LED source with a tunable laser at telecom
wavelengths, that can tune the wavelength in the C-band with a spectral
resolution of 1pm, and an InGaAs camera that images the output facet of the
MMF (Fig.1d). We fix the laser at a specific wavelength and apply the same
optimization process described above, to enhance the intensity at a chosen
square region of interest (ROI) on the camera. We set the width of the ROI to
be equal to the FWHM of the intensity autocorrelation of the speckle pattern
recorded by the camera. After the optimization, we scan the wavelength of the
laser and observe the transmitted spectrum. The FWHM of the measured
transmission peaks is inversely proportional to the fiber’s length. Using a
150m long fiber, we realize bandpass filters with a FWHM of 5 pm, and
enhancement factors in the range of 5 to 7 (inset of Fig.3)

Figure 3: Realization of narrow bandpass filters. The measured FWHM of the
optimized transmission peaks versus the fiber’s length $L$ exhibits inverse
linear scaling. The black curve is the predicted $1/L$ scaling [28]. Inset:
the transmission spectrum for $L=15$m (green), $L=50$m (yellow), and $L=150$m
(blue). All spectra are normalized by the average background intensity.
Errorbars in both graphs correspond to the standard deviation of different
optimization runs.

### Numerical simulations

To compare the performance of our system with its optimal limit, we
numerically simulate the operation of the fiber-piano by computing the
transmission spectra through a MMF with controlled local macro bends. We model
the fiber-piano by numerically propagating scalar waves through curved
segments of a MMF, separated by straight segments. Each curved segment
simulates a macro bend induced by an actuator, where the travel of the
actuator corresponds to a change in the radius of curvature of the segment. We
use 44 discrete values for the radii of curvatures in the range of 1cm to 8cm,
which roughly corresponds to the range of curvatures induced by the actuators
in the experiment.

To find the field at the output of the fiber, we calculate the transmission
matrix (TM) of each segment and propagate the input field by multiplying it by
the TM’s of all the segments. To calculate the TM of the curved segments, we
use ”BPM-matlab” [39], an open source optical propagation simulation in
MATLAB, which implements a finite difference beam propagation method for
monochromatic light. We first compute the TM of the 44 curved segments to be
used in the simulation, by propagating each guided mode of the fiber and
projecting the output field on the guided mode basis. Propagation in the
straight fiber segment is modeled by a diagonal TM, whose elements, that
represent the phase accumulated by each guided mode, are computed by solving
the Helmholtz equation for the fiber’s index profile. The goal of the
simulation is to find the radius of curvature of each bend, out of the 44
possible radii, that focuses the light at the distal end of the MMF, at a
chosen wavelength. To this end, we use the same SA optimization algorithm used
in the experiment. Once the optimal radii of curvature are found, we obtain
the output spectrum by simulating propagation through the same configuration
of bends and the same input field, at different wavelengths.

We start by simulating propagation through a fiber-piano with the same number
of actuators as in our experiments, 28 curved segments separated by straight
segments. The fiber we simulate is a 2m-long step-index MMF, with a numerical
aperture of 0.22 and a core diameter of 50$\mu$m. We excite the 50 first modes
of the fiber, corresponding to the roughly 50 modes we excite in the
experiment (see Fig.S4), with equal amplitudes and random relative phases. We
achieve an enhancement of 15 and a FWHM of 350pm (blue curve in Fig.4a). Since
we simulate the propagation over a bandwidth that is only slightly wider than
the bandwidth of the fiber, here we define the enhancement as the spatial
peak-to-background, instead of the spectral peak-to-background measured in the
experiments. We attribute the difference from the experimental values obtained
for the same fiber and the same wavelength range (enhancement of 7 and FWHM of
650pm) to mode mixing that may increase the fiber’s spectral bandwidth [40]
and system instabilities that may decrease the obtained peak-to-background
ratios [41].

It is instructive to compare the enhancement obtained by the fiber piano to
the optimal enhancement one expects to achieve, for example using a perfect
spatial light modulator at the proximal end of the fiber. We therefore look
for the input field that maximizes the peak-to-background ratio using the same
bend configuration that was found by the optimization algorithm for some
arbitrary input. To this end, we first calculate the TM of the bent fiber as
described above. Next, by multiplying the Hermitian conjugate of the TM by the
target pattern (a focused spot in the desired ROI), we get the input field
that maximizes the intensity in the ROI [41]. Then, to find the output
spectrum in the ROI, we numerically propagate this input field at different
wavelengths. The optimal input field yields an enhancement of 45 and a FWHM of
320pm (red curve in Fig.4a). We therefore conclude that with 28 actuators we
could achieve $\approx 30\%$ of the optimal enhancement.

To further explore the performance of the fiber-piano under optimal
conditions, we simulate the achievable enhancement in a loss-free system, for
an increasing number of fiber modes and actuators. Intuitively, the
enhancement should rise as these two parameters are increased since they
provide more degrees of freedom for the optimization process. The number of
curved segments in the simulation is set by the number of actuators we
simulate. The number of guided modes in the fiber is set by changing the core
diameter while keeping the numerical aperture of the fiber fixed. We fix the
total fiber length to 20m, and excite all the guided modes of the fiber with
equal amplitudes and random relative phases. To neglect loss in the
simulation, we force the transmission matrix of each segment to be unitary.

In Fig.4b, we depict the enhancement versus the number of actuators, for
fibers with different number of modes. As expected, the enhancement increases
with the number of actuators (Fig.4b). The saturated enhanced values obtained
by the SA optimization, are close to the values expected by an optimal
wavefront control, as depicted in Fig.4c, confirming that with enough
actuators the fiber-piano can approach the optimal values. However, saturation
is expected when the number of degrees of control, the actuators, approaches
the number of guided modes of the fiber. Here we see the saturation is
obtained at a much higher number of actuators, which indicates that the
degrees of control in the simulation are not independent. Noticeably, as the
number of modes grows the simulation performance deteriorates compared with
the theoretical limit. This difference is most probably due to the increased
probability that the SA algorithm finds a local minimum when the dimension of
the optimization problem increases.

Figure 4: Numerical simulation results. (a) Simulation of the system with a 2m
long fiber. We first run the SA optimization algorithm to search for the radii
of curvature of the curved segments that maximize the intensity in the region
of interest (ROI), at wavelength $\lambda=1550$ nm. Then for the same bend
configuration and the same input modes, we propagate the field at different
wavelengths and obtain the wavelength-dependent intensity in the ROI. For each
wavelength, we present the enhancement, defined by the spatial peak-to-
background ratio. The blue curve corresponds to the enhancements found by the
optimization algorithm, while the red curve depicts the predicted enhancement
for a perfect wavefront control (see text). (b) Simulated enhancements for
different number of modes and actuators, showing that the enhancement
increases with each of these parameters until saturation is observed. (c) The
enhancement is achieved by the optimization algorithm (blue) and the optimal
enhancements for perfect wavefront control (red), as a function of the number
of guided modes in the fiber. The simulation data points are taken for the
highest number of actuators that we tested for each fiber.

## Conclusions and Outlook

We experimentally realized all-fiber spectral shaping by applying computer-
controlled deformations to a multimode fiber. As a proof of concept of this
approach, we demonstrated a tunable bandpass filter with FWHM of 245pm, and a
dual-wavelength filter with the same spectral width. Next, we showed the
possibility to achieve narrower bandpass filters, using longer MMF, and
achieved FWHM bandwidths as low as 5pm. Finally, we ran numerical simulations
to test the performance of the fiber under optimal conditions. We show that
with the fiber-piano it is possible to achieve peak-to-background ratios that
are above $70\%$ of the ratios obtained for optimal wavefront control.

In the high resolution configuration, where a camera that images the distal
end of the fiber is used to select one spatial channel, one can simplify the
detection method by replacing the camera with a SMF that is spliced at the
distal end of the MMF along with a telecom photodiode. Our method demonstrates
a first step towards realizing an all-fiber modulator for tailoring the
spatial-spectral waveform at the output of multimode fibers, a key ingredient
in a wide range of applications that are based on multimode fibers.

## Funding Information

This research was supported by the Zuckerman STEM Leadership Program and the
ISF-NRF Singapore joint research program (grant No. 3538/20).

## Disclosures

The authors declare no conflicts of interest.

## References

  * [1] D.. Richardson, J.. Fini and L.. Nelson  “Space-division multiplexing in optical fibres”  In _Nature Photonics_ 7.5, 2013, pp. 354–362
  * [2] Martin Plöschner, Tomáš Tyc and Tomáš Čižmár  “Seeing through chaos in multimode fibres”  In _Nature Photonics_ 9.8, 2015, pp. 529–535
  * [3] Benjamin J Puttnam, Georg Rademacher and Ruben S Luı́s  “Space-division multiplexing for optical fiber communications”  In _Optica_ 8.9 Optical Society of America, 2021, pp. 1186–1203
  * [4] Logan G Wright, Demetrios N Christodoulides and Frank W Wise  “Spatiotemporal mode-locking in multimode fiber lasers”  In _Science_ 358.6359 American Association for the Advancement of Science, 2017, pp. 94–97
  * [5] Uğur Teğin et al.  “Spatiotemporal self-similar fiber laser”  In _Optica_ 6.11 Optical Society of America, 2019, pp. 1412–1415
  * [6] Henry Haig et al.  “Multimode Mamyshev oscillator”  In _Optics Letters_ 47.1 Optica Publishing Group, 2022, pp. 46–49
  * [7] Hugo Defienne et al.  “Two-photon quantum walk in a multimode fiber”  In _Science advances_ 2.1 American Association for the Advancement of Science, 2016, pp. e1501054
  * [8] Natalia Herrera Valencia et al.  “Unscrambling entanglement through a complex medium”  In _Nature Physics_ 16.11 Nature Publishing Group, 2020, pp. 1112–1116
  * [9] Saroch Leedumrongwatthanakun et al.  “Programmable linear quantum networks with a multimode fibre”  In _Nature Photonics_ 14.3 Nature Publishing Group, 2020, pp. 139–142
  * [10] Huan Cao et al.  “Distribution of high-dimensional orbital angular momentum entanglement over a 1 km few-mode fiber”  In _Optica_ 7.3 Optical Society of America, 2020, pp. 232–237
  * [11] Kfir Sulimany and Yaron Bromberg  “All-fiber source and sorter for multimode correlated photons”  In _npj Quantum Information_ 8.1 Nature Publishing Group, 2022, pp. 1–5
  * [12] Arash Mafi  “Pulse propagation in a short nonlinear graded-index multimode optical fiber”  In _Journal of Lightwave Technology_ 30.17 IEEE, 2012, pp. 2803–2811
  * [13] Logan G Wright, Demetrios N Christodoulides and Frank W Wise  “Controllable spatiotemporal nonlinear effects in multimode fibres”  In _Nature photonics_ 9.5 Nature Publishing Group, 2015, pp. 306–310
  * [14] Katarzyna Krupa et al.  “Spatial beam self-cleaning in multimode fibres”  In _Nature Photonics_ 11.4 Nature Publishing Group, 2017, pp. 237–241
  * [15] Omer Tzang, Antonio M Caravaca-Aguirre, Kelvin Wagner and Rafael Piestun  “Adaptive wavefront shaping for controlling nonlinear multimode interactions in optical fibres”  In _Nature Photonics_ 12.6 Nature Publishing Group, 2018, pp. 368–374
  * [16] Katarzyna Krupa et al.  “Multimode nonlinear fiber optics, a spatiotemporal avenue”  In _APL Photonics_ 4.11 AIP Publishing LLC, 2019, pp. 110901
  * [17] Uğur Teğin et al.  “Controlling spatiotemporal nonlinearities in multimode fibers with deep neural networks”  In _Apl Photonics_ 5.3 AIP Publishing LLC, 2020, pp. 030804
  * [18] Hamed Pourbeyram et al.  “Direct observations of thermalization to a Rayleigh–Jeans distribution in multimode optical fibres”  In _Nature Physics_ Nature Publishing Group, 2022, pp. 1–6
  * [19] Tomáš Čižmár and Kishan Dholakia  “Exploiting multimode waveguides for pure fibre-based imaging”  In _Nature communications_ 3.1 Nature Publishing Group, 2012, pp. 1–9
  * [20] Youngwoon Choi et al.  “Scanner-free and wide-field endoscopic imaging by using a single multimode optical fiber”  In _Physical review letters_ 109.20 APS, 2012, pp. 203901
  * [21] Ruo Yu Gu, Reza Nasiri Mahalati and Joseph M Kahn  “Design of flexible multi-mode fiber endoscope”  In _Optics express_ 23.21 Optical Society of America, 2015, pp. 26905–26918
  * [22] Antonio M Caravaca-Aguirre and Rafael Piestun  “Single multimode fiber endoscope”  In _Optics express_ 25.3 Optical Society of America, 2017, pp. 1656–1665
  * [23] Navid Borhani, Eirini Kakkava, Christophe Moser and Demetri Psaltis  “Learning to see through multimode fibers”  In _Optica_ 5.8 Optica Publishing Group, 2018, pp. 960–966
  * [24] Ivo T Leite, Sergey Turtaev, Dirk E Boonzajer Flaes and Tomáš Čižmár  “Observing distant objects with a multimode fiber-based holographic endoscope”  In _APL Photonics_ 6.3 AIP Publishing LLC, 2021, pp. 036112
  * [25] Shachar Resisi, Sebastien M Popoff and Yaron Bromberg  “Image Transmission Through a Dynamically Perturbed Multimode Fiber by Deep Learning”  In _Laser & Photonics Reviews_ 15.10 Wiley Online Library, 2021, pp. 2000553
  * [26] Szu-Yu Lee, Vicente J Parot, Brett E Bouma and Martin Villiger  “Confocal 3D reflectance imaging through multimode fiber without wavefront shaping”  In _Optica_ 9.1 Optical Society of America, 2022, pp. 112–120
  * [27] Hui Cao and Yaniv Eliezer  “Harnessing disorder for photonic device applications”  In _Applied Physics Reviews_ 9.1 AIP Publishing LLC, 2022, pp. 011309
  * [28] Brandon Redding and Hui Cao  “Using a multimode fiber as a high-resolution, low-loss spectrometer”  In _Optics letters_ 37.16 Optical Society of America, 2012, pp. 3384–3386
  * [29] Brandon Redding, Mansoor Alam, Martin Seifert and Hui Cao  “High-resolution and broadband all-fiber spectrometers”  In _Optica_ 1.3 Optical Society of America, 2014, pp. 175–180
  * [30] Yaron Bromberg et al.  “Remote key establishment by random mode mixing in multimode fibers and optical reciprocity”  In _Optical Engineering_ 58.1 SPIE, 2019, pp. 016105
  * [31] Lyubov V Amitonova, Tristan BH Tentrup, Ivo M Vellekoop and Pepijn WH Pinkse  “Quantum key establishment via a multimode fiber”  In _Optics express_ 28.5 Optica Publishing Group, 2020, pp. 5965–5981
  * [32] Uğur Teğin et al.  “Scalable optical learning operator”  In _Nature Computational Science_ 1.8 Nature Publishing Group, 2021, pp. 542–549
  * [33] Shachar Resisi, Yehonatan Viernik, Sebastien M Popoff and Yaron Bromberg  “Wavefront shaping in multimode fibers by transmission matrix engineering”  In _APL Photonics_ 5.3 AIP Publishing LLC, 2020, pp. 036103
  * [34] Philipp Del Hougne, Mathias Fink and Geoffroy Lerosey  “Optimally diverse communication channels in disordered environments with tuned randomness”  In _Nature Electronics_ 2.1 Nature Publishing Group, 2019, pp. 36–41
  * [35] Philipp Del Hougne, K Brahima Yeo, Philippe Besnier and Matthieu Davy  “Coherent wave control in complex media with arbitrary wavefronts”  In _Physical Review Letters_ 126.19 APS, 2021, pp. 193903
  * [36] David J McCabe et al.  “Spatio-temporal focusing of an ultrafast pulse through a multiply scattering medium”  In _Nature communications_ 2.1 Nature Publishing Group, 2011, pp. 1–5
  * [37] Ori Katz, Eran Small, Yaron Bromberg and Yaron Silberberg  “Focusing and compression of ultrashort pulses through scattering media”  In _Nature photonics_ 5.6 Nature Publishing Group, 2011, pp. 372–377
  * [38] Rob A Rutenbar  “Simulated annealing algorithms: An overview”  In _IEEE Circuits and Devices magazine_ 5.1 IEEE, 1989, pp. 19–26
  * [39] Madhu Veettikazhy et al.  “BPM-Matlab: an open-source optical propagation simulation tool in MATLAB”  In _Optics Express_ 29.8 Optical Society of America, 2021, pp. 11819–11832
  * [40] Keang-Po Ho and Joseph M Kahn  “Statistics of group delays in multimode fiber with strong mode coupling”  In _Journal of lightwave technology_ 29.21 IEEE, 2011, pp. 3119–3128
  * [41] Ivo Micha Vellekoop and AP Mosk  “Phase control algorithms for focusing light through turbid media”  In _Optics communications_ 281.11 Elsevier, 2008, pp. 3071–3080

