# Dehydration as a Universal Mechanism for Ion Selectivity in Graphene and
Other Atomically Thin Pores – Supporting Information

Subin Sahu  Center for Nanoscale Science and Technology, National Institute of
Standards and Technology, Gaithersburg, Maryland 20899  Maryland Nanocenter,
University of Maryland, College Park, Maryland 20742  Department of Physics,
Oregon State University, Corvallis, Oregon 97331  Massimiliano Di Ventra
Department of Physics, University of California, San Diego, California 92093
Michael Zwolak  Center for Nanoscale Science and Technology, National
Institute of Standards and Technology, Gaithersburg, Maryland 20899
<EMAIL_ADDRESS>
###### Contents

  1. I Methods
    1. I.1 Molecular Dynamics
    2. I.2 Pore radius and area
    3. I.3 Current definition and electric field
  2. II Current behavior at the subnanoscale
  3. III Selectivity
    1. III.1 Experimental observation of selectivity
    2. III.2 Dehydration and selectivity
  4. IV Model for ion transport
  5. V The effect of the ion concentration
  6. VI Tables

##  I Methods

###  I.1 Molecular Dynamics

Graphene has carbon atoms located at the points
$\hat{r}_{nm}=m\hat{a}_{1}+n\hat{r}_{2}$ for $m,\,n\in\mathbb{Z}$, where
$\hat{a}_{1}=a(3,\sqrt{3})/2$ and $\hat{a}_{2}=a(3,-\sqrt{3})/2$ are the 2D
lattice vectors and $a\approx 0.14$ nm is the C-C bond length. We open a pore
of nominal radius $r_{n}$ at the center of each membrane by removing carbon
atoms satisfying the condition $x^{2}+y^{2}<r_{n}^{2}$, with $x,\,y$ the
coordinates of the atom in the $z=0$ plane. We then immerse the membrane in an
aqueous ionic solution, typically with 1 mol/L salt concentration, consistent
with experiments. We use the CHARMM27 force field to model the atoms. The
carbon atoms are type CA and water molecules are TIP3P from the CHARMM27 force
field.

We perform all-atom molecular dynamics (MD) simulations using NAMD
phillips2005  with a time step of 1 fs and periodic boundary conditions in all
directions. We use a cutoff of 1.2 nm for non-bonded interactions, i.e., van
der Waals and electrostatics. However, we use full electrostatic calculations
every 4 fs via the particle-mesh Ewald (PME) method darden1993 . We first
minimize the energy of the system for 4000 steps (4 ps) and then heat it to
295 K in another 4 ps. A 0.5 ns NPT (constant number of particles, pressure
and temperature) equilibration using the Nose-Hoover Langevin piston method
Martyna1994  – to raise the pressure to 101 325 Pa (i.e., 1 atm) – followed by
1.5 ns of NVT (constant number of particles, volume and temperature)
equilibration generates the initial atomic configuration. An electric field
perpendicular to the plane of the membrane drives the ionic current through
the pore. We set the Langevin damping rate to 0.2/ps for carbon and water (via
its oxygen atoms) during these runs. Test runs show that damping the hydrogen
atoms does not affect the results. Damping the ions, however, affects the
current as it changes the ionic mobility.

We fix the outer edge of the graphene membrane, but the bulk of the membrane
has no confinement other than the C-C bonds of graphene. The production runs
vary from 100 ns to 1.1 $\mu$s based on the convergence of the current and
other properties of interest. When calculating the water density around an ion
fixed at the origin (e.g., in the center of the pore when it is not
fluctuating), the parameters are the same except there was no external
electric field present.

We use the adaptive biasing force method (ABF) Darve2008 ; Henin2004  in the
colvar module of NAMD to perform the free energy calculation. In this method
the reaction coordinate, $\zeta$ ($z$ for the setup here), is divided into
equally spaced bins and the free-energy difference along $\zeta$ is calculated
by integrating the equation

$\left\langle f_{\zeta}\right\rangle_{k}=-\left\langle\frac{\partial
U(X)}{\partial\zeta}-\frac{\partial
ln|J|}{\partial\zeta}\right\rangle_{k}\equiv-\frac{dF(k)}{d\zeta},$ (1)

where $\left\langle f_{\zeta}\right\rangle_{k}$ and $F(k)$ are the mean force
and free energy at bin $k$, $X$ are the cartesian coordinates, and $|J|$ is
the Jacobian of the transformation to cartesian coordinates. The ABF method
applies an iterative biasing force, $\bar{f_{n}}(k)$, which is the average of
all force samples after $n$ MD steps in the bin $k$. This force enables the
system to overcome free-energy barriers during an unconstrained MD run and
allows for a more uniform sampling along the reaction coordinate. In our
simulation, we calculate the one-dimensional free energy profile along the
$z$-axis in bins of width 0.01 nm from $z=-1.5$ nm to $z=1.5$ nm. We perform
ABF calculation on three windows, ($-1.5$ nm $\leqslant z\leqslant-0.5$ nm),
($-0.5$ nm $\leqslant z\leqslant 0.5$ nm), and ($0.5$ nm $\leqslant z\leqslant
1.5$ nm), and also symmetrize the final result about $z=0$ for better
sampling. Also, we confine the ion within a cylinder of radius 0.2 nm centered
at the origin, where a bounding potential with force constant $\approx 43$
eV/nm2 turns on outside of the 0.2 nm cylinder. In each bin, 800 samples of
the instantaneous force are accrued prior to the full application of the ABF.
This biasing force scales up linearly by a factor of 0 to 1 from 400 samples
to 800 samples, and no biasing force is applied below 400 samples. The total
simulation consists of about 120 runs of 10 ns each for each window.

We use the block standard error (BSE) method grossfield2009  to compute the
error bars for all plots. The BSE is given by

$\mathrm{BSE}=\frac{s_{\tau}\sqrt{\tau}}{\sqrt{T}},$ (2)

where $T$ is the total simulation time (the time of the MD trajectory), $\tau$
is a length of time used to partition the simulation into many contiguous
blocks, and $s_{\tau}=\sqrt{\frac{\sum_{i}(\langle I_{\tau}\rangle_{i}-\langle
I_{T}\rangle)^{2}}{(N_{b}-1)}}$ is the standard deviation of the mean current,
$\langle I_{\tau}\rangle$, within each of the $N_{b}$ blocks. The BSE depends
on $\tau$ when the latter is very small (i.e., when $\tau$ is smaller than the
timescale required to get independent reads of the current) or very large
($\tau\approx T$). In the first case, the dependence is due to the
fluctuations in the mean being correlated and, in the later case, the estimate
of the standard deviation having too few data points. However, the BSE is
fairly constant over a broad range of $\tau$ in between, which is the value we
used to estimate errors. The error bars in the plots are $\pm 1$ BSE unless
otherwise indicated.

###  I.2 Pore radius and area

Figure S-1:  Membrane and pore structure. The images show a small section of the graphene membrane (2.1 nm by 2.1 nm) containing the pore. (a-d) The top panels show the nominal radii, $r_{n}=$ 0.38 nm, 0.51 nm, 0.62 nm, and 0.71 nm. (e-h) The bottom panels show the corresponding effective radii, $r_{p}=$ 0.21 nm, 0.34 nm, 0.45 nm, and 0.54 nm. $r_{n}$ (nm) |  0.38 |  0.51 |  0.62 |  0.71 |  0.86 |  0.93 |  1.11 |  1.24 |  1.36 |  1.48
---|---|---|---|---|---|---|---|---|---|---
$r_{p}$ (nm) |  0.21 |  0.34 |  0.45 |  0.54 |  0.69 |  0.76 |  0.94 |  1.07 |  1.19 |  1.31
Table S-1:  Radii for various pores. Here, $r_{n}$ is the nominal radius of
the pore (it defines the construction of the pore for the simulations) and
$r_{p}=r_{n}-\sigma_{C}$ is the effective radius, where $\sigma_{c}$ is the
vdW radius of carbon. The quantity $r_{p}=r_{n}-\sigma_{C}$ also gives the
radius that would be observed in experiments due to electron density around
the carbon atoms and bonds.

As noted above, we open pores by removing carbon atoms with coordinates
satisfying $x^{2}+y^{2}<r_{n}^{2}$. There is a range of $r_{n}$ that give the
same pore size due to the discrete nature of the membrane. We choose $r_{n}$
to be the maximum of this range. This $r_{n}$ also gives the distance of the
carbon atoms at the pore edge to the center of the pore, see Fig. S-1 (in
other words, the radius of the largest circle that will fit into the pore).

When comparing to experimental results, however, these nominal radii may or
may not correspond to the values reported. For instance, Ref. OHern2014,
defines the pore area by where electron density is not observed in
transmission electron microscopy (TEM) images. This will roughly correspond to
$r_{n}-\sigma_{C}$ where $\sigma_{C}\approx 0.17$ nm is the van der Waals
(vdW) radius of carbon. This is approximately where the electron density
vanishes. Moreover, the actual area available for transport is smaller than
nominal area due to hydration and the finite size of atoms (note, however,
that the flexibility of the pore edge will tend to slightly increase the area
available). These factors have negligible effect in larger pores but are
significant for subnanoscale pores. We found that in general the maximum
radial spread of the ion inside the pore is given by: $\rho_{max}\approx
r_{n}-\sigma_{C}$ (see Fig. S-2). Thus, we define the effective pore radius as
$r_{p}=r_{n}-\sigma_{C}$ and effective area of the pore as $A_{p}=\pi
r_{p}^{2}$. This definition should roughly correspond to experimentally
observed values. We note that to find the actual accessible area for
transport, one should also account for the spatial dependence of the free
energy. Table S-5 reports the nominal and effective radii.

Figure S-2:  Transport area and current definition. (a) The maximum (cutoff of
99 %) radial spread ($\rho^{2}=x^{2}+y^{2}$) of ions inside the pore is
roughly equal to $r_{p}$ nm. That is, when looking at the integrated density
of translocation events from 0 to $\rho$, 99 % of the events fall between 0
and $\rho_{max}$. (b) Current calculated from the two different definitions
$I_{p}$ (solid line) and $I_{z}$ (dashed line). Connecting lines are shown as
a guide to the eye.

###  I.3 Current definition and electric field

Figure S-3:  Voltage drop. (a) Potential drop along the $z$-axis for various
pore sizes and (b) a map of the potential for a pore of radius 0.21 nm. For
larger pores, most of the potential drop (about 80% shown by the two dotted
horizontal lines) occurs within a distance $r_{p}$ from the center of pore.
The electric field can thus be approximated by $E\approx 0.8\,V/2r_{p}$.
However, for small pores, the entire potential drop occurs over 1 nm due to
vdW repulsion of the ions by the graphene membrane. In Fig. 2 of the main
text, we use the approximate electric field for larger pores in the expression
$ez_{\nu}n_{\nu}\mu_{\nu}A_{p}E$ for the whole range of values reported. This
overestimates the field for smaller pores, but will approach the right value
as the pore size increases. Note, as well, that $\mathrm{K}^{+}$ tends to come
closer to the graphene membrane due to its smaller vdW repulsion. This results
in the potential drop occurring mostly on the anion side and, when the voltage
is increased enough to substantially polarize water, in the effective barrier
for $\mathrm{K}^{+}$ decreasing before that for $\mathrm{Cl}^{-}$. Connecting
lines are shown as a guide to the eye. The potential maps were produced using
the method described in Ref. 8.

