# Strong Purcell Effect in All-Dielectric Epsilon-Near-Zero Microcavities

Ali Panahpour<EMAIL_ADDRESS>Jussi Kelavuori  Mikko Huttunen
corresponding author<EMAIL_ADDRESS>Photonics Laboratory, Physics
Unit, Tampere University, FI-33014 Tampere, Finland

###### Abstract

We theoretically study the quality and Purcell enhancement factors in epsilon-
near-zero (ENZ) optical microcavities with ideally lossless or slightly lossy
structures. In the case of ENZ rectangular cavities with perfect electric
conductor (PEC) walls, our results supported by numerical simulations, show
that both the quality and Purcell factors grow with increasing the length of
cavity, having respective functional dependencies of $n^{-3}$ and $n^{-2}$ on
the effective index of refraction, $n$. Further, we study extremely low-loss
all-dielectric Bragg reflection waveguide cavities, near their cutoff
frequencies. In such ENZ structures, the respective quality and Purcell
factors scale as $(n\kappa)^{-1}$ and $n/\kappa$, in terms of real ($n$) and
imaginary ($\kappa$) parts of the cavity effective index of refraction. We
believe such structures exhibiting remarkably high quality ($\sim 10^{6}$) and
Purcell factors ($\sim 10^{3}$), can drastically enhance various linear and
nonlinear optical processes, and thus find use in applications ranging from
laser development to optical sensing and metrology.

††preprint: APS/123-QED

Epsilon-near-zero (ENZ) photonics provides a novel platform for developing new
types of integrated photonic devices with unconventional functionalities [1,
2, 3, 4, 5, 6, 7]. This is due to variety of extraordinary effects, offered by
ENZ materials and structures, such as energy squeezing [8], tunneling through
subwavelength channels and bends [9], enhanced optical nonlocality and
additional waves [10], control of spectral and directional emissivity [11],
enhancement of optical nonlinearities [4], and boosting of the Purcell factor
(PF) in radiative systems [7, 12, 13, 14].

Conventional ENZ materials or nanostructures have been mainly based on
conducting oxide or nitride materials as well as metamaterials consisting of
metal–dielectric or semiconductor–dielectric components [2, 3]. Perfect
electric conductor (PEC) waveguides, and more realistic but lossy metallic
waveguides, near their cutoff frequencies have also been utilized to emulate
the properties of an ENZ medium [13]. However, most of the unique features of
ENZ media are considerably deteriorated near optical frequencies due to
internal dissipations caused by ohmic losses in conducting materials. In fact,
conventional ENZ materials/structures usually exhibit rather low figure-of-
merits (FoMs), defined as the ratio of real to imaginary part of effective
permittivity, within their ENZ spectral range [15].

Several all-dielectric epsilon-mu-near-zero (EMNZ) architectures have been
recently demonstrated to realize low-loss near-zero-index (NZI) metamaterials
or photonic crystals [5, 16]. However, little work seems to exist on all-
dielectric ENZ structures. Despite known challenges, such as poor transmission
due to impedance mismatch between conventional and ENZ media, they also
exhibit unique and highly advantageous properties. Notably, it has been shown
that among the various subcategories of NZI media including ENZ, mu-near-zero,
and EMNZ (meta)materials, the PF diverges solely in the one-dimensional (1D)
ENZ media, due to substantial growth of electromagnetic density of states
(DOS) in such structures [17].

The ENZ-induced PF enhancement has been demonstrated both theoretically and
experimentally in linear as well as nonlinear systems [18, 19, 20]. However,
full exploitation of the Purcell enhancement of radiative processes is
severely restricted by optical losses of the ENZ structures. Importantly, this
problem prevents further enhancement of spontaneous emission rate (SER) in ENZ
media through coupling of embedded emitters to high-quality-factor
($Q$-factor) optical resonators. In fact, as we will show, such couplings in
metal or semiconductor structures can result in suppression of high-$Q$
resonances with no significant enhancement of radiative processes.

In this letter, we address the issue of conductor-induced ohmic losses by
studying the Purcell effect and Q-factor in ENZ waveguides and microcavities
formed by using all-dielectric Bragg reflection waveguides (BRWs). We show
that these structures, operated near their cut-off frequencies, can provide a
viable platform for experimental realization of low-loss ENZ microcavities.
Furthermore, the studied ENZ cavities are predicted to result in scaling laws
for $Q$\- and Purcell factors that are different from those associated with
conventional cavities. Our numerical results indicate the potential
realization of cavities with Q-factors on the order of $10^{6}$. More
importantly, our theoretical analysis suggests that such ENZ cavities could
simultaneously exhibit high Purcell factors exceeding $10^{3}$.

The perturbative effect of structured vacuum on radiative processes, known as
Purcell effect was originally demonstrated by coupling an emitter to a
resonant cavity [21]. When the emission wavelength of the emitter is tuned to
the resonant mode of a nondispersive cavity, the emission is enhanced by a
factor of

${F}_{\mathrm{P}}=\frac{6}{\pi^{2}}\left(\frac{\lambda}{2n}\right)^{3}\frac{Q}{V}\,,$
(1)

where $\lambda$ is the photon wavelength in vacuum, $n$ is the refractive
index of medium inside the cavity, $Q$ is the $Q$-factor of the cavity and $V$
is the mode volume. The conventional approaches for enhancing the PF are
mainly based on the development of three different classes of photonic
structures: ultra-high-$Q$-factor dielectric microcavities with $Q$-factors in
the range of $10^{8}-10^{9}$ and mode volumes much larger than
$\left(\lambda/2n\right)^{3}$ [22], photonic crystal nanocavities of moderate
$Q$-factors $(10^{4}-10^{5})$ with mode volumes on the order of
$\left(\lambda/2n\right)^{3}$ [23] and finally plasmonic nanoresonators having
rather small $Q$-factors ($Q\leq 100$) but with mode volumes much smaller than
$\left(\lambda/2n\right)^{3}$ [24]. Microcavities loaded with highly
dispersive materials or structures have also been implemented for improving
the cavity $Q$-factor [25, 26].

An intriguing idea to promote the Purcell effect is to load a cavity with
high-FoM ENZ media. This presents a dual advantage by amplifying the unique
features of ENZ media through cavity resonance as well as offering an
additional mechanism for general enhancement of PF in optical resonators,
based on ENZ characteristics rather than cavity features. Interestingly, due
to the elongation of wavelengths in the ENZ region, ENZ cavities can
potentially mitigate scattering-based losses caused by structural stress,
imperfections, or defects. This property is of great practical importance as
it alleviates a fundamental limitation frequently impeding the performance of
high $Q$-factor microresonators [27].

To study the Purcell effect in ENZ structures in the weak interaction regime,
we apply Fermi’s golden rule giving the SER of a quantum emitter:

$\mathrm{\Gamma}=\frac{2\pi}{\hbar^{2}}\left|M_{12}\right|^{2}g(\omega)$ (2)

in terms of transition matrix element $M_{12}$ and optical DOS $g(\omega)$.
Applying the quantization procedure for a homogeneous and lossless dispersive
medium [28, 17], the transition matrix element is given by:

$\left|M_{12}\right|^{2}=\frac{\hbar\omega}{2\varepsilon_{0}V}\frac{\left|\mathbf{p}\right|^{2}}{{n(\omega)n}_{g}(\omega)}\,,$
(3)

with $n$, $n_{g}$, $V$ and $\mathbf{p}$ being the (real part of) medium phase
index, group index, quantization volume and emitter’s dipole moment,
respectively. A straightforward method to realize low-dimensional (in
$k$-space) ENZ structures is to use metallic waveguides with planar or
rectangular geometries, operating near their cutoff frequencies. In the case
of a 1D rectangular waveguide with transverse dimensions of $L_{x}$ and
$L_{y}$, and with idealized PEC walls, the DOS for an arbitrary quantization
length of $L_{z}$ along $z$-axis is given by
$g\left(\omega\right)=\left(L_{z}/\pi c\right)n_{g}(\omega)$. Thus, by using
Eqs. (2) and (3), the SER of an emitter in the waveguide is given by:

$\Gamma^{\mathrm{W}}=\frac{\left|\mathbf{p}\right|^{2}}{\varepsilon_{0}\hbar}\frac{1}{L_{x}L_{y}}\frac{\omega}{c}\frac{1}{n}\,,$
(4)

and by normalizing it to the SER in free space, the PF turns into

$F_{\mathrm{P}}^{\mathrm{W}}=\frac{3\pi}{L_{x}L_{y}}\frac{c^{2}}{\omega^{2}}\frac{1}{n}\,.$
(5)