## Supplementary Material

### Optimization Algorithm

The propagation of each wavelength in the fiber is determined by the curvature
of the bends, set by the travel of each actuator. Therefore, the spectrum at
each spot on the distal end of the MMF is determined by the travel of each
actuator. The optimization algorithm searches for the travels that maximize
the intensity in a desired wavelength band (in the low resolution
configuration) or in a desired ROI on the camera (in the high resolution
configuration).

Since the propagation of the light in each curved segment depends on the
deformations induced by all the actuators prior to it, the ideal way to deal
with such an optimization problem is using a nonlinear optimization algorithm.
Here we use the simulated annealing (SA) algorithm [38], which we found to
yield the best results in terms of both the cost values and convergence times,
which is about 10 minutes.

The optimization parameters we used are initial temperature T of 20, with a
cooling function that scales as T$\alpha^{i}$, where $i$ is the iteration
number and $\alpha$ is 0.97 (0.99) in the experiments (simulations). The cost
function maximizes the intensity in the spectral (spatial) window in the
spectrum (image), normalized by the total intensity. We run the algorithm
until we observe a sequence of 30 iterations without any improvement in the
cost function. To decrease the probability of converging to local minima, at
each iteration we add with some probability $p$ a weak perturbation to the
optimal configuration of actuators and increase $p$ as the temperature
decreases.

### System response time

To measure the response time of the fiber-piano, we use the low resolution
configuration and optimize the system for a specific spectral window. Then we
switch between the optimal configuration of actuators and some other
configuration. The response time of the system is measured by the time it
takes to reach from 10 to 90 percent of the maximal intensity. We find that
the rise time is 45ms, whereas the fall time is 10ms, as shown in Fig.S1.

### System stability

To test the stability of the system, once we find the optimal configuration,
we keep the actuators in the same position and record the output spectrum over
time. We find that the spectrum maintains its shape and the overall
transmissions fall by approximately $30\%$ after 10 hours (see Fig.S2).

Figure S1: System response time. Normalized intensity obtained when switching
between the optimized configuration of the actuators and a random
configuration. The rise time (10-90 percent from the optimal intensity) is
45ms (top right inset), and the fall time is 10ms (bottom right inset). Figure
S2: System stability. The measured spectra right after the optimization (blue
curve), 5 hours after optimization (red curve), and 10 hours after the
optimization (yellow curve). The peak intensity dropped by $\approx$$30\%$
after 10 hours.

### Simulation

#### Transmission matrix computation

To simulate the operation on the fiber-piano, we chose 44 values for the radii
of curvature, which model the bends induced by the piezoelectric actuators.
The inverse of the radii of curvature are equally spaced between the maximal
and minimal values. We compute the 44 transmission matrices (TM),
${C_{1},...C_{44}}$, that correspond to the 44 values of the radii of
curvature, by propagating the fiber’s guided modes in the curved segment and
projecting the output fields on the fiber mode basis. We then compute the TM
of the fiber, which is deformed by $M$ actuators, by alternatively multiplying
each one of the $M$ transmission matrices of the $M$ curved segments,
$T_{i}\in{C_{1},..C_{44}};i\in{1,..,M}$, with the diagonal matrix $D$:
$T_{fiber}=DT_{M}D...T_{2}DT_{1}D$. $D$ represents the phase accumulated by
each guided mode in a straight fiber. Fig.S3 depicts the fiber’s TM for an
increasing number of actuators $M$, showing that as $M$ increases the TM
evolves from a band-diagonal matrix to a fully randomized matrix.

Figure S3: Transmission matrix of the fiber for an increasing number of
actuators $M$. Each image represents the absolute square of the TM of the
fiber, supporting 129 modes

#### Evaluation of the number of excited modes

For the simulations that model the experimental configuration with 28
actuators, we excite at the input only the first 50 modes, and not all the
modes supported by the fiber, since by inspection of the speckle pattern
obtained experimentally, we conclude that in our experiments we excite roughly
50 modes. The estimation of the number of excited modes is based on measuring
the number of speckle grains at the distal end of the fiber, defined by the
ratio of the core area and the speckle correlation area. Indeed, the
experimentally measured speckle pattern has a similar grain size and number of
grains to the speckle pattern simulated assuming excitation of the first 50
guided modes Fig.S4.

Figure S4: Estimation of the number of excited modes. (a) Experimental and (b)
simulated output of the MMF. In the simulation we found that we need to
excited 50 modes in order to obtain roughly the same number of speckle grains
and the same grain size as in the experiment.


# Diameter two properties in some vector-valued function spaces

Han Ju Lee  Department of Mathematics Education, Dongguk University - Seoul,
04620 (Seoul), Republic of Korea<EMAIL_ADDRESS>and  Hyung-Joon Tag
Department of Mathematics Education, Dongguk University - Seoul, 04620
(Seoul), Republic of Korea<EMAIL_ADDRESS>
(Date: 2021. February 20.)

###### Abstract.

We introduce a vector-valued version of a uniform algebra, called the vector-
valued function space over a uniform algebra. The diameter two properties of
the vector-valued function space over a uniform algebra on an infinite compact
Hausdorff space are investigated. Every nonempty relatively weakly open subset
of the unit ball of a vector-valued function space $A(K,(X,\tau))$ over an
infinite dimensional uniform algebra has the diameter two, where $\tau$ is a
locally convex Hausdorff topology on a Banach space $X$ compatible to a dual
pair. Under the assumption on $X$ being uniformly convex with norm topology
$\tau$ and the additional condition that $A\otimes X\subset A(K,X)$, it is
shown that Daugavet points and $\Delta$-points on $A(K,X)$ over a uniform
algebra $A$ are the same, and they are characterized by the norm-attainment at
a limit point of the Shilov boundary of $A$. In addition, a sufficient
condition for the convex diametral local diameter two property of $A(K,X)$ is
also provided. As a result, the similar results also hold for an infinite
dimensional uniform algebra.

###### Key words and phrases:

diameter two property, uniform algebra, Urysohn-type lemma, Shilov boundary

###### 2010 Mathematics Subject Classification:

Primary 46B04; Secondary 46B20, 46E30, 47B38

The first author was supported by Basic Science Research Program through the
National Research Foundation of Korea(NRF) funded by the Ministry of
Education, Science and Technology [NRF-2020R1A2C1A01010377]. The second author
was supported by Basic Science Research Program through the National Research
Foundation of Korea(NRF) funded by the Ministry of Education, Science and
Technology [NRF-2020R1A2C1A01010377].

##  1\. Introduction

In this article, we study the diameter two properties of a uniform algebra and
the vector-valued function space $A(K,(X,\tau))$ for a complex Banach space
$X$ with a locally convex Hausdorff topology $\tau$ compatible with a dual
pair by using the Urysohn-type lemma. The diameter two properties have gained
a lot of attention recently and been studied in various Banach spaces.
Exploring these properties involves weakly open subsets and slices of the unit
ball $B_{X}$ of a Banach space $X$. Let $S_{X}$ be the unit sphere of a Banach
space $X$ and $X^{*}$ be the dual space of $X$. For $x^{*}\in S_{X^{*}}$ and
$\epsilon>0$, a slice $S(x^{*},\epsilon)$ of the unit ball $B_{X}$ is defined
by $S(x^{*},\epsilon)=\\{x\in B_{X}:Re\,x^{*}x>1-\epsilon\\}$. Here we recall
the definitions of the “classical” diameter two properties introduced in [3].

######  Definition 1.1.

(“Classical” D2Ps)

  1. (i)

A Banach space $X$ has the strong diameter two property (SD2P) when every
convex combination of slices has diameter two.

  2. (ii)

A Banach space $X$ has the diameter two property (D2P) when every nonempty
relatively weakly open subset of the unit ball $B_{X}$ has diameter two.

  3. (iii)

A Banach space $X$ has the local diameter two property (LD2P) when every slice
of the unit ball $B_{X}$ has diameter two.

Since every weakly open subset of the unit ball contains a convex combination
of slices [12, Lemma II.1], the SD2P implies the D2P. The D2P implies the LD2P
because every slice is a relatively weakly open subset of the unit ball. It is
well-known that these properties are weaker than the Daugavet property [3,
Theorem 4.4] and that these properties are the other extremes of the Radon-
Nikodým property which states, geometrically, that there exists a nonempty
slice of the unit ball with arbitrarily small diameter. The classical D2Ps are
examined in various spaces. It is well-known that $c_{0}$ and $\ell_{\infty}$
has the SD2P. The infinite dimensional uniform algebra over a compact
Hausdorff space also has the SD2P [3, 20], in fact, even stronger property
called the symmetric strong diameter two property (SSD2P) [4]. Other known
examples with the D2P are interpolation spaces $L_{1}+L_{\infty}$ and
$L_{1}\cap L_{\infty}$ equipped with certain norms [2]. These spaces do not
satistfy the Daugavet property but the D2P. For a Banach space $X$, the space
$C(K,(X,\tau))$ of $X$-valued continuous function over a compact Hausdorff
space $K$, where $\tau$ a locally convex Hausdorff topology compatible to a
dual pair, is also known to have the D2P [5].

Not only the “classical” D2Ps mentioned above, but we also explore the
“diametral” version of the D2Ps. Studying these properties requires us to have
a firm grasp on the $\Delta$-points and the Daugavet points. Let
$\Delta_{\epsilon}(x)=\\{y\in B_{X}:\|x-y\|\geq 2-\epsilon\\}$. Then a point
$x\in S_{X}$ is said to be a $\Delta$-point if
$x\in\overline{\text{conv}}\Delta_{\epsilon}(x)$ for every $\epsilon>0$. We
denote the set of all $\Delta$-points by $\Delta_{X}$. Similarly, a point
$x\in S_{X}$ is said to be a Daugavet point if $B_{X}$ =
$\overline{\text{conv}}\Delta_{\epsilon}(x)$ for every $\epsilon>0$. Notice
that every Daugavet point is a $\Delta$-point. We recall the definitions of
the diametral diameter two properties introduced in [1, 6].