The ionic current was calculated using two definitions:

$I_{z}(t)=\frac{1}{\Delta tL_{z}}\sum_{i=1}^{N}q_{i}[z_{i}(t+\Delta
t)-z_{i}(t)],$ (3)

and

$I_{p}(t)=\frac{1}{\Delta t}\sum_{i=1}^{N}q_{i}(\Theta[Z_{i}(t+\Delta
t)]-\Theta[Z_{i}(t)]),$ (4)

where $\Theta$ is the Heaviside step function and $\Delta t=1$ ps is the
measurement time (we record the atomic configuration every $\Delta t$
increment). The first definition takes into account the motion of all ions in
the $z$-direction and the second definition counts ions crossing the pore.
These definitions give the same value for the current so long as the
simulation is converged with respect to the total simulation time (see Fig.
S-2(b)).

We found that about 80 % of the potential drops within a sphere of radius
$r_{p}$ from the center of the pore for larger pores (see Fig. S-3). Thus, the
electric field inside the pore can be estimated as $E\approx 0.8\,V/2r_{p}$.
For the smallest pore, however, the electric field is more accurately
determined by $E\approx V/L$ with $L=1$ nm. This is because vdW repulsion of
the ions by the carbon prevents the charge layers from getting closer than
about 1 nm.

We tested the effect of box size on the current by comparing different box
sizes, a large box (with fixed cross-sectional area 7.4 nm $\times$ 7.4 nm and
relaxed height 6.9 nm), a small box (with fixed cross-sectional area 3.7 nm
$\times$ 3.7 nm and relaxed height 3.4 nm), and a extended small box (with
fixed cross-sectional area 3.7 nm $\times$ 3.7 nm and relaxed height 6.9 nm).
The current and selectivity are in agreement (to within the errors reported)
for all box sizes when the pore is small and at low voltages, as the current
is dominated by the high pore resistance. The latter allows well-defined
charge layers to develop and persist. However, for larger pores, the current
was smaller by about 20 % for the larger box size. Since both box sizes have
the same voltage and same concentration of ions, the difference in current is
likely due to a smaller access resistance and lack of well-defined charge
layers in the boxes with smaller cross-sectional area, as the pore diameter
approaches the edge of the box. For the smallest pore size calculations, we
use the extended small box, as it considerably reduces errors due to
convergence in time, i.e., we can run microsecond long simulations, which are
necessary when the currents are so small.

Figure S-4:  Estimates of the free energy barrier and noise for various pores.
Sharp rise in (a) the free energy barrier and (b) the noise in the current at
the subnanoscale. An “average” free energy barrier is estimated as $\Delta
F_{\nu}=k_{B}T\,\log\left(\frac{q_{\nu}nA_{p}\mu_{\nu}E}{I_{\nu}}\right)$,
where $q_{\nu}$, $\mu_{\nu}$, $n$ are the charge, average mobility (of
$\mathrm{K}^{+}$ and $\mathrm{Cl}^{-}$), and particle concentration. The error
in $\Delta F_{\nu}$ is thus $k_{B}T\delta I_{\nu}/I_{\nu}$, where $\delta
I_{\nu}$ is BSE in $I_{\nu}$. The noise in current, $\Delta I_{\nu}(\tau)$ is
measured by standard deviation of the mean current within blocks of length
$\tau=10$ ns. The relative noise in the current, $\Delta
I_{\nu}(\tau)/I_{\nu}$ increases sharply due to the fluctuations in hydration
layer configurations that allow or prohibit passage of ions through the pore
zwolak10-1 . The errors in $\Delta I_{\nu}$ is estimated as $\frac{\Delta
I_{\nu}}{\sqrt{2(N_{b}-1)}}$ ahn2003 . Connecting lines are shown as a guide
to the eye.

##  II Current behavior at the subnanoscale

The ionic current begins to show abnormal behavior as the radius of the pore
decreases to the sub-nanometer scale. This behavior is seen in the sharp rise
in the pore resistance and noise in the current, Fig. S-4. At these length
scales, the pore edge begins to deform the hydration layers around the ions,
which increases the energy barrier for ions to cross the pore.

##  III Selectivity

###  III.1 Experimental observation of selectivity

$A_{p}$ (nm2) |  0.2 |  0.123 |  0.108 |  0.321 |  0.368
---|---|---|---|---|---
$r_{p}$ (nm) |  0.25 |  0.20 |  0.19 |  0.32 |  0.34
$S_{p}$ |  S |  S |  S |  1 |  1
Table S-2: Current-carrying pores from the selectivity measurement of Ref. 7. We assign a selectivity $S$ for all pores with $r_{p}\approx 0.2$ nm and 1 for larger pores. There are also many pores with smaller radii, but we expect these pores to carry negligible current (and thus, even though they will be selective, they do not contribute to the observed selectivity). We do not include them in this table. Salt (XCl) |  CsCl |  RbCl |  KCl |  NaCl |  LiCl
---|---|---|---|---|---
$\sigma$ (pS) |  67 |  70 |  64 |  42 |  27
$\kappa$ (10-3 S m-1) |  1.42 |  1.42 |  1.3 |  1.19 |  0.95
$E_{\mathrm{X}}$ (eV) |  3.1 |  3.4 |  3.7 |  4.6 |  5.7
$\mu_{\mathrm{X}}$ (10-8 m2V-1s-1) |  8.01 |  8.06 |  7.62 |  5.19 |  4.01
$\mu_{\mathrm{eff}}$ (10-8 m2V-1s-1) |  4.25 |  3.59 |  4.29 |  1.63 |  1.03
$\sigma_{\mathrm{XCl}}$ (pS) |  67 |  61-67 |  65-67 |  44-55 |  39-50
$\sigma_{\mathrm{X}}$ (pS) |  67 |  57-66 |  64-68 |  26-43 |  16-34
Table S-3:  Leakage conductance ($\sigma$), bulk conductivity of cations and anions together ($\kappa$), and hydration energy ($E_{\mathrm{X}}$) of the cation $\mathrm{X}$, as reported by Ref. 11. Note that the reported bulk conductivity is four orders of magnitude smaller than that in their nanopore current measurements. We take the cation mobilities ($\mu_{\mathrm{X}}$) from bulk hille2001  and effective cation mobilities ($\mu_{\mathrm{eff}}$) in a biological pore from Ref. 13. The bulk mobility of $\mathrm{Cl}^{-}$ ion is $7.92\times 10^{-8}$ m-2 V-1 s-1 and its effective pore mobility (from Ref. 13) is $3.44\times 10^{-8}$ m-2 V-1 s-1. We make two estimates of the range of the conductance: One for $\sigma_{\mathrm{XCl}}$ (assuming both the cation and anion contribute) and the other for $\sigma_{\mathrm{X}}$ (assuming only the cation contributes). The range in each case is set by the bulk mobility and the effective pore mobility. We note that even though graphene is atomically thick (and effective mobilities are not well defined), the defect channel structure is not known – those channels could be long channels through gaps in the device. We use the effective mobilities of Ref. 13 for a biological pore only as a very rough estimate. These ranges show that the deviation of the relative conductance from that predicted by bulk mobilities can easily be due to cation-only conductance (e.g., due to local charge-based selectivity) and/or effective mobilities through the defect channels responsible for the leakage conductance. Note that the conductance of CsCl and Cs only in the last two lines are the same. This is because they are assuming two different hypotheses about the origin of the conductance and we take them both to be the experimentally determined conductance. Also, for clarity, the estimated range is color coded with blue indicating the estimate using the bulk mobility and red the estimate from the effective mobility. cation(X) |  K+ |  Li+ |  Ba2+ |  Ca2+ |  Mg2+
---|---|---|---|---|---
$\mu_{\mathrm{X}}$ (10-8 m2V-1s-1) |  7.62 |  4.01 |  6.60 |  6.17 |  5.50
| device 3, $r_{p}=(0.36\pm 0.10)$ nm
$\sigma$ (nS) |  2.3 $\pm$ 1.2 |  1.1$\pm$ 1.1 |  1.0$\pm$0.8 |  0.7$\pm$ 0.5 |  0.4$\pm$ 0.03
$S_{\mathrm{XCl/KCl}}$ |  1 |  0.00 - 2.61 |  0.06 - 1.75 |  0.06 - 1.25 |  0.11 - 0.41
$S_{\mathrm{X/K}}$ |  1 |  0.00 - 3.80 |  0.07 - 1.89 |  0.07 - 1.40 |  0.13 - 0.49
| device 4, $r_{p}=(0.50\pm 0.10)$ nm
$\sigma$ (nS) |  4.2 $\pm$ 0.3 |  2.2$\pm$ 1.3 |  1.4 $\pm$0 |  1.5$\pm$0.1 |  1.3$\pm$ 0.1
$S_{\mathrm{XCl/KCl}}$ |  1 |  0.25 - 1.17 |  0.34 - 0.40 |  0.34 - 0.45 |  0.31 - 0.39
$S_{\mathrm{X/K}}$ |  1 |  0.37 - 1.71 |  0.37 - 0.43 |  0.38 - 0.51 |  0.37 - 0.46
| device 8, $r_{p}=(0.39\pm 0.06)$ nm
$\sigma$ (nS) |  2.6$\pm$1.5 |  1.3$\pm$0.1 |  2.3$\pm$1 |  2$\pm$1 |  1$\pm$1
$S_{\mathrm{XCl/KCl}}$ |  1 |  0.39 - 1.65 |  0.32 - 3.21 |  0.28 - 3.01 |  0.00 - 2.00
$S_{\mathrm{X/K}}$ |  1 |  0.57 - 2.42 |  0.35 - 3.46 |  0.31 - 3.37 |  0.00 - 2.39
Table S-4:  Chloride salt conductance ($\sigma$) of three different devices
holding the chloride concentration constant at 100 mM Jain2015 . The quantity
$S_{\mathrm{XCl/KCl}}$ is the selectivity quantified by assuming both cation
and anion contribute, Eq. 12, and $S_{\mathrm{X/K}}$ by assuming only cations
contribute, Eq. 13. The selectivity is shown as a range based on the error in
$\sigma$ (the actual range – the range of the data measured in Ref. 14 – is
larger than shown here). Only bivalent ions in device 4 and Mg2+ in device 3
potentially show selective behavior. However, selectivity of this magnitude
was observed in Ref. 15 for large pores, where dehydration can not be playing
a role.

Ref. OHern2014,  found that graphene membranes with a distribution of pore
sizes (at the subnanoscale) display selective behavior for K+ over Cl-. In
that work, this is indicated by a nonzero membrane potential $E_{m}$. We use
the relation

$E_{m}=\frac{k_{B}T}{e}\ln\left(\frac{P_{\mathrm{K}}[\mathrm{K}]_{o}+P_{\mathrm{Cl}}[\mathrm{Cl}]_{i}}{P_{\mathrm{K}}[\mathrm{K}]_{i}+P_{\mathrm{Cl}}[\mathrm{Cl}]_{o}}\right)$
(5)

or, rearranging,

$\quad
e^{eE_{m}/k_{B}T}=\frac{\frac{P_{\mathrm{K}}}{P_{\mathrm{Cl}}}[\mathrm{K}]_{o}+[\mathrm{Cl}]_{i}}{\frac{P_{\mathrm{K}}}{P_{\mathrm{Cl}}}[\mathrm{K}]_{i}+[\mathrm{Cl}]_{o}}$
(6)

to estimate the selectivity from the reported membrane potential ($E_{m}=3.3$
mV $\pm 1$ mV, where we keep the second digit to not introduce rounding error,
$[\mathrm{K}]_{i}=[\mathrm{Cl}]_{i}=0.17$ mol/L,
$[\mathrm{K}]_{o}=[\mathrm{Cl}]_{o}=0.5$ mol/L , $T=297$ K, and $e$ is the
magnitude of the electron charge). We obtain
$\tilde{S}=P_{\mathrm{K}}/P_{\mathrm{Cl}}=1.3\pm 0.1$ as an “average”
selectivity for their distribution of pore sizes. The ratio
$P_{\mathrm{K}}/P_{\mathrm{Cl}}$ is the concentration-imbalance equivalent of
$I_{\mathrm{K}}/I_{\mathrm{Cl}}$.

To compare with our numbers, we need to extract the selectivity for particular
pore sizes or to use the MD results to compute the average for the
distribution of pore sizes in experiment. We will do both. We first note that
the average selectivity from experiment is the same as the selectivity we find
for $r_{p}=0.34$ nm (see Fig. 2 in the main text and Table S-7). Although the
experimental selectivity is for the distribution of pore radii from
$r_{p}\approx 0.1$ nm to $r_{p}\approx 0.3$ nm, it is likely that most of the
current and hence the average selectivity is dominated by pores with
$r_{p}\approx 0.3$ nm. In fact, we find that current for $r_{p}=0.34$ nm is an
order of magnitude larger than that for $r_{p}=0.2$ nm, supporting that the
slightly larger pore may be dominating the average.

To go further, though, we will first use a rough estimate to extract the
selectivity for the $r_{p}\approx 0.2$ nm from experiment and then separately
show that, when using the areas and free energies from our MD calculations, we
get an average selectivity similar to experiment. Both of these calculations
confirm that dehydration-only selectivity yields results in agreement with
experiment.

For a membrane with a distribution of pore sizes, the observed selectivity
$\tilde{S}$ can be roughly estimated as

$\tilde{S}=\frac{\sum_{p}I_{pK}}{\sum_{p}I_{pCl}}=\frac{\sum_{p}S_{p}I_{pCl}}{\sum_{p}I_{pCl}}\approx\frac{\sum_{p}S_{p}A_{p}}{\sum_{p}A_{p}}$
(7)

or

$\tilde{S}=\frac{\sum_{p}I_{pK}}{\sum_{p}I_{pCl}}=\frac{\sum_{p}I_{pK}}{\sum_{p}(1/S_{p})I_{pK}}\approx\frac{\sum_{p}A_{p}}{\sum_{p}(1/S_{p})A_{p}},$
(8)

where $S_{p}$ ($A_{p}$) is the selectivity (area) for a pore of radius $r_{p}$
and the sum over $p$ goes over individual pores. The approximate expressions
in each equation assume that the current is proportional to area, but without
a free energy barrier (see below for the average computed with free energy
barriers). This is a strong approximation. It requires, at the least, that the
smallest pore sizes (i.e., with radii of about 0.15 nm and below) to be
dropped from the sum (as their current contribution is negligible and not
necessarily proportional to area). The sum is thus over the data in Table S-2,
which is from the selective membrane (5 min etch time) of Ref. OHern2014, .
This gives $S=1.8\pm 0.3$ from Eq. (7) and $S=2.5\pm 1$ from Eq. (8), where we
again keep the second digit to not introduce rounding error. The difference in
the selectivity estimated from the two equations is due to the fact that they
did not take into account the free energy barrier at smaller radii. The error
is calculated based on the error in $\tilde{S}$ only, as the error in area is
not reported in Ref. OHern2014, . However, the error due to area is expected
to be much smaller than the approximations in the equations. In the main text,
we report the range $1.8$ to $2.5$ for the selectivity of the $r_{p}\approx
0.2$ nm pores.

A more accurate calculation requires the value of free energy barrier and
selectivity for each of the pores in the distribution. We can, however,
estimate the average selectivity for their pore size distribution (Table S-2)
using the free-energy barriers in Fig S-4,

$\tilde{S}_{\mathrm{estimate}}=\frac{\sum_{p}I_{pK}}{\sum_{p}I_{pCl}}=\frac{\sum_{p}\mu_{\mathrm{K}}\,A_{p}e^{-\Delta
F_{\mathrm{K}}/K_{B}T}_{p}}{\sum_{p}\mu_{\mathrm{Cl}}\,A_{p}e^{-\Delta
F_{\mathrm{Cl}}/K_{B}T}_{p}}\approx 1.5,$ (9)

which is very close to the $\approx 1.3$ result from experiment. Here, we
again dropped the smallest pores (ones smaller than $r_{p}\approx 0.2$ nm)
from the average, as the free energy barrier will be more substantial and
suppress their contribution to the selectivity.

We note a few important limitations of this comparison between the experiments
and our calculations. The lowest voltage in our simulations is still order of
magnitude larger than the equivalent chemical potential difference in the
experiments. The regime below 0.25 V is very difficult to reach using all-atom
MD simulation for these pore sizes. Moreover, current experimental techniques
cannot control the functionalization of graphene nanopores (and
functionalization/surface species of the graphene membrane) which depends on
the fabrication method and other factors. Since it is not clear what groups
will be present and where, we choose the simplest pore – the one with no
functionalization. As well, some functionalization will introduce
charges/dipoles to the pore region. However, if these are close (either on the
pore rim or nearby), they will not give rise to the weak selectivity observed,
but rather a strong selectivity, unless the charge is very small in magnitude.

Garaj et al. garaj2010  report that the leakage current (the current through a
graphene membrane when the pores have yet to be constructed) deviates from
what the bulk conductivity would predict, which they conjecture may be due to
dehydration. Taking the experimental conductance of CsCl to be the reference
value, we can estimate the conductance of other salts XCl as

$\sigma_{\mathrm{XCl}}=\frac{\sigma_{\mathrm{CsCl}}}{(\mu_{\mathrm{Cs}}+\mu_{\mathrm{Cl}})}(\mu_{\mathrm{X}}+\mu_{\mathrm{Cl}}).$
(10)

As shown in Table S-3, the difference in effective mobilities of ions inside
of pores explains some of the deviation. In fact, since the leakage
conductance varies widely from membrane to membrane (by a factor of two
garaj2010 ), different effective mobilities alone may explain the deviation to
within experimental uncertainties.

An alternative (or complementary) explanation is that the defect channels –
the structure of which is unknown – are cation selective due to the presence
of negative (partial) charges. In this case, the estimated conductance is

$\sigma_{\mathrm{X}}=\frac{\sigma_{\mathrm{Cs}}}{\mu_{\mathrm{Cs}}}\mu_{\mathrm{X}}.$
(11)

This estimate accurately explains the observations by itself. Relying on
dehydration requires a convoluted explanation – or minuscule dehydration,
e.g., a 1/100th fractional removal of water – to account for the differences
between K+, Na+, and Li+ when comparing to Cs+. For instance, Na+ and Li+ have
a much larger hydration energies than Cs+, Rb+, or K+, and thus one expects
that, if dehydration is a factor, the membrane conductance would go from
cation and anion both contributing (for Cs+, Rb+, and K+) to just anion
contributing (for Na+ and Li+). In other words, NaCl and LiCl would have the
same conductance, but they differ by 50 %.

We also note that ion transport for different cations was measured in Ref.
Jain2015, , where the authors claim to have observed hydration-based
selectivity. However, their selectivity can be explained based on difference
in cation mobilities. Selectivity with respect to KCl can be captured via the
normalized conductance

$S_{\mathrm{XCl/KCl}}=\frac{\sigma_{\mathrm{XCl}}/(\mu_{\mathrm{X}}+\mu_{\mathrm{Cl}})}{\sigma_{\mathrm{KCl}}/(\mu_{\mathrm{K}}+\mu_{\mathrm{Cl}})}.$
(12)

As shown in Table S-4, the selectivity has a large range due to a large
variation of the conductance. This makes it difficult to determine if there is
any selectivity at all. Only the bivalent ions in device 4 and Mg2+ in device
3 have the entire range of $S_{\mathrm{XCl/KCl}}$ less than 1. Moreover, the
experimental results seem to indicate the presence of negative charges near
the pore (see Fig. 3d in Ref. Jain2015, ), which would mean that most of the
current is carried by the cations. Thus, selectivity should be quantified as

$S_{\mathrm{X/K}}=\frac{\sigma_{\mathrm{XCl}}/\mu_{\mathrm{X}}}{\sigma_{\mathrm{KCl}}/\mu_{\mathrm{K}}}.$
(13)

This shows that (12) overestimates the deviation from non-selective behavior.
As we mention in the main text, even if one ignores the large variation in the
measured conductance in individual devices (and from device to device), the
selectivity reported is consistent with the charge-based selectivity observed
in Ref. rollings2016, . Those latter results were in pores much bigger than
hydrated ions, indicating that the selective behavior is likely due to
differences in how bivalent versus monovalent ions screen the charged
pore/membrane. We finally note that the difference in hydration energy of
monovalent and divalent ions is around 10 eV. It seems unlikely that this
would give selectivity on the order of a factor of 2.

Figure S-5:  Water density quantified by the oxygen density. (a) The contour
plots show either a potassium or a chloride ion fixed at the center of the
pore. The pore radius is shown in the upper left corner of each panel. The
hydration layers are visible as the high-density region around the ion. As the
pore size gets smaller, the hydration layers around the ion are distorted by
the pore edge. The ions remain fairly well hydrated. (b) The radial
distribution function of oxygen atoms around $\mathrm{K}^{+}$ and
$\mathrm{Cl}^{-}$ in bulk and at the center of a $r_{p}=0.21$ nm pore (with a
counter ion fixed at the edge of the simulation cell) for zero bias showing
partial dehydration. The number of water molecules in first hydration layer of
$\mathrm{K}^{+}$ and $\mathrm{Cl}^{-}$ are 6.8 and 7.4 in bulk and 5.2 and 5.8
at the center of pore, respectively, i.e., a loss of about 1/4 of the water
molecules from first hydration layer. Connecting lines are shown as a guide to
the eye.

###  III.2 Dehydration and selectivity

Figure S-5 shows the water density around the ion in the pore and the radial
distribution function of oxygen, $g_{O}$. In our simulation, the number of
oxygen atoms in first hydration layer around $\mathrm{K}^{+}$ and
$\mathrm{Cl}^{-}$ in bulk water are 7.4 and 6.8, respectively, and, when
placed in the center of a $r_{p}=0.21$ nm pore, 5.8 and 5.2. This is loss of
about 1/4 of the water molecules from the innermost layer. Even though the
fractional dehydration for $\mathrm{K}^{+}$ and $\mathrm{Cl}^{-}$ are nearly
equal, the energy penalty is higher for Cl- as its inner hydration layer is
more strongly bound compare to K+. As we increase the applied voltage, both
ions were able to remain more hydrated while crossing the pore due to
polarization-induced chaperoning of the ions. This results in a lower free
energy barrier for transport, as seen from Fig. S-6.

##  IV Model for ion transport

The model for ion transport is discussed in detail in the main text. We use
the method of least squares, with the first data voltage point constrained to
reflect behavior at lower voltage, to fit the data. We only present data up to
1.5 V in the main text as water starts to dissociate at high fields. However,
all-atom MD simulations allow us to apply much higher voltages without
dissociating water. We thus looked at the IV characteristics up to 3 V to
check consistency with our model, which fits well up to that voltage when
accounting for a change in free energy barrier for $\mathrm{Cl}^{-}$, as seen
in Fig. S-6.

Figure S-6:  Model for ion transport. (a) Current-voltage characteristics for
a pore of radius 0.21 nm and 1 mol/L KCl versus voltage (top panels). The
relative selectivity of $\mathrm{K}^{+}$ over $\mathrm{Cl}^{-}$ in the same
pore (bottom panel). Data points give the MD results and the solid line gives
the piece-wise linear model we fitted taking each region to be linearly
related to the differential conductance $g_{\nu}=ez_{\nu}n_{\nu}e^{(-\Delta
F_{\nu}/k_{B}T)}\mu_{\nu}A_{p}/L=g_{0\nu}e^{(-\Delta F_{\nu}/k_{B}T)}$. Around
$(1.1\pm 0.1)$ V the energy barrier for potassium drops from $(0.09\pm 0.004)$
eV to $(0.06\pm 0.002)$ eV, whereas that of chloride drops from $(0.12\pm
0.001)$ eV to $(0.08\pm 0.008)$ eV around $(2.38\pm 0.04)$ V. This change in
conductance with voltage results in a rise and fall of selectivity. (b) The
distance ($|\vec{r}_{C}-\vec{r}_{\nu}|$) between an ion $\nu$ (or an
oxygen/hydrogen from water) crossing the pore and the nearest carbon atom
versus voltage. At higher voltage ions are able to apparently enter the
repulsive zone ($r<\sigma$) of vdW potential of carbon atom, likely due to
larger forces that can take advantage of the flexibility of the pore rim. This
effectively increases the area of transport. This is partly responsible for
the increased current at higher voltages. However, the main cause of the
latter is the chaperoning of ions across the pore by polarized water, which
happens at lower voltage for K+ compare to Cl- due to the charge layer of the
former being closer to the graphene membrane, see Fig. S-3(a).

Figure S-7:  Selectivity and pore resistance for three different
concentrations. (a) Selectivity and (b) pore resistance for 0.1 mol/L, 0.5
mol/L, and 1.0 mol/L KCl. The solid line is the fit of resistance of the form,
$R=\frac{a}{r_{p}}+\frac{b}{r_{p}^{2}}$ Hall1975 ; Schneider2010 . Here, the
1/$r_{p}$ term is due to access resistance and the 1/$r_{p}^{2}$ term is due
to the pore resistance. Note that this deviation from normal behavior is quite
large as the axis is on a logarithmic scale. The persistence of selectivity
and the sharp rise in pore resistance at lower concentrations confirms that
the behavior is not due to ion-ion interactions or some other many-body
effect. Connecting lines are shown as a guide to the eye.

##  V The effect of the ion concentration

In order to confirm that the anomalous behavior of current and weak
selectivity is not due to a many-body effect but rather to single ion
behavior, we repeat the selectivity calculations for lower concentrations of
KCl. Both the anomalous behavior of current and weak selectivity are still
present in concentrations as low as 0.1 mol/L, at which point there were only
a few ions of each type in the smallest simulation box size (hence, box size
errors start to become significant).

##  VI Tables

$r_{p}$ (nm) |  0.21 |  0.34 |  0.54 |  0.76 |  0.94 |  1.19 |  1.39 |  1.83 |  2.36
---|---|---|---|---|---|---|---|---|---
$l$ (nm) |  3.7 |  3.7 |  3.7 |  3.7 |  3.7 |  7.4 |  7.4 |  7.4 |  7.4
$I_{\mathrm{K}}$ (nA) |  0.032 |  0.23 |  1.13 |  2.40 |  3.57 |  5.7 |  8.1 |  13.6 |  21.8
$I_{\mathrm{Cl}}$ (nA) |  0.009 |  0.17 |  1.13 |  2.32 |  3.61 |  5.5 |  8.0 |  13.9 |  20.8
$I_{\mathrm{K}}/I_{\mathrm{Cl}}$ |  3.7 |  1.3 |  1.0 |  1.0 |  1.0 |  1.0 |  1.0 |  1.0 |  1.0
Table S-5:  Current for various radii. The table shows the current for 1.0 mol/L KCl solution for the box of height $h=6.9$ nm for various radii. The edge of the box is $l=b=7.4$ nm for larger pores ($r_{p}>1$ nm) and $l=b=3.7$ nm for smaller pores. The latter allows for much longer simulations, which are needed to achieve convergence. The error in the current is shown in Fig. 2 in the main text. The block standard error determines the number of significant digits in this and the following tables. $V$ (V) |  0.25 |  0.5 |  0.75 |  1.0 |  1.25 |  1.5 |  1.75 |  2.0 |  2.25 |  2.5 |  2.75 |  3.0
---|---|---|---|---|---|---|---|---|---|---|---|---
$I_{\mathrm{K}}$ (pA) |  2.3 |  7.0 |  15 |  32 |  52 |  87 |  113 |  153 |  167 |  197 |  210 |  234
$I_{\mathrm{Cl}}$ (pA) |  1.0 |  2.2 |  4.5 |  9 |  12 |  15 |  16 |  17 |  21 |  29 |  41 |  55
$I_{\mathrm{K}}/I_{\mathrm{Cl}}$ |  2.3 |  3.0 |  3.2 |  3.7 |  4.4 |  5.6 |  7.1 |  9.1 |  8.0 |  6.8 |  5.1 |  4.2
Table S-6:  Current for various voltages. The table shows the current for 1.0 mol/L KCl solution through the smallest pore $r_{p}=0.21$ nm for various voltages. For these calculations, we use the box height $h=6.9$ nm and the smaller cross section, $l=b=3.7$ nm. The error in the current is shown in Fig. S-6 | pore | bulk
---|---|---
|  $\langle p_{r}\rangle$ (D) |  $\langle n\rangle$ |  $\langle p_{r}\rangle$ (D) |  $\langle n\rangle$
K+ |  2.02 |  5.2 |  1.60 |  6.8
Cl- |  1.81 |  5.8 |  1.54 |  7.4
Table S-7:  Dipole orientation. The average radial component of individual water dipole $\langle p_{r}\rangle$ and average number of water dipoles $\langle n\rangle$ in the first hydration layer of K+ an Cl- ions in the smallest pore ($r_{p}=0.21$ nm) and in the bulk. The dipole moment of water in our model is 2.35 in units of Debye (0.021 e nm). The dipoles are strongly oriented in the pore compare to bulk but have fewer dipoles and hence there is a decrease in total dipole moment. Atoms ($\mathrm{X}$) |  K |  Cl |  H |  O |  C
---|---|---|---|---|---
$\epsilon_{\mathrm{X}}$ (meV) |  3.773 |  6.505 |  1.89 |  6.596 |  3.036
$r_{\mathrm{X}}$ (nm) |  0.176 |  0.227 |  0.022 |  0.177 |  0.199
Table S-8:  Lennard-Jones parameters for individual elements. The vdW
potential between two atoms at a distance $d$, is calculated using the
Lennard-Jones relation,
$V_{\mathrm{LJ}}=\epsilon_{m}\left[\left(\frac{r_{m}}{d}\right)^{12}-2\left(\frac{r_{m}}{d}\right)^{6}\right]$,
where the parameter $r_{m}=r_{1}+r_{2}$ is the equilibrium distance and
$\epsilon_{m}=\sqrt{\epsilon_{1}\epsilon_{2}}$ is the well depth of the
interaction.

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$(\mathrm{id}\otimes\vartheta|_{\mathrm{C}_{0}(\mathbb{G})^{\overline{\Psi}\boxtimes\dot{\Psi}}})\circ\Delta^{\Psi}$,
$\vartheta\in(\mathrm{C}_{0}(\mathbb{G})\rtimes_{\rho}\Gamma^{2})^{*}$. By the
definition of $\Delta^{\Psi}$ in Theorem 3.3, it suffices to show that
$\tilde{S}$ commutes with the operators
$(\mathrm{id}\otimes\vartheta)\circ\operatorname{Ad}((\lambda^{R}\otimes\lambda^{L})(\Upsilon))\circ\tilde{\Delta}$,
$\vartheta\in(\mathrm{C}_{0}(\mathbb{G})\rtimes_{\rho}\Gamma^{2})^{*}$ (see
Observation 1.3). This amounts to the equality

(4.10)
$(\tilde{S}\otimes\mathrm{id})\circ\operatorname{Ad}((\lambda^{R}\otimes\lambda^{L})(\Upsilon))\circ\tilde{\Delta}=\operatorname{Ad}((\lambda^{R}\otimes\lambda^{L})(\Upsilon))\circ\tilde{\Delta}\circ\tilde{S}$

(which makes sense because $\tilde{S}$, unlike $S^{\Psi}$, is already known to
be non-degenerate).

The unitary $(\lambda^{R}\otimes\lambda^{L})(\Upsilon)$ discussed in (3.8)
satisfies
$(\tilde{S}\otimes\mathrm{id})\big{(}(\lambda^{R}\otimes\lambda^{L})(\Upsilon)\big{)}=(\lambda^{R}\otimes\lambda^{L})(\Upsilon)$;
thus it is in the multiplicative domain of $\tilde{S}\otimes\mathrm{id}$, and
it suffices in fact to show that

$(\tilde{S}\otimes\mathrm{id})\circ\tilde{\Delta}=\tilde{\Delta}\circ\tilde{S}.$

Let then
$f\in\mathrm{C}_{c}(\Gamma^{2},\mathrm{C}_{0}(\mathbb{G}))\subseteq\mathrm{C}_{0}(\mathbb{G})\rtimes_{\rho}\Gamma^{2}$
and compute on one hand

$\displaystyle(\tilde{\Delta}\circ\tilde{S})(f)$
$\displaystyle=\int_{(\gamma_{1},\gamma_{2})\in\Gamma^{2}}(\iota\otimes\iota)(\Delta(S(f(\gamma_{1},\gamma_{2}))))\cdot(\lambda_{\gamma_{1}}^{L}\otimes\lambda_{\gamma_{2}}^{R})\,\mathrm{d}(\gamma_{1},\gamma_{2})$
$\displaystyle=\int_{(\gamma_{1},\gamma_{2})\in\Gamma^{2}}((\iota\circ
S)\otimes\iota)(\Delta(f(\gamma_{1},\gamma_{2})))\cdot(\lambda_{\gamma_{1}}^{L}\otimes\lambda_{\gamma_{2}}^{R})\,\mathrm{d}(\gamma_{1},\gamma_{2})$

and on the other

$(\tilde{S}\otimes\mathrm{id})\circ\tilde{\Delta}(f)=\int_{(\gamma_{1},\gamma_{2})\in\Gamma^{2}}(\tilde{S}\otimes\mathrm{id})\big{(}(\iota\otimes\iota)(\Delta(f(\gamma_{1},\gamma_{2})))\cdot(\lambda_{\gamma_{1}}^{L}\otimes\lambda_{\gamma_{2}}^{R})\big{)}\,\mathrm{d}(\gamma_{1},\gamma_{2}),$

which are equal.

(2) (2)b. By part (1) applied to $\mathbb{G}^{\Psi}$ in place of $\mathbb{G}$,
we should show that the operator
$\pi^{\Psi}\circ\mathscr{R}_{\mu}^{\Psi}\circ{(\pi^{\Psi})}^{-1}$ on
$\mathrm{C}_{0}(\mathbb{G}^{\Psi})$ commutes with the left-right translation
action
$\Gamma^{2}\overset{}{\curvearrowright}\mathrm{C}_{0}(\mathbb{G}^{\Psi})$,
which by Proposition 3.8 is equal to
$\pi^{\Psi}\circ\rho_{\gamma_{1},\gamma_{2}}^{\overline{\Psi}\boxtimes\dot{\Psi}}\circ{(\pi^{\Psi})}^{-1}$
at $(\gamma_{1},\gamma_{2})\in\Gamma^{2}$. But
$\rho_{\gamma_{1},\gamma_{2}}^{\overline{\Psi}\boxtimes\dot{\Psi}}$ is just
the restriction to
$\mathrm{C}_{0}(\mathbb{G})^{\overline{\Psi}\boxtimes\dot{\Psi}}$ of the
automorphism of $\mathrm{C}_{0}(\mathbb{G})\rtimes_{\rho}\Gamma^{2}$ given by
$\operatorname{Ad}(\lambda_{\gamma_{1},\gamma_{2}})$, i.e. induced by the
automorphism $\rho_{\gamma_{1},\gamma_{2}}$ of $\mathrm{C}_{0}(\mathbb{G})$.
The required commutation is now clear because of the definition of
$\mathscr{R}_{\mu}^{\Psi}$ and the assumption that $\mathscr{R}_{\mu}$
commutes with $\rho_{\gamma_{1},\gamma_{2}}$.

(2) (2)c. This follows from the second (last) assertion in Proposition 3.9. We
leave the technical verification to the reader. ∎

######  Remark 4.11.

The proof of Proposition 4.10 (2) (2)a. would become technically simpler if we
knew in advance that $S^{\Psi}=\mathscr{R}_{\mu}^{\Psi}$ is non-degenerate
(Question 4.7), for then we could say immediately that our goal is to show
that
$(T\otimes\mathrm{id})\circ\Delta_{\mathrm{C}_{0}(\mathbb{G}^{\Psi})}=\Delta_{\mathrm{C}_{0}(\mathbb{G}^{\Psi})}\circ
T$ (see (4.1)), which is equivalent to
$(S^{\Psi}\otimes\mathrm{id})\circ\Delta^{\Psi}=\Delta^{\Psi}\circ S^{\Psi}$,
and for this last equality it suffices to verify (4.10).

######  Remark 4.12.

The condition of Proposition 4.10 (1) holds trivially in the following
particular cases:

  1. (1)

When $\mathbb{G}$ is a locally compact group $G$ and the support of the
measure $\mu$ is contained in the centraliser $Z_{G}(\Gamma)$ of $\Gamma$
inside $G$. (See more on this case in Subsection 4.4 below.)

  2. (2)

When $\Gamma\leq\mathbb{G}$ factors through the centre
$\mathscr{Z}(\mathbb{G})$, hence $\mathbb{G}^{\Psi}=\mathbb{G}$ by Proposition
3.4. In this case $R_{\gamma}=L_{-\gamma}$, thus
$\rho_{\gamma}^{R}=\rho_{-\gamma}^{L}$, for all $\gamma\in\Gamma$, hence
$\mathrm{ad}^{\mathrm{u}}$ is trivial.

Another trivial case is when $\mu$ is ‘supported by $\Gamma$’. In this
situation $\mu^{\Psi}$ too is ‘supported by $\Gamma$’ as the next corollary
shows.

######  Corollary 4.13.

Suppose that $\mathbb{G}$ is co-amenable and write
$\Pi:\mathrm{C}_{0}(\mathbb{G})\to\mathrm{C}_{0}(\Gamma)$ for the strong
quantum homomorphism associated to $\Gamma$ being a closed quantum subgroup of
$\mathbb{G}$. Let $\nu$ be a probability measure on $\Gamma$ and consider the
state $\mu:=\nu\circ\Pi$ of $\mathrm{C}_{0}(\mathbb{G})$. Then $\mu$ satisfies
the condition of Proposition 4.10 (1) and the ‘deformed’ state $\mu^{\Psi}$ of
$\mathrm{C}_{0}(\mathbb{G}^{\Psi})$ is also ‘supported by $\Gamma$’, i.e., it
factors through the strong quantum homomorphism associated to $\Gamma$ being a
closed quantum subgroup of $\mathbb{G}^{\Psi}$.

Remark that as before, we believe that the co-amenability condition is not
required.

###### Proof of Corollary 4.13.

The first assertion is obvious.

Using Proposition 4.10 as well as Proposition 3.5 and its notation we have

(4.11)
$\begin{split}\mu^{\Psi}&=\epsilon_{\mathbb{G}^{\Psi}}\circ\pi^{\Psi}\circ\mathscr{R}_{\mu}^{\Psi}\circ{(\pi^{\Psi})}^{-1}=\epsilon_{\Gamma}\circ\pi_{\Gamma}^{\Psi}\circ\Pi^{\Psi}\circ{(\pi^{\Psi})}^{-1}\circ\pi^{\Psi}\circ\mathscr{R}_{\mu}^{\Psi}\circ{(\pi^{\Psi})}^{-1}\\\
&=\epsilon_{\Gamma}\circ\pi_{\Gamma}^{\Psi}\circ\Pi^{\Psi}\circ\mathscr{R}_{\mu}^{\Psi}\circ{(\pi^{\Psi})}^{-1}=\epsilon_{\Gamma}\circ\pi_{\Gamma}^{\Psi}\circ(\Pi\circ\mathscr{R}_{\mu})^{\Psi}\circ{(\pi^{\Psi})}^{-1}.\end{split}$

On the other hand,

$\Pi\circ\mathscr{R}_{\mu}=(\mu\otimes\Pi)\circ\Delta_{\mathbb{G}}=(\nu\otimes\mathrm{id})\circ(\Pi\otimes\Pi)\circ\Delta_{\mathbb{G}}=(\nu\otimes\mathrm{id})\circ\Delta_{\Gamma}\circ\Pi=\mathscr{R}_{\Gamma,\nu}\circ\Pi,$

where $\mathscr{R}_{\Gamma,\nu}$ is the right convolution operator on
$\mathrm{C}_{0}(\Gamma)$ associated with $\nu$. In conclusion, the state

$\mu^{\Psi}=\epsilon_{\Gamma}\circ\pi_{\Gamma}^{\Psi}\circ(\mathscr{R}_{\Gamma,\nu}\circ\Pi)^{\Psi}\circ{(\pi^{\Psi})}^{-1}=\epsilon_{\Gamma}\circ\pi_{\Gamma}^{\Psi}\circ\mathscr{R}_{\Gamma,\nu}^{\Psi}\circ\Pi^{\Psi}\circ{(\pi^{\Psi})}^{-1}$

indeed factors through
$\pi_{\Gamma}^{\Psi}\circ\Pi^{\Psi}\circ{(\pi^{\Psi})}^{-1}$, as the second
assertion claims. ∎

######  Remark 4.14.

If $\Gamma$ is compact, we can construct states of
$\mathrm{C}_{0}^{\mathrm{u}}(\mathbb{G})$ satisfying the condition of
Proposition 4.10 (1) by averaging: if $\nu$ is an arbitrary state of
$\mathrm{C}_{0}^{\mathrm{u}}(\mathbb{G})$, then
$\mu(x):=\int_{\Gamma}(\nu\circ\mathrm{ad}_{\gamma}^{\mathrm{u}})(x)\,\mathrm{d}\gamma$,
$x\in\mathrm{C}_{0}^{\mathrm{u}}(\mathbb{G})$, defines a state $\mu$ of
$\mathrm{C}_{0}^{\mathrm{u}}(\mathbb{G})$ that is invariant under the adjoint
action. Furthermore, if $\nu$ is faithful, then so is $\mu$.

We now arrive at the first main result of this paper, asserting that under a
mild condition, a convolution semigroup of states on $\mathbb{G}$ can be
canonically deformed into a convolution semigroup of states on the Rieffel
deformation $\mathbb{G}^{\Psi}$.

######  Theorem 4.15.

Suppose that ${(\mu_{t})}_{t\geq 0}$ is a $w^{*}$-continuous convolution
semigroup of states of $\mathrm{C}_{0}^{\mathrm{u}}(\mathbb{G})$ invariant
under the adjoint action
$\Gamma\overset{\mathrm{ad}^{\mathrm{u}}}{\curvearrowright}\mathrm{C}_{0}^{\mathrm{u}}(\mathbb{G})$.
Applying the construction of Proposition 4.5 to the operators in the
$C_{0}$-semigroup ${(\mathscr{R}_{\mu_{t}})}_{t\geq 0}$ of right convolution
operators on $\mathrm{C}_{0}(\mathbb{G})$ yields a $C_{0}$-semigroup
${(\mathscr{R}_{\mu_{t}}^{\Psi})}_{t\geq 0}$ on
$\mathrm{C}_{0}(\mathbb{G})^{\overline{\Psi}\boxtimes\dot{\Psi}}$. The
$C_{0}$-semigroup
${(\pi^{\Psi}\circ\mathscr{R}_{\mu_{t}}^{\Psi}\circ{(\pi^{\Psi})}^{-1})}_{t\geq
0}$ on $\mathrm{C}_{0}(\mathbb{G}^{\Psi})$ is induced by a $w^{*}$-continuous
convolution semigroup of states ${(\mu_{t}^{\Psi})}_{t\geq 0}$ of
$\mathrm{C}_{0}^{\mathrm{u}}(\mathbb{G}^{\Psi})$ (via the correspondence in
Theorem 4.2 (2)). The states ${(\mu_{t}^{\Psi})}_{t\geq 0}$ are invariant
under the adjoint action
$\Gamma\overset{}{\curvearrowright}\mathrm{C}_{0}^{\mathrm{u}}(\mathbb{G}^{\Psi})$.

Repeating the above construction with
$\mathbb{G}^{\Psi},\overline{\Psi},{(\mu_{t}^{\Psi})}_{t\geq 0}$ instead of
$\mathbb{G},\Psi,{(\mu_{t})}_{t\geq 0}$, the new convolution semigroup of
states of
$\mathrm{C}_{0}^{\mathrm{u}}((\mathbb{G}^{\Psi})^{\overline{\Psi}}\cong\mathbb{G})$
equals ${(\mu_{t})}_{t\geq 0}$.

###### Proof.

Proposition 4.10 implies that we are in the situation where we can apply
Proposition 4.5 to each of the operators ${(\mathscr{R}_{\mu_{t}})}_{t\geq
0}$, yielding a family ${(\mathscr{R}_{\mu_{t}}^{\Psi})}_{t\geq 0}$ of
operators on $\mathrm{C}_{0}(\mathbb{G})^{\overline{\Psi}\boxtimes\dot{\Psi}}$
such that each of the operators in
${(\pi^{\Psi}\circ\mathscr{R}_{\mu_{t}}^{\Psi}\circ{(\pi^{\Psi})}^{-1})}_{t\geq
0}$ is the right convolution operator of some state $\mu_{t}^{\Psi}$ of
$\mathrm{C}_{0}^{\mathrm{u}}(\mathbb{G}^{\Psi})$, which is invariant under the
adjoint action
$\Gamma\overset{\mathrm{ad}^{\mathrm{u}}}{\curvearrowright}\mathrm{C}_{0}^{\mathrm{u}}(\mathbb{G}^{\Psi})$.
By the defining formula (4.2), ${(\mathscr{R}_{\mu_{t}}^{\Psi})}_{t\geq 0}$ is
a semigroup:
$\mathscr{R}_{\mu_{0}}^{\Psi}=\mathrm{id}_{\mathrm{C}_{0}(\mathbb{G})^{\overline{\Psi}\boxtimes\dot{\Psi}}}$
and
$\mathscr{R}_{\mu_{s}}^{\Psi}\circ\mathscr{R}_{\mu_{t}}^{\Psi}=\mathscr{R}_{\mu_{s+t}}^{\Psi}$
for all $s,t\geq 0$.

The semigroup
${(\pi^{\Psi}\circ\mathscr{R}_{\mu_{t}}^{\Psi}\circ{(\pi^{\Psi})}^{-1})}_{t\geq
0}$ on $\mathrm{C}_{0}(\mathbb{G}^{\Psi})$ is induced by a convolution
semigroup ${(\mu_{t}^{\Psi})}_{t\geq 0}$ on $\mathbb{G}^{\Psi}$, and we have
to show that ${(\mu_{t}^{\Psi})}_{t\geq 0}$ is $w^{*}$-continuous. This will
be done via a short detour to the von Neumann algebraic setting, where the
topological considerations are simpler. Recall from Subsection 4.1 that each
map
$\mathscr{R}_{\mu}:\mathrm{C}_{0}(\mathbb{G})\to\mathrm{C}_{0}(\mathbb{G})$,
$\mu\in\mathrm{C}_{0}^{\mathrm{u}}(\mathbb{G})^{*}$, is actually a restriction
of a normal map on $L^{\infty}(\mathbb{G})$, and from Subsection 3.2 that the
left and right translation actions
$\Gamma\curvearrowright\mathrm{C}_{0}(\mathbb{G})$ are restrictions of (von
Neumann algebraic) actions on $L^{\infty}(\mathbb{G})$ and hence so is the
left-right translation action
$\Gamma^{2}\overset{\rho}{\curvearrowright}\mathrm{C}_{0}(\mathbb{G})$. We
denote these extensions in the same way. The (extended) operators
${(\mathscr{R}_{\mu_{t}})}_{t\geq 0}$ on $L^{\infty}(\mathbb{G})$ form a
$C_{0}^{*}$-semigroup and commute with the left-right translation action
$\Gamma^{2}\overset{\rho}{\curvearrowright}L^{\infty}(\mathbb{G})$. This
$C_{0}^{*}$-semigroup induces – by elementary von Neumann algebra theory – a
$C_{0}^{*}$-semigroup ${(\overline{\mathscr{R}}_{\mu_{t}})}_{t\geq 0}$ on the
von Neumann algebraic crossed product
$L^{\infty}(\mathbb{G})\rtimes_{\rho}\Gamma^{2}$, which canonically contains
$\mathrm{M}(\mathrm{C}_{0}(\mathbb{G})\rtimes_{\rho}\Gamma^{2})$ and thus also
$\mathrm{C}_{0}(\mathbb{G})^{\overline{\Psi}\boxtimes\dot{\Psi}}$. Evidently,
the maps ${(\overline{\mathscr{R}}_{\mu_{t}})}_{t\geq 0}$ on
$L^{\infty}(\mathbb{G})\rtimes_{\rho}\Gamma^{2}$ extend the maps
${(\mathscr{R}_{\mu_{t}}^{\Psi})}_{t\geq 0}$ on
$\mathrm{C}_{0}(\mathbb{G})^{\overline{\Psi}\boxtimes\dot{\Psi}}$. By Lemma
4.4 (with $\mathbb{G}$ being $\mathbb{G}^{\Psi}$, $A$ being
$\mathrm{C}_{0}(\mathbb{G})^{\overline{\Psi}\boxtimes\dot{\Psi}}$ and $M$
being $L^{\infty}(\mathbb{G})\rtimes_{\rho}\Gamma^{2}$), the convolution
semigroup ${(\mu_{t}^{\Psi})}_{t\geq 0}$ is $w^{*}$-continuous.

Finally the last statement follows immediately from Proposition 4.10 (2) (2)c.
∎

######  Remark 4.16.

A more transparent way to show the $C_{0}$-continuity of
${(\mathscr{R}_{\mu_{t}}^{\Psi})}_{t\geq 0}$ is to move everything to the von
Neumann algebraic setting, especially identifying
$L^{\infty}(\mathbb{G}^{\Psi})$ with a subalgebra of
$L^{\infty}(\mathbb{G})\rtimes_{\rho}\Gamma^{2}$, and then use [SkV1, Lemma
2.3] implication (c)$\implies$(a) in lieu of Lemma 4.4. The realisation of
$L^{\infty}(\mathbb{G}^{\Psi})$ inside
$L^{\infty}(\mathbb{G})\rtimes_{\rho}\Gamma^{2}$ is given in [FiV, Proposition
5], which is the von Neumann analogue of Theorem 3.2.

Our next goal is to discuss the _symmetry_ of the convolution semigroups of
states on $\mathbb{G}^{\Psi}$ constructed in Theorem 4.15. We call a state of
the universal C∗-algebra of a locally compact quantum group _symmetric_ if it
is invariant under the unitary antipode.

For a state $\mu\in\mathrm{C}_{0}^{\mathrm{u}}(\mathbb{G})^{*}$ that is
invariant under the adjoint action of $\Gamma$, we know from Proposition 4.10
that the convolution operators $\mathscr{R}_{\mu},\mathscr{L}_{\mu}$ on
$\mathrm{C}_{0}(\mathbb{G})$ induce maps
$\mathscr{R}_{\mu}^{\Psi},\mathscr{L}_{\mu}^{\Psi}$ on
$\mathrm{C}_{0}(\mathbb{G})^{\overline{\Psi}\boxtimes\dot{\Psi}}$, and the
maps $\pi^{\Psi}\circ\mathscr{R}_{\mu}^{\Psi}\circ{(\pi^{\Psi})}^{-1}$ and
$\pi^{\Psi}\circ\mathscr{L}_{\mu}^{\Psi}\circ{(\pi^{\Psi})}^{-1}$ on
$\mathrm{C}_{0}(\mathbb{G}^{\Psi})$ are the right and left convolution
operators, respectively, associated with two, _a priori_ possibly different,
states on $\mathrm{C}_{0}^{\mathrm{u}}(\mathbb{G}^{\Psi})$. The next
proposition shows that, as expected, these two states coincide, and they are
symmetric if $\mu$ is.

######  Proposition 4.17.

Assume that $\mathbb{G}$ is co-amenable. Let $\mu$ be a state of
$\mathrm{C}_{0}^{\mathrm{u}}(\mathbb{G})$ that is invariant under the adjoint
action
$\Gamma\overset{\mathrm{ad}^{\mathrm{u}}}{\curvearrowright}\mathrm{C}_{0}^{\mathrm{u}}(\mathbb{G})$.
Consider the non-degenerate completely positive maps
$\pi^{\Psi}\circ\mathscr{R}_{\mu}^{\Psi}\circ{(\pi^{\Psi})}^{-1}$ and
$\pi^{\Psi}\circ\mathscr{L}_{\mu}^{\Psi}\circ{(\pi^{\Psi})}^{-1}$ on
$\mathrm{C}_{0}(\mathbb{G}^{\Psi})$ constructed in Proposition 4.10. Then

(4.12)
$\epsilon_{\mathbb{G}^{\Psi}}\circ\pi^{\Psi}\circ\mathscr{L}_{\mu}^{\Psi}\circ{(\pi^{\Psi})}^{-1}=\epsilon_{\mathbb{G}^{\Psi}}\circ\pi^{\Psi}\circ\mathscr{R}_{\mu}^{\Psi}\circ{(\pi^{\Psi})}^{-1}$

and

(4.13)
$\epsilon_{\mathbb{G}^{\Psi}}\circ\pi^{\Psi}\circ\mathscr{R}_{\mu}^{\Psi}\circ{(\pi^{\Psi})}^{-1}\circ\text{\scalebox{1.15}{$\mathtt{R}$}}_{\mathbb{G}^{\Psi}}=\epsilon_{\mathbb{G}^{\Psi}}\circ\pi^{\Psi}\circ\mathscr{L}_{\mu\circ\text{\scalebox{1.15}{$\mathtt{R}$}}_{\mathbb{G}}}^{\Psi}\circ{(\pi^{\Psi})}^{-1}=\epsilon_{\mathbb{G}^{\Psi}}\circ\pi^{\Psi}\circ\mathscr{R}_{\mu\circ\text{\scalebox{1.15}{$\mathtt{R}$}}_{\mathbb{G}}}^{\Psi}\circ{(\pi^{\Psi})}^{-1}.$

In particular, if $\mu$ is symmetric, then

(4.14)
$\begin{split}\epsilon_{\mathbb{G}^{\Psi}}\circ\pi^{\Psi}\circ\mathscr{R}_{\mu}^{\Psi}\circ{(\pi^{\Psi})}^{-1}\circ\text{\scalebox{1.15}{$\mathtt{R}$}}_{\mathbb{G}^{\Psi}}&=\epsilon_{\mathbb{G}^{\Psi}}\circ\pi^{\Psi}\circ\mathscr{R}_{\mu}^{\Psi}\circ{(\pi^{\Psi})}^{-1}\\\
&=\epsilon_{\mathbb{G}^{\Psi}}\circ\pi^{\Psi}\circ\mathscr{L}_{\mu}^{\Psi}\circ{(\pi^{\Psi})}^{-1}=\epsilon_{\mathbb{G}^{\Psi}}\circ\pi^{\Psi}\circ\mathscr{L}_{\mu}^{\Psi}\circ{(\pi^{\Psi})}^{-1}\circ\text{\scalebox{1.15}{$\mathtt{R}$}}_{\mathbb{G}^{\Psi}}.\end{split}$

###### Proof.

We consider the following maps:

  * •

the non-degenerate completely positive maps
$\widetilde{\mathscr{R}_{\mu}},\widetilde{\mathscr{L}_{\mu}}$ on
$\mathrm{C}_{0}(\mathbb{G})\rtimes_{\rho}\Gamma^{2}$ induced by
$\mathscr{R}_{\mu},\mathscr{L}_{\mu}$, which restrict (after extension) to the
non-degenerate completely positive maps
$\mathscr{R}_{\mu}^{\Psi},\mathscr{L}_{\mu}^{\Psi}$ on
$\mathrm{C}_{0}(\mathbb{G})^{\overline{\Psi}\boxtimes\dot{\Psi}}$ and yield
the convolution operators
$\pi^{\Psi}\circ\mathscr{R}_{\mu}^{\Psi}\circ{(\pi^{\Psi})}^{-1}$ and
$\pi^{\Psi}\circ\mathscr{L}_{\mu}^{\Psi}\circ{(\pi^{\Psi})}^{-1}$ on
$\mathrm{C}_{0}(\mathbb{G}^{\Psi})$, respectively (Proposition 4.10);

  * •

the morphism
$\tilde{\Pi}\in\operatorname{Mor}(\mathrm{C}_{0}(\mathbb{G})\rtimes_{\rho}\Gamma^{2},\mathrm{C}_{0}(\Gamma)\rtimes_{\rho_{\Gamma}}\Gamma^{2})$
induced by the strong quantum homomorphism
$\Pi:\mathrm{C}_{0}(\mathbb{G})\to\mathrm{C}_{0}(\Gamma)$, which restricts to
a morphism
$\Pi^{\Psi}\in\operatorname{Mor}(\mathrm{C}_{0}(\mathbb{G})^{\overline{\Psi}\boxtimes\dot{\Psi}},\mathrm{C}_{0}(\Gamma)^{\overline{\Psi}\boxtimes\dot{\Psi}})$
such that
$\epsilon_{\mathbb{G}^{\Psi}}=\epsilon_{\Gamma}\circ\pi_{\Gamma}^{\Psi}\circ\Pi^{\Psi}\circ{(\pi^{\Psi})}^{-1}$
(Proposition 3.5);

  * •

the anti-automorphism $\tilde{\text{\scalebox{1.15}{$\mathtt{R}$}}}$ of
$\mathrm{C}_{0}(\mathbb{G})\rtimes_{\rho}\Gamma^{2}$, which satisfies
$\tilde{\text{\scalebox{1.15}{$\mathtt{R}$}}}(\mathrm{C}_{0}(\mathbb{G})^{\overline{\Psi}\boxtimes\dot{\Psi}})=\mathrm{C}_{0}(\mathbb{G})^{\overline{\Psi}\boxtimes\dot{\Psi}}$
and
$\tilde{\text{\scalebox{1.15}{$\mathtt{R}$}}}|_{\mathrm{C}_{0}(\mathbb{G})^{\overline{\Psi}\boxtimes\dot{\Psi}}}={(\pi^{\Psi})}^{-1}\circ\text{\scalebox{1.15}{$\mathtt{R}$}}_{\mathbb{G}^{\Psi}}\circ\pi^{\Psi}$
(Proposition 3.7).

As mentioned in Subsection 3.2, $\rho^{L},\rho^{R}$ are the left and right,
respectively, quantum homomorphisms associated with $\Gamma$ being a closed
quantum subgroup of $\mathbb{G}$. The formulas that relate these quantum
homomorphisms to $\Pi$ [BCV, (2.1) and (2.2)] and the assumption that $\mu$ is
invariant under the adjoint action of $\Gamma$ imply that
$\Pi\circ\mathscr{R}_{\mu}=\Pi\circ\mathscr{L}_{\mu}$. Therefore,
$\Pi^{\Psi}\circ\mathscr{R}_{\mu}^{\Psi}=(\Pi\circ\mathscr{R}_{\mu})^{\Psi}=(\Pi\circ\mathscr{L}_{\mu})^{\Psi}=\Pi^{\Psi}\circ\mathscr{L}_{\mu}^{\Psi}$
. Just as in (4.11), by the formula
$\epsilon_{\mathbb{G}^{\Psi}}=\epsilon_{\Gamma}\circ\pi_{\Gamma}^{\Psi}\circ\Pi^{\Psi}\circ{(\pi^{\Psi})}^{-1}$
we deduce formula (4.12) because

$\begin{split}\epsilon_{\mathbb{G}^{\Psi}}\circ\pi^{\Psi}\circ\mathscr{L}_{\mu}^{\Psi}\circ{(\pi^{\Psi})}^{-1}&=\epsilon_{\Gamma}\circ\pi_{\Gamma}^{\Psi}\circ\Pi^{\Psi}\circ\mathscr{L}_{\mu}^{\Psi}\circ{(\pi^{\Psi})}^{-1}\\\
&=\epsilon_{\Gamma}\circ\pi_{\Gamma}^{\Psi}\circ\Pi^{\Psi}\circ\mathscr{R}_{\mu}^{\Psi}\circ{(\pi^{\Psi})}^{-1}=\epsilon_{\mathbb{G}^{\Psi}}\circ\pi^{\Psi}\circ\mathscr{R}_{\mu}^{\Psi}\circ{(\pi^{\Psi})}^{-1}.\end{split}$

Also $\mu\circ\text{\scalebox{1.15}{$\mathtt{R}$}}_{\mathbb{G}}$ is invariant
under the adjoint action of $\Gamma$ because
$\text{\scalebox{1.15}{$\mathtt{R}$}}_{\mathbb{G}}$ intertwines the actions
$\rho^{L}$ and $\rho^{R}$. Applying (4.12) to
$\mu\circ\text{\scalebox{1.15}{$\mathtt{R}$}}_{\mathbb{G}}$ proves the second
equality in (4.13).

Observe that
$\mathscr{R}_{\mu}\circ\text{\scalebox{1.15}{$\mathtt{R}$}}_{\mathbb{G}}=\text{\scalebox{1.15}{$\mathtt{R}$}}_{\mathbb{G}}\circ\mathscr{L}_{\mu\circ\text{\scalebox{1.15}{$\mathtt{R}$}}_{\mathbb{G}}}$.
This implies that
$\widetilde{\mathscr{R}_{\mu}}\circ\tilde{\text{\scalebox{1.15}{$\mathtt{R}$}}}=\tilde{\text{\scalebox{1.15}{$\mathtt{R}$}}}\circ\widetilde{\mathscr{L}_{\mu\circ\text{\scalebox{1.15}{$\mathtt{R}$}}_{\mathbb{G}}}}$,
because for each $f\in\mathrm{C}_{c}(\Gamma^{2},\mathrm{C}_{0}(\mathbb{G}))$
we have

$\begin{split}(\widetilde{\mathscr{R}_{\mu}}\circ\tilde{\text{\scalebox{1.15}{$\mathtt{R}$}}})\left(\int_{\Gamma^{2}}\iota(f(\gamma_{1},\gamma_{2}))\lambda_{\gamma_{1},\gamma_{2}}\,\mathrm{d}(\gamma_{1},\gamma_{2})\right)&=\widetilde{\mathscr{R}_{\mu}}\left(\int_{\Gamma^{2}}\lambda_{-\gamma_{2},-\gamma_{1}}\iota(\text{\scalebox{1.15}{$\mathtt{R}$}}_{\mathbb{G}}(f(\gamma_{1},\gamma_{2})))\,\mathrm{d}(\gamma_{1},\gamma_{2})\right)\\\
&=\int_{\Gamma^{2}}\lambda_{-\gamma_{2},-\gamma_{1}}\iota((\mathscr{R}_{\mu}\circ\text{\scalebox{1.15}{$\mathtt{R}$}}_{\mathbb{G}})(f(\gamma_{1},\gamma_{2})))\,\mathrm{d}(\gamma_{1},\gamma_{2})\\\
&=\int_{\Gamma^{2}}\lambda_{-\gamma_{2},-\gamma_{1}}\iota((\text{\scalebox{1.15}{$\mathtt{R}$}}_{\mathbb{G}}\circ\mathscr{L}_{\mu\circ\text{\scalebox{1.15}{$\mathtt{R}$}}_{\mathbb{G}}})(f(\gamma_{1},\gamma_{2})))\,\mathrm{d}(\gamma_{1},\gamma_{2})\\\
&=\tilde{\text{\scalebox{1.15}{$\mathtt{R}$}}}\left(\int_{\Gamma^{2}}\iota(\mathscr{L}_{\mu\circ\text{\scalebox{1.15}{$\mathtt{R}$}}_{\mathbb{G}}}(f(\gamma_{1},\gamma_{2})))\lambda_{\gamma_{1},\gamma_{2}}\,\mathrm{d}(\gamma_{1},\gamma_{2})\right)\\\
&=(\tilde{\text{\scalebox{1.15}{$\mathtt{R}$}}}\circ\widetilde{\mathscr{L}_{\mu\circ\text{\scalebox{1.15}{$\mathtt{R}$}}_{\mathbb{G}}}})\left(\int_{\Gamma^{2}}\iota(f(\gamma_{1},\gamma_{2}))\lambda_{\gamma_{1},\gamma_{2}}\,\mathrm{d}(\gamma_{1},\gamma_{2})\right).\end{split}$

By restricting to
$\mathrm{C}_{0}(\mathbb{G})^{\overline{\Psi}\boxtimes\dot{\Psi}}$ we obtain
$\mathscr{R}_{\mu}^{\Psi}\circ\tilde{\text{\scalebox{1.15}{$\mathtt{R}$}}}|_{\mathrm{C}_{0}(\mathbb{G})^{\overline{\Psi}\boxtimes\dot{\Psi}}}=\tilde{\text{\scalebox{1.15}{$\mathtt{R}$}}}|_{\mathrm{C}_{0}(\mathbb{G})^{\overline{\Psi}\boxtimes\dot{\Psi}}}\circ\mathscr{L}_{\mu\circ\text{\scalebox{1.15}{$\mathtt{R}$}}_{G}}^{\Psi}$.
Consequently, using the fact that $\epsilon_{\mathbb{G}^{\Psi}}$ is invariant
under $\text{\scalebox{1.15}{$\mathtt{R}$}}_{\mathbb{G}^{\Psi}}$, we get

$\begin{split}\epsilon_{\mathbb{G}^{\Psi}}\circ\pi^{\Psi}\circ\mathscr{R}_{\mu}^{\Psi}\circ{(\pi^{\Psi})}^{-1}\circ\text{\scalebox{1.15}{$\mathtt{R}$}}_{\mathbb{G}^{\Psi}}&=\epsilon_{\mathbb{G}^{\Psi}}\circ\pi^{\Psi}\circ\mathscr{R}_{\mu}^{\Psi}\circ\tilde{\text{\scalebox{1.15}{$\mathtt{R}$}}}|_{\mathrm{C}_{0}(\mathbb{G})^{\overline{\Psi}\boxtimes\dot{\Psi}}}\circ{(\pi^{\Psi})}^{-1}\\\
&=\epsilon_{\mathbb{G}^{\Psi}}\circ\pi^{\Psi}\circ\tilde{\text{\scalebox{1.15}{$\mathtt{R}$}}}|_{\mathrm{C}_{0}(\mathbb{G})^{\overline{\Psi}\boxtimes\dot{\Psi}}}\circ\mathscr{L}_{\mu\circ\text{\scalebox{1.15}{$\mathtt{R}$}}_{G}}^{\Psi}\circ{(\pi^{\Psi})}^{-1}\\\
&=\epsilon_{\mathbb{G}^{\Psi}}\circ\text{\scalebox{1.15}{$\mathtt{R}$}}_{\mathbb{G}^{\Psi}}\circ\pi^{\Psi}\circ\mathscr{L}_{\mu\circ\text{\scalebox{1.15}{$\mathtt{R}$}}_{G}}^{\Psi}\circ{(\pi^{\Psi})}^{-1}\\\
&=\epsilon_{\mathbb{G}^{\Psi}}\circ\pi^{\Psi}\circ\mathscr{L}_{\mu\circ\text{\scalebox{1.15}{$\mathtt{R}$}}_{\mathbb{G}}}^{\Psi}\circ{(\pi^{\Psi})}^{-1}.\end{split}$

This completes the proof of (4.13). Now (4.14) follows easily if $\mu$ is
symmetric, that is,
$\mu=\mu\circ\text{\scalebox{1.15}{$\mathtt{R}$}}_{\mathbb{G}}$. ∎

The next theorem, which is the second main result of this paper, follows as a
direct corollary of Proposition 4.17. We stress again that the co-amenability
assumption might be unnecessary.

######  Theorem 4.18.

Assume that $\mathbb{G}$ is co-amenable. Under the assumptions of Theorem
4.15, if each state in the convolution semigroup ${(\mu_{t})}_{t\geq 0}$ on
$\mathbb{G}$ is symmetric (i.e.,  $\mathtt{R}$ -invariant), then each state in
the induced convolution semigroup ${(\mu_{t}^{\Psi})}_{t\geq 0}$ on
$\mathbb{G}^{\Psi}$ is symmetric (i.e.,
$\text{\scalebox{1.15}{$\mathtt{R}$}}_{\mathbb{G}^{\Psi}}$-invariant).

###  4.4. The commutative case

In this subsection we focus on the particular case that $\mathbb{G}=G$ is a
locally compact _group_ ; as before, $\Gamma$ is an abelian closed subgroup of
$G$ and $\Psi$ is a normalised continuous $2$-cocycle on $\hat{\Gamma}$. We
write $G^{\Psi}$ for $\mathbb{G}^{\Psi}$ and begin with a few observations.

######  Observation 4.19.

In order for the locally compact quantum group $G^{\Psi}$ to be of interest it
must not be commutative, equivalently, its dual
$\widehat{G^{\Psi}}=\hat{G}_{\Psi}$ should not be co-commutative. This means
that, with $\Omega:=(L\otimes
L)(\Psi)\in\operatorname{VN}(G)\mathbin{\overline{\otimes}}\operatorname{VN}(G)$
as usual, we should not have
$\hat{\Delta}_{\Omega}=\sigma\circ\hat{\Delta}_{\Omega}$, where $\sigma$ is
the flip operator and
$\hat{\Delta}_{\Omega}=(\operatorname{Ad}\Omega)\circ\hat{\Delta}$ as in
Subsection 2.2. Equivalently, the unitary
$\Omega\sigma(\Omega)^{*}\in\operatorname{VN}(G)\mathbin{\overline{\otimes}}\operatorname{VN}(G)$
should not belong to the commutant of
$\left\\{\lambda_{g}^{G}\otimes\lambda_{g}^{G}\in\operatorname{VN}(G)\mathbin{\overline{\otimes}}\operatorname{VN}(G):g\in
G\right\\}$.

######  Observation 4.20.

Denote again by $Z_{G}(\Gamma)$ the centraliser of $\Gamma$ inside $G$. A
computation similar to (3.13) shows that the canonical embedding
$L_{Z_{G}(\Gamma)}$ of
$L^{\infty}(\widehat{Z_{G}(\Gamma)})=\operatorname{VN}(Z_{G}(\Gamma))$ inside
$L^{\infty}(\hat{G})=\operatorname{VN}(G)$ turns $Z_{G}(\Gamma)$ into a closed
quantum subgroup of $G^{\Psi}$. Furthermore, one can repeat the proof of
Proposition 3.5 to show that the restriction map
$\Pi_{Z_{G}(\Gamma)}:\mathrm{C}_{0}(G)\to\mathrm{C}_{0}(Z_{G}(\Gamma))$
induces a morphism
$\Pi_{Z_{G}(\Gamma)}^{\Psi}\in\operatorname{Mor}(\mathrm{C}_{0}(G)^{\overline{\Psi}\boxtimes\dot{\Psi}},\mathrm{C}_{0}(Z_{G}(\Gamma))^{\overline{\Psi}\boxtimes\dot{\Psi}})$
and the morphism
$\pi_{Z_{G}(\Gamma)}^{\Psi}\circ\Pi_{Z_{G}(\Gamma)}^{\Psi}\circ{(\pi^{\Psi})}^{-1}\in\operatorname{Mor}(\mathrm{C}_{0}(G^{\Psi}),\mathrm{C}_{0}(Z_{G}(\Gamma)^{\Psi}=Z_{G}(\Gamma)))$
is the strong quantum homomorphism associated to $Z_{G}(\Gamma)$ being a
closed quantum subgroup of $G^{\Psi}$, where the equality
$Z_{G}(\Gamma)^{\Psi}=Z_{G}(\Gamma)$ follows from Proposition 3.4. In
addition, the co-unit $\epsilon_{G^{\Psi}}$ of $G^{\Psi}$ equals
$\epsilon_{Z_{G}(\Gamma)}\circ\pi_{Z_{G}(\Gamma)}^{\Psi}\circ\Pi_{Z_{G}(\Gamma)}^{\Psi}\circ{(\pi^{\Psi})}^{-1}$.

Consequently, repeating the proof of Corollary 4.13 shows that for every
probability measure $\mu$ on $G$ that is supported by $Z_{G}(\Gamma)$, the
‘deformed’ state $\mu^{\Psi}$ of $\mathrm{C}_{0}(G^{\Psi})$ given by
Proposition 4.10 (2) is also ‘supported by $Z_{G}(\Gamma)$’, i.e., it factors
through the strong quantum homomorphism associated to $Z_{G}(\Gamma)$ being a
closed quantum subgroup of $G^{\Psi}$. This strengthens Remark 4.12 (1), and
means that when seeking in the commutative case examples of the results of the
previous subsection in which the convolution semigroup of measures cannot be
‘transported directly’ to the Rieffel deformation, we should avoid measures
supported by the centraliser $Z_{G}(\Gamma)$.

Remark that if $\Gamma$ is compact, then as a particular case of Remark 4.14
we can take an arbitrary probability measure $\nu$ on $G$ and integrate
$\Gamma\ni\gamma\mapsto\nu\circ\mathrm{ad}_{\gamma}$, where
$\Gamma\overset{\mathrm{ad}}{\curvearrowright}\mathrm{C}_{0}(G)$ is the
adjoint action, to get an $\mathrm{ad}$-invariant probability measure $\mu$ on
$G$, which can be later used to generate a convolution semigroup of measures
on $G$ associated with the relevant compound Poisson process and satisfying
the assumptions of Corollary B. In addition, if $\nu$ has full support, then
so has $\mu$. If $\Gamma$ contains the centre $Z=Z(G)$ of $G$, we can replace
the condition that $\Gamma$ is compact by the slightly more general condition
that the quotient $\Gamma/Z$ is compact, in which case the fact that
$\Gamma\overset{\mathrm{ad}}{\curvearrowright}\mathrm{C}_{0}(G)$ factors
through the quotient map $\Gamma\to\Gamma/Z$ allows us to consider the induced
map $\Gamma/Z\ni\left[\gamma\right]\mapsto\nu\circ\mathrm{ad}_{\gamma}$ and
integrate it (with respect to the Haar measure on $\Gamma/Z$) to obtain an
$\mathrm{ad}$-invariant probability measure on $G$.

######  Example 4.21.

Take $G:=\mathrm{\tilde{E}}(2)=\left\\{\left(\begin{smallmatrix}w&z\\\
0&w^{-1}\end{smallmatrix}\right):z\in\mathbb{C},w\in\mathbb{T}\right\\}$ as a
topological subgroup of $\mathrm{GL}(2,\mathbb{C})$ and
$\Gamma:=\left\\{\left(\begin{smallmatrix}w&0\\\
0&w^{-1}\end{smallmatrix}\right):w\in\mathbb{T}\right\\}\cong\mathbb{T}.$ Then
$Z_{G}(\Gamma)=\Gamma$. As explained in the previous paragraph, averaging
allows constructing probability measures on $G$ that are invariant under the
adjoint action
$\Gamma\overset{\mathrm{ad}}{\curvearrowright}\mathrm{C}_{0}(G)$ and have full
support and so, in particular, are not supported by $Z_{G}(\Gamma)$.

Fixing $w_{1},w_{2}\in\mathbb{T}$, $w_{1}\neq\pm w_{2}$, consider the
bicharacter $\Psi:=\Psi_{w_{1},w_{2}}$ on $\hat{\mathbb{T}}\cong\mathbb{Z}$
given by $\Psi_{w_{1},w_{2}}(n,m):=w_{1}^{n}w_{2}^{m}$,
$(n,m)\in\mathbb{Z}^{2}$, and form the Rieffel deformation locally compact
quantum group $G^{\Psi}$. To prove that $G^{\Psi}$ is not commutative we
should show that $\Omega\sigma(\Omega)^{*}$ does not belong to the commutant
$\left\\{\lambda_{g}^{G}\otimes\lambda_{g}^{G}\in\operatorname{VN}(G)\mathbin{\overline{\otimes}}\operatorname{VN}(G):g\in
G\right\\}^{\prime}$, see Observation 4.19. But
$\Omega=\lambda_{j(w_{1})}^{G}\otimes\lambda_{j(w_{2})}^{G}$ where
$j(w):=\left(\begin{smallmatrix}w&0\\\ 0&w^{-1}\end{smallmatrix}\right)$ for
$w\in\mathbb{T}$, and so
$\Omega\sigma(\Omega)^{*}=\lambda_{j(w_{1}w_{2}^{-1})}^{G}\otimes\lambda_{j(w_{2}w_{1}^{-1})}^{G}$,
which is indeed not in the above commutant because
$w_{1}w_{2}^{-1}\notin\left\\{1,-1\right\\}$.

At the moment we do not have an example of a locally compact group $G$ with a
non-compact (or more generally, non-compact modulo the centre of $G$) abelian
subgroup $\Gamma$ admitting a probability measure on $G$ which is invariant
under the adjoint action of $\Gamma$ and not supported by the centraliser
$Z_{G}(\Gamma)$. The question of whether or not such an example exists seems
to be related to deep topological group considerations. Uri Bader has informed
us that results of [BDL] imply that when $G$ is an algebraic group and
$\Gamma$ satisfies mild non-compactness assumptions, such an invariant measure
cannot exist.

## Acknowledgments

We are grateful to Kenny De Commer and to Sergey Neshveyev for helpful
conversations and correspondences about the subject of this paper; to Stefaan
Vaes for asking a question that prompted us to further extend the results we
had at the time; to Orr M. Shalit and to Michael Skeide for answering Question
4.9 affirmatively; to Uri Bader for his enlightening remarks concerning
Subsection 4.4; to Mariusz Tobolski and to Jacek Krajczok for their assistance
on Observation 1.2; and to Alcides Buss for an interesting correspondence on
crossed products and in particular for providing the references in Lemma 3.6.
Finally we thank Paweł Kasprzak, our Rieffel deformation guru, for the ongoing
inspiration.

## Funding

The first author was partially supported by the National Science Centre (NCN)
grant no. 2020/39/I/ST1/01566.

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