Now, we consider a single mode rectangular cavity, consisting of a vacuum-
filled PEC waveguide with transverse dimensions of $L_{x}$ and $L_{y}$, and a
limited length of $L_{z}$. The resonance feedback along $L_{z}$ is provided by
reflections at the waveguide ports due to impedance mismatch between the ports
and free space. We assume the cavity resonance mode is tuned to the emission
frequency ($\omega_{0}$) of a dipole emitter at the center of the cavity. In
this condition, the DOS around the resonant mode can be written as a
normalized Lorentzian function of
$g\left(\omega\right)=(2\delta\omega/\pi)/({4(\omega-\omega_{0})}^{2}+{\delta\omega}^{2})$,
which exactly at resonance, reduces to
$g\left(\omega_{0}\right)=(2/\pi)(Q/\omega_{0})$. Here,
$Q=\omega_{0}/\delta\omega$ is the $Q$-factor of the cavity with the resonance
width of $\delta\omega$, which in terms of cavity finesse $\mathcal{F}$, is
given by $Q=\mathcal{F}L_{z}\omega_{0}/\pi c$. Taking into account the cavity
dispersion, particularly when resonance is near the cutoff frequency of the
waveguide, the free spectral range is given by $c/2n_{g}L_{z}$ and the cavity
$Q$-factor is scaled by the group index as $Qn_{g}$ [25, 26]. Therefore, the
DOS is accordingly determined by
$g\left(\omega_{0}\right)=(2/\pi)(Q/\omega_{0})n_{g}(\omega_{0})$ and the SER
in a dispersive single mode rectangular cavity is obtained as:

$\Gamma^{\mathrm{C}}=\frac{2\left|\mathbf{p}\right|^{2}}{\hbar\varepsilon_{0}V}\frac{Q}{n}\,.$
(6)

After normalizing to the SER in vacuum, the PF in terms of $\lambda_{0}=2\pi
c/\omega_{0}$ fulfills:

${F}^{\mathrm{C}}_{\mathrm{P}}=\frac{3}{4\pi^{2}}\frac{\lambda_{0}^{3}}{V}\frac{Q}{n}\,.$
(7)

Assuming $L_{y}=\lambda_{0}$ and $L_{x}$ slightly larger than $\lambda_{0}/2$,
which implies that the cavity resonance wavelength
($\lambda_{0}=2L_{x}/\sqrt{1+(L_{x}/L_{z})^{2}}$) is close to the cutoff
wavelength ($\lambda_{c}=2L_{x}$), and by using
$Q=(2L_{z}/\lambda_{0})\mathcal{F}$, we see from relation (7) that PF is
proportional to the ratio $\mathcal{F}/n$. In the absence of material losses,
the cavity finesse $\mathcal{F}$ is determined by the reflectivity finesse,
$\mathcal{F}_{R}=\pi\sqrt{R}/(1-R)$ in terms of reflectivity of the ports,
$R$. It can easily be shown that for NZI cavities, reflectivity finesse is
proportional to $n^{-1}$. Therefore, the PF scales as $n^{-2}$ and regarding
the cavity fundamental resonance condition $n=\lambda_{0}/2L_{z}$, we find
${F}^{\mathrm{C}}_{\mathrm{P}}\propto(L_{z}/\lambda_{0})^{2}$ and
$Q\propto(L_{z}/\lambda_{0})^{3}$. This is in contrast to the case of ordinary
microcavities, where PF is reduced by increasing the mode volume.

Although relations (4) and (5) imply that SER and PF grow to infinity in
lossless 1D structures at the frequency of zero effective index, the
enhancement of radiative processes and PF is strongly limited by losses in
real structures consisting of metallic or semiconductor components [17, 18,
19, 20]. This deterioration is much more pronounced in resonant cavities when
light propagates multiple times through the cavity structure. To demonstrate
this effect in metallic cavities and compare it to the case of PEC-wall
cavities, we conduct COMSOL simulations in wavelength domain, where we exploit
the mathematical analogy of quantum and classical treatments of the SER.
According to Poynting’s theorem the radiated power of any current distribution
$\mathbf{j}$($\mathbf{r})$ with a harmonic time dependence in a linear medium
must be equal to the rate of energy dissipation given by
$\mathrm{d}W/\mathrm{d}t=-1/2\int_{V}\mathrm{Re}\left\\{\mathbf{j}^{\ast}\cdot\mathbf{E}\right\\}\mathrm{d}V$.
For a current distribution corresponding to a point dipole emitter
$\mathbf{j}$($\mathbf{r})=-\mathrm{i}\omega\mathbf{p}\delta(\mathbf{r}-\mathbf{r}_{0})$
with dipole moment $\mathbf{p}$ located at $\mathbf{r}_{0}$, the equation can
be rewritten in terms of the electric dyadic Green function $\mathbf{G}$, as
$\mathrm{d}W/\mathrm{d}t=\omega^{3}/2c^{2}\varepsilon_{0}\varepsilon\left[\mathrm{Im}\,\left\\{\mathbf{p}\cdot\mathbf{G}(\mathbf{r}_{0},\mathbf{r}_{0};\omega)\cdot\mathbf{p}\right\\}\right]$,
which establishes a link between the quantum and classical formalisms for the
relative SER and PF calculations [29]. Following this approach, we place a
gold nanoparticle carrying a harmonic current density, at the center of the
cavity, as an oscillating dipole emitter. Then we compute the power
dissipation and normalize it to its corresponding value in free space,
allowing us to determine the PF within the structures [30]. In Fig. 1, the PF
curves are calculated for cavities formed by either PEC (solid curve) or gold
(dashed curve) walls. The dimensions of the cavities are $L_{x}=0.5\;\upmu$m,
$L_{y}=1\;\upmu$m, and $L_{z}=8\;\upmu$m and the dipole moment of the emitting
nanoparticle is along $y$-axis, exciting the lowest order TE mode of the
cavity. For the case of gold cavity, $L_{x}$ is reduced by the value of skin
depth to keep the cutoff at $\lambda_{c}=1\;\upmu$m (the effective thickness
of the waveguide is larger than the distance between the walls due to some
penetration of waves into metallic walls). The resonance feedback is provided
by impedance mismatch through incorporating two waveguide sections filled with
materials of refractive index $n=2$ at the cavity ports.

Figure 1:  Numerically calculated PFs for cavities with $L_{x}=0.5\;\upmu$m,
$L_{y}=1\;\upmu$m, $L_{z}=8\;\upmu$m and cutoff wavelength of
$\lambda_{c}=1\;\upmu$m, for PEC (solid curve) and gold (dashed curve) cavity
walls.

The curves show that the free spectral range and spectral width of the
resonances are reduced in both PEC and gold-wall resonators, when approaching
the cutoff wavelength. This is due to increased dispersion inside the cavities
near the cutoff wavelength. However, the $Q$-factor and PF of the gold cavity
are considerably lower than those of the PEC cavity, due to the ohmic losses
of gold. The losses also detrimentally affect the dispersion profile of modes
in a waveguide. As depicted in Fig. 2a, the dispersion of the real part of
effective index, $n(\omega)$ and consequently the group index (being
proportional to the derivative of $n(\omega)$ or $n^{-1}(\omega)$) in a PEC
waveguide tends to infinity when approaching the cutoff wavelength of
$\lambda_{c}=1\;\upmu$m. However, for a gold-wall waveguide (Fig. 2b), the
group index is maximized already before the cutoff and gradually tends to zero
at longer wavelengths. In this case, losses (corresponding to imaginary part
of effective index, $\kappa(\omega)$) grow gradually before the cutoff and
increase abruptly by passing through the cutoff wavelength.

Figure 2:  Dispersion curves of (a) a PEC waveguide, (b) a gold-wall waveguide
and (c) a BRW around the cutoff wavelength of $\lambda_{c}=1\;\upmu$m. The
transverse dimensions of the PEC and gold wall waveguides are the same as in
Fig. 1. (d) A magnified view of the BRW dispersion curve near the cutoff
wavelength, shown in logarithmic scale.

In Figs. 3a and 3b, the results of COMSOL simulations are demonstrated, to
show the functional dependence of PF and $Q$-factor (blue dots) on the
effective index of refraction ($n=\lambda_{0}/2L_{z}$) in a PEC-wall cavity,
through changing the cavity length, $L_{z}$. The resonance feedback is
provided by the same method as in Fig. 1. The computed PF values differ from
those obtained from Eq. 7 by a factor of around six, mainly due to the smaller
value of the effective cavity mode volume, compared to the geometrical volume
used in Eq. 7. The straight lines in Fig. 3, correspond to different
functional dependence on $1/n$. Clearly, the simulation results confirm the
scaling of PF and $Q$-factor as $n^{-2}$ and $n^{-3}$, respectively.

Figure 3:  Simulation results (blue dots) showing (a) PF and (b) $Q$-factor
values in PEC-wall cavities with $L_{x}=0.5\;\upmu$m, $L_{y}=1~{}\upmu$m, and
$L_{z}=4,6,8,10,12\;\upmu$m, corresponding to $1/n\approx 8,12,16,20$ and
$24$, respectively. The lines represent different functional dependence of PF
and $Q$-factor on $1/n$.

In practice, perfectly lossless counterparts of ENZ PEC waveguides and
cavities do not exist. However, by utilizing all-dielectric planar BRWs [31]
operating near their cutoff frequencies, one can potentially realize extremely
low-loss ENZ structures. Enhanced group index has been studied in near-cutoff
BRWs with focus on the associated slow-light features [32, 33]. The guided
waves in such structures propagate parallel to the Bragg walls with very
little losses compared to common ENZ structures. In principle, the radiative
losses through the Bragg walls can be arbitrarily reduced by increasing the
number of cladding layers. Therefore, the supported low-loss waveguide modes
show strong dispersion and large group indices ($n_{g}\approx n^{-1}$) near
the cutoff frequencies [34, 35, 36], behaving thus very similar to PEC
waveguides (Figs. 2a and 2c). The dispersion curves in Fig. 2c are obtained by
two-dimensional (2D) COMSOL mode analysis simulation of a planar quarter-wave
BRW with infinite lateral dimensions. The electric field polarization of the
mode is parallel to the Bragg reflectors and perpendicular to the propagation
direction. The waveguide core material is assumed to be a Borosilicate glass
and the cladding structures on each side was taken to consist of 20 pairs of
alternative Borosilicate and TiO2 layers, with phase thicknesses of $\pi/2$
and refractive indices of $n_{\mathrm{BK7}}=1.51+\mathrm{i}9.93\times 10^{-9}$
[37] and $n_{\mathrm{TiO}_{2}}=2.31+\mathrm{i}10^{-6}$ [38]. The magnified
view of the BRW dispersion curve in Fig. 2d shows that the FoM of the
waveguide at $\lambda=999.5$ nm is as high as
$\epsilon^{\prime}/\epsilon^{\prime\prime}=n^{2}-\kappa^{2}/2n\kappa\approx
2677$, with $n=0.053$ and $\kappa=9.9\times 10^{-6}$ as the real and imaginary
parts of the effective index. The $Q$-factor of a waveguide with negligible
dispersion would be also given by the same above ratio
($Q=\epsilon^{\prime}/\epsilon^{\prime\prime}$). However, for a dispersive and
low-loss medium, such as the high-FoM BRW, near its cutoff wavelength, the
$Q$-factor is estimated as [36, 39]:

$Q^{\mathrm{W}}\approx\frac{1}{2\epsilon^{\prime\prime}}\frac{\partial[\omega\epsilon^{\prime}(\omega)]}{\partial\omega}\approx\frac{1}{2n\kappa}\,,$
(8)

To derive the above relation, the Drude model is used (representing a
dispersive and absorptive medium) in the limit of negligible damping constant
($\gamma<<\omega$). For the above-described BRW, this estimation yields:
$Q^{\mathrm{W}}\approx 9.53\times 10^{5}$ at $\lambda=999.5$ nm.

In planar ENZ BRWs with finite lateral dimensions (Fig. 4a), there are
additional radiative losses through the lateral sides of the waveguide (along
$y$\- and $z$-directions).

Figure 4: (a) Schematic view of a BRW with finite lateral sizes and (b) its
transverse cross-section view, showing the $y$-component of electric field,
corresponding to an ENZ mode of the waveguide, computed by 2D mode-analysis
simulation.

In ENZ waveguides, due to lower effective index of the core with respect to
cladding and free space, the guiding mechanism is different from that in
conventional dielectric waveguides based on total internal reflection. In
contrast, the structure supports leaky modes and guidance is achieved by
partial reflections from the Bragg or lateral walls. While the reflection from
the Bragg walls is caused by destructive interference of waves in the caldding
layers, reflection from the side walls is due to large impedance mismatch
between the core and surrounding free space. Thus, a planar BRW with finite
lateral dimensions acts like a 3D cavity.

We assume that the propagation of $y$-polarized waves corresponding to the
cavity ENZ mode is along the $z$-axis shown in Fig. 4a. Therefore, the cavity
output coupling is taken to be along $z$-axis, and optical leakage along $x$
and $y$ directions are considered as the radiative losses of the waveguide.
Our mode-analysis simulations approve the reduction of radiative losses along
$y$-direction by increasing the cavity width, $L_{y}$. Namely, for a BRW with
the same parameters used in Fig. 2c, but with the limited lateral width of
$L_{y}=10$ $\upmu$m, the mode analysis simulation gives an effective index of
$n=0.047+\mathrm{i}1.29\times 10^{-5}$, at $\lambda=999.5$ nm, corresponding
to FoM of $1828$ (Fig. 4b). By increasing the width to $15$ $\upmu$m and $20$
$\upmu$m, the FoM grows to 2483 and 2621 correspondingly, approaching to the
value obtained for the laterally infinite structure of Fig. 2c. This implies
that by increasing the cavity width, the associated lateral dissipation
reduces to amounts comparable or even smaller than leakage through the finite
Bragg walls. The cavity $Q$-factor, corresponding to output coupling,
represented by $Q^{\mathrm{out}}$, can be enhanced by increasing the cavity
length and subsequent reduction of the effective index
($n=\lambda_{0}/2L_{z}$). However, the overall $Q$-factor of the cavity is
also affected by the waveguiding structure through the relation
$1/Q=1/Q^{\mathrm{W}}+1/Q^{\mathrm{out}}$. Regarding the waveguide radiative
losses through the finite Bragg reflectors and side walls, as well as material
losses of the structure, the overall Q-factor is limited by $Q^{\mathrm{W}}$,
determined in equation (8).

For numerical assessment of $Q^{\mathrm{W}}$, we consider the same BRW
structure related to Fig. 2c and calculate the transmission through the
structure, when illuminated by an incident wave propagating perpendicular to
the Bragg walls. The incident wave is spectrally centered around the waveguide
cutoff frequency with its electric field polarization parallel to the Bragg
walls. Regarding the wave normal incidence, a mode with a zero effective index
is developed in the core, at the waveguide cutoff wavelength. This mode
exhibits no propagation in the waveguide core along the Bragg walls, resulting
in zero group velocity. The BRW acts like a quarter-wave-shifted distributed
feedback resonator or simply as a Fabry–Pérot resonator, composed of two Bragg
reflectors with a half-wave-thickness core in between. The core serves as a
local defect in a crystal lattice, generating a local state at the center of
the stop-band (bandgap) (Fig. 5a). A transmission peak emerges as a result of
multiple wave reflections from the Bragg reflectors. The waveguide $Q$-factor
is calculated by measuring the spectral width of the transmission peak. The
ENZ modes in the BRW, associated with wavelengths smaller than the cutoff
wavelength are established in a similar manner, when the structure is
illuminated by waves having slightly non-zero incidence angles. In this
condition, the waves successively reflect off the Bragg walls and exhibit a
slow propagation characterized by a very small wave-vector component along the
direction of propagation (small phase index $n$, and large group index,
$n_{g}\;{\approx}\;n^{-1}$). As the angle of incidence increases, the
transmission peak experiences a blue shift, while the waveguide Q-factor is on
the same order of magnitude as in the case of normal incidence.

Figure 5:  Perpendicular transmission through the BRW with parameters as in
Fig. 2c, with electric field polarization parallel to the Bragg walls. (a)
Transmission corresponding to the number of cladding pairs of $N=15$, (b) and
a zoom up view of transmission peak for $N=15$, 16 and 20, represented by
dashed, dotted and solid curves, respectively.

The COMSOL simulation results representing the resonant transmission peaks for
the condition of normal incidence are shown in Fig. 5b, for three different
number of alternative Borosilicate and TiO2 layers. We obtain
$Q^{\mathrm{W}}\approx 4.5\times 10^{5},8.5\times 10^{5}$ and $2.5\times
10^{6}$, corresponding to $N=16,18$ and $20$ pairs of layers. The $Q$-factor
is enhanced by increasing the number of layers, but the visibility of the
transmitted wave is dramatically reduced. As expected, the $Q$-factor
corresponding to $N=20$ is of the same order of magnitude estimated in
relation (8).

For estimating the PF of a BRW cavity, we apply Eq. (7) and take $L_{x}$
slightly larger than $\lambda_{0}/2n_{\mathrm{core}}$ ($n_{\mathrm{core}}$ is
the real part of the BRW core material), $L_{z}=\lambda_{0}/2n$ and
$L_{y}\approx\lambda_{0}/2n$ (not exactly equal to $\lambda_{0}/2n$ to ensure
a single resonant mode along $z$-axis). We obtain
${F}^{\mathrm{BRWC}}_{\mathrm{P}}\approx(6n^{2}n_{\mathrm{core}}/\pi^{2})(Q/n)$,
where $Q/n$ (almost equivalent to $Qn_{g}$) represents the Q-factor of the
dispersive cavity. Replacing the $Q/n$ by the limiting Q-factor of the cavity
from relation (8) we obtain:

${F}^{\mathrm{BRWC}}_{\mathrm{P}}\approx\frac{3n_{\mathrm{core}}}{\pi^{2}}\frac{n}{\kappa}\,,$
(9)

which for $n=0.053$ and $\kappa=9.9\times 10^{-6}$, corresponding to the
assumed BRW cavity, and $n_{\mathrm{core}}=1.51$ associated with the
borosilicate core, results in ${F}^{\mathrm{BRWC}}_{\mathrm{P}}\approx 2457$.
This typical value is two orders of magnitude greater than the PF we obtained
for the gold-wall cavity in Fig. 1, and almost three orders of magnitude
higher than radiation enhancement factors experimentally achieved in plasmonic
ENZ structures [19, 20]. However, numerical assessment of relation (9) through
3D simulation of BRW microcavities, having dimensions much larger than
wavelength (in free space), poses challenges due to the substantial
computational workload and more critically, serious convergence issues.