######  Definition 1.2.

(“Diametral” D2Ps)

  1. (i)

A Banach space $X$ has the convex diametral local diameter two property
(convex-DLD2P) if $B_{X}=\overline{\text{conv}}\Delta_{X}$, that is, the unit
ball is the closed convex hull of the set of all $\Delta$-points.

  2. (ii)

A Banach space $X$ has the diametral local diameter two property (DLD2P) if
for every slice $S$ of $B_{X}$, every $x\in S_{X}\cap S$, and every
$\epsilon>0$ there exists $y\in S$ such that $\|x-y\|\geq 2-\epsilon$.

  3. (iii)

A Banach space $X$ has the diametral diameter two property (DD2P) if for every
relative weakly open subset $U$ of $B_{X}$, every $x\in S_{X}\cap U$, and
every $\epsilon>0$, there exists $y\in U$ such that $\|x-y\|\geq 2-\epsilon$.

There is also a stronger version of the DD2P called the diametral strong
diameter two property (DSD2P), which uses a convex combination of slices, but
this property is now known to be equivalent to the Daugavet property [14]. It
was shown in [6] that the Daugavet property implies the DD2P and that the DD2P
implies the DLD2P. The convex-DLD2P is stronger than the LD2P and weaker than
DLD2P [1]. Moreover, we can see that the D2P is weaker than the DD2P from
their definitions. If one wishes to see a clearer picture, we refer to a nice
visualization of the relationship between various diameter two properties in
the dissertation of Pirk (see [23, pg 82-85]). These properties have been also
examined on several Banach spaces. For example, the spaces $c$ of convergent
sequences and $l_{\infty}$ do not have the DLD2P but the convex-DLD2P [1,
Corollary 5.4, Remark 5.5]. Moreover, $c_{0}$ even fails to have the convex-
DLD2P [1, pg 18].

The DLD2P and the Daugavet property have equivalent definitions in terms of
$\Delta$-points and Daugavet points. A Banach space $X$ having the DLD2P is
equivalent to say that every point in $S_{X}$ is a $\Delta$-point (see [13,
Theorem 1.4] and [24, Open problem (7)]). Similarly, a Banach space $X$ has
the Daugavet property if and only if every point in $S_{X}$ is a Daugavet
point [24, Corollary 2.3].

The organization of this article consists of four parts. In section 2, we
provide necessary information about the uniform algebra $A$, the space
$A(K,(X,\tau))$, and the Urysohn-type lemma, which is the key ingredient to
prove our results throughout this article. In section 3, we make a few remarks
on the symmetric strong diameter two property of a scalar-valued function
algebra. In section 4, we show that the vector-valued function space
$A(K,(X,\tau))$ over a uniform algebra also satisfies the D2P. By the Gelfand
transform we show that a function space $A(\Omega,X)$ on a Hausdorff space
$\Omega$ is isometrically isomorphic to a function space
$A(M_{A},(X^{**},w^{*}))$ over a uniform algebra, so it is shown that
$A(\Omega,X)$ also has the D2P if the base (function) algebra is infinite
dimensional. This result shows D2P of some function spaces like
$A_{b}(B_{X},Y)$ and (resp. $A_{u}(B_{X},Y)$) consisting of $Y$-valued
functions which are bounded (resp. uniformly continuous) on $B_{X}$ and are
holomorphic on the interior of $B_{X}$ when $X$ and $Y$ are Banach spaces. In
section 5, we present some results on the Daugavet and $\Delta$\- points in a
vector-valued function space $A(K,X)$ over a uniform algebra. Under the
condition of $X$ being uniformly convex and the additional assumption that
$A\otimes X\subset A(K,X)$, we show that these points are the same and
characterized by norm-attainment at a limit point of the Shilov boundary. Also
under the same assumptions, a sufficient condition for the convex-DLD2P of
$A(K,X)$ is provided.

##  2\. Preliminaries

In this article, we assume that $K$ is a compact Hausdorff space and $X$, $Y$
are nontrivial complex Banach spaces unless specified. Let $X^{*}$ be the dual
space of $X$. We work with a Banach space $X$ equipped with a locally convex
Hausdorff (LCH) topology $\tau$ compatible to the dual pair $(X,X^{*})$ or
$(X^{*},X)$. Recall that $(X,\tau)$ is said to be compatible to a dual pair
$(X,X^{*})$ if $(X,\tau)^{*}=X^{*}$ and a LCH topology $\tau$ on $X^{*}$ is
said to be compatible to a dual pair $(X^{*},X)$ if $(X^{*},\tau)^{*}=X$. In
this paper, we say that a LCH topology $\tau$ on a Banach space $X$ is
compatible to a dual pair if $(X,\tau)^{*}=X^{*}$ or $(X,\tau)^{*}=X_{*}$,
where $X_{*}$ is the predual of $X$.

The weak topology on $X$ is denoted by $(X,w)$, the weak∗ topology on $X^{*}$
by $(X^{*},w^{*})$, and the norm topology on $X$ by $(X,\|\cdot\|)$. The
Mackey topology for $X$ (resp. $X^{*}$) is the finest topology where every
linear functional $x^{*}\in X^{*}$ (resp. linear functional defined by $x\in
X$) is continuous. The Mackey-Arens theorem [19, Theorem 8.7.4] says that a
locally convex topology $\tau$ for $X$ (resp. $X^{*}$) compatible to a dual
pair includes the weak (resp. weak∗-) topology and is included in the Mackey
topology on $X$ (resp. $X^{*}$) and that is known to be coarser than the norm
topology on $X$ [22, IV.3.4] (resp. the norm topology on $X^{*}$ [19, Example
8.5.5 and Example 8.8.9]). Hence for any LCH topology $\tau$ compatible to a
dual pair $(X,X^{*})$ (resp. $(X^{*},X)$), we have
$(X,w)\subset\tau\subset(X,\|\cdot\|)$ (resp.
$(X^{*},w^{*})\subset\tau\subset(X^{*},\|\cdot\|)$). Also for such topology
$\tau$ on $X$ compatible to $(X,X^{*})$, $\tau$-bounded sets are weakly
bounded and so norm-bounded. Similarly, for a LCH topology $\tau$ on $X^{*}$
compatible to $(X^{*},X)$, $\tau$-bounded sets are weakly∗-bounded and so
norm-bounded in $X^{*}$. For more details on the theory of topological vector
spaces, we refer to [19, 22].

Now we show that the space $C(K,(X,\tau))$, which consists of all continuous
functions from a compact Hausdorff space $K$ to $(X,\tau)$, equipped with the
supremum norm is a Banach space. Even though this fact is already known in
[5], we include the proof for completeness.

######  Proposition 2.1.

Let $K$ be a compact Hausdorff space, let $X$ be either a Banach space and let
$\tau$ be a locally convex Hausdorff topology compatible to a dual pair. Then
the space $C(K,(X,\tau))$ equipped with the supremum norm is a Banach space.

###### Proof.

Let us consider when $\tau$ is a LCH topology compatible to either a dual pair
$(X,X^{*})$ or $(X,X_{*})$. Since $f(K)$ is $\tau$-compact for every $f\in
C(K,(X,\tau))$, the range $f(K)$ is bounded in the norm topology of $X$. This
makes the norm $\|f\|=\sup\\{\|f(t)\|_{X}:t\in K\\}$ to be well-defined.

To show the completeness of $C(K,(X,\tau))$, let
$(f_{n})_{n=1}^{\infty}\subset C(K,(X,\tau))$ be a Cauchy sequence. Then for
any $\epsilon>0$, there exists $N$ such that for every $n,m\geq N$, we have
$\|f_{n}-f_{m}\|<\epsilon$. Notice that for every $t\in K$,

(1) $\displaystyle\|f_{n}(t)-f_{m}(t)\|_{X}\leq\|f_{n}-f_{m}\|<\epsilon,$

and so $(f_{n}(t))_{n=1}^{\infty}$ is a Cauchy sequence in $X$ with respect to
the norm topology. Since $X$ is complete with the norm topology,
$(f_{n}(t))_{n=1}^{\infty}$ converges in norm. So for each $t\in K$, let
$f(t)=\lim_{n\rightarrow\infty}f_{n}(t)$. Then by (1), for any $\epsilon>0$
there exists $N$ such that for all $n\geq N$, $\sup_{t\in
K}\|f_{n}(t)-f(t)\|_{X}<\frac{\epsilon}{2}$.

Now we have to show that $f\in C(K,(X,\tau))$. First, let $V$ be a
$\tau$-neighborhood of $0$. Choose a balanced, convex $\tau$-neighborhood $W$
of $0$ such that $W+W+W\subset V$. Since $W$ is also open in the norm
topology, there exists $\epsilon B_{X}\subset W$ and $N\in\mathbb{N}$ such
that for all $n\geq N$ and $t\in K$, $f(t)-f_{n}(t)\in\epsilon B_{X}$. Hence

$\displaystyle f(t)-f(t_{0})$
$\displaystyle=(f(t)-f_{N}(t))+(f_{N}(t)-f_{N}(t_{0}))+(f_{N}(t_{0})-f(t_{0}))$
$\displaystyle\in\epsilon B_{X}+(f_{N}(t)-f_{N}(t_{0}))+\epsilon B_{X}$
$\displaystyle\subset W+(f_{N}(t)-f_{N}(t_{0}))+W.$

Moreover, $f_{N}$ is continuous with respect to $\tau$. So there exists a
neighborhood $U$ of $t_{0}$ such that for every $t\in U$,
$f_{N}(t)-f_{N}(t_{0})\in W$. Therefore, $f(t)-f(t_{0})\in W+W+W\subset V$ for
all $t\in U$. This shows that $f\in C(K,(X,\tau))$. Therefore, $C(K,(X,\tau))$
is a Banach space. ∎

Let $\Omega$ be a Hausdorff space and $C_{b}(\Omega)$ be the Banach algebra of
all bounded continuous functions over $\Omega$ equipped it the supremum norm.
A function algebra $A(\Omega)$ on $\Omega$ is a closed subalgebra of
$C_{b}(\Omega)$ that separates the points of $\Omega$ and contains constant
functions. Here separating points means for each pair
$(s,t)\in\Omega\times\Omega$ with $s\neq t$, there exists $f\in A(\Omega)$
such that $f(s)\neq f(t)$. If $\Omega$ is a compact Hausdorff space, a
function algebra is called a uniform algebra.

We introduce $(X,\tau)$-valued function space over a uniform algebra for a
Banach space $X$ as follows.

######  Definition 2.2.

Let $X$ be a Banach space and $\tau$ be a LCH topology on $X$ compatible to a
dual pair. A closed subspace $A(K,(X,\tau))$ of $C(K,(X,\tau))$ is said to be
the $(X,\tau)$-valued function space over a uniform algebra $A$ if the
following two conditions are satisfied:

  1. (i)

The base algebra, defined by $A:=\\{y^{*}\circ f:f\in
A(K,(X,\tau)),y^{*}\in(X,\tau)^{*}\\}$, is a uniform algebra over $K$.

  2. (ii)

$\phi f\in A(K,(X,\tau))$ for every $\phi\in A$ and $f\in A(K,(X,\tau))$ where
$(\phi f)(t)=\phi(t)f(t)$ for $t\in K$.

Notice that if $X=\mathbb{C}$, we have the usual uniform algebra $A$ on $K$.
When $\tau$ is the norm topology, we call the space $A(K,(X,\tau))$ an
$X$-valued function space over a uniform algebra and denote the space by
$A(K,X)$. Similar vector-valued version of a uniform algebra has been studied
in [7].

A point $t_{0}\in K$ is said to be a strong boundary point for a uniform
algebra $A$ if for every neighborhood $U$ containing $t_{0}$, there exists
$f\in A$ such that $f(t_{0})=\|f\|=1$ and $\sup_{t\in K\setminus U}|f(t)|<1$.
A subset $S\subset X$ is said to be a boundary if for each $f\in A$, there
exists an element $t\in S$ such that $|f(t)|=\|f\|_{\infty}$. The Shilov
boundary $\Gamma(A)$ of $A$ is the smallest closed boundary of $A$ and it is
known that it is the intersection of all closed boundaries of $A$. For a
uniform algebra, let $K_{A}=\\{\lambda\in
A^{*}:\|\lambda\|=\lambda(1_{A})=1\\}$ and denote the set of its extreme
points by $\text{ex}\,K_{A}$. The set $\text{ex}\,K_{A}$ is called the Choquet
boundary of $A$. It is well-known that each $\lambda\in\text{ex}\,K_{A}$ are
associated with some elements $x\in K$ and uniquely represented by the Dirac
measure $\delta_{x}$ [9, Lemma 4.3.2]. For this reason, we use $\Gamma_{0}(A)$
to denote the set of such elements $x\in K$ corresponding to the elements in
the Choquet boundary. Also, the set of strong boundary points and
$\Gamma_{0}(A)$ coincide with each other when we consider a uniform algebra
$A$ over a compact Hausdorff space $K$ [9, Theorem 4.3.5]. It is known that
the Shilov boundary $\Gamma(A)$ of a uniform algebra $A$ is the closure of its
Choquet boundary ${\Gamma_{0}(A)}$ [9, Corollary 4.3.7.a].

Now we recall the Urysohn-type lemma that is extensively used throughout this
article.

######  Lemma 2.3.

[8, Lemma 2.5] Let $A\subset C(K)$ be a uniform algebra and $\Gamma_{0}$ be
its Choquet boundary. Then, for every open set $U\subset K$ with
$U\cap\Gamma_{0}\neq\emptyset$ and $0<\epsilon<1$, there exists $f\in A$ and
$t_{0}\in U\cap\Gamma_{0}$ such that $f(t_{0})=\|f\|_{\infty}=1$,
$|f(t)|<\epsilon$ for every $t\in K\setminus U$ and

(2) $|f(t)|+(1-\epsilon)|1-f(t)|\leq 1\,\,\,\text{for all}\,\,\,t\in K.$

The following observation is useful for later.

######  Lemma 2.4.

Let $X$ be a Banach space and let $\tau$ be a LCH topology compatible to a
dual pair. Suppose that $A(K,(X,\tau))$ is a $(X,\tau)$-valued function space
over the base algebra $A$ and $L$ is a closed boundary for $A$. The space of
restrictions of elements of $A(K,(X,\tau))$ to $L$ is denoted by
$A(L,(X,\tau))$ and the restrictions of elements of $A$ to $L$ is denoted by
$A(L)$. Then $A(L,(X,\tau))$ is a $(X,\tau)$-valued function space over the
base algebra $A(L)$ and it is isometrically isomorphic to $A(K,(X,\tau))$.

###### Proof.

Given $f\in A(K,(X,\tau))$, let $f_{|L}$ be the restriction of $f$ to $L$.
Then we have

$\displaystyle\|f\|$ $\displaystyle=\sup\\{\|f(t)\|:t\in
K\\}=\sup\\{|x^{*}f(t)|:t\in K,x^{*}\in B_{(X,\tau)^{*}}\\}$
$\displaystyle=\sup\\{|x^{*}f(t)|:t\in L,x^{*}\in
B_{(X,\tau)^{*}}\\}=\sup\\{\|f(t)\|:t\in L\\}=\|f_{|L}\|.$

So $A(L,(X,\tau))$ is a surjective isometric image of $A(K,(X,\tau))$ by the
map $f\mapsto f_{|L}$ and it is a closed subspace of $C(L,(X,\tau))$. It is
easy to see that $A(L,(X,\tau))$ satisfies the condition (i) and (ii) in the
Definition 2.2 and its base algebra is $A(L)$. This completes the proof. ∎

Since the Shilov boundary $\Gamma$ of a uniform algebra is a closed boundary
of $K$, by Lemma 2.4 or [17, Theorem 4.1.6]), a uniform algebra is isometric
to a uniform algebra of its restrictions to $\Gamma$. We need another useful
lemma on the isolated point of the Shilov boundary.

######  Lemma 2.5.

Let $A$ be a uniform algebra on a compact Hausdorff space $K$ and let $t_{0}$
be an isolated point of the Shilov boundary $\Gamma$ of $A$. Then there exists
a function $\phi\in A$ such that $\phi(t_{0})=\|\phi\|=1$ and $\phi(t)=0$ for
$t\in\Gamma\setminus\\{t_{0}\\}$.

###### Proof.

Since $t_{0}$ is an isolated point of $\Gamma$, there is an open set such that
$U\cap\Gamma=\\{t_{0}\\}$. The set of strong boundary points $\Gamma_{0}$ is
dense in $\Gamma$. So the point $t_{0}\in\Gamma$ is in fact a strong boundary
point of $A$. By definition of a strong boundary point, there exists a
function $\psi\in A$ such that $\psi(t_{0})=\|\psi\|=1$ and $\sup_{t\in
K\setminus U}|\psi(t)|<1$. Note that
$\\{(\psi_{|\Gamma})^{n}\\}_{n=1}^{\infty}$ converges uniformly in $C(L)$. By
the isometry of the map from $f\in A$ to its restriction $f_{|L}$ to $L$,
$\\{\psi^{n}\\}_{n=1}^{\infty}$ is Cauchy in $A$ and converges uniformly to
$\phi$ in $A$. Now it is clear that $\phi(t)=0$ for
$t\in\Gamma\setminus\\{t_{0}\\}$ and $\phi(t_{0})=\|\phi\|=1$. Thus, we obtain
the desired function. ∎

##  3\. A short remark on the SSD2P of function algebras

First we start with what is known as the symmetric strong diameter two
property (SSD2P) that is stronger than the SD2P.

######  Definition 3.1.

[4, Definition 1.3] A Banach space $X$ has the symmetric strong diameter two
property (SSD2P) if whenever $n\in\mathbb{N}$, $S_{1},\dots,S_{n}$ are slices
of $B_{X}$, and $\epsilon>0$, there exists $x_{i}\in S_{i}$,
$i=1,2,3,\dots,n$, and $y\in B_{X}$ such that $x_{i}\pm y\in S_{i}$ for every
$i\in\\{1,2,3,\dots,n\\}$ and $\|y\|>1-\epsilon$.

It has been already known from [3, Theorem 4.2] that the infinite dimensional
uniform algebra over a compact Hausdorff space has the SSD2P and their proof
method is extended to show the SSD2P of somewhat regular subspaces of the
space $C_{0}(L)$ of continuous functions over a locally compact Hausdorff
space $L$ vanishing at infinity [4, Theorem 2.2].

######  Definition 3.2.

[4, Definition 2.1] Let $L$ be a locally compact Hausdorff space and
$C_{0}(L)$ be the space of continuous functions over $L$ vanishing at
infinity. A linear subspace $Y$ of $C_{0}(L)$ is somewhat regular, if whenever
$V$ is a non-empty open subset of $L$ and $0<\epsilon<1$, there is an $f\in Y$
such that

$\|f\|=1\,\,\,\text{and}\,\,\,|f(x)|\leq\epsilon\,\,\,\text{for
every}\,\,\,x\in L\setminus V.$

We make a few remarks here. In view of Lemma 2.3 we can easily verify that the
infinite dimensional uniform algebra $A$ over a compact Hausdorff space $K$ is
an example of somewhat regular subspace of $C(\Gamma)$ due to the fact that
$A$ is isometric to $A(\Gamma)$ and that $\Gamma_{0}=\Gamma_{0}(A)$ is dense
in $\Gamma=\Gamma(A)$ because $K$ is compact. In addition, Lemma 2.3 is a
special case of the Urysohn-type lemma for somewhat regular subspaces (see [4,
Lemma 2.4]) that targets on uniform algebras.

######  Theorem 3.3.

[4, Theorem 2.2] The infinite dimensional uniform algebra $A$ over a compact
Hausdorff space $K$ has the SSD2P.

Now let $C_{b}(B_{X})$ be the space of bounded, complex-valued, continuous
functions on the unit ball $B_{X}$ of a Banach space $X$. The space
$A_{b}(B_{X})$ is a closed subalgebra of $C_{b}(B_{X})$ that consists of
holomorphic functions on the interior of $B_{X}$. The space $A_{u}(B_{X})$ is
a closed subalgebra of $C_{b}(B_{X})$ that consists of functions in
$A_{b}(B_{X})$ that are uniformly continuous on $B_{X}$. From the fact that
function algebras $A_{b}(B_{X})$ and $A_{u}(B_{X})$ are isometric to a uniform
algebra on a compact Hausdorff space via the Gelfand transformation (see [16,
Proposition 2] or Theorem 4.5), we can deduce the following fact on the SSD2Ps
of $C_{b}(B_{X})$, $A_{b}(B_{X})$ and $A_{u}(B_{X})$.

######  Corollary 3.4.

Let $X$ be a Banach space. The function algebras $C_{b}(B_{X})$,
$A_{b}(B_{X})$ and $A_{u}(B_{X})$ have the SSD2P.

##  4\. The D2P of vector-valued function spaces over a uniform algebra

Now we consider the space $A(K,(X,\tau))$ where $X$ is a Banach space with a
locally convex Hausdorff topology $\tau$ compatible with a dual pair. Here the
norm of the space $A=A(K)$ is denoted by $\|\cdot\|_{\infty}$. First we recall
a useful fact about the D2P of a direct sum of two Banach spaces.

######  Lemma 4.1.

[5, Lemma 2.2] Let $X$ be a Banach space satisfying the diameter two property.
Then for any arbitrary Banach space $Y$, $X\oplus_{\infty}Y$ has the diameter
two property.

Here we present the main result of this section.

######  Theorem 4.2.

Let $K$ be a compact Hausdorff space and $X$ be a Banach space endowed with a
locally convex Hausdorff topology $\tau$ compatible to a dual pair. If the
base algebra $A$ is infinite dimensioanl, then the space $A(K,(X,\tau))$ has
the diameter two property, i.e. every nonempty weakly open subset of the unit
ball has the diameter two.

###### Proof.

In view of Lemma 2.4, we only have to show the D2P for $A(\Gamma,(X,\tau))$
where $\Gamma=\Gamma(A)$ is a Shilov boundary of a uniform algebra. Since $A$
is infinite dimensional, $\Gamma$ is infinite. We write $A(\Gamma,X)$ instead
of $A(\Gamma,(X,\tau))$ for convenience.

Now let $W$ be a nonempty weakly open subset of $A(\Gamma,X)$. Three cases
will be considered.

  1. (i)

Assume that $\Gamma$ is perfect. Let $a\in W\cap S_{A(\Gamma,X)}$. Then for
every $\delta>0$, there exists $t_{0}\in\Gamma$ such that
$\|a(t_{0})\|_{X}>1-\delta$. Also we can find $f\in S_{(X,\tau)^{*}}$ such
that $\text{Re}\,f(a(t_{0}))>1-\delta$ because $(X,\tau)^{*}=X^{*}$ or
$(X,\tau)^{*}=X_{*}$, where $X_{*}$ is a predual of $X$.

Define $U=\\{t\in\Gamma:\text{Re}\,f(a(t))>1-\delta\\}$. Notice that $U$
contains $t_{0}$. Since $t_{0}$ is a limit point of $\Gamma$, we can construct
pairwise disjoint open subsets $U_{n}\subset U$ for $n\in\mathbb{N}$. From the
fact that $\Gamma_{0}=\Gamma_{0}(A)$ is dense in $\Gamma$, we know
$U_{n}\cap\Gamma_{0}\neq\emptyset$ for all $n\in\mathbb{N}$. So by Lemma 2.3,
for each $n\in\mathbb{N}$ there exist $\phi_{n}\in A$ and $t_{n}\in
U_{n}\cap\Gamma_{0}$ such that $\phi_{n}(t_{n})=\|\phi_{n}\|_{\infty}=1$,
$|\phi_{n}(t)|<\frac{\delta}{2^{n+2}}$ for $t\in\Gamma\setminus U_{n}$, and

$|\phi_{n}(t)|+\left(1-\frac{\delta}{2^{n+2}}\right)|1-\phi_{n}(t)|\leq
1\,\,\,\text{for}\,\,\,t\in\Gamma.$

Define $g_{n}=1-2\phi_{n}$. First we want to show that $\|g_{n}\|\leq
1+\frac{\delta}{2^{n}}$. For every $t\in\Gamma$,

$\displaystyle|1-2\phi_{n}(t)|$ $\displaystyle=$
$\displaystyle\left|1-2\phi_{n}(t)+\frac{\delta}{2^{n+2}}-\frac{\delta}{2^{n+2}}+\frac{\delta}{2^{n+1}}\phi_{n}(t)-\frac{\delta}{2^{n+1}}\phi_{n}(t)\right|$
$\displaystyle\leq$
$\displaystyle\left|\left(1-\frac{\delta}{2^{n+2}}\right)(1-2\phi_{n}(t))\right|+\frac{\delta}{2^{n+2}}+\left|\frac{\delta}{2^{n+1}}\phi_{n}(t)\right|$
$\displaystyle\leq$
$\displaystyle\left|\left(1-\frac{\delta}{2^{n}}\right)(1-\phi_{n}(t))-\left(1-\frac{\delta}{2^{n}}\right)\phi_{n}(t)\right|+\frac{\delta}{2^{n}}$
$\displaystyle\leq$ $\displaystyle
1-|\phi_{n}(t)|+|\phi_{n}(t)|+\frac{\delta}{2^{n}}$ $\displaystyle\leq$
$\displaystyle 1+\frac{\delta}{2^{n}}.$

Hence $\|g_{n}\|_{\infty}\leq 1+\frac{\delta}{2^{n}}$. Define
$h_{n}=\frac{g_{n}}{1+\delta/2^{n}}$. Notice that $h_{n}\in B_{A}$ for all
$n\in\mathbb{N}$ and that $h_{n}(t)$ converges to $1$ pointwise.

Now, let $F\in A(\Gamma,X)^{*}$ be a bounded linear functional on
$A(\Gamma,X)$. If we define $\varphi(h)=F(ha)$ for $h\in A$, $\varphi$ is a
bounded linear functional on $A$ because
$|\varphi(h)|\leq\|F\|_{A(\Gamma,X)^{*}}\|h\|_{\infty}\|a\|<\|h\|_{\infty}\|F\|_{A(\Gamma,X)^{*}}<\infty$
for all $h\in A$. By the Hahn-Banach extension theorem and the Riesz
representation theorem, $\varphi$ is represented by a regular complex Borel
measure $\mu$ and $\varphi(h_{n})=\int_{\Gamma}h_{n}d\mu$. Moreover,
$\varphi(h_{n})=\int_{\Gamma}h_{n}d\mu\rightarrow\int_{\Gamma}1d\mu=\varphi(1)$
by the bounded convergence theorem. Hence
$F(h_{n}a)=\varphi(h_{n})\rightarrow\varphi(1)=F(a)$ and so the functions
$h_{n}a$ converges to $a$ weakly in $A(\Gamma,X)$. Furthermore, we have

$\displaystyle\left\|a-h_{n}a\right\|\geq\left\|a(t_{n})-h_{n}(t_{n})a(t_{n})\right\|_{X}$
$\displaystyle=$ $\displaystyle\left|1-h_{n}(t_{n})\right|\|a(t_{n})\|_{X}$
$\displaystyle\geq$
$\displaystyle\frac{2+\delta/2^{n}}{1+\delta/2^{n}}\cdot\text{Re}\,f(a(t_{n}))$
$\displaystyle\geq$
$\displaystyle\left(2-\frac{\delta/2^{n}}{1+\delta/2^{n}}\right)(1-\delta)$
$\displaystyle\geq$ $\displaystyle(2-\delta)(1-\delta)$ $\displaystyle\geq$
$\displaystyle 2-3\delta.$

Since $\delta>0$ is arbitrary, the diameter of $W\cap B_{A(\Gamma,X)}$ is $2$.
The proof is finished.

  2. (ii)

Now we consider when $\Gamma$ has infinitely many isolated points. Let
$(t_{n})_{n=1}^{\infty}$ be a sequence of isolated points and $t_{0}$ be the
accumulation point of the sequence. Let $a\in W\cap S_{A(\Gamma,X)}$. Let $U$
be a $\tau$-open neighborhood around $0$ and let $V$ be an $\tau$-open
neighborhood around $a(t_{0})\neq 0$ such that $U\cap V=\emptyset$. Notice
that every $\tau$-open neighborhood is also an open neighborhood in the norm
topology on $X$. So by the continuity of $a$, $a^{-1}(V)$ contains infinitely
many $t_{n}$’s. Since $U$ is norm-open, for each $n\in\mathbb{N}$ there is
$\delta>0$ such that $\delta B_{X}\subset U$. Hence $a(t_{n})\in V\subset
X\setminus\delta B_{X}$. That is, $\|a(t_{n})\|_{X}>\delta$ infinitely many
$n\in\mathbb{N}$. So we may assume that $\|a(t_{n})\|_{X}>\delta$ for all
$n\in\mathbb{N}$. Since $\Gamma_{0}$ is dense in $\Gamma$, every isolated
point $t_{n}\in\Gamma$ is a strong boundary point. Hence by Lemma 2.5, there
exists $\phi_{n}\in A$ such that $\phi_{n}(t_{n})=\|\phi_{n}\|=1$ and
$\phi_{n}(t)=0$ for all $t\in\Gamma\setminus\\{t_{n}\\}$.

Define
$g_{n}=1+\left(\frac{1}{\|a(t_{n})\|_{X}}-1\right)\phi_{n}+\left(\frac{-1}{\|a(t_{n+1})\|_{X}}-1\right)\phi_{n+1}$.
We see that $g_{n}(t)$ converges to $1$ pointwise,
$\sup_{n\in\mathbb{N}}\|g_{n}\|_{\infty}<1+\frac{1}{\delta}$ and
$\|g_{n}a\|=1$ for every $n\in\mathbb{N}$. Hence $g_{n}\rightarrow 1$ weakly
in $A$. Moreover, $g_{n}a\rightarrow a$ weakly in $A(\Gamma,X)$ by the same
argument in (i). When we compute the diameter of the relatively weakly open
set of $B_{A(\Gamma,X)}$, we have

$\displaystyle\text{diam}\,(W\cap B_{A(\Gamma,X)})\geq\|g_{n+1}a-g_{n}a\|$
$\displaystyle\geq$
$\displaystyle\|g_{n+1}(t_{n+1})a(t_{n+1})-g_{n}(t_{n+1})a(t_{n+1})\|_{X}$
$\displaystyle=$
$\displaystyle\frac{2}{\|a(t_{n+1})\|_{X}}\|a(t_{n+1})\|_{X}=2.$

  3. (iii)

Finally, let us consider when $\Gamma$ has only finitely many isolated points
$t_{1},t_{2},\dots,t_{n}$. Denote $\Gamma_{1}=\\{t_{1},t_{2},\dots,t_{n}\\}$
and the perfect subset of $\Gamma$ by $\Gamma_{2}=\Gamma\setminus\Gamma_{1}$.
Since the set of strong boundary points are dense in $\Gamma$, we see that
$t_{1},t_{2},\dots,t_{n}$ are also strong boundary points and for each
$i=1,2,\dots,n$, once again by Lemma 2.5, there exist $\phi_{i}\in A(\Gamma)$
such that $\phi_{i}(t_{i})=1$ and $\phi_{i}(t)=0$ for all $t\neq t_{i}$.
Define a bounded linear operator $P:A(\Gamma,X)\rightarrow A(\Gamma,X)$ by
$Pf=\sum_{i=1}^{n}\phi_{i}f$. Then it is easy to check that $P$ is a norm-one
projection. Let $B=P(A(\Gamma,X))$ and $C$ be the restrictions of $f\in
A(\Gamma,X)$ to $\Gamma_{2}$. Then $C$ is an $(X,\tau)$-valued function space
over an infinite dimensional uniform algebra on $\Gamma_{2}$. We will show
that $A(\Gamma,X)$ is isometrically isomorphic to $B\oplus_{\infty}C$. Then by
(i), $C$ has diameter two property and Lemma 4.1 completes the proof.

For $f\in A(\Gamma,X)$, define $\Phi(f)=(P(f),f|_{\Gamma_{2}})$. Then
$\Phi:A(\Gamma,X)\to B\oplus_{\infty}C$ is well-defined and an isometry as
follows

$\|\Phi(f)\|=\max\\{\|P(f)\|,\|f_{\Gamma_{2}}\|\\}=\max\\{\sup_{t\in\Gamma_{1}}\|f(t)\|_{X},\sup_{t\in{\Gamma_{2}}}\|f(t)\|_{X}\\}=\sup_{t\in\Gamma}\|f(t)\|_{X}=\|f\|.$

Given $(f,g)\in B\oplus_{\infty}C$, there exist $f_{1},f_{2}\in A(\Gamma,X)$
such that $f=P(f_{1})$ and $g=f_{2}|_{\Gamma_{2}}$. Let
$h=P(f_{1})+f_{2}-P(f_{2})\in A(\Gamma,X)$ and it is clear that
$\Phi(h)=(P(f_{1}),f_{2}|_{\Gamma_{2}})=(f,g)$. So $\Phi$ is surjective. This
shows that $A(\Gamma,X)$ is isometrically isomorphic to $B\oplus_{\infty}C$
and completes the proof.

∎

The norm, weak and weak-* toplogies are compatible to a dual pair and we get
the following.

######  Corollary 4.3.

Let $X$ be a Banach space and let $K$ be an infinite compact Hausdorff space.
Then the spaces $A(K,X)$, $A(K,(X,w))$, and $A(K,(X^{*},w^{*}))$ have the D2P
if their corresponding base algebras are infinite dimensional.

Now we introduce a Banach space $X$-valued function space (over a function
space) on a Hausdorff space and we show that this pace is isometrically
isomorphic to a $(X^{**},w^{*})$-valued function space over a uniform algebra.
Let $\Omega$ be a Hausdorff space and let $X$ be a Banach space equipped with
the norm topology. The space $C_{b}(\Omega,X)$ is the Banach space of all
bounded $X$-valued continuous functions over $\Omega$ equipped it the supremum
norm.

######  Definition 4.4.

Let $X$ be a Banach space. An $X$-valued function space over a function
algebra on $\Omega$, denoted by $A(\Omega,X)$, is a closed subspace of
$C_{b}(\Omega,X)$ satisfying

  1. (i)

the base algebra $A:=\\{x^{*}\circ f:f\in A(\Omega,X)\\}$ is a function
algebra on $\Omega$.

  2. (ii)

$fg\in A(\Omega,X)$ for every $f\in A$ and $g\in A(\Omega,X)$.

Given a function algebra $A$ on a Hausdorff space $\Omega$, let $M_{A}$ be its
maximal ideal space consisting of all nonzero algebraic homomorphisms from $A$
to $\mathbb{C}$. $M_{A}$ is a compact Hausdorff space with the Gelfand
topology [21, Theorem 11.9]. The Gelfand transform
$\hat{f}:M_{A}\rightarrow\mathbb{C}$ of $f\in A$ is defined by
$\hat{f}(\phi)=\phi(f)$. Now for $g\in A(\Omega,X)$ and $x^{*}\in X^{*}$ let
$\hat{g}(\phi)(x^{*})=\phi(x^{*}\circ g)$. From the fact that

$|\hat{g}(\phi)(x^{*})|=|\phi(x^{*}\circ g)|\leq\|x^{*}\circ
g\|_{\infty}=\sup_{t\in\Omega}\\{|x^{*}(g(t))|\\}\leq\|g\|\|x^{*}\|_{X^{*}},$

the mapping $\hat{g}(\phi)$ is a bounded linear functional on $X^{*}$.
Moreover, $\hat{g}:M_{A}\rightarrow X^{**}$ is continuous on $M_{A}$ if we
consider the weak∗-topology on $X^{**}$. Let $A(M_{A},(X^{**},w^{*}))$ be the
set of such $\hat{g}$’s.

######  Theorem 4.5.

Let $X$ be a Banach space and $A(\Omega,X)$ be an $X$-valued function space
over a base algebra $A$ on a Hausdorff space $\Omega$. Then $A(\Omega,X)$ is
isometrically isomorphic to an $(X^{**},w^{*})$-valued function space over a
base algebra $\hat{A}$ on the compact Hausdorff space $M_{A}$. In fact,
$M_{A}$ is the maximal ideal space of $A$ with the Gelfand topology.

We split the proof of Theorem 4.5 into several parts for readability. First we
show that the mapping $g\mapsto\hat{g}$ is an isometry.

######  Proposition 4.6.

Let $X$ be a Banach space, let $A(\Omega,X)$ be an $X$-valued function space
over a base algebra $A$ on $\Omega$ and let $M_{A}$ be the maximal ideal space
of $A$. Then for every $g\in A(\Omega,X)$ and $x^{*}\in X^{*}$, the mapping
$g\mapsto\hat{g}$ where $\hat{g}\in A(M_{A},(X^{**},w^{*}))$ is defined by
$\hat{g}(\phi)(x^{*})=\phi(x^{*}\circ g)$ is an isometry.

###### Proof.

By the fact that $\|\phi\|_{\infty}=1$ we obtain

$\displaystyle\|\hat{g}\|=\sup_{\phi\in M_{A}}\\{\|\hat{g}(\phi)\|\\}$
$\displaystyle=\sup_{\phi\in M_{A}}\sup_{x^{*}\in
B_{X^{*}}}\\{|\hat{g}(\phi)(x^{*})|\\}=\sup_{\phi\in M_{A}}\sup_{x^{*}\in
B_{X^{*}}}\\{|\phi(x^{*}\circ g)\\}$ $\displaystyle\leq\sup_{x^{*}\in
B_{X^{*}}}\\{\|x^{*}\circ g\|_{\infty}\\}$ $\displaystyle=\sup_{x^{*}\in
B_{X^{*}}}\sup_{t\in\Omega}\\{|(x^{*}\circ g)(t)|\\}$
$\displaystyle\leq\sup_{t\in\Omega}\\{\|g(t)\|_{X}\\}=\|g\|.$

To show the reverse inequality, let $\phi_{t}\in M_{A}$ be an evaluation
functional at $t\in\Omega_{0}=\\{t\in\Omega:g(t)\neq 0\\}$. Then we have

$\displaystyle\|g\|=\sup_{t\in\Omega_{0}}\\{\|g(t)\|_{X}\\}$
$\displaystyle=\sup_{t\in\Omega_{0}}\sup_{x^{*}\in B_{X^{*}}}\\{|(x^{*}\circ
g(t)|\\}=\sup_{t\in\Omega_{0}}\sup_{x^{*}\in B_{X^{*}}}\\{|\phi_{t}(x^{*}\circ
g)\\}$ $\displaystyle\leq\sup_{\phi\in M_{A}}\sup_{x^{*}\in
B_{X^{*}}}\\{|\phi(x^{*}\circ g)\\}$ $\displaystyle=\sup_{\phi\in
M_{A}}\sup_{x^{*}\in B_{X^{*}}}\\{|\hat{g}(\phi)(x^{*})|\\}$
$\displaystyle\leq\sup_{\phi\in M_{A}}\\{\|\hat{g}(\phi)\|\\}=\|\hat{g}\|,$

so the mapping $g\mapsto\hat{g}$ is an isometry. ∎

Now we show that the space $A(M_{A},(X^{**},w^{*}))$ satisfies the condition
(i) and (ii) in the Definition 2.2 for a $(X,\tau)$-valued function space over
a uniform algebra on a compact Hausdorff space.

######  Proposition 4.7.

Let $X$ be a Banach space, let $A(\Omega,X)$ be a $X$-valued function space
over a base algebra $A$ on $\Omega$, and let $M_{A}$ be the maximal ideal
space of $A$. Then the space

$A_{1}=\\{x^{*}\circ\hat{g}:\hat{g}\in
A(M_{A},(X^{**},w^{*}))\,\,\,\text{and}\,\,\,x^{*}\in X^{*}\\}$

is a uniform algebra over $M_{A}$.

###### Proof.

From the definition of the Gelfand transformation, notice that
$x^{*}\circ\hat{g}(\phi)=\phi(x^{*}\circ g)=\widehat{x^{*}\circ g}(\phi)$.
Hence we can rewrite $A_{1}=\\{\widehat{x^{*}\circ g}:g\in
A(\Omega,X)\,\,\,\text{and}\,\,\,x^{*}\in X^{*}\\}$. We see that the set
$A_{1}$ is, in fact, the image of the Gelfand transformation on $A$. Thus $A$
is isometrically isomorphic to $A_{1}$ and so $A_{1}$ is a closed subalgebra
of $C(M_{A})$ that separates the points of $M_{A}$ and contains constant
functions. This shows that $A_{1}$ is a uniform algebra over $M_{A}$. ∎

######  Proposition 4.8.

Suppose that $X$ is a Banach space and $A(\Omega,X)$ is an $X$-valued function
space over a base algebra $A$ on $\Omega$. Let $M_{A}$ be the maximal ideal
space of $A$ and let

$A_{1}=\\{x^{*}\circ\hat{g}:\hat{g}\in
A(M_{A},(X^{**},w^{*}))\,\,\,\text{and}\,\,\,x^{*}\in X^{*}\\}.$

Then $\phi\cdot\hat{g}\in A(M_{A},(X^{**},w^{*}))$ for every $\phi\in A_{1}$
and $\hat{g}\in A(M_{A},(X^{**},w^{*}))$.

###### Proof.

Let $\hat{f},\hat{g}\in A(M_{A},(X^{**},w^{*}))$ and let $x^{*}\circ\hat{f}\in
A_{1}$ where $x^{*}\in(X^{**},w^{*})^{*}=X^{*}$. We claim that
$(x^{*}\circ\hat{f})\cdot\hat{g}\in A(M_{A},(X^{**},w^{*}))$. Note first that
$h=(x^{*}\circ f)\cdot g\in A(\Omega,X)$. We need to show that
$\hat{h}=(x^{*}\circ\hat{f})\cdot\hat{g}$.

Since $\widehat{x^{*}\circ f}(\phi)=x^{*}\circ\hat{f}(\phi)=\phi(x^{*}\circ
f)$ for every nonzero algebra homomorphism $\phi\in M_{A}$, we have

$\hat{h}(\phi)(y^{*})=(\widehat{x^{*}\circ
f}\cdot\hat{g})(\phi)(y^{*})=(x^{*}\circ\hat{f})(\phi)\cdot\hat{g}(\phi)(y^{*})=((x^{*}\circ\hat{f})\cdot\hat{g})(\phi)(y^{*})$

for $y^{*}\in X^{*}$. Therefore, $\hat{h}=(x^{*}\circ\hat{f})\cdot\hat{g}$ and
this proves our claim. ∎

Theorem 4.2 and Theorem 4.5 prove the following.

######  Corollary 4.9.

Suppose that $X$ is a Banach space and $A(\Omega,X)$ is an $X$-valued function
space over a function algebra $A$ on a Hausdorff space $\Omega$. If the base
algebra $A$ is infinite dimensional, then $A(\Omega,X)$ has the D2P.

For Banach spaces $X$ and $Y$, let $C_{b}(B_{X},Y)$ be the bounded $Y$-valued
continuous functions on the unit ball $B_{X}$. The space $A_{b}(B_{X},Y)$ is
defined to be a closed subspace of $C_{b}(B_{X},Y)$ that consists of
holomorphic funcions on the interior of $B_{X}$ and the space $A_{u}(B_{X},Y)$
is a closed subspace of $A_{b}(B_{X},Y)$ consisting of uniformly continuous
functions on $B_{X}$. It is easy to check that $C_{b}(B_{X},Y)$,
$A_{b}(B_{X},Y)$ and $A_{u}(B_{X},Y)$ are $Y$-valued function spaces in
Definition 4.4. Hence we prove the following.

######  Corollary 4.10.

Let $X$ and $Y$ be Banach spaces. Then $C_{b}(B_{X},Y)$, $A_{b}(B_{X},Y)$ and
$A_{u}(B_{X},Y)$ have the D2P.

##  5\. Daugavet points and $\Delta$-points on vector-valued function spaces
over a uniform algebra

In this section, we present another application of Lemma 2.3 to the Daugavet
points and $\Delta$-points of $A(K,X)$ with respect to its Shilov boundary of
the base algebra. Recall that a Banach space $X$ has the Daugavet property if
for every rank-one operator $T:X\rightarrow X$ satisfies $\|I+T\|=1+\|T\|$.
Well-known examples are $L_{1}(\mu)$ and $L_{\infty}(\mu)$ where $\mu$ is a
nonatomic measure [11, 18] and $C(K)$ where $K$ does not have isolated points
[10]. An infinite dimensional uniform algebra over a compact Hausdorff space
satisfies the Daugavet property if the Shilov boundary does not contain
isolated points [25, 26]. More generally, if a locally compact Hausdorff space
$L$ does not have isolated points, somewhat regular subspaces of $C_{0}(L)$
also have the Daugavet property [4]. Similar results are extended to vector-
valued spaces. The space $C(K,X)$ has the Daugavet property if $K$ does not
have isolated points [24]. A subspace $A^{X}$ of $C(K,X)$, defined by
$A^{X}=\\{f\in C(K,X):x^{*}\circ f\in A\\}$ where $A$ is a uniform algebra, is
known to satisfy the Daugavet property from the fact that it satisfies the
polynomial Daugavet property [7].

We mentioned earlier that we can investigate the Daugavet property and the
DLD2P of Banach spaces through the Daugavet and $\Delta$-points. So it is
clear that the nonexistence of isolated points in the Shilov boundary of an
infinite dimensional uniform algebra implies that every point on the unit
sphere $S_{A}$ is a Daugavet point. However, a characterization of Daugavet
points and $\Delta$-points on an infinite dimensional uniform algebra has not
been known. In [1], the $\Delta$-points and the Daugavet points for $C(K)$ are
determined by norm-attainment at a limit point of $K$. Motivated by this
result, we provide a characterization for the Daugavet points and the
$\Delta$-points on the vector-valued function space $A(K,X)$ over a uniform
algebra when $X$ is uniformly convex. First we have the following observation
on the $\Delta$-points.

######  Lemma 5.1.

[1, Lemma 2.1] Let $X$ be a Banach space. The following are equivalent:

  1. (i)

$x\in S_{X}$ is a $\Delta$-point.

  2. (ii)

for every $x^{*}\in X^{*}$ with $x^{*}x=1$, the projection $P=x^{*}\otimes x$
satisfies $\|I-P\|\geq 2$.

We show that if $f\in S_{A(K,(X,\tau))}$ attains its norm at a limit point on
the Shilov boundary of its base algebra with the additional condition
$A\otimes X\subset A(K,(X,\tau))$, it is a Daugavet point. For a function
algebra $A$ on $\Omega$ and a Banach space $X$, $A\otimes X$ means the set of
all functions $f\otimes x$ ($f\in A$, $x\in X$) defined by $(f\otimes
x)(t)=f(t)x$ ($t\in\Omega$).

######  Theorem 5.2.

Let $K$ be a compact Hausdorff space and let $X$ be a Banach space endowed
with a locally convex Hausdorff topology $\tau$ compatible to a dual pair.
Suppose that a function space $A(K,(X,\tau))$ over a base algebra $A$
satisfies the additional condition that $A\otimes X\subset A(K,(X,\tau))$ and
let $\Gamma$ be the Shilov boundary of $A$. If $f$ is a norm-one element of
$A(K,(X,\tau))$ and there is a limit point $t_{0}$ of $\Gamma$ such that
$\|f\|=\|f(t_{0})\|_{X}$, then $f$ is a Daugavet point.

###### Proof.

Suppose that $\|f\|=\|f(t_{0})\|_{X}=1$ for a limit point $t_{0}$ of $\Gamma$.
Since $\Gamma$ is closed, $t_{0}$ is an element of $\Gamma$. Let
$A(\Gamma,(X,\tau))$ be set of all restrictions of $f\in A(K,(X,\tau))$ to the
Shilov boundary $\Gamma$. By Lemma 2.4, the mapping, defined by $g\in
A(K,(X,\tau))\mapsto g|_{\Gamma}$, is an isometry. So it is enough to show
that $f|_{\Gamma}$ is a Daugavet point of $A(\Gamma,(X,\tau))$ and we may
assume that $K=\Gamma$. Denote the base algebra of $A(K,(X,\tau))$ as $A$.

Now fix $g\in B_{A(K,(X,\tau))}$ and $\epsilon>0$. Let $U$ be an open set
containing $t_{0}$ such that $\|f(t)-f(t_{0})\|_{X}<\epsilon$ for every $t\in
U$. By the Hausdorff condition and the fact that $t_{0}$ is a limit point of
$\Gamma$, we can find pairwise disjoint nonempty open subsets
$U_{1},U_{2},\dots$ such that $U_{n}\subset U$ for each $n\in\mathbb{N}$.
Since the set of strong boundary points is dense in $\Gamma$, each $U_{n}$
contains a strong boundary point. By Lemma 2.3, for each $n\in\mathbb{N}$,
there exist $\phi_{n}\in A$ and $t_{n}\in U_{n}\cap\Gamma_{0}$ such that
$\phi_{n}(t_{n})=\|\phi_{n}\|_{\infty}=1$,
$|\phi_{n}(t)|<\frac{\epsilon}{3\cdot 2^{n}}$ for all $t\in K\setminus U_{n}$,
and

$|\phi_{n}(t)|+\left(1-\frac{\epsilon}{3\cdot 2^{n}}\right)|1-\phi_{n}(t)|\leq
1\,\,\,\text{for all}\,\,\,t\in K.$

Now for each $n\in\mathbb{N}$, define a function
$g_{n}(t)=(1-\frac{\epsilon}{3\cdot
2^{n}})(1-\phi_{n}(t))g(t)-\phi_{n}(t)f(t_{0})$. Then $g_{n}$ is an element of
$A(K,(X,\tau))$ and we show that $g_{n}\in B_{A(K,(X,\tau))}$ for all
$n\in\mathbb{N}$. It is easy to see that for all $t\in K$,

$\|g_{n}(t)\|_{X}=\left\|(1-\frac{\epsilon}{3\cdot
2^{n}})(1-\phi_{n}(t))g(t)-\phi_{n}(t)f(t_{0})\right\|_{X}\leq\left(1-\frac{\epsilon}{3\cdot
2^{n}}\right)|1-\phi_{n}(t)|+|\phi_{n}(t)|\leq 1.$

Hence $\|g_{n}\|\leq 1$. Notice that $g_{n}(t_{n})=-f(t_{0})$. This implies
that

$\|f-g_{n}\|\geq\|f(t_{n})-g_{n}(t_{n})\|_{X}=\|f(t_{n})+f(t_{0})\|_{X}>2\|f(t_{0})\|_{X}-\|f(t_{n})-f(t_{0})\|_{X}\geq
2-\epsilon,$

and so $g_{n}\in\Delta_{\epsilon}(f)$. Furthermore, for every $t\in U_{n}$ we
see that

$\displaystyle\|g(t)-g_{n}(t)\|_{X}$ $\displaystyle=$
$\displaystyle\left\|g(t)-\left(1-\frac{\epsilon}{3\cdot
2^{n}}\right)(1-\phi_{n}(t))g(t)+\phi_{n}(t)f(t_{0})\right\|_{X}$
$\displaystyle=$ $\displaystyle\left\|\frac{\epsilon}{3\cdot
2^{n}}g(t)+\left(1-\frac{\epsilon}{3\cdot
2^{n}}\right)\phi_{n}(t)g(t)+\phi_{n}(t)f(t_{0})\right\|_{X}$
$\displaystyle\leq$ $\displaystyle\frac{\epsilon}{3\cdot
2^{n}}+\left(1-\frac{\epsilon}{3\cdot 2^{n}}\right)+1=2.$

Since $|\phi_{n}(t)|<\frac{\epsilon}{3\cdot 2^{n}}$ for $t\in K\setminus
U_{n}$, we also have

$\displaystyle\|g(t)-g_{n}(t)\|_{X}$
$\displaystyle=\left\|\frac{\epsilon}{3\cdot
2^{n}}g(t)+\left(1-\frac{\epsilon}{3\cdot
2^{n}}\right)\phi_{n}(t)g(t)+\phi_{n}(t)f(t_{0})\right\|_{X}$
$\displaystyle\leq\frac{\epsilon}{3\cdot 2^{n}}+\left(1-\frac{\epsilon}{3\cdot
2^{n}}\right)\frac{\epsilon}{3\cdot 2^{n}}+\frac{\epsilon}{3\cdot
2^{n}}<\frac{\epsilon}{2^{n}}.$

From the fact that each $U_{n}$’s are pairwise disjoint, we obtain, for each
$t\in\cup_{n=1}^{m}U_{n}$,

$\left\|g(t)-\frac{1}{m}\sum_{n=1}^{m}g_{n}(t)\right\|_{X}\leq\frac{1}{m}\left(2+\left(\frac{\epsilon}{2}+\frac{\epsilon}{4}+\dots+\frac{\epsilon}{2^{m}}\right)\right)=\frac{1}{m}\left(2+\left(1-\frac{1}{2^{m}}\right)\epsilon\right)$

and for each $t\in K\setminus\cup_{n=1}^{m}U_{n}$, we have

$\left\|g(t)-\frac{1}{m}\sum_{n=1}^{m}g_{n}(t)\right\|_{X}\leq\frac{1}{m}\left(\frac{\epsilon}{2}+\frac{\epsilon}{4}+\dots+\frac{\epsilon}{2^{m}}\right)=\frac{1}{m}\left(1-\frac{1}{2^{m}}\right)\epsilon.$

Hence, we get

$\left\|g-\frac{1}{m}\sum_{n=1}^{m}g_{n}\right\|\leq\frac{1}{m}\left(2+\left(1-\frac{1}{2^{m}}\right)\epsilon\right).$

By taking the limit $m\to\infty$, we show that
$g\in\overline{\text{conv}}\Delta_{\epsilon}(f)$. Hence
$B_{X}=\overline{\text{conv}}\Delta_{\epsilon}(f)$, in other words, $f$ is a
Daugavet point. ∎

From this, we obtain a sufficient condition for the Daugavet property of
$A(K,X)$ since the map $t\mapsto\|f(t)\|$ is continuous on a compact space and
attains its maximum on $K$ if $\tau$ is norm topology and $f\in A(K,X)$.

######  Corollary 5.3.

Let $K$ be a compact Hausdorff space and let $X$ be a Banach space. Suppose
that a function space $A(K,X)$ over a base algebra $A$ satisfies the
additional condition that $A\otimes X\subset A(K,X)$. If the Shilov boundary
$\Gamma$ of the base algebra $A$ does not have isolated points, then $A(K,X)$
has the Daugavet property.

Furthermore, when $X$ is uniformly convex, Daugevet points and $\Delta$-points
are the same in $A(K,X)$ and characterized by the norm-attainment at a limit
point of $\Gamma$. Recall that the modulus of convexity $\delta_{X}(\epsilon)$
of a Banach space $X$ is defined by for $0<\epsilon<2$

$\delta_{X}(\epsilon)=\inf\left\\{1-\left\|\frac{x+y}{2}\right\|:x,y\in
B_{X},\|x-y\|\geq\epsilon\right\\}.$

A Banach space $X$ is said to be uniformly convex if $\delta_{X}(\epsilon)>0$
for each $0<\epsilon<2$.

######  Theorem 5.4.

For a uniformly convex Banach space $X$ and a compact Hausdorff space $K$, let
$\Gamma$ be the Shilov boundary of the base algebra $A$ of a function space
$A(K,X)$ and let $f\in S_{A(K,X)}$. Then (i)$\implies$ (ii) $\implies$ (iii)
holds.

  1. (i)

$f$ is a Daugavet point.

  2. (ii)

$f$ is a $\Delta$-point.

  3. (iii)

there is a limit point $t_{0}$ of $\Gamma$ such that $\|f\|=\|f(t_{0})\|_{X}$.

Moreover, if we assume the additional condition that $A\otimes X\subset
A(K,X)$, then (i), (ii) and (iii) are equivalent.

###### Proof.

In view of Lemma 2.4, $A(K,X)$ is isometric to $A(\Gamma,X)$ and so it is
enough to prove our claim with respect to $f_{|\Gamma}\in S_{A(\Gamma,X)}$. So
we may assume that $K=\Gamma$. (i) $\implies$ (ii) is clear from their
definitions. (iii) $\implies$ (i) is already shown in Theorem 5.2 with the
additional assumption that $A\otimes X\subset A(K,X)$. Hence we only need to
show (ii) $\implies$ (iii).

Assume to the contrary that $f\in S_{A(K,X)}$ is a $\Delta$-point but
$\|f\|\neq\|f(t)\|_{X}$ for every limit point of $K$. Let $F=\\{t\in
K:\|f(t)\|_{X}=1\\}$. Then $F$ only contains isolated points of $K$ by the
assumption and is nonempty. Since $K$ is compact, we also see that
$|F|<\infty$. For each $t\in F$ we can always find $x_{t}^{*}\in S_{X^{*}}$
such that $x_{t}^{*}(f(t))=1$.

Since $X$ is uniformly convex, let
$\delta=\delta_{X}\left(\frac{1}{2|F|}\right)>0$, where $|F|$ is the number of
elements of $F$. If $\text{Re}\,x_{t}^{*}(x)\geq 1-\delta$ and $x\in B_{X}$,
then
$\left\|\frac{x+f(t)}{2}\right\|\geq\text{Re}\,x_{t}^{*}\left(\frac{x+f(t)}{2}\right)\geq
1-\frac{1}{2}\delta>1-\delta_{X}\left(\frac{1}{2|F|}\right)$. This means that
$\|x-f(t)\|<\frac{1}{2|F|}$ by the definition of $\delta_{X}$. Hence

(3) $\text{if}\,\,\,\text{Re}\,x_{t}^{*}(x)\geq 1-\delta\,\,\text{and}\,x\in
B_{X}\,\,\text{then}\,\,\,\|x-f(t)\|_{X}\leq\frac{1}{2|F|}\,\,\,\text{for
every}\,\,\,t\in F.$

Let $\epsilon=1-\max_{t\in K\setminus F}\|f(t)\|_{X}>0$. Define a bounded
linear functional $\psi$ on $A(K,X)$ by

$\psi(g)=\frac{1}{|F|}\sum_{t\in F}x_{t}^{*}(g(t)),$

where $g\in A(K,X)$. Notice that $\|\psi\|=\psi(f)=1$ and this defines a
bounded projection $P:A(K,X)\rightarrow A(K,X)$ by $P(g)=\psi(g)f$ for $g\in
A(K,X)$. It is clear that $\|P\|=\|P(f)\|=\|f\|=1$. We see that $P$ is in fact
a norm-one projection on $A(K,X)$. Then in view of Lemma 5.1, we should have
$\|I-P\|\geq 2$. But we show that in fact $\|I-P\|<2$.

For a given $g\in B_{A(K,X)}$, if $t\in K\setminus F$ we have

(4) $\|g(t)-Pg(t)\|_{X}=\|g(t)-\psi(g)f(t)\|_{X}\leq 1+\|f(t)\|_{X}\leq
2-\epsilon.$

Now we divide $F$ into two parts. Let $t_{0}\in F_{1}=\\{t\in
F:|x_{t}^{*}(g(t))|\geq 1-\delta\\}$. Then there exists a scalar
$\lambda_{0}\in B_{\mathbb{C}}$ such that
$|x_{t_{0}}^{*}(g(t_{0}))|=\lambda_{0}x_{t_{0}}^{*}(g(t_{0}))=x_{t_{0}}^{*}(\lambda_{0}g(t_{0}))\geq
1-\delta$. Then by (3), we have $\|\lambda
g(t_{0})-f(t)\|_{X}\leq\frac{1}{2|F|}$. Then

$\displaystyle\|g(t_{0})-Pg(t_{0})\|_{X}$ $\displaystyle=$
$\displaystyle\left\|g(t_{0})-\frac{1}{|F|}\sum_{t\in
F}x_{t}^{*}(g(t))f(t_{0})\right\|_{X}\leq\left\|g(t_{0})-\frac{1}{|F|}x_{t_{0}}^{*}(g(t_{0}))f(t_{0})\right\|_{X}+\frac{|F|-1}{|F|}$
$\displaystyle=$
$\displaystyle\left\|\lambda_{0}g(t_{0})-\frac{1}{|F|}x_{t_{0}}^{*}(\lambda_{0}g(t_{0}))f(t_{0})\right\|_{X}+\frac{|F|-1}{|F|}.$

From the fact that $x_{t_{0}}^{*}(\lambda_{0}g(t_{0}))\leq 1$, we have
$1-\frac{1}{|F|}x_{t_{0}}^{*}(\lambda_{0}g(t_{0}))\geq 0$. Hence

$\displaystyle\left\|\lambda_{0}g(t_{0})-\frac{1}{|F|}x_{t_{0}}^{*}(\lambda_{0}g(t_{0}))f(t_{0})\right\|_{X}$
$\displaystyle\leq$
$\displaystyle\|\lambda_{0}g(t_{0})-f(t_{0})\|_{X}+\|f(t_{0})\|_{X}\left(1-\frac{1}{|F|}x_{t_{0}}^{*}(\lambda_{0}g(t_{0}))\right)$
$\displaystyle\leq$ $\displaystyle\frac{1}{2|F|}+1-\frac{1}{|F|}(1-\delta),$

and so

(5)
$\|g(t_{0})-Pg(t_{0})\|_{X}\leq\frac{1}{2|F|}+1-\frac{1}{|F|}(1-\delta)+\frac{|F|-1}{|F|}\leq
2-\frac{1}{2|F|}.$

When $t_{0}\in F_{2}=F\setminus F_{1}$, we have
$|x_{t_{0}}^{*}(g(t_{0}))|<1-\delta$. So we see that

$\displaystyle\|g(t_{0})-Pg(t_{0})\|_{X}$ $\displaystyle\leq$
$\displaystyle\left\|\lambda_{0}g(t_{0})-\frac{1}{|F|}x_{t_{0}}^{*}(\lambda_{0}g(t_{0}))f(t_{0})\right\|_{X}+\frac{|F|-1}{|F|}.$
$\displaystyle\leq$ $\displaystyle
1+\frac{1}{|F|}(1-\delta)+\frac{|F|-1}{|F|}\leq 2-\frac{\delta}{|F|}.$

Hence, with (4) and (5) we obtain
$\|g-Pg\|\leq\max\\{2-\frac{1}{2|F|},2-\frac{\delta}{|F|},2-\epsilon\\}<2$.
Since $g\in B_{A(K,X)}$ is chosen arbitrarily, $\|I-P\|<2$, and we get a
contradiction. ∎

Since $\mathbb{C}$ is uniformly convex, we have the following consequences for
a scalar-valued uniform algebra.

######  Corollary 5.5.

Let $K$ be a compact Hausdorff space and let $\Gamma$ be the Shilov boundary
of $A(K)$. Then for $f\in S_{A(K,X)}$, the following are equivalent:

  1. (i)

$f$ is a Daugavet point.

  2. (ii)

$f$ is a $\Delta$-point.

  3. (iii)

there is a limit point $t_{0}$ of $\Gamma$ such that
$\|f\|_{\infty}=|f(t_{0})|$.

By the characterization given in Theorem 5.4, we show when the Daugavet
property, DD2P, and DLD2P of $A(K,X)$ are equivalent to each other.

######  Theorem 5.6.

Let $X$ be a uniformly convex Banach space, let $K$ be a compact Hausdorff
space, and let $\Gamma$ be the Shilov boundary of the base algebra of
$A(K,X)$. Then (i) $\implies$ (ii) $\implies$ (iii) $\implies$ (iv) holds:

  1. (i)

$A(K,X)$ has the Daugavet property.

  2. (ii)

$A(K,X)$ has the DD2P.

  3. (iii)

$A(K,X)$ has the DLD2P.

  4. (iv)

The Shilov boundary $\Gamma$ does not have isolated points.

Moreover, if we assume the additional condition that $A\otimes X\subset
A(K,X)$, then (i), (ii), (iii) and (iv) are equivalent.

###### Proof.

Since $A(K,X)$ is isometric to $A(\Gamma,X)$ by Lemma 2.4, showing the
equivalence for $A(\Gamma,X)$ is enough. It is a well-known that (i)
$\implies$ (ii) $\implies$ (iii) [6]. By Corollary 5.3, we have (iv)
$\implies$ (i) with the additional assumption that $A\otimes X\subset A(K,X)$.

Finally we prove (iii) $\implies$ (iv). Assume that $A(\Gamma,X)$ has the
DLD2P but the Shilov boundary $\Gamma$ of the base algebra of $A(K,X)$ has an
isolated point $t_{0}$. Since the set of strong boundary points $\Gamma_{0}$
is dense in $\Gamma$, this $t_{0}$ is also in $\Gamma_{0}$. Then by Lemma 2.5
there is $\phi\in A$ such that $\phi(t_{0})=1=\|\phi\|$ and $\phi(t)=0$ for
all $t\in\Gamma\setminus\\{t_{0}\\}$. Since $\phi$ is in the base algebra $A$,
there is $g\in A(\Gamma,X)$ such that $\phi=x^{*}\circ g$ for some $x^{*}\in
X^{*}$. Then $g(t_{0})\neq 0$ and let $h=\frac{\phi}{\|g(t_{0})\|_{X}}g$. So
$h$ is an element of $A(K,X)$. Since $\|h\|=\|h(t_{0})\|=1$ and $\|h(t)\|=0$
for every point $t$ of $\Gamma$ other than $t_{0}$. So $\|h(t)\|=0$ for every
limit point $t$ of $\Gamma$. Hence in view of Theorem 5.4, we see that $h\in
S_{A(K,X)}$ is not a $\Delta$-point, and this is a contradiction to our
assumption that $A(K,X)$ has the DLD2P, which states that every function in
$S_{A(K,X)}$ is a $\Delta$-point. Therefore (iii) $\implies$ (iv) is proved. ∎

As we mentioned before, it is well-known that the nonexistence of isolated
points on the Shilov boundary implies that the scalar-valued uniform algebra
has the Daugavet property [25, 26]. Here we have something even further that
the Daugavet property, DD2P, and DLD2P are equivalent to each other.

######  Corollary 5.7.

Let $K$ be a compact Hausdorff space and let $\Gamma$ be the Shilov boundary
of a uniform algebra $A$. Then the following are equivalent:

  1. (i)

$A$ has the Daugavet property.

  2. (ii)

$A$ has the DD2P.

  3. (iii)

$A$ has the DLD2P.

  4. (iv)

The Shilov boundary $\Gamma$ does not have isolated points.

We conclude this article with showing when the space $A(K,X)$ has the convex-
DLD2P. But here we take a slightly different approach. To use Lemma 2.3 we
need to find a limit point of $\Gamma$ that is also a strong boundary point,
but the existence of such point does not seem to be guaranteed in general. So
here we use a different version of the Urysohn-type lemma for strong peak
points. For a uniform algebra $A$ over a compact Hausdorff space $K$, an
element $t_{0}\in K$ is said to be a strong peak point if there exists a
function $f\in A$ such that $|f(t_{0})|=\|f\|_{\infty}$ and
$\|f\|_{\infty}>\sup\\{|f(t)|:t\in K\setminus U\\}$ for every open
neighborhood $U$ of $t_{0}$. Let us denote the set of all strong peak points
for $A$ by $\rho(A)$.

######  Lemma 5.8.

[15, Lemma 3] Let $\Omega$ be a Hausdorff space and $A$ be a subalgebra of
$C_{b}(\Omega)$. Suppose that $U$ is an open subset of $\Omega$ and $t_{0}\in
U\cap\rho(A)$. Then given $0<\epsilon<1$, there exists $\phi\in A$ such that
$\|\phi\|=1=\phi(t_{0})$, $\sup_{t\in K\setminus U}|\phi(t)|<\epsilon$, and
for all $t\in\Omega$,

$|\phi(t)|+(1-\epsilon)|1-\phi(t)|\leq 1.$

The original statement of Lemma 5.8 is given with $\|\phi\|=1=|\phi(t_{0})|$.
However, if we observe the proof of Lemma 3 in [15] carefully, it is easy to
see that the statement without the absolute value also holds.

######  Theorem 5.9.

For a compact Hausdorff space $K$ and a uniformly convex Banach space $X$,
suppose that a function space $A(K,X)$ over a base algebra $A$ satisfies the
additional condition that $A\otimes X\subset A(K,X)$. Let $\Gamma^{\prime}$ be
the set of limit points in the Shilov boundary $\Gamma$ of the base algebra
$A$ and let $\rho(A)$ the set of all strong peak points for $A$. If
$\rho(A)\cap\Gamma^{\prime}\neq\emptyset$, then the space $A(K,X)$ has the
convex-DLD2P.

###### Proof.

By the isometry in Lemma 2.4, we may assume that $K=\Gamma$. We need to show
that $S_{A(K,X)}\subset\overline{\text{conv}}\Delta$, where $\Delta$ is the
set of all $\Delta$-points of $A(K,X)$.

Fix $f\in S_{A(K,X)}$. Pick $t_{0}\in\rho(A)\cap K^{\prime}$ and let
$\lambda=\frac{1+\|f(t_{0})\|_{X}}{2}$. For $\epsilon>0$, let $U$ be an open
neighborhood containing $t_{0}$ such that $\|f(t)-f(t_{0})\|_{X}<\epsilon$. By
Lemma 5.8, there exists a function $\phi\in A$ such that
$\phi(t_{0})=\|\phi\|_{\infty}=1$, $\sup_{t\in K\setminus
U}|\phi(t)|<\epsilon$ and

$|\phi(t)|+(1-\epsilon)|1-\phi(t)|\leq 1.$

for every $t\in K$. Choose a norm-one vector $v_{0}\in X$ and set

$x_{0}=\begin{cases}\frac{f(t_{0})}{\|f(t_{0})\|_{X}}&\mbox{if }f(t_{0})\neq
0\\\ v_{0}&\mbox{if }f(t_{0})=0.\end{cases}$

Now define two functions

$\displaystyle f_{1}(t)$
$\displaystyle=(1-\epsilon)(1-\phi(t))f(t)+\phi(t)x_{0}$ $\displaystyle
f_{2}(t)$ $\displaystyle=(1-\epsilon)(1-\phi(t))f(t)-\phi(t)x_{0},\ \ \ t\in
K.$

Since $A\otimes X\subset A(K,X)$, we see that the functions $f_{1},f_{2}\in
A(K,X)$. Notice that

$\displaystyle\|f_{1}(t)\|_{X}$
$\displaystyle=\left\|(1-\epsilon)(1-\phi(t))f(t)+\phi(t)x_{0}\right\|_{X}$
$\displaystyle\leq(1-\epsilon)|1-\phi(t)|+|\phi(t)|\leq 1,$

for every $t\in K$, in particular, $\|f_{1}(t_{0})\|_{X}=1$. Hence
$\|f_{1}\|=\|f_{1}(t_{0})\|_{X}=1$. We can also show
$\|f_{2}\|=\|f_{2}(t_{0})\|_{X}=1$ by the same argument. Thus by Theorem 5.4,
we see that $f_{1},f_{2}\in\Delta$. Let $g(t)=\lambda
f_{1}(t)+(1-\lambda)f_{2}(t)$. We consider two cases.

  1. (i)

First, consider when $f(t_{0})\neq 0$. Then
$g=(1-\epsilon)(1-\phi(t))f(t)+\phi(t)f(t_{0})$. We see that

$\displaystyle\|g(t)-f(t)\|_{X}$ $\displaystyle=$
$\displaystyle\|(1-\epsilon)(1-\phi(t))f(t)+\phi(t)f(t_{0})-f(t)\|_{X}$
$\displaystyle=$
$\displaystyle\|(1-\epsilon)(1-\phi(t))f(t)+\phi(t)f(t_{0})-(1-\epsilon)f(t)-\epsilon
f(t)\|_{X}$ $\displaystyle=$
$\displaystyle\|(1-\epsilon)(-\phi(t))f(t)+(1-\epsilon)\phi(t)f(t_{0})+\epsilon\phi(t)f(t_{0})-\epsilon
f(t)\|_{X}$ $\displaystyle=$
$\displaystyle\|(1-\epsilon)\phi(t)(f(t_{0})-f(t))+\epsilon\phi(t)f(t_{0})-\epsilon
f(t)\|_{X}$ $\displaystyle\leq$
$\displaystyle(1-\epsilon)|\phi(t)|\cdot\|f(t)-f(t_{0})\|_{X}+\epsilon|\phi(t)|\cdot\|f(t_{0})\|_{X}+\epsilon\|f(t)\|_{X}$
$\displaystyle\leq$
$\displaystyle(1-\epsilon)|\phi(t)|\cdot\|f(t)-f(t_{0})\|_{X}+2\epsilon.$

For $t\in U$, we have
$(1-\epsilon)|\phi(t)|\cdot\|f(t)-f(t_{0})\|_{X}\leq(1-\epsilon)\epsilon<\epsilon$.
Since $|\phi(t)|<\epsilon$ for $t\in K\setminus U$, we have
$(1-\epsilon)|\phi(t)|\cdot\|f(t)-f(t_{0})\|_{X}\leq
2(1-\epsilon)\epsilon<2\epsilon$ by the triangle inequality. Therefore
$\|g-f\|<4\epsilon$, and this implies that $f\in\overline{\text{conv}}\Delta$.

  2. (ii)

Now consider when $f(t_{0})=0$. Then for all $t\in U$,
$\|f(t)\|_{X}<\epsilon$. Notice that $\lambda=\frac{1}{2}$ and
$g(t)=(1-\epsilon)(1-\phi(t))f(t)$. Hence

$\displaystyle\|g(t)-f(t)\|_{X}$ $\displaystyle=$
$\displaystyle\|(1-\epsilon)(1-\phi(t))f(t)-(1-\epsilon)f(t)-\epsilon
f(t)\|_{X}$ $\displaystyle\leq$
$\displaystyle(1-\epsilon)|\phi(t)|\cdot\|f(t)\|_{X}+\epsilon\|f(t)\|_{X}\leq(1-\epsilon)|\phi(t)|\cdot\|f(t)\|_{X}+\epsilon.$

For $t\in U$, we have
$(1-\epsilon)|\phi(t)|\cdot\|f(t)\|_{X}\leq(1-\epsilon)\epsilon<\epsilon$. In
view of Lemma 5.8, since $\sup_{t\in K\setminus U}|\phi(t)|<\epsilon$ for
$t\in K\setminus U$, we have
$(1-\epsilon)|\phi(t)|\cdot\|f(t)\|_{X}\leq(1-\epsilon)\epsilon<\epsilon$.
Therefore $\|g-f\|<2\epsilon$, and this also implies that
$f\in\overline{\text{conv}}\Delta$.

Since $f\in S_{A(K,X)}$ is arbitrary, we see that
$S_{A(K,X)}\subset\overline{\text{conv}}(\Delta)$, and so $A(K,X)$ has the
convex-DLD2P. ∎

######  Corollary 5.10.

Let $K$ be a compact Hausdorff space and $\Gamma^{\prime}$ be the set of limit
points in the Shilov boundary $\Gamma$ of a uniform algebra $A$. If
$\rho(A)\cap\Gamma^{\prime}\neq\emptyset$, then the uniform algebra $A$ has
the convex-DLD2P.

######  Remark 5.11.

The sufficient condition $\rho(A)\cap\Gamma^{\prime}\neq\emptyset$ in
Corollary 5.10 does not guarantee the DLD2P. For example, let
$K=\\{\frac{1}{n}:n\in\mathbb{N}\\}\cup\\{0\\}$ in the real line and let
$A=C(K)$ be the space of continuous functions over $K$ equipped with the
supremum norm. Notice that $K=\rho(A)$ is its Shilov boundary and
$\rho(A)\cap\Gamma^{\prime}=\\{0\\}$. So in view of Theorem 5.7 and Theorem
5.10, the space $C(K)$ has the convex-DLD2P, while it does not have DLD2P. In
fact, $C(K)$ is isometric to the space $c$ of convergent sequences. The same
results in the space $c$ were shown in [1, Corollary 5.4, Remark 5.5].

## References

  * [1] T. Abrahamsen, R. Haller, V. Lima, and K. Pirk, _Delta- and Daugavet points in Banach Spaces_ , Proc. Edin. Math. Soc., Vol. 63, Issue 2, 2020 pp. 475–496.
  * [2] M. D. Acosta and A. Kamińska, _Weak neighborhoods of the unit ball in interpolation spaces $L^{1}+L^{\infty}$ and $L^{1}\cap L^{\infty}$_, Indiana Univ. Math. J., 57, No. 1 (2008), 77–96.
  * [3] T. Abrahamsen, V. Lima and O. Nygaard, _Remarks on Diameter $2$ Properties_, J. Convex Analysis, 20 (2013), pp. 439–452.
  * [4] T. Abrahamsen, O. Nygaard, and Märt Põldvere, _New applications of extremely regular function spaces_ , Pacific Journal of Mathematics, Vol. 301, No.2 (2019), pg 385–394.
  * [5] J. Becerra Guerrero and G. López Pérez, _Relatively Weakly Open Subsets of the Unit Ball in Function Spaces_ , J. Math. Anal. Appl. 315 (2006), no.2, 544-554.
  * [6] J. Becerra Guerrero, G. López Pérez, and A. Rueda Zoca, _Diametral Diameter Two Properties in Banach Spaces_ , J. Convex Anal. 25 (2018), 817–840.
  * [7] Y. S. Choi, D. García, S. K. Kim and M. Maestre,_Some geometric properties of disk algebras_ , J. Math. Anal. Appl. 409 (2014), no. 1, 147–157.
  * [8] B. Cascales, A. Guirao, and V. Kadets, _A Bishop-Phelps-Bollobás type Theorem for Uniform Algebras_ , Advances in Mathematics, vol 240, 2013, pp. 370-382
  * [9] H.G. Dales, _Banach Algebras and Automatic Continuity_ , vol.24, London Mathematical Society Monographs. New Series, the Clarendon Press, Oxford University Press, New York, NY, USA, 2000
  * [10] I.K Daugavet, _On a property of compltely continuous operaotrs in the space $C$_, Uspekhi Mat. Nauk 18.5 (1963) 157–158 (Russian)
  * [11] C.Foia̧ and I. Singer _Points of diffusion of linear operators and almost diffuse operators in spaces of continuous functions_ , Math. Z. 87 (1965), 434–450.
  * [12] N. Ghoussoub, G. Godefroy, B. Maurey, W. Schachermayer, _Some Topological and Geometrical Structures in Banach Spaces_ , Mem. Amer. Math. Soc. 70 (1987), no.378, iv +116. MR 912637 (89h:46024)
  * [13] Y. Ivakhno and V. Kadets, _Unconditional Sums of Spaces with Bad Projections_ , Visn. Khark. Univ., Ser. Mat. Pryki. Mat. Mekh 645 (2004), 30–35.
  * [14] V. Kadets, _The Diametral Strong Diameter 2 Property of Banach spaces Is the Same as the Daugavet Property_ , Proc. Amer. Math. Soc, 2020, https://doi.org/10.1090/proc/15448
  * [15] S.K Kim, H.J Lee, _A Urysohn-type Theorem and the Bishop-Phelps-Bollobás Theorem for Holomorphic Functions_ , J. Math. Anal. and Appl, Volume 480, Issue 2, 2019, 123393, ISSN 0022-247X, https://doi.org/10.1016/j.jmaa.2019.123393.
  * [16] H.J Lee, _Generalized Numerical Index of Function Algebras_ , Journal of Function Spaces, vol. 2019, Article ID 9080867, 6 pages, 2019. https://doi.org/10.1155/2019/9080867
  * [17] G.M Leibowitz, _Lectures on Complex Function Algebras_. Scott, Foresman and Company 1970
  * [18] G. Ya, Lozanovskii, _On almost integral operators in KB-spaces_ , Vestnik Leningrad Univ. Mat. Mekh. Astr, 21.7 (1966), 35–44 (Russian)
  * [19] L. Narici, E, Beckenstein, _Topological Vector Spaces_ , 2nd ed., CRC Press, 2011.
  * [20] O. Nygaard and D. Werner, _Slices in the Unit Ball of a Uniform Algebra_ , Arch. Math. (Basel) 76 (2001), no.6, pp 441-444.
  * [21] W. Rudin, _Functional Analysis_ , 2nd ed, McGraw Hill, 1991.
  * [22] H. Schaefer, M. Wolff, _Topological Vector spaces_ , 2nd ed., Springer-Verlag, 1999.
  * [23] K. Pirk, _Diametral Diameter Two Properties, Daugavet-, and $\Delta$-points in Banach Spaces_, PhD dissertation, Dissertationes Mathematicae Universitatis Tartuensis, 133, 2020
  * [24] D. Werner, _Recent Progress on the Daugavet property_ , Irish Math. Soc. Bulletin, 46, 2001, pg 77–97
  * [25] D. Werner, _The Daugavet equation for operators not fixing a copy of $C[0,1]$_, J. Operator Theory 39 (1998), 89–98.
  * [26] P. Wojtaszczyk, _Some remarks on the Daugavet equation_ , Proc. Amer. Math. Soc 115 (1992), 1047–1052.