# Effects of Field Fluctuations on Driven Autoionizing Resonances

G. Mouloudakis1<EMAIL_ADDRESS>P. Lambropoulos1,2 1Department of
Physics, University of Crete, P.O. Box 2208, GR-71003 Heraklion, Crete, Greece
2Institute of Electronic Structure and Laser, FORTH, P.O.Box 1527, GR-71110
Heraklion, Greece

###### Abstract

The excitation of an autoionizing resonance by intense radiation requires a
theoretical description beyond the transition probability per unit time. This
implies a time-dependent formulation incorporating all features of the
radiation source, such as pulse temporal shape and duration, as well as
stochastic properties, for pulses other than Fourier limited. The radiation
from short wavelength free electron lasers is a case in point, as it is the
only source that can provide the necessary intensity. In view of ongoing
experiments with such sources, we present a systematic study for an isolated
autoionizing resonance. We find that intensity, pulse duration and field
fluctuations conspire in producing unexpected excitation profiles, not
amenable to a description in terms of the usual Fano profile. In particular,
the role of intensity fluctuations turns out to pose challenging theoretical
problems part of which have been addressed herein.

PACS numbers

32.80.Aa, 32.80.Hd, 32.70.Jz, 32.80.Rm

###### pacs:

Valid PACS appear here

††preprint: APS/123-QED

##  I Introduction

Autoionization is a process by which an atom in an excited state spontaneously
ejects one of its electrons due to its interaction with a continuum ref1 . A
common example of an autoionizing state (AIS) - also referred to as AI
resonance - is a discrete state involving the excitation of two electrons with
total energy larger than the one-electron ionization threshold. Such states
are unstable and will eventually decay non-radiatevely due to their
interaction with the continuum. If the width of the excitation profile of an
AIS, is much smaller than the energy distance from the nearest AIS, it is
usually referred to as an isolated AIS. The paradigm of an isolated AIS is
provided by the doubly excited 2s2p state of Helium which can be excited by
radiation in the XUV range; with photon energy around 60 eV.

In most traditional photoabsorption studies of AIS’, the resonances were
excited by synchrotron sources with low intensities and practically
monochromatic radiation. The availability of strong radiation sources, such as
lasers, motivated the exploration of the behavior of resonant transitions,
driven by strong, pulsed and possibly non-monochromatic radiation. Over the
last 35 years or so, a plethora of related studies have addressed issues such
as AC Stark splitting in strongly driven bound states in double optical
resonance, including the effect of field fluctuations ref2 ; ref3 . Those
studies were limited to the optical or near UV spectral region, in which
sources of sufficient intensity were at the time available. The strong driving
of AI states, such as the 2s2p in Helium, which requires radiation in the XUV
range, were beyond the reach of those sources. Nevertheless, some initial
theoretical exploration of the expected behavior of a strongly driven AI
state, as well as the case of double resonance, involving the strong coupling
of two AI states were published as early as 1980 ref4 ; ref5  and revisited
much later ref6 , when short wavelength Free Electron Lasers (FEL) began
delivering strong radiation in the XUV and beyond. Extension of that work to
triply excited hollow states followed a few years later ref7 ; ref8 ; ref9 .
To the best of our knowledge, there are two examples ref10 ; ref11  of
experimental work providing some evidence of the strong coupling of two AI
states.

At this point, we need to define the notion of strong coupling in AI states. A
formulation of an AI state, particularly valid for an isolated resonance,
involves the superposition of a discrete state and the continuum to which it
is coupled via intra-atomic interaction. Diagonalization of the relevant part
of the Hamiltonian leads to a modification of the position of the discrete
part and a decay rate, referred to as AI width, as it corresponds to the width
of the profile of the resonance. As discussed in detail in the next section,
the dipole matrix element coupling a bound state to the discrete part of the
AI resonance multiplied by the electric field amplitude, to within some
coefficients, represents an effective Rabi frequency characterizing the
strength of the coupling. Under traditional synchrotron radiation experiments,
that Rabi frequency in much smaller than the AI width. That is what we shall
call weak coupling, in which case the interaction is describable in terms of a
transition probability per unit time (rate), as given by Fermi’s golden rule.
The coupling is strong when the above condition is reversed, with the Rabi
frequency being larger than the AI width. When an EM field couples two AI
states, as for example in refs. ref4 ; ref5 ; ref6 , the field is strong when
the Rabi frequency between the two AI states is larger than the AI width of at
least one of the resonances.

The availability of strong, short pulse duration, short wavelength (XUV and
beyond) radiation through the recent Free Electron Laser (FEL) sources ref12
provides the opportunity to explore the behavior of strongly driven AI
resonances. For a pulse of high peak intensity, the pulse duration enters as
an important parameter paired to the intensity. Clearly, under any intensity,
given sufficiently long time exposure of any system to the radiation, complete
ionization will ensue. When the intensity is high, even a seemingly short
pulse duration can cause significant ionization, to the extent of distorting
the resonance profile. That effect noted in the early paper by Lambropoulos
and Zoller ref5  and referred to as ”time saturation”, has until now been of
only academic interest. But as shown in this paper, it should be expected to
be of crucial influence in experiments under FEL radiation.

In addition to the high intensity and short pulse duration, at least at the
present time, FEL’s exhibit strong intensity fluctuations ref12 . This means
that the radiation seen by the atom has a non-zero bandwidth, while the Rabi
frequency will undergo stochastic fluctuations. The influence of field
fluctuations has not received much attention in the theoretical literature,
until now, simply because AI resonances tend to be much broader than the
bandwidths of synchrotron sources. Under FEL radiation, however, that is no
longer the case, which requires a formulation that accounts for the stochastic
fluctuations of the field. Depending on the strength of the coupling to the
radiation, it may be that only the bandwidth is of importance. In the most
general case of strong driving, in the sense defined above, accounting for the
intensity fluctuations becomes imperative.

As demonstrated in the sections that follow, hitherto ”academic” effects such
as time broadening are expected to cause dramatic distortion of the profile.
Moreover the complete theory of an AI resonance driven strongly by a
stochastic field, with intensity fluctuations, poses challenging problems not
readily amenable to the theoretical tools developed for strongly driven
transitions between bound states ref2 . In a recent short paper ref13 , we
have reported a first assessment of the basic effects to be expected. The
present paper provides a more detailed exposition of the theory, as well as a
partial account of the effect of intensity fluctuations. In the light of our
treatment and results, a number of important aspects, outlined in the
concluding remarks, pose challenging open problems.