In summary, we derived scaling laws for quality and Purcell factors in
lossless microcavities, revealing their growth with increasing the length of
cavity. In addition, all-dielectric BRWs, operating near their cutoff
frequencies are introduced as building blocks of high-FoM ENZ microcavities to
harness and amplify the unique ENZ features, through significant enhancement
of both quality and Purcell factors in such microcavities.

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# Yosida duality

Bas Westerbaan
Radboud Universiteit Nijmegen
<EMAIL_ADDRESS>
###### Abstract

In this note we prove Yosida duality — that is: the category of compact
Hausdorff spaces with continuous maps is dually equivalent to the category of
uniformly complete Archimedean Riesz spaces with distinguished units and unit-
preserving Riesz homomorphisms between them.

For a compact Hausdorff space $X$, write $C(X)$ for the vector space of real-
valued continuous functions on it. The space $C(X)$ contains sufficient
additional structure to reconstruct $X$ from it. In fact, there are several
ways to do it.

One particularly well-known method is using Gel’fand-duality: it states that
any commutative unital C∗-algebra is isomorphic to the algebra of _complex
valued_ continuous functions on some up-to-isomorphism unique compact
Hausdorff spaces. The categorical corollary is that the category of unital
C∗-algebras with unital $*$-homomorphisms is dually equivalent to that of
compact Hausdorff spaces.

The proof of Gel’fand’s Theorem is rather involved — in this note we will
present a lesser known method due to Yosida[Yos42] which will lead to a
different duality. Gel’fand uses the product on $C(X)$ to reconstruct $X$.
Yosida instead uses the lattice order. At the end of this note, we will have
shown that the category of compact Hausdorff spaces is dually equivalent to
the category of uniformly complete Archimedean unitary Riesz spaces.

The present proof is based on a lecture series given by A. van Rooij in 2011
of which notes can be found here[vR11]. The present text takes a short path to
Yosida duality; for a broad treatment of Riesz spaces, see e.g. [vR11, DJR77,
Zaa12].

##  1 Riesz spaces

We will first define Riesz spaces and derive their elementary theory.

######  Definition 1.

  1. 1.

A real ordered vector space $E$ is a real vector space with a partial order
$\leq$ such that for all $x,y\in E$ with $x\leq y$ we have

    1. (a)

$x+a\leq y+a$ for any $a\in E$ and

    2. (b)

$\lambda\cdot x\leq\lambda\cdot y$ for all scalars $\lambda\geq 0$.

  2. 2.

A Riesz space is a real ordered vector space that is a lattice (i.e. all
finite infima and suprema exist.)

  3. 3.

A linear map $\varphi\colon E\to E^{\prime}$ between Riesz spaces is called a
Riesz homomorphism if $\varphi$ preserves finite infima and suprema.

######  Example 2.

  1. 1.

For any topological space $X$, the vector space $C(X)$ of continuous real-
valued functions on $X$ is a Riesz space.

  2. 2.

Let $\leq_{\mathrm{lex}}$ be the lexicographic order on $\mathbbm{R}^{2}$ —
that is

$(x,y)\leq_{\mathrm{lex}}(x^{\prime},y^{\prime})\quad\iff\quad
x<x^{\prime}\text{ or }(x=x^{\prime}\text{ and }y\leq y^{\prime}).$

Then $\mathbbm{R}^{2}$ ordered by $\leq_{\mathrm{lex}}$ is a Riesz space.

In a Riesz space the lattice operations interact nicely with other structure:

######  Lemma 3.

For a Riesz space $E$ with elements $x,y,a\in E$ it holds

  1. 1.

$\lambda(x\vee y)=(\lambda x)\vee(\lambda y)$ for scalars $\lambda\geq 0$

  2. 2.

$-(x\vee y)=(-x)\wedge(-y)$

  3. 3.

$(x+a)\vee(y+a)=(x\vee y)+a$

  4. 4.

$x\vee y+x\wedge y=x+y$

  5. 5.

$(x\wedge y)\vee a=(x\vee a)\wedge(y\vee a)$

###### Proof.

The maps $x\mapsto\lambda x$ and $x\mapsto x+a$ are order isomorphisms (if
$\lambda>0$) and these preserve suprema, hence 1 and 3. Negation is an order
anti-isomorphism and so 2. Point 4 follows from the previous two:

$(x\vee y)-x-y\stackrel{{\scriptstyle
3}}{{=}}(x-x-y)\vee(y-x-y)=(-y)\vee(-x)\stackrel{{\scriptstyle
2}}{{=}}-(x\wedge y).$

In every lattice $(x\wedge y)\vee a\leq(x\vee a)\wedge(y\vee a)$. For the
other inequality, note

$(x\vee a)\wedge(y\vee a)\leq x\vee a\stackrel{{\scriptstyle
4}}{{=}}x+a-x\wedge a\leq x+a-x\wedge y\wedge a$

and so $(x\vee a)\wedge(y\vee a)+x\wedge y\wedge a-a\leq x$. Similarly $(x\vee
a)\wedge(y\vee a)+x\wedge y\wedge a-a\leq y$. And so $(x\vee a)\wedge(y\vee
a)+x\wedge y\wedge a-a\leq x\wedge y$.

$(x\vee a)\wedge(y\vee a)\leq x\wedge y+a-x\wedge y\wedge
a\stackrel{{\scriptstyle 4}}{{=}}(x\wedge y)\vee a.\qed$

The lattice-order on a Riesz space gives quite some structure.

######  Definition 4.

For any Riesz space $E$, define

  1. 1.

the absolute value $|x|:=x\vee(-x)$ for $x,y\in E$;

  2. 2.

the positive cone $E^{+}:=\\{e\in E;e\geq 0\\}$;

  3. 3.

orthogonality $x\perp y:\Leftrightarrow|x|\wedge|y|=0$ for $x,y\in E$ _and_

  4. 4.

the positive $x^{+}:=x\vee 0$ and negative parts $x^{-}:=(-x)\vee 0$.

Note that in $C(X)$ the absolute value is point-wise; $C(X)^{+}$ are the
positive valued functions _and_ $f\perp g$ iff $f$ and $g$ have disjoint
support.

######  Lemma 5.

For any Riesz space $E$ with $x,y\in E$ we have

  1. 1.

$x=x^{+}-x^{-}$

  2. 2.

$|x|=x^{+}+x^{-}$

  3. 3.

$|x|\geq 0$

  4. 4.

$|x|=x^{+}\vee x^{-}$

  5. 5.

$x^{+}\perp x^{-}$

  6. 6.

$|\lambda x|=|\lambda||x|$

  7. 7.

$|x|=0\implies x=0$

  8. 8.

$(x+y)^{+}\leq x^{+}+y^{+}$

  9. 9.

$|x+y|\leq|x|+|y|$

  10. 10.

$x\vee y=\frac{x+y}{2}+\left|\frac{x-y}{2}\right|$

###### Proof.

As $x=x+0=x\vee 0+x\wedge 0=x\vee 0-(-x)\vee 0=x^{+}-x^{-}$ we have 1. So

$x^{+}+x^{-}=2x^{+}-x=2(x\vee 0)-x=(2x)\vee 0-x=(2x-x)\vee(-x)=|x|$

hence 2. Clearly $0\leq 0\vee x=x^{+}$ and similarly $0\leq x^{-}$. So by the
previous point $0\leq x^{+}+x^{-}=|x|$ hence 3. So $|x|=|x|\vee
0=x\vee(-x)\vee 0=(x\vee 0)\vee((-x)\vee 0)=x^{+}\vee x^{-}$, which is point
4. Also $x^{+}+x^{-}=x^{+}\vee x^{-}+x^{+}\wedge x^{-}$ thus we must have
$x^{+}\wedge x^{-}=0$ viz point 5. For $\lambda\geq 0$ we have
$|\lambda||x|=\lambda(x\vee(-x))=(\lambda x)\vee(-\lambda x)=|\lambda x|$ and
as $|x|=|-x|$ we establish 6. Note $-|x|\leq x\leq|x|$ and so if $|x|=0$ we
have $x=0$ i.e. point 7. Always $x\leq x^{+}$. So $x+y\leq x^{+}+y^{+}$. Hence
$(x+y)^{+}\leq(x^{+}+y^{+})^{+}=x^{+}+y^{+}$ so 8. Similarly $(x+y)^{-}\leq
x^{-}+x^{-}$. So 9 follows: $|x+y|\leq|x|+|y|$. From Lemma 3 point 3 it
follows

$\begin{split}x+y+\left|x-y\right|&=x+y+(x-y)\vee(y-x)\\\
&=(x+y+x-y)\vee(x+y+y-x)=2(x\vee y),\end{split}$

which (together with pt. 2) shows pt. 10 and completes the proof. ∎

######  Lemma 6 (Riesz Decomposition Lemma).

Let $E$ be any Riesz space. For any positive $x,a,b\in E^{+}$ with $x\leq a+b$
there is a decomposition $x=a^{\prime}+b^{\prime}$ with $0\leq a^{\prime}\leq
a$ and $0\leq b^{\prime}\leq b$.

###### Proof.

Define $a^{\prime}:=x\wedge a$ and $b^{\prime}:=x-x\wedge a$. Clearly
$x=a^{\prime}+b^{\prime}$ and $a^{\prime}\leq a$. We have to show
$b^{\prime}\leq b$. Note $b^{\prime}=x-x\wedge a=x\vee a-a$. By assumption
$x\leq a+b$ and so $x\vee a\leq(a+b)\vee a=a+b$ hence $b^{\prime}\equiv x\vee
a-a\leq b$. ∎

##  2 Riesz subspaces and ideals

To construct Riesz homomorphisms into $\mathbbm{R}$ we study Riesz ideals.

######  Definition 7.

Let $E$ be a Riesz space. A subset $D\subseteq E$ is called a Riesz ideal if
it is the kernel of some Riesz homomorphism. The ideal $D$ is said to be
proper when $D\neq E$. The set $D\subseteq E$ is called a Riesz subspace if
$x\vee y,x\wedge y\in D$ whenever $x,y\in D$.

######  Lemma 8.

A linear subspace $D\subseteq E$ is a Riesz subspace if and only if $x^{+}\in
D$ whenever $x\in D$.

###### Proof.

If $D$ is a Riesz subspace, then clearly $x^{+}=x\vee 0\in D$ whenever $x\in
D$. The converse follows from $x\vee
y=\frac{x+y}{2}+\left|\frac{x-y}{2}\right|$ and $|x|=x^{+}+(-x)^{+}$. ∎

######  Lemma 9.

Given a Riesz space $E$ and a $D\subseteq E$, then TFAE:

  1. 1.

$D$ is a Riesz ideal.

  2. 2.

$D$ is a linear subspace and for all $a\in D$ and $x\in E$:

if $|x|\leq|a|$, then $x\in D$.

  3. 3.

$D$ is a Riesz subspace and for all $a\in D^{+}$ and $x\in E^{+}$:

if $x\leq a$, then $x\in D$.

###### Proof.

1 $\Rightarrow$ 2\. Assume $D$ is a Riesz ideal, say it is the kernel of a
Riesz homomorphism $f$. Clearly $D$ is a linear subspace. Assume $|x|\leq|a|$
for some $x\in E$ and $a\in D$. Note Riesz homomorphisms preserve absolute
values as $|x|\equiv x\vee(-x)$ and so $|f(x)|=f(|x|)\leq f(|a|)=|f(a)|=0$
hence $f(x)=0$. Thus $x\in D$, as desired.

2 $\Rightarrow$ 3\. Assume $D$ is a linear subspace for which $x\in D$
whenever $|x|\leq|a|$ for some $a\in D$. We only need to show $D$ is a Riesz
subspace. Assume $a\in D$. We have to show $a^{+}\in D$. Recall $a^{+}\leq
a^{+}+a^{-}=|a|$ and so $a^{+}\in D$, as desired.

3 $\Rightarrow$ 1\. Consider the linear quotient $q\colon E\to E/_{D}$. We
will turn $E/_{D}$ into a Riesz space. We want to use $q(E^{+})$ as positive
cone for a partial order on $E/_{D}$; that is we define $q(x)\leq q(y)\iff
q(y)-q(x)\in q(E^{+})$. This turns $E/_{D}$ into a ordered vector space if
$q(E^{+})$ is closed under addition, scalar multiplication by positive reals
and if we have $q(a),-q(a)\in q(E^{+})\Rightarrow q(a)=q(0)$. Clearly
$q(E^{+})$ is closed under addition and scalar multiplication by positive
reals. Note $q(a)\in q(E^{+})$ iff there is $d\in D$ with $d\leq a$. To show
the last condition, assume $q(a),-q(a)\in q(E^{+})$. Then $d\leq a$ and
$d^{\prime}\leq-a$ for some $d,d^{\prime}\in D$. Hence $0\leq
a-d\leq-d^{\prime}-d$. By assumption on $D$ we find $a-d\in D$. Thus $d\in D$
and $q(a)=q(0)$. Indeed $q(E^{+})$ is a positive cone and we can order
$E/_{D}$ using it. Automatically $q$ is order preserving.

To show $E/_{D}$ is a Riesz space assume $q(x),q(y)\in E/_{D}$. Clearly
$q(x\wedge y)\leq q(x)$ and $q(x\wedge y)\leq q(y)$. Assume there is a
$q(a)\in E/_{D}$ with $q(a)\leq q(x)$ and $q(a)\leq q(y)$. Then $x-c\geq d$
and $y-c\geq d^{\prime}$ for some $d,d^{\prime}\in D$. Consequently $x\wedge
y-c=(x-c)\wedge(y-c)\geq d\wedge d^{\prime}$. As $d\wedge d^{\prime}\in D$ we
see $q(c)\leq q(x\wedge y)$. We have shown $E/_{D}$ has pairwise infima. As
$x\mapsto-x$ is an order anti-isomorpism we see $E/_{D}$ has pairwise suprema
as well. We also saw $q$ is a Riesz homomorphism and by construction $D$ is
its kernel. ∎

######  Lemma 10.

For any $a\in E$ the set $a^{\perp}:=\\{x;\ x\in E;\ x\perp a\\}$ is a Riesz
ideal.

###### Proof.

We will first prove that $a^{\perp}$ is a linear subspace. Let $x\in
a^{\perp}$ and $\lambda\in\mathbbm{R}$. Assume $|\lambda|\leq 1$. Then
$0\leq|\lambda x|\wedge|a|=|\lambda||x|\wedge|a|\leq|x|\wedge|a|=0$ and so
$\lambda x\perp a$. In the other case $|\lambda|\geq n$ for some
$n\in\mathbbm{N}$ with $n\neq 0$ and so $|\frac{\lambda}{n}|\leq 1$. By the
previous $|\frac{\lambda}{n}|\wedge|a|=0$. Hence
$0=|\lambda|\wedge(n|a|)\geq|\lambda|\wedge|a|$, as desired.

Let $x,y\in a^{\perp}$. By distributivity
$(|x|\vee|y|)\wedge|a|=(|x|\wedge|a|)\vee(|y|\wedge|a|)=0$ so $|x|\vee|y|\perp
a$. By the previous $2(|x|\vee|y|)\perp a$. Thus

$|x+y|\wedge|a|\leq(|x|+|y|)\wedge|a|\leq(2(|x|\vee|y|))\wedge|a|=0.$

Thus $a^{\perp}$ is a linear subspace. To show $a^{\perp}$ is a Riesz ideal,
we will demonstrate condition 3 from Lemma 9. Assume $0\leq x\leq y\in
a^{\perp}$. Then $0\leq x\wedge|a|\leq y\wedge|a|=0$ and so $x\in a^{\perp}$ —
it remains to be shown that $a^{\perp}$ is a Riesz subspace. It is sufficient
to show $x^{+}\in a^{\perp}$ whenever $x\in a^{\perp}$. Indeed from

$0=|x|\wedge|a|=(x^{+}\vee
x^{-})\wedge|a|=(x^{+}\wedge|a|)\vee(x^{-}\wedge|a|)$

it follows $x^{+}\perp a$ as desired. ∎

##  3 Units and their norms

######  Definition 11.

  1. 1.

An element $u\in E^{+}$ is called an (order) unit if for every $x\in E$ there
exists a $n\in\mathbbm{N}$ such that $|x|<n\cdot u$. A Riesz space is called
unitary if it has a unit.

  2. 2.

An element $\varepsilon\in E$ is called infinitesimal if there is a $b\in E$
such that $n\cdot\varepsilon\leq b$ for all $n\in\mathbbm{Z}$. A Riesz space
is called Archimedean if its only infinitesimal is $0$.

######  Definition 12.

Given an Riesz space $E$ with unit $u\in E$ write

$\|x\|_{u}:=\inf\\{\lambda;\ \lambda\in\mathbbm{R}^{+};\ |x|\leq\lambda u\\}.$

######  Lemma 13.

Let $E$ be a Riesz space with units $e,u$.

  1. 1.

$\|\ \|_{u}$ is a seminorm.

  2. 2.

If $\|x\|_{u}=0$, then $x$ is infinitesimal.

  3. 3.

If $E$ is Archimedean, then $\|\ \|_{u}$ is a norm and $|x|\leq\|x\|_{u}u$.

  4. 4.

$\|\ \|_{e}$ and $\|\ \|_{u}$ are equivalent.

###### Proof.

First we’ll show $\|\ \|_{u}$ is a seminorm. Let $\varepsilon>0$ be given. For
any $x\in E$ we have $|x|\leq(\|x\|_{u}+\varepsilon)u$. So
$|a+b|\leq|a|+|b|\leq(\|a\|_{u}+\|b\|_{u}+2\varepsilon)u$. Hence
$\|a+b\|_{u}\leq\|a\|_{u}+\|b\|_{u}+2\varepsilon$. Thus, as $\varepsilon>0$
was arbitrary, the triangle inequality holds for $\|\ \|_{u}$.

Now we’ll show the absolute homogeneity. Let $\mu>0$ be given. With $|\mu
a|=|\mu||a|$ and similar reasoning as before we see $\|\mu
a\|_{u}\leq|\mu|\|a\|_{u}$. To prove the other inequality first note that for
any $\|a\|_{u}\geq\varepsilon>0$ we have $|a|\nleq(\|a\|_{u}-\varepsilon)u$.
Thus $|\mu a|=|\mu||a|\nleq(|\mu|\|a\|_{u}-|\mu|\varepsilon)u$. As we can make
$|\mu|\varepsilon$ arbitrarily small we see $\|\mu a\|_{u}\geq|\mu|\|a\|_{u}$.
This gives absolute homogeneity except for the case $\mu=0$, which is
trivially true. Finally, by definition $\|\ \|_{u}$ is positive, so it’s a
seminorm.

Assume $\|x\|_{u}=0$. Then $|x|\leq\frac{1}{n}u$ for any $n\in\mathbbm{N}$
with $n>0$. Hence $n|x|\leq u$ Now, for any $m\in\mathbbm{Z}$ we have
$mx\leq|mx|=|m||x|\leq u$ and so $x$ is infinitesimal.

If $E$ is Archimedean then directly by the previous we see $\|\ \|_{u}$ is a
norm. Now we show $|a|\leq\|a\|_{u}u$. Suppose $n\in\mathbbm{N}$ with $n\neq
0$. Then $|a|\leq(\|a\|_{u}+\frac{1}{n})u=\|a\|_{u}u+\frac{1}{n}u$. By taking
the infimum over $n\in\mathbbm{N}_{>0}$, we see it suffices to show
$\inf\\{\frac{1}{n}u;\ n\in\mathbbm{N}_{>0}\\}=0$. Clearly $0$ is a lower-
bound. Assume $b\in E$ is another lower-bound, i.e. $b\leq\frac{1}{n}u$ for
all $n\in\mathbbm{N}_{>0}$. Then $b^{+}\leq(\frac{1}{n}u)^{+}=\frac{1}{n}u$ so
$nb^{+}\leq\frac{1}{n}u$ and $-nb^{+}\leq 0\leq\frac{1}{n}u$ for all
$n\in\mathbbm{N}$. Thus $b^{+}$ is infinitesimal: so $b^{+}=0$ and $b\leq 0$,
as desired.

Finally we show point 4. As for any $\varepsilon>0$ we have
$|x|\leq(\|x\|_{u}+\varepsilon)u$ we get
$\|x\|_{e}\leq(\|x\|_{u}+\varepsilon)\|u\|_{e}$ and so
$\|x\|_{e}\leq\|x\|_{u}\|u\|_{e}$. Similarly$\|x\|_{u}\leq\|x\|_{e}\|e\|_{u}$
and so $\frac{1}{\|e\|_{u}}\|x\|_{u}\leq\|x\|_{e}\leq\|u\|_{e}\|x\|_{u}$,
which shows $\|\ \|_{u}$ and $\|\ \|_{e}$ are equivalent norms. ∎

######  Lemma 14.

Any non-zero Riesz homomorphism $f\colon E\to E^{\prime}$ between unital
Archimedean Riesz spaces is continuous with respect to the norms induced by
any unit.

###### Proof.

Pick any unit $u\in E$ and $u^{\prime}\in E^{\prime}$. As all unit norms are
equivalent it is sufficient to show $f$ is continuous with respect to $\|\
\|_{u}$ and $\|\ \|_{u^{\prime}}$. If $\|f(u)\|_{u^{\prime}}=0$, then $f(u)=0$
and so $f=0$ as $-\|x\|_{u}u\leq x\leq\|x\|_{u}u$ for any $x\in E$. Thus
$\|f(u)\|_{u^{\prime}}>0$. Let $\varepsilon>0$ be given. If
$\|x-y\|_{u}\leq\delta$ for some $\delta>0$ then $|x-y|\leq\delta u$ and so
$|f(x)-f(y)|=f(|x-y|)\leq\delta
f(u)=\delta|f(u)|\leq\delta\|f(u)\|_{u^{\prime}}u^{\prime}$. Thus it is
sufficient to choose $\delta\leq\frac{\varepsilon}{\|f(u)\|_{u^{\prime}}}$. ∎

######  Definition 15.

An Archimedean unitary Riesz space is called uniformly complete if $\|\
\|_{u}$ is complete for some unit $u\in E$.

As order-unit norms are equivalent we know that if $E$ is complete with
respect to one unit, then it is complete with respect to all units.

######  Definition 16.

We write $\mathsf{CAURiesz}$ for the category of uniformly complete
Archimedean Riesz spaces with distinguished unit as objects and Riesz
homomorphisms that preserve the selected unit as arrows between.

##  4 Maximal Riesz ideals

The real numbers form a totally ordered Archimedean Riesz space and is, in
fact, the only one. This Proposition will play the same rôle as the
Banach–Mazur Theorem in the development of Gel’fand Duality.

######  Proposition 17.

If $E$ is a Riesz space with more than one element, then TFAE.

  1. 1.

$E\cong\mathbbm{R}$

  2. 2.

The only ideals of $E$ are $\\{0\\}$ and $E$.

  3. 3.

$E$ is totally ordered and Archimedean.

###### Proof.

1 $\Rightarrow$ 2\. Direct.

2 $\Rightarrow$ 3\. First we will prove $E$ is totally ordered. Assume $x,y\in
E$. Then $J=\\{v\in E;\ v\perp(x-y)^{+}\\}$ is a Riesz ideal by Lemma 10. If
$J=\\{0\\}$ then since $(x-y)^{-}\perp(x-y)^{+}$, we know $(x-y)^{-}=0$ and
thus $0\leq x-y$. Hence $x\geq y$. In the other case: if $J=E$, then
$(x-y)^{+}\perp(x-y)^{+}$ and thus $(x-y)^{+}=0$. Hence $x-y=-(x-y)^{-}\leq 0$
and thus $x\leq y$.

Now we will prove $E$ is Archimedean. Assume $\varepsilon$ is an infinitesimal
of $E$. We will show $\varepsilon=0$. Let $b\in E$ such that
$n|\varepsilon|\leq b$ for all $n\in\mathbbm{N}$. Note $J:=\\{x\in E;\
n|x|\leq b\text{ for all }n\in\mathbbm{N}\\}$ is a Riesz ideal. If $J=\\{0\\}$
we are done. In the other case $J=E$. Then $b\in J$ and $n|b|\leq b\leq n|b|$
for all $n\neq 0$. Thus $2|b|=b=|b|$ so $|\varepsilon|\leq b=0$ and we are
done.

3 $\Rightarrow$ 1\. Pick any $a>0$. The map $\lambda\mapsto\lambda a$ is a
linear order isomorphism of $\mathbbm{R}$ onto a $1$-dimensional subspace of
$E$. We are done if this subspace is $E$ itself. Let $b\in E^{+}$. Define
$r:=\sup\\{r;\ r\in\mathbbm{R}^{+};\ rb\leq a\\}$. As $E$ is Archimedean,
there is a $n$ with $nb\nleq a$. Thus $nb\geq a$ and so $r<\infty$. For any
$\varepsilon>0$ we must have $(r+\varepsilon)b>a$ as $E$ is totally ordered.
Taking infimum over $\varepsilon>0$ we see $a=rb$ thus indeed $E$ is
1-dimensional. ∎

An important corollary is the following:

######  Proposition 18.

Given an Archimedean Riesz space $E$ with unit $e$. For every $a\in E$, $a\neq
0$, there is a Riesz homomorphism $\varphi\colon E\to\mathbbm{R}$ such that
$\varphi(e)=1$ and $\varphi(a)\neq 0$.

###### Proof.

It is sufficient to find such $\varphi$ for $|a|$ as
$0\neq\varphi(|a|)=|\varphi(a)|$ implies $\varphi(a)\neq 0$. Thus assume
$a>0$. As $E$ is Archimedean, there is an $n\in\mathbbm{N}$ with $na\nleq e$.
That is $na-e\nleq 0$. But $na-e\leq(na-e)^{+}$, so $(na-e)^{+}\neq 0$.

Consider the proper Riesz ideal $J:=\\{x;\ x\perp(na-e)^{+}\\}$. Find using
Zorn’s Lemma a maximal proper Riesz ideal $J_{0}\supseteq J$. Then
$E/_{J_{0}}$ contains precisely two ideals and must be isomorphic to
$\mathbbm{R}$. Define $\varphi_{0}$ as the composition of this isomorphism
with the quotient-map $E\to E/_{J_{0}}\cong\mathbbm{R}$. Note
$\varphi_{0}(e)>0$ as otherwise $\varphi_{0}=0$. Define
$\varphi:=\frac{\varphi_{0}}{\varphi_{0}(e)}$. Clearly $\varphi(e)=1$.

Note $(na-e)^{-}\in J$ and so
$0=\varphi((na-e)^{-})=(n\varphi(a)-\varphi(e))^{-}$ which implies
$n\varphi(a)-\varphi(e)=(n\varphi(a)-\varphi(a))^{+}\geq 0$. Thus
$n\varphi(a)\geq\varphi(e)>0$ and so $\varphi(a)\neq 0$. ∎

##  5 Function spaces

######  Example 19.

Let $X$ be a compact Hausdorff space. Then $C(X)$ with the obvious pointwise
operations and order, is a Riesz space. Since $X$ is compact, all these
functions are bounded, $\mathbbm{1}(x):=1$ is a unit and $\|\
\|_{\mathbbm{1}}$ is a supnorm for which $C(X)$ is complete. It is also
obvious that the zero function is the only infinitesimal. Thus $C(X)$ is
Archimedean. Thus $C(X)$ with unit $\mathbbm{1}$ is an object of
$\mathsf{CAURiesz}$.

Let $X$ be a compact Hausdorff space and $x\in X$. Then  _point evaluation_
$\delta_{x}\colon C(X)\to\mathbbm{R}$ defined by $\delta_{x}(f)=f(x)$ is a
Riesz homomorphism into $\mathbbm{R}$ with $\delta_{x}(\mathbbm{1})=1$. Every
unit-preserving Riesz homomorphism $C(X)\to\mathbbm{R}$ is of this form:

######  Proposition 20.

Let $X$ be a compact Hausdorff space. For any Riesz homomorphism
$\varphi\colon C(X)\to\mathbbm{R}$ with $\varphi(\mathbbm{1})=1$ there is a
point $x\in X$ with $\varphi=\delta_{x}$.

###### Proof.

For any $f\in C(X)$ define $\tilde{f}:=|f-\varphi(f)\mathbbm{1}|$. If we can
find an $x_{0}\in X$ with $x_{0}\notin\mathop{\mathrm{supp}}\tilde{f}:=\\{x;\
\tilde{f}(x)\neq 0\\}$ for all $f\in C(X)$, then we are done as
$\tilde{f}(x_{0})=0$ implies $\varphi(f)=f(x_{0})=\delta_{x_{0}}(f)$. Note
that by definition $\tilde{f}\geq 0$ and $\varphi(\tilde{f})=0$. For a single
$f\in C(X)$ there must be an $x\in X$ with
$x\notin\mathop{\mathrm{supp}}\tilde{f}$ (i.e. $\tilde{f}(x)=0$) for otherwise
$\tilde{f}>0$ and so by compactness $\tilde{f}>\varepsilon\mathbbm{1}$ for
some $\varepsilon>0$ which gives $0=\varphi(\tilde{f})>\varepsilon$ quod non.
Now, let $f_{1},\ldots,f_{n}\in C(X)$ be given. Write
$g:=\tilde{f}_{1}\vee\ldots\vee\tilde{f}_{n}$. Note $\varphi(g)=0$ and so
there must also be an $x\in X$ with $g(x)=0$. As
$\mathop{\mathrm{supp}}g=\mathop{\mathrm{supp}}\tilde{f}_{1}\cup\ldots\cup\mathop{\mathrm{supp}}\tilde{f}_{n}$
we see $x\notin\mathop{\mathrm{supp}}\tilde{f}_{i}$ for every $1\leq i\leq n$.
Thus no finite subset of the family of open sets
$\mathcal{A}:=\\{\mathop{\mathrm{supp}}\tilde{f};\ f\in C(X)\\}$ can cover
$X$. Thus by contraposition of compactness we see that $\cup\mathcal{A}$
cannot cover $X$. Hence there is an $x_{0}\in X$ with
$x_{0}\notin\mathop{\mathrm{supp}}\tilde{f}$ for any $f\in C(X)$ as desired. ∎

This Proposition is a cornerstone of the duality. Before we will look into
that, we will proof a variant of the Stone-Weierstrass theorem in the language
of Riesz spaces.

######  Theorem 21 (Stone-Weierstrass).

For any compact Hausdorff space $X$ and Riesz subspace $D$ of $C(X)$ such
that:

  1. 1.

$D$ is unital — that is: $\mathbbm{1}\in D$

  2. 2.

$D$ separates the points — that is: for all $x,y\in X$ with $x\neq y$ there is
an $f\in D$ with $f(x)\neq f(y)$.

Then: $D$ is $\|\ \|_{\mathbbm{1}}$-dense in $C(X)$.

###### Proof.

Let $g\in C(X)$ and $\varepsilon>0$ be given. We will find a $g_{0}\in D$ with
$\|g-g_{0}\|_{\mathbbm{1}}\leq\varepsilon$. By approximating $g^{+}$ and
$-g^{-}$ separately we see we may assume without loss of generality $g\geq 0$.
We will approximate $g$ progressively in three steps.

Let $y,z\in X$ be any points with $y\neq z$. By assumption there is an $f\in
D$ with $f(z)\neq f(y)$. Define
$f_{y}:=\frac{g(z)}{f(z)-f(y)}(f-f(y)\mathbbm{1})$. Note $f_{y}(y)=0$,
$f_{y}(z)=g(z)$ and still $f_{y}\in D$.

Write $U_{f_{y}}:=\\{x;\ x\in X;\ f_{y}(x)<g(x)+\frac{\varepsilon}{2}\\}$.
Clearly both $z,y\in U_{f_{y}}$. Thus the $U_{f_{y}}$ are an open cover of
$X$. By compactness we can find $n\in\mathbbm{N}$ and $y_{1},\ldots,y_{n}$
such that $U_{y_{1}}\cup\cdots\cup U_{y_{n}}=X$. Define
$f^{\prime}_{z}:=f_{y_{1}}\wedge\cdots\wedge f_{y_{n}}$. The map
$f^{\prime}_{z}\in D$ is a better approximation: $f^{\prime}_{z}(z)=g(z)$ and
$f^{\prime}_{z}\leq g+\frac{\varepsilon}{2}\mathbbm{1}$.

Write $V_{f^{\prime}_{z}}:=\\{x;\
g(x)-\frac{\varepsilon}{2}<f^{\prime}_{z}(x)\\}$, which are also open sets
with $z\in V_{f^{\prime}_{z}}$. By compactness we can find $m\in\mathbbm{N}$
and $z_{1},\ldots,z_{m}$ with $V_{f^{\prime}_{z_{1}}}\cup\cdots\cup
V_{f^{\prime}_{z_{m}}}=X$. Now define $g_{0}:=f^{\prime}_{z_{1}}\vee\cdots\vee
f^{\prime}_{z_{m}}$. Note $g-\frac{\varepsilon}{2}\mathbbm{1}\leq g_{0}\leq
g+\frac{\varepsilon}{2}\mathbbm{1}$ and so
$\|g_{0}-g\|_{\mathbbm{1}}\leq\varepsilon$. ∎

##  6 The spectrum

We can turn a compact Hausdorff space $X$ into an Archimedean uniformly
complete Riesz space $C(X)$ — how do we go the other way?

######  Definition 22.

Let $E$ be an Archimedean Riesz space with a unit $u$. The spectrum of $E$ is
defined to be the set

$\Phi(E):=\\{\varphi;\ \varphi\colon E\to\mathbbm{R}\ \text{Riesz homomorphism
with }\varphi(u)=1\\}.$

which is a topological space with induced topology of the inclusion
$\Phi(E)\subseteq\mathbbm{R}^{E}$ where we take the product topology on the
latter.

At first glance, the spectrum seems to depend on the choice of unit. However:

######  Claim 23.

Let $E$ be an Archimedean Riesz space with units $e,u\in E$. We will write
$\Phi_{e}(E)$ and $\Phi_{u}(E)$ for the spectra respective to $e$ and $u$.
Then: $\Phi_{e}(E)\cong\Phi_{u}(E)$ via
$\varphi\mapsto(x\mapsto\frac{\varphi(x)}{\varphi(u)})$.

######  Lemma 24.

The spectrum $\Phi(E)$ is a compact Hausdorff space.

###### Proof.

As for any $x\in E$ it holds $-\|x\|_{u}\leq\varphi(x)\leq\|x\|_{u}$ we have
the following inclusions of topological spaces

$\Phi(E)\ \subseteq\ \prod_{x\in E}[-\|x\|_{u},\|x\|_{u}]\ \subseteq\
\mathbbm{R}^{E}.$

The middle one is compact by Tychonoff’s Theorem theorem. Thus if we can show
$\Phi(E)$ is closed in $\mathbbm{R}^{E}$, we know $\Phi(E)$ is compact and
Hausdorff.

Let $(\varphi_{\alpha})_{\alpha}$ in $\Phi(E)$ be a net converging to some
$\varphi$ in $\mathbbm{R}^{E}$. As addition, infimum, supremum and scalar
multiplication are continuous the map $\varphi$ must be a Riesz homorphism as
well. Trivially $\varphi(u)=\lim_{\alpha}\varphi_{\alpha}(u)=1$. Thus indeed
$\Phi(E)$ is closed in $\mathbbm{R}^{E}$ as desired. ∎

##  7 Yosida duality

We are almost ready to state and prove Yosida duality. First a notational
matter: recall the objects of $\mathsf{CAURiesz}$ are pairs $(E,u)$ of a
uniformly complete Archimedean Riesz space $E$ together with a distinguished
unit $u\in E$. For brevity we’ll simply write $E$ and denote the implicit
distinguished unit by $u_{E}$.

######  Theorem 25 (Yosida).

$C$ and $\Phi$ are categorically dual:

  1. 1.

The map $X\mapsto C(X)$ extends to a functor
$\mathsf{CHsdf}\to\mathsf{CAURiesz}^{\partial}$ taking $\mathbbm{1}$ as unit
on $C(X)$ and sending a continuous map $f\colon X\to Y$ to the Riesz
homomorphism $C(f)\colon C(Y)\to C(X)$ given by $C(f)(g)=g\mathbin{\circ}f$.

  2. 2.

The map $E\mapsto\Phi(E)$ extends to a functor
$\mathsf{CAURiesz}\to\mathsf{CHsdf}^{\partial}$ by mapping a Riesz
homomorphism $f\colon E\to E^{\prime}$ to
$\Phi(f)\colon\Phi(E^{\prime})\to\Phi(E)$ given by
$\Phi(f)(\varphi)=\varphi\mathbin{\circ}f$.

  3. 3.

We have $C\Phi\cong\mathrm{id}$ (every uniformly complete Archimedean Riesz
space is naturally isomorphic to a continuous function space) and $\Phi
C\cong\mathrm{id}$ (every compact Hausdorff space is naturally isomorphic to
the spectrum of an Archimedean Riesz space) and so $\mathsf{CHsdf}$ and
$\mathsf{CAURiesz}$ are dually equivalent.

###### Proof.

We first show 1 and 2.

  1. 1.

We already saw $C(X)$ is in $\mathsf{CAURiesz}$ (Example 19 and that $\Phi(E)$
is in $\mathsf{CHsdf}$ (Lemma 24). Maps on arrows defined by pre-composition
(such as $C$ and $\Phi$) are automatically functorial.

  2. 2.

Clearly $C(f)(g)\equiv g\mathbin{\circ}f$ is continuous as it is the
composition of two continuous maps. To finish 1, we still have to check $C(f)$
is a Riesz homomorphism. It is additive:
$C(f)(g+g^{\prime})(x)=(g+g^{\prime})\mathbin{\circ}f(x)=g(f(x))+g^{\prime}(f(x))=C(f)(g)(x)+C(f)(g)(x)$.
In a similar fashion one checks $C(f)$ preserves scalar multiplication, $\vee$
and $\wedge$. Unit-preservation is different, but also simple:
$C(f)(\mathbbm{1})(x)=\mathbbm{1}\mathbin{\circ}f(x)=1=\mathbbm{1}(x)$.

  3. 3.

To wrap up 2 we need to check $\Phi(f)$ is a continuous map for all $f\colon
E\to E^{\prime}$. Let $(\varphi_{\alpha})_{\alpha}$ be a net in
$\Phi(E^{\prime})$ converging to $\varphi$. In particular we have for any
$x\in E$ that $\varphi_{\alpha}(f(x))\to\varphi(f(x))$. Hence
$\Phi(f)(\varphi_{\alpha})\equiv\varphi_{\alpha}\mathbin{\circ}f\to\varphi\mathbin{\circ}f\equiv\Phi(f)(\varphi)$
in $\Phi(E)$, as desired.

Now we turn to the proof of $C\Phi\cong\mathrm{id}$. Let $E$ be in CAURiesz.

  1. 4.

First we define what will turn out to be an isomorphism. For $x\in E$, define
$\hat{x}$ (initially in $\mathbbm{R}^{\Phi(E)}$) by
$\hat{x}(\varphi):=\varphi(x)$. To prove $\hat{x}$ is continuous, assume
$\varphi_{\alpha}\to\varphi$ is a converging net in $\Phi(E)$. Then
$\hat{x}(\varphi_{\alpha})=\varphi_{\alpha}(x)\rightarrow\varphi(x)=\hat{x}(\varphi)$,
thus $\hat{x}\in C\Phi(E)$. For the moment, write $\hat{E}:=\\{\hat{x};\ x\in
E\\}$.

  2. 5.

Note that
$(\hat{x}+\hat{y})(\varphi)=\varphi(x+y)=\varphi(x)+\varphi(y)=\hat{x}(\varphi)+\hat{y}(\varphi)$
and thus $\hat{x}+\hat{y}=\widehat{x+y}\in\hat{E}$. The same reasoning applies
to $\cdot$, $\wedge$ and $\vee$. Thus $x\mapsto\hat{x}$ is a Riesz-
homomorphism and $\hat{E}$, being its image, is a Riesz subspace of
$C\Phi(E)$.

  3. 6.

Suppose $x\in E$ and $x\neq 0$. Then by Proposition 18, there is a
$\varphi\in\Phi(E)$ such that $\varphi(x)\neq 0$. Thus $\hat{x}\neq 0$. Thus
$x\mapsto\hat{x}$ is injective. Consequently $E\cong\hat{E}$.

  4. 7.

Given $\varphi,\psi\in\Phi(E)$ with $\varphi\neq\psi$. By definition there is
an $x\in E$ such that $\varphi(x)\neq\psi(x)$. Together with
$\hat{u}_{E}(\varphi)=\varphi(u_{E})=\mathbbm{1}$, we conclude with the
theorem of Stone-Weierstrass (see Thm. 21) that $\hat{E}$ is $\|\
\|_{\mathbbm{1}}$-dense in $C\Phi(E)$. Since $E\cong\hat{E}$ is uniformly
complete (i.e. $\|\ \|_{\mathbbm{1}}$-complete) we find $C\Phi(E)\cong E$.

  5. 8.

We finish with naturality. Assume $f\colon E\to E^{\prime}$ in
$\mathsf{CAURiesz}$, $x\in E$ and $\varphi\in\Phi(E^{\prime})$. Expanding
definitions we find
$C\Phi(f)(\hat{x})(\varphi)=\hat{x}(\Phi(f)(\varphi))=\hat{x}(\varphi\circ
f)=\varphi\circ f(x)=\widehat{f(x)}(\varphi)$. Thus $C\Phi\cong\mathrm{id}$.

Now we prove the other direction: $\Phi C\cong\mathrm{id}$. Let
$X\in\textsf{CHsdf}$ be given.

  1. 9.

Recall $\delta_{x}\in\Phi C(X)$ defined by $\delta_{x}(f)\equiv f(x)$. Write
$\Delta(X):=\\{\delta_{x};\ x\in X\\}$.

  2. 10.

Suppose $x_{\alpha}\rightarrow x$ in $X$. Then for $f\in C(X)$ we have
$\delta_{x_{\alpha}}(f)=f(x_{\alpha})\rightarrow f(x)=\delta_{x}(f)$. Thus
$x\mapsto\delta_{x}$ is continuous.

  3. 11.

Let $x,y\in X$ with $x\neq y$ be given. By Urysohn’s Lemma there is an $f\in
C(X)$ such that $f(x)\neq f(y)$. Hence $\delta_{x}\neq\delta_{y}$. Thus
$x\mapsto\delta_{x}$ is injective.

  4. 12.

Assume $\varphi\in\Phi C(X)$. By Proposition 20 there is an $x\in X$ such that
$\varphi=\delta_{x}$. Thus the map $x\mapsto\delta_{x}$ is surjective.
Together with the previous two points we see $x\mapsto\delta_{x}$ is a
continuous bijection between compact Hausdorff spaces and thus a
homeomorphism.

  5. 13.

The demonstration of naturality is the same as in the previous Proposition.
Thus $\Phi C\cong\mathrm{id}$ as desired. ∎

### Acknowledgements

Robert Furber kindly encouraged me to publish these notes. I am grateful to
Abraham Westerbaan for his suggestions.

## References

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  * [Yos42] Kôsaku Yosida.  70\. on the representation of the vector lattice.  Proceedings of the Imperial Academy, 18(7):339–342, 1942.
  * [Zaa12] Adriaan C Zaanen.  Introduction to operator theory in Riesz spaces.  Springer Science & Business Media, 2012.