A few comments on the features of the FEL radiation are in order, at this
point. The source is pulsed. The term pulse duration means the full width at
half maximum of the temporal profile, while the peak intensity will often be
referred to simply as the intensity. If in comparing theory to experimental
data, we were to be concerned with accuracy of 1$\%$ or better, then the exact
temporal profile would be of importance. But given the present state of
uncertainty in the source parameters, long experience has shown that assuming
a Gaussian temporal profile is more than adequate. Moreover, for illustrative
purposes of various aspects of the theory, again experience has shown that
assuming a square pulse of roughly the same duration does encapsulate the
essential physics. Its occasional use in this paper is limited to the cases in
which analytic feasibility of the theory is deemed informative.

##  II Theoretical Formulation

We begin by considering a two-level atom whose ground state
$\left|1\right\rangle$ is coupled to an isolated autoionizing state (AIS)
$\left|2\right\rangle$ via a single-photon transition, in the presence of an
external electric field $E(t)=\frac{1}{2}[\mathcal{E}(t){e^{i\omega
t}}+c.c.]$. Its frequency $\omega$ is tuned around the resonant frequency
${\omega_{21}}\equiv{\omega_{2}}-{\omega_{1}}$ and
$\mathcal{E}(t)=\left|{\mathcal{E}(t)}\right|\exp[i\varphi(t)]$ is in general
assumed to undergo stochastic fluctuations. The coupling to the resonance is
characterized by the complex Rabi frequency
$\tilde{\Omega}(t)=\Omega(t)(1-\frac{i}{q})=\frac{1}{2}\mathcal{E}(t){d_{21}}(1-\frac{i}{q})$,
where ${d_{21}}$ is the electric dipole matrix element between the ground
state and the discrete part of the resonance and q the asymmetry parameter
ref14 . The ground state is also coupled directly to the continuum via a
dipole matrix element accounting for ionization into the smooth continuum,
leading to an ionization width denoted by $\gamma(t)$. The autoionization
width $\Gamma$ represents the rate of decay of the AI state due to the Coulomb
interaction between the two excited electrons. Spectroscopically, it appears
as the width of the excitation profile of the resonance and is equal to the
inverse of the lifetime of the AIS. The q parameter accounts for the
interference between the two paths to the continuum; the direct and the one
via the discrete part, and is related to the Rabi frequency and the two widths
through the strict relation $4{\Omega^{2}}={q^{2}}\gamma\Gamma$.

Depending on the aspects of the problem to be addressed, the theory can be
cast either in terms of the time-dependent Schrondinger equation or the
density matrix. In what follows, we treat the problem in terms of the density
matrix $\rho(t)$ because it allows the distinction between phase and intensity
fluctuations, which is instructive. It provides in addition a convenient tool
for useful approximations, such as the decorrelation of atomic and field
dynamics.

The dynamical evolution of the slowly varying part ${\sigma}(t)$ of the
density matrix is governed by the following equations ref15 ; ref16 :

${\partial_{t}}{\sigma_{11}}(t)=-\gamma(t){\sigma_{11}}(t)+2Im\left\\{{\Omega(t)\left({1-\frac{i}{q}}\right){\sigma_{21}}(t)}\right\\}$
(1)
${\partial_{t}}{\sigma_{22}}(t)=-\Gamma{\sigma_{22}}(t)-2Im\left\\{{\Omega(t)\left({1+\frac{i}{q}}\right){\sigma_{21}}(t)}\right\\}$
(2)
$\left[{{\partial_{t}}-i\Delta+\frac{1}{2}\left({\gamma(t)+\Gamma}\right)}\right]{\sigma_{21}}(t)=-i\Omega(t)\left({1-\frac{i}{q}}\right){\sigma_{11}}(t)$
$+i\Omega(t)\left({1+\frac{i}{q}}\right){\sigma_{22}}(t)$ (3)

where we have introduced the slowly varying matrix elements
${\sigma_{ij}}(t)$, defined by ${\rho_{ii}}(t)={\sigma_{ii}}(t)$ , $i=1,2$ and
${\rho_{21}}(t)={\sigma_{21}}(t)exp[i\omega t]$. The detuning $\Delta$ of the
photon frequency from resonance is defined by
$\Delta\equiv\omega-{\omega_{21}}$. Note that the left side of equation (3)
may in general contain additional coherence (off-diagonal) relaxation
constants which are of no relevance to our problem, in the case of
monochromatic field. However, a coherence relaxation constant appears below,
as we introduce field fluctuations.

The matrix elements of the density matrix in the above equations are generally
fluctuating variables owing to the stochastic character of the field which
imparts fluctuations to the Rabi frequency and the ionization width. We are
therefore dealing with stochastic differential equations. The observed
quantities which refer to the atom are given by the average over the
stochastic fluctuations of the field. This requires a realistic model of the
stochastic properties of the field, or a brute force numerical integration
over trajectories imitating the fluctuations of the field. We have here
adopted the first approach.

To this end, we solve equation (3) for ${\sigma_{21}}(t)$ formally, and
substitute into equations (1) and (2). Denoting the stochastic averages of the
resulting equations by angular brackets, we obtain:

${\partial_{t}}\left\langle\sigma_{11}(t)\right\rangle=-\left\langle\gamma(t)\sigma_{11}(t)\right\rangle+2{\rm
Im}\left\\{\left(1-\frac{i}{q}\right)\int_{0}^{t}-i\left(1-\frac{i}{q}\right)\left\langle{\Omega(t)\Omega(t^{\prime})\sigma_{11}(t^{\prime})}\right\rangle
e^{-\kappa(t-t^{\prime})}dt^{\prime}\right.$
$+\left.\left(1-\frac{i}{q}\right)\int_{0}^{t}i\left(1+\frac{i}{q}\right)\left\langle\Omega(t)\Omega(t^{\prime})\sigma_{22}(t^{\prime})\right\rangle
e^{-\kappa(t-t^{\prime})}dt^{\prime}\right\\}$ (4)

${\partial_{t}}\left\langle\sigma_{22}(t)\right\rangle=-\Gamma\left\langle\sigma_{22}(t)\right\rangle-2{\rm
Im}\left\\{\left(1+\frac{i}{q}\right)\int_{0}^{t}-i\left(1-\frac{i}{q}\right)\left\langle{\Omega(t)\Omega(t^{\prime})\sigma_{11}(t^{\prime})}\right\rangle
e^{-\kappa(t-t^{\prime})}dt^{\prime}\right.$
$+\left.\left(1+\frac{i}{q}\right)\int_{0}^{t}i\left(1+\frac{i}{q}\right)\left\langle\Omega(t)\Omega(t^{\prime})\sigma_{22}(t^{\prime})\right\rangle
e^{-\kappa(t-t^{\prime})}dt^{\prime}\right\\}$ (5)

where to compress notation somewhat,we have introduced
$\kappa\equiv-i\Delta+\frac{1}{2}(\gamma+\Gamma)$.

Equations (4) and (5) involve atom-field correlation functions of the form
$\left\langle{\Omega(t)\Omega(t^{\prime}){\sigma_{ii}}(t^{\prime})}\right\rangle$,
$i=1,2$. Generally such correlation functions cannot be evaluated without
knowing the specific form of the fluctuations of the field. As an
approximation valid under certain conditions, one could decorrelate the
atomic-field dynamics ref2  by taking
$\left\langle{\Omega(t)\Omega(t^{\prime}){\sigma_{ii}}(t^{\prime})}\right\rangle=\left\langle{\Omega(t)\Omega(t^{\prime})}\right\rangle\left\langle{{\sigma_{ii}}(t^{\prime})}\right\rangle$.
There are, however, specific models of fluctuating fields where the
decorrelation is mathematically rigorous. In the following, we describe
briefly two widely used models and apply them to the context of our problem;
namely, the phase-diffusion and the chaotic field model.

In the phase-diffusion (PD) model, the field has a non-fluctuating amplitude
but its phase is a Wiener-Levy stochastic process [R]. In that case the nth-
order correlation function of the field obeys the relation ref18

$\left\langle{{\varepsilon^{*}}({t_{1}})\varepsilon({t_{2}})...{\varepsilon^{*}}({t_{2n-1}})\varepsilon({t_{2n}})}\right\rangle=\prod\limits_{j\to
odd}^{2n-1}{\left\langle{{\varepsilon^{*}}({t_{j}})\varepsilon({t_{j+1}})}\right\rangle}$
(6)

with ${t_{j}}>{t_{j+1}}$. This represents a Markovian process, with an
exponential first-order correlation function given by ref17

$\left\langle{{\varepsilon^{*}}({t_{1}})\varepsilon({t_{2}})}\right\rangle=\left\langle{{{\left|{\varepsilon(t)}\right|}^{2}}}\right\rangle\exp[-\frac{1}{2}{\gamma_{L}}\left|{{t_{1}}-{t_{2}}}\right|]$
(7)

where ${\gamma_{L}}$ is the bandwidth of the field.

It has been established ref2  that, in the case of the phase-diffusion model,
the decorrelation of the atom-field dynamics is rigorous without any
approximation. Physically, this is easy to understand, because for a constant
amplitude, the fluctuations of the phase of the field cannot affect the
evolution of the populations, but only the coherence, which means the relative
phase of the coefficients representing the superposition of the states coupled
by the field. And it is the correlation between the time evolution of
populations that is factorized out in the process of decorrelation. As we will
discuss shortly, this is not the case when also the amplitude undergoes random
fluctuations.

Formally, in our problem, the decorrelation implies the relation
$\left\langle{\Omega(t)\Omega(t^{\prime}){\sigma_{ii}}(t^{\prime})}\right\rangle=\left\langle{\Omega(t)\Omega(t^{\prime})}\right\rangle\left\langle{{\sigma_{ii}}(t^{\prime})}\right\rangle$.
Note that ${\Omega(t)}$ is real. The same argument as above justifies the
decorrelation of the quantities in the term
$\left\langle{\gamma(t){\sigma_{11}}(t)}\right\rangle$, because ${\gamma(t)}$,
representing the direct single-photon ionization from the ground state, is
proportional to the intensity which does not undergo fluctuations .

In the chaotic field model the field undergoes both amplitude and phase
fluctuations. Its amplitude is a complex Gaussian stochastic process with its
nth order correlation function obeying ref18 :

$\left\langle{{\varepsilon^{*}}({t_{1}})\varepsilon({t_{2}})...{\varepsilon^{*}}({t_{2n-1}})\varepsilon({t_{2n}})}\right\rangle=\sum\limits_{P}{\prod\limits_{j\to
odd}^{2n-1}{\left\langle{{\varepsilon^{*}}({t_{j}})\varepsilon({t_{P(j+1)}})}\right\rangle}}$
(8)

where the sum is over all possible permutations P, with ${t_{j}}>{t_{j+1}}$.
For the sake of simplicity we will assume the chaotic field to be Markovian,
something that it is not necessarily satisfied for a general chaotic field. In
that case the first-order correlation function of the field is given by
equation (7).

In contrast to the case of the phase diffusion model, the decorrelation
$\left\langle{\Omega(t)\Omega(t^{\prime}){\sigma_{ii}}(t^{\prime})}\right\rangle=\left\langle{\Omega(t)\Omega(t^{\prime})}\right\rangle\left\langle{{\sigma_{ii}}(t^{\prime})}\right\rangle$
is not mathematically rigorous for a chaotic field, but as an approximation,
it is valid in the weak field regime. The relative errors of this
approximation have been evaluated recently, as a function of the ratio
$\Omega/\Gamma$ for various field bandwidths ref19 . It has been shown that
the error of the decorrelation approximation (DA) becomes significant for
increasing intensities and laser bandwidths comparable to the autoionization
width. The variables $\gamma(t)$ and ${\sigma_{11}}(t)$ are also decorrelated
within the DA since $\gamma(t)$ is proportional to the intensity which is
approximately replaced by its averaged value.

In view of the above, we proceed with the decorrelation of the atomic-field
dynamics, with the resulting equations being exact for the PD model and valid
in the weak to moderate field limit for the chaotic model. After the
decorrelation, equations (4) and (5) become:

${\partial_{t}}\left\langle{{\sigma_{11}}(t)}\right\rangle=-\left\langle\gamma(t)\right\rangle\left\langle{{\sigma_{11}}(t)}\right\rangle+d_{12}^{2}I(t){\mathop{\rm
Im}\nolimits}\left\\{\left({1-\frac{i}{q}}\right)\int\limits_{0}^{t}-i\left({1-\frac{i}{q}}\right)\left\langle{{\sigma_{11}}(t^{\prime})}\right\rangle{e^{-\tilde{\kappa}(t-t^{\prime})}}dt^{\prime}\right.$
$+\left.\left({1-\frac{i}{q}}\right)\int\limits_{0}^{t}{i\left({1+\frac{i}{q}}\right)\left\langle{{\sigma_{22}}(t^{\prime})}\right\rangle{e^{-\tilde{\kappa}(t-t^{\prime})}}dt^{\prime}}\right\\}$
(9)

${\partial_{t}}\left\langle{{\sigma_{22}}(t)}\right\rangle=-\Gamma\left\langle{{\sigma_{22}}(t)}\right\rangle-
d_{12}^{2}I(t){\mathop{\rm
Im}\nolimits}\left\\{\left({1+\frac{i}{q}}\right)\int\limits_{0}^{t}-i\left({1-\frac{i}{q}}\right)\left\langle{{\sigma_{11}}(t^{\prime})}\right\rangle{e^{-\tilde{\kappa}(t-t^{\prime})}}dt^{\prime}\right.$
$+\left.\left({1+\frac{i}{q}}\right)\int\limits_{0}^{t}{i\left({1+\frac{i}{q}}\right)\left\langle{{\sigma_{22}}(t^{\prime})}\right\rangle{e^{-\tilde{\kappa}(t-t^{\prime})}}dt^{\prime}}\right\\}$
(10)

where we have substituted the complete expression of the Rabi frequency
$\Omega(t)=\frac{1}{2}\mathcal{E}(t){d_{21}}$ and defined
$\tilde{\kappa}\equiv\kappa+\frac{1}{2}{\gamma_{L}}=-i\Delta+\frac{1}{2}(\gamma+\Gamma+{\gamma_{L}})$.
In the RWA, the intensity appearing in equations (9) and (10) is expressed in
terms of the field amplitude as
$I(t)=\frac{{\left\langle{{{\left|{\mathcal{E}(t)}\right|}^{2}}}\right\rangle}}{2}$.

Let us consider, for the moment, the case of constant intensity
$I(t)={I_{0}}$, which leads to considerable simplification enabling analytical
solutions. Since the integrals appearing in (8) and (9) are with respect to
the time t, which is a real variable, the interchange of the Imaginary (or
Real) part and integration are mathematically rigorous. Using this fact and
expanding the exponential functions in the integrands in terms of Cosine and
Sine functions, we obtain:

${\partial_{t}}\left\langle{{\sigma_{11}}(t)}\right\rangle=-\gamma\left\langle{{\sigma_{11}}(t)}\right\rangle+d_{12}^{2}{I_{0}}\left({-1+\frac{1}{{{q^{2}}}}}\right)\int\limits_{0}^{t}{\left\langle{{\sigma_{11}}(t^{\prime})}\right\rangle{e^{-\frac{1}{2}(\gamma+\Gamma+{\gamma_{L}})(t-t^{\prime})}}}\cos[\Delta(t-t^{\prime})]dt^{\prime}$
$+d_{12}^{2}{I_{0}}\left({-\frac{2}{q}}\right)\int\limits_{0}^{t}{\left\langle{{\sigma_{11}}(t^{\prime})}\right\rangle{e^{-\frac{1}{2}(\gamma+\Gamma+{\gamma_{L}})(t-t^{\prime})}}}\sin[\Delta(t-t^{\prime})]dt^{\prime}+d_{12}^{2}{I_{0}}\left({1+\frac{1}{{{q^{2}}}}}\right)\int\limits_{0}^{t}{\left\langle{{\sigma_{22}}(t^{\prime})}\right\rangle{e^{-\frac{1}{2}(\gamma+\Gamma+{\gamma_{L}})(t-t^{\prime})}}}\cos[\Delta(t-t^{\prime})]dt^{\prime}$
(11)
${\partial_{t}}\left\langle{{\sigma_{22}}(t)}\right\rangle=-\Gamma\left\langle{{\sigma_{22}}(t)}\right\rangle-
d_{12}^{2}{I_{0}}\left({-1-\frac{1}{{{q^{2}}}}}\right)\int\limits_{0}^{t}{\left\langle{{\sigma_{11}}(t^{\prime})}\right\rangle{e^{-\frac{1}{2}(\gamma+\Gamma+{\gamma_{L}})(t-t^{\prime})}}}\cos[\Delta(t-t^{\prime})]dt^{\prime}$
$-d_{12}^{2}{I_{0}}\left({-\frac{2}{q}}\right)\int\limits_{0}^{t}{\left\langle{{\sigma_{22}}(t^{\prime})}\right\rangle{e^{-\frac{1}{2}(\gamma+\Gamma+{\gamma_{L}})(t-t^{\prime})}}}\sin[\Delta(t-t^{\prime})]dt^{\prime}-d_{12}^{2}{I_{0}}\left({1-\frac{1}{{{q^{2}}}}}\right)\int\limits_{0}^{t}{\left\langle{{\sigma_{22}}(t^{\prime})}\right\rangle{e^{-\frac{1}{2}(\gamma+\Gamma+{\gamma_{L}})(t-t^{\prime})}}}\cos[\Delta(t-t^{\prime})]dt^{\prime}$
(12)

Note that for constant intensity, the width of the direct ionization to the
continuum is also constant, i.e $\gamma(t)=\gamma$. The integrals appearing in
equations (10) and (11) are now convolutions of
$\left\langle{{\sigma_{ii}}(t^{\prime})}\right\rangle$ , $i=1,2$ and the
Sin/Cosine functions. Taking now the Laplace transforms of the above equations
we obtain:

$s{F_{1}}(s)=-\gamma{F_{1}}(s)+d_{12}^{2}{I_{0}}\left[\left(-1+\frac{1}{{{q^{2}}}}\right){F_{1}}(s){G_{1}}(s)\right.$
$\left.-\frac{2}{q}{F_{1}}(s){G_{2}}(s)+\left(1+\frac{1}{{{q^{2}}}}\right){F_{2}}(s){G_{1}}(s)\right]$
(13)
$s{F_{2}}(s)=-\Gamma{F_{2}}(s)-d_{12}^{2}{I_{0}}\left[\left(-1-\frac{1}{{{q^{2}}}}\right){F_{1}}(s){G_{1}}(s)\right.$
$\left.-\frac{2}{q}{F_{2}}(s){G_{2}}(s)+\left(1-\frac{1}{{{q^{2}}}}\right){F_{2}}(s){G_{1}}(s)\right]$
(14)

where with ${{F_{1}}(s)}$ and ${{F_{2}}(s)}$ we denote the Laplace transforms
of ${\left\langle{{\sigma_{11}}(t)}\right\rangle}$ and
${\left\langle{{\sigma_{22}}(t)}\right\rangle}$, while with ${{G_{1}}(s)}$ and
${{G_{2}}(s)}$ the Laplace transforms of the functions
${{g_{1}}(t)={e^{-\frac{1}{2}(\gamma+\Gamma+{\gamma_{L}})t}}\cos(\Delta t)}$
and ${{g_{2}}(t)={e^{-\frac{1}{2}(\gamma+\Gamma+{\gamma_{L}})t}}sin(\Delta
t)}$, respectively. It is a matter of straightforward algebraic manipulations
to show that

${{G_{1}}(s)=\frac{{s+\frac{1}{2}(\gamma+\Gamma+{\gamma_{L}})}}{{{{\left[{s+\frac{1}{2}(\gamma+\Gamma+{\gamma_{L}})}\right]}^{2}}+{\Delta^{2}}}}}$
(15)

and

${{G_{2}}(s)=\frac{\Delta}{{{{\left[{s+\frac{1}{2}(\gamma+\Gamma+{\gamma_{L}})}\right]}^{2}}+{\Delta^{2}}}}}$
(16)

The system of equations (12) and (13) can be solved easily for ${{F_{1}}(s)}$
and ${{F_{2}}(s)}$, the inverse Laplace transform of which provide the exact
time dependence of $\left\langle{{\sigma_{11}}(t)}\right\rangle$ and
$\left\langle{{\sigma_{22}}(t)}\right\rangle$. The expressions are too lengthy
and complicated to be visually enlightening, but the results are discussed in
later sections.

Returning now to the more general case of time-dependent intensity $I(t)$, the
above analytical treatment using the Laplace transform does not lead to
helpful expressions and even the numerical solution of equations (8) and (9)
tends to be a very cumbersome task. However, useful insight can be gained
through the approximation,

$\begin{array}[]{*{20}{c}}{\left\langle{{\sigma_{ii}}(t^{\prime})}\right\rangle\simeq\left\langle{{\sigma_{ii}}(t^{\prime}=t)}\right\rangle}&{}\hfil&{i=1,2}\end{array}$
(17)

which is valid in the weak field limit. Its validity rests upon the
realization that, under weak driving, the populations do not change
significantly, over times of rapid oscillations of the rest of the integrand.
As a result they can be evaluated at times $t^{\prime}=t$ and factored out of
the integral. In that case, the integration with respect to $t^{\prime}$ in
equations (8) and (9) can be performed, leading to a somewhat simplified
system of differential equations governing the time evolution of the
populations, known as rate equations. They are:

${\partial_{t}}\left\langle{{\sigma_{11}}(t)}\right\rangle=-\gamma(t)\left\langle{{\sigma_{11}}(t)}\right\rangle+d_{12}^{2}I(t){\mathop{\rm
Im}\nolimits}\left\\{{\frac{{1-{e^{-\tilde{\kappa}t}}}}{{\tilde{\kappa}}}\left[{\left({-i}\right){{\left({1-\frac{i}{q}}\right)}^{2}}\left\langle{{\sigma_{11}}(t)}\right\rangle+i\left({1-\frac{i}{q}}\right)\left({1+\frac{i}{q}}\right)\left\langle{{\sigma_{22}}(t)}\right\rangle}\right]}\right\\}$
(18)
${\partial_{t}}\left\langle{{\sigma_{22}}(t)}\right\rangle=-\Gamma\left\langle{{\sigma_{22}}(t)}\right\rangle-
d_{12}^{2}I(t){\mathop{\rm
Im}\nolimits}\left\\{{\frac{{1-{e^{-\tilde{\kappa}t}}}}{{\tilde{\kappa}}}\left[{\left({-i}\right)\left({1-\frac{i}{q}}\right)\left({1+\frac{i}{q}}\right)\left\langle{{\sigma_{11}}(t)}\right\rangle+i{{\left({1+\frac{i}{q}}\right)}^{2}}\left\langle{{\sigma_{22}}(t)}\right\rangle}\right]}\right\\}$
(19)

The system of equations (17) and (18) can now be solved numerically in the
limit of weak to moderate fields, as long as the Rabi frequency is less than
or comparable to the autoionization width $\Gamma$. Actually, for special
forms of the time dependent $I(t)$, analytical solutions may also be obtained.

##  III Results and Discussion

In this section we present and discuss the main results of this work on an AIS
driven by stochastically fluctuating fields. For a quantitative analysis, we
apply our theory to the case of Helium 2s2p ${}^{1}P$ AIS which offers a
perfect example of an isolated autoionizing resonance.

We begin by solving the system of equations (13) and (14) and inverting the
Laplace transforms to obtain the expressions for
$\left\langle{{\sigma_{11}}(t)}\right\rangle$ and
$\left\langle{{\sigma_{22}}(t)}\right\rangle$. The ionization probability at
the end of the square pulse (constant intensity) at a time T would normally be
given by
${P_{ion}}(T)=1-\left\langle{{\sigma_{11}}(T)}\right\rangle-\left\langle{{\sigma_{22}}(T)}\right\rangle$.
However, at time T there will be population in the excited state that will
decay to the continuum with a rate $\Gamma$ due to the configuration
interaction (the spontaneous decay rate is negligible compared to $\Gamma$).
Therefore, in order to account for this population we should express the
ionization probability at times $t>T$ as ref5 :

${P_{ion}}(t)=1-\left\langle{{\sigma_{11}}(t)}\right\rangle-\left\langle{{\sigma_{22}}(t)}\right\rangle{e^{-\Gamma(t-T)}}$
(20)

We can now plot the ionization probability calculated at times $t>T$ as a
function of the driving frequency around the resonance, for various
intensities, laser bandwidths and interaction times T. However, for this
particular AIS, in addition to the ionization of the neutral, the radiation
can ionize the ${He(1s)}^{+}$ ions produced from autoionization. This process
involves the absorption of one additional photon, producing $He^{2+}$, i.e.
$\alpha$-particles. If it is electrons or He ions that are counted, the
resulting $\alpha$-particles do not influence the observation. But in
transmission, those additional photon absorptions do contribute to the
counting. The calculation must therefore include that additional channel of
photon absorption, for which the cross section is ${1.2}\times{10^{-18}}$cm2;
about the same as the one for the single-photon ionization of the neutral, at
the smooth part of the continuum, away from the resonance. For the sake of
completeness, we have included that additional channel in our calculations, by
writing

$\dot{P}_{ion}^{\diamondsuit}(t)={\dot{P}_{ion}}(t)-P_{ion}^{\diamondsuit}(t){\gamma_{DI}}$
(21)

where ${P_{ion}^{\diamondsuit}(t)}$ is the ionization probability including
the double ionization of Helium and ${{\gamma_{DI}}}$ is the rate with which
the Helium ions produced from autoionization, absorb one additional photon.
This rate can be expressed as the product of the relevant cross section of the
process and the photon flux. The results will be compared with the numerical
results of Appendix A, in which we use equations (18) and (19) under the
assumption of a Gaussian shaped pulse.

For the Helium 2s2p ${}^{1}P$ AIS, the parameters involved in the theory
(expressed in atomic units) are ref20 ; ref21 ; ref22 : $q=-2.79$,
$\Gamma=1.37\times{{10}^{-3}}$, $\Omega=0.025\frac{{{E_{0}}}}{2}$,
$\gamma=0.1775{{\rm I}_{0}}$. We also set ${\omega_{g}}=0$, therefore the
energy difference ${\omega_{ag}}$ ($\hbar=1$) is equal to the energy of the
2s2p ${}^{1}P$ AIS, namely ${\omega_{ag}}=65.40eV\simeq 2.211$a.u. The
autoionization lifetime is approximately 18fs.

Figure 1: Probability of ionization as a function of the driving frequency for
various intensities and $T=120fs$ , ${\gamma_{L}}=0.0018$a.u. Blue Line:
${I_{0}}={10^{13}}$W/cm2, Red Line: ${I_{0}}=5\times{10^{13}}$W/cm2, Purple
Line: ${I_{0}}={10^{14}}$W/cm2 and Green Line:
${I_{0}}=5\times{10^{14}}$W/cm2.

Figure 1 illustrates the effects of the intensity on the AI profile. Notice
that the Rabi frequency becomes comparable to the AI width at intensities
around $2\times{10^{14}}$W/cm2. For an interaction time around 120fs we see
that, for strong intensities the ionization profile is almost flat. For weaker
intensities the profile has an asymmetric form with its peak around the
resonant frequency. The minimum, that is also barely visible in the Green
curve, arises due to the interference between the direct ionization channel
and the indirect channel via the configuration interaction, i.e. the Coulomb
interaction for the problem at hand. The position of the minimum is intensity-
dependant, a result that has also been noted before ref5 , due to the
modification of the interference as the intensity changes. In contrast to the
non-fluctuating weak field case ref14 , we can see that the ionization
probability is non-zero at the minimum. This result is due to the presence of
a finite laser bandwidth, which samples signal from the wings although the
driving frequency is tuned exactly at the minimum. As a result, the minimum is
”filled out”. Even for weak intensities the profile bears no resemblance to
the usual textbook Fano profile ref14  and an attempt to fit the weak field
curves with the standard Fano parameters $q$ and $\varepsilon$, leads to
totally irrelevant values of $q$.

Figure 2: Probability of ionization as a function of the driving frequency for
various intensities and $T=20$fs, ${\gamma_{L}}=0.0018$a.u. Blue Line:
${I_{0}}={10^{13}}$W/cm2, Red Line: ${I_{0}}=5\times{10^{13}}$W/cm2, Purple
Line: ${I_{0}}={10^{14}}$W/cm2 and Green Line:
${I_{0}}=5\times{10^{14}}$W/cm2.

It might seem reasonable to infer that the distortion of the profile in high
intensities is due to power broadening. Although the intensity does play a
significant role in the modifications of the AI profile, it is the combination
of intensity and interaction time that truly determines the profile shape. In
figure 2, we choose an interaction time that is 6 times smaller, i.e. $T=20$fs
for different values of the intensity. At intensities such that the Rabi
frequency is comparable to the AI width, we can see that the profile is not
completely flat if the interaction time is chosen so that it is comparable to
the AI lifetime.

Figure 3: Probability of ionization as a function of the driving frequency for
various interaction times T and ${I_{0}}={10^{13}}$W/cm2,
${\gamma_{L}}=0.0018$a.u. Blue Line: $T=120$fs, Red Line: $T=240$fs, Purple
Line: $T=480$fs and Green Line: $T=960$fs.

In figure 3 we explore the effects of the interaction time on the AI profile
for a weak field of constant intensity. The interactions times are chosen
sufficiently larger compared to the AIS lifetime (18fs). As the interaction
time increases, the ionization probability generally increases, as there is
more time available for the atom to be ionized. From figures 1 to 3 we can
safely assume that the ionization is mainly determined by whether the system
is time saturated, i.e. if for a given intensity, the time that the field is
present is sufficient for the atom to be ionized completely.

Figure 4: Probability of ionization as a function of the driving frequency for
various interaction times T and ${I_{0}}={10^{14}}$W/cm2,
${\gamma_{L}}=0.0018$a.u. Blue Line: $T=5$fs, Red Line: $T=10$fs, Purple Line:
$T=15$fs and Green Line: $T=20$fs.

If on the other hand, the interaction time becomes sufficiently short, we will
begin to observe a broadening of the profile, with the asymmetry obscured. As
an example, in figure 4 the interaction times are chosen such that they
introduce a visible Fourier broadening. The main reason of this distortion is
not the Fourier broadening itself, but a combination of power broadening and
Fourier broadening due to short pulse durations, with the last being the
dominant broadening mechanism.

Figure 5: Probability of ionization as a function of the driving frequency for
various laser bandwidths and ${I_{0}}={10^{13}}$W/cm2, $T=150$fs. Blue Line:
${\gamma_{L}}=0.0001$a.u. , Red Line: ${\gamma_{L}}=0.0005$a.u., Black Line:
${\gamma_{L}}=0.003$a.u. and Green Line: ${\gamma_{L}}=0.001$a.u.

In figure 5 we illustrate the effects of the field bandwidth on the AI
profile. We can clearly see that when the bandwidth of the field becomes
sufficiently larger than the autoionization width
$\Gamma=1.37\times{{10}^{-3}}$ a.u., the FWHM of the profile is mainly
determined by ${\gamma_{L}}$. As the bandwidth increases the profile broadens
and both the minimum and the maximum tend to be smoothed out. For very large
bandwidths the asymmetry tends to become less visible since the profile
becomes almost flat. If the bandwidth is sufficiently smaller than the AI
width then the FWHM is determined mainly by $\Gamma$ or by the interaction
time if it is shorter than the AI lifetime.

##  IV Closing Remarks

The effects discussed in the previous sections were of no importance in
experiments under synchrotron sources which have small bandwidths and low
intensity. However, the theory shows that when the laser bandwidth is
substantial and is included in the calculations, the resulting profiles can be
distorted dramatically. At the same time parameters such as pulse duration and
intensity are interwoven in a non-linear fashion, leading to further
distortion of the profile. For intensities in the strong field regime, as
defined above, the presence of intensity fluctuations ushers in problems that
we have only glimpsed at in this paper. The theoretical techniques developed
and employed in previous work ref2  in the context of bound states are not
directly applicable here. One way of approaching that case is through
numerical simulation, as employed in ref19  for Auger resonances. An AI
resonance, however, presents additional difficulties due to the interference
between the discreet-discreet and discreet-continuum transitions. Although one
might reasonably expect the qualitative behavior to be similar to that found
for Auger resonances, substantial quantitative differences are to be expected.
The strong driving of AI resonances viewed over the last 30 years is never
free of surprises. The strong coupling between AI resonances ref4 ; ref5 ;
ref6  in the presence of intensity fluctuations promises to be even more
challenging. In view of ongoing and planned related experiments under FEL
radiation, these theoretical issues have now come to center stage.

## Acknowledgements

The authors would like to thank G. M. Nikolopoulos for his careful reading of
the manuscript and helpful remarks. We also gratefully acknowledge many and
ongoing discussions with Drs. Thomas Pfeifer and Christian Ott, on
experimental as well as theoretical issues related to this work.

## Appendix: Gaussian Pulse Shape Results

In this appendix we present our results using equations (18) and (19) for a
Gaussian shaped pulse. For the sake of comparison with the analytical results
of constant intensity, we use the same combinations of the relevant parameters
appearing in figures 1 to 5. Note that for a Gaussian pulse we refer to
${I_{0}}$ as the peak intensity.

Figure 6: Probability of ionization as a function of the driving frequency for
various intensities and $T=120fs$ , ${\gamma_{L}}=0.0018$a.u. under a Gaussian
pulse. Blue Line: ${I_{0}}={10^{13}}$W/cm2, Red Line:
${I_{0}}=5\times{10^{13}}$W/cm2, Purple Line: ${I_{0}}={10^{14}}$W/cm2 and
Green Line: ${I_{0}}=5\times{10^{14}}$W/cm2. Figure 7: Probability of
ionization as a function of the driving frequency for various intensities and
$T=20$fs, ${\gamma_{L}}=0.0018$a.u. under a Gaussian pulse. Blue Line:
${I_{0}}={10^{13}}$W/cm2, Red Line: ${I_{0}}=5\times{10^{13}}$W/cm2, Purple
Line: ${I_{0}}={10^{14}}$W/cm2 and Green Line:
${I_{0}}=5\times{10^{14}}$W/cm2. Figure 8: Probability of ionization as a
function of the driving frequency for various interaction times T and
${I_{0}}={10^{13}}$W/cm2, ${\gamma_{L}}=0.0018$a.u. under a Gaussian pulse.
Blue Line: $T=120$fs, Red Line: $T=240$fs, Purple Line: $T=480$fs and Green
Line: $T=960$fs. Figure 9: Probability of ionization as a function of the
driving frequency for various interaction times T and
${I_{0}}={10^{14}}$W/cm2, ${\gamma_{L}}=0.0018$a.u. under a Gaussian pulse.
Blue Line: $T=5$fs, Red Line: $T=10$fs, Purple Line: $T=15$fs and Green Line:
$T=20$fs.

Comparing the above diagrams to the ones presented in the previous section we
can safely accept that the shape of the pulse generally does not have a very
important impact on the determination of the AI profile. This result is
generally known and is widely used for analytical and numerical
simplifications. However, the comparison between figures 2 and 7 as well as 4
and 9 respectively, reveals that if the pulse duration is of the order of the
AI lifetime, a Gaussian pulse produces a lineshape different from that
produced by a square pulse, under the same combination of the relevant
parameters. The general picture that arises is that the Gaussian pulse for
such interaction times is less effective in AIS ionization than a square
pulse. This seems quite logical since the square pulse forces the atom to be
driven by a field whose intensity is ${I_{0}}$ over the whole duration of the
pulse, whereas the Gaussian pulse has a peak value of ${I_{0}}$ and its wings
at smaller intensities. The above results have also been tested with
trapezoidal and Lorentzian pulse shapes and the picture doesn’t differ a lot.
However, we should always be aware that equations (18) and (19) are rate
equations and the validity of the results becomes questionable for very strong
fields where the approximation (17) breaks down.

Figure 10: Probability of ionization as a function of the driving frequency
for various laser bandwidths and ${I_{0}}={10^{13}}$W/cm2, $T=150$fs under a
Gaussian pulse. Blue Line: ${\gamma_{L}}=0.0001$a.u. , Red Line:
${\gamma_{L}}=0.0005$a.u., Black Line: ${\gamma_{L}}=0.003$a.u. and Green
Line: ${\gamma_{L}}=0.001$a.u.

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# Homotopy type of the unitary group of the uniform Roe algebra on
$\mathbb{Z}^{n}$

Tsuyoshi Kato  Department of Mathematics, Kyoto University, Kyoto, 606-8502,
Japan<EMAIL_ADDRESS>,  Daisuke Kishimoto  Department of
Mathematics, Kyoto University, Kyoto, 606-8502, Japan
<EMAIL_ADDRESS>and  Mitsunobu Tsutaya  Faculty of Mathematics,
Kyushu University, Fukuoka 819-0395, Japan<EMAIL_ADDRESS>
###### Abstract.

We study the homotopy type of the space of the unitary group
$\operatorname{U}_{1}(C^{\ast}_{u}(|\mathbb{Z}^{n}|))$ of the uniform Roe
algebra $C^{\ast}_{u}(|\mathbb{Z}^{n}|)$ of $\mathbb{Z}^{n}$. We show that the
stabilizing map
$\operatorname{U}_{1}(C^{\ast}_{u}(|\mathbb{Z}^{n}|))\to\operatorname{U}_{\infty}(C^{\ast}_{u}(|\mathbb{Z}^{n}|))$
is a homotopy equivalence. Moreover, when $n=1,2$, we determine the homotopy
type of $\operatorname{U}_{1}(C^{\ast}_{u}(|\mathbb{Z}^{n}|))$, which is the
product of the unitary group
$\operatorname{U}_{1}(C^{\ast}(|\mathbb{Z}^{n}|))$ (having the homotopy type
of $\operatorname{U}_{\infty}(\mathbb{C})$ or $\mathbb{Z}\times
B\operatorname{U}_{\infty}(\mathbb{C})$ depending on the parity of $n$) of the
Roe algebra $C^{\ast}(|\mathbb{Z}^{n}|)$ and rational Eilenberg–MacLane
spaces.

###### Key words and phrases:

uniform Roe algebra, Roe algebra, unitary group, homotopy type, operator
$K$-theory

###### 2010 Mathematics Subject Classification:

55Q52 (Primary), 46L80 (Secondary)

Kato was supported by JSPS KAKENHI 17K18725 and 17H06461. Kishimoto was
supported by JSPS KAKENHI 17K05248 and 19K03473. Tsutaya was supported by JSPS
KAKENHI 19K14535

##  1\. Introduction

For a $C^{\ast}$-algebra $A$, let $\operatorname{GL}_{d}(A)$ and
$\operatorname{U}_{d}(A)$ denote the space of the invertible and unitary
matrices with entries in $A$, respectively. It it well-known that they always
have the same homotopy type. We will often refer only to
$\operatorname{U}_{d}(A)$ but most statements are valid for
$\operatorname{GL}_{d}(A)$ as well. There have been a lot of works on the
homotopy theory of $\operatorname{U}_{d}(A)$ and some of them have important
applications. For finite-dimensional case, the complex-valued unitary matrices
$\operatorname{U}_{d}(\mathbb{C})$ is just the usual unitary group acting
linearly on $\mathbb{C}^{d}$. For inifinite-dimensional case, Kuiper [Kui65]
proved that the space of all unitary operators on an inifinite-dimensional
Hilbert space is contractible. This result is basic in the Atiyah–Singer index
theory. This kind of contractibility result has been extended to
$\operatorname{U}_{d}(A)$ of some other algebras $A$ while $A$ is all the
bounded operators on a infinite dimensional Hilbert space in the original
result. Of course it is not always the case for $\operatorname{U}_{d}(A)$ of
other infinite-dimensional $C^{\ast}$-algebras $A$. In general, it is hard to
determine the homotopy type of $\operatorname{U}_{d}(A)$.

Let us use the notation

$\operatorname{GL}_{\infty}(A)=\lim_{d\to\infty}\operatorname{GL}_{d}(A)\quad\text{and}\quad\operatorname{U}_{\infty}(A)=\lim_{d\to\infty}\operatorname{U}_{d}(A).$

It is well-known that $\operatorname{U}_{\infty}(A)$ has the same homotopy
type as $\operatorname{U}_{1}(A\otimes\mathcal{K})$ where $\mathcal{K}$ is the
space of compact operators. The $K$-theory $K_{i}(A)$ ($i=0,1$) is a basic
homotopy invariant of $A$, which is characterized as

$K_{0}(A)=\pi_{1}(\operatorname{U}_{\infty}(A))\quad\text{and}\quad
K_{1}(A)=\pi_{0}(\operatorname{U}_{\infty}(A)).$

Since $\operatorname{U}_{d}(A)$ is not necessarily homotopy equivalent to
$\operatorname{U}_{\infty}(A)$, $K_{i}(A)$ is not a so strong invariant in
general. But sometimes the natural map
$\operatorname{U}_{d}(A)\to\operatorname{U}_{\infty}(A)$, which we will call
the stabilizing map, becomes a homotopy equivalence. Study on such stability
seems to trace back to the work of Bass [Bas64]. There have been a various
works on this kind of stability. Rieffel introduced the topological stable
rank in [Rie83] and applied it to show the stability of the non-commutative
torus in [Rie87], which is a key tool in the present work. It is difficult in
general to determine how stable a given $C^{\ast}$-algebra is.

In the present paper, we study the stability of the uniform Roe algebra
$C^{\ast}_{u}(|\mathbb{Z}^{n}|)$ on $\mathbb{Z}^{n}$ and investigate its
homotopy type. The uniform Roe algebra $C^{\ast}_{u}(X)$ of a metric space $X$
introduced by Roe in [Roe88] to establish an index theory on open manifolds,
where the index lives in the $K$-theory $K_{\ast}(C^{\ast}_{u}(X))$. The
algebra $C^{\ast}_{u}(X)$ itself is also important since it encodes a kind of
“large scale geometry” of $X$. Studying the homotopy type of
$\operatorname{U}_{d}(C^{\ast}_{u}(X))$ will provide more insights from a
homotopy theoretic viewpoint, which cannot be obtained only from its
$K$-theory. But there are only a few works on the homotopy type of
$\operatorname{U}_{d}(C^{\ast}_{u}(X))$ yet. For example, Manuilov and
Troitsky [MT21] studied some condition for
$\operatorname{U}_{d}(C^{\ast}_{u}(X))$ being contractible. In the present
work, we observe the other extreme, that is,
$\operatorname{U}_{d}(C^{\ast}_{u}(|\mathbb{Z}^{n}|))$ has a highly nontrivial
homotopy type.

We give some comment on the relation with our previous work [KKT] on the space
$\mathcal{U}$ of finite propagation unitary operators on $\mathbb{Z}$. Note
that $\operatorname{U}_{1}(C^{\ast}_{u}(|\mathbb{Z}|))$ can be viewed as a
kind of completion of $\mathcal{U}$. We determined the homotopy type of
$\mathcal{U}$ there. But it is not clear whether $\mathcal{U}$ has the same
homotopy type as $\operatorname{U}_{1}(C^{\ast}_{u}(|\mathbb{Z}|))$. Actually,
they turn out to have the same homotopy type (Theorem 1.2). Also, the method
there does not seem to be extended to $\mathbb{Z}^{n}$ when $n\geq 2$. We
employ rather operator algebraic technique in the present paper. Our method
here reduces the problem to determine the homotopy type of
$\operatorname{U}_{d}(C^{\ast}_{u}(|\mathbb{Z}^{n}|))$ to the one to show the
surjectivity of the homomorphism on $K$-theory
$K_{\ast}(C^{\ast}_{u}(|\mathbb{Z}^{n}|))\to
K_{\ast}(C^{\ast}(|\mathbb{Z}^{n}|))$ induced from the inclusion (Proposition
7.4), where $C^{\ast}(|\mathbb{Z}^{n}|)$ denotes the Roe algebra of
$\mathbb{Z}^{n}$.

For stability, we show the following theorem in Section 3.

######  Theorem 1.1.

For any integer $n\geq 1$, the stabilizing maps

$\displaystyle\operatorname{GL}_{1}(C^{\ast}_{u}(|\mathbb{Z}^{n}|))\to\operatorname{GL}_{\infty}(C^{\ast}_{u}(|\mathbb{Z}^{n}|)),$
$\displaystyle\operatorname{U}_{1}(C^{\ast}_{u}(|\mathbb{Z}^{n}|))\to\operatorname{U}_{\infty}(C^{\ast}_{u}(|\mathbb{Z}^{n}|)),$
$\displaystyle\operatorname{GL}_{1}(C^{\ast}(|\mathbb{Z}^{n}|))\to\operatorname{GL}_{\infty}(C^{\ast}(|\mathbb{Z}^{n}|)),$
$\displaystyle\operatorname{U}_{1}(C^{\ast}(|\mathbb{Z}^{n}|))\to\operatorname{U}_{\infty}(C^{\ast}(|\mathbb{Z}^{n}|))$

between the spaces of invertible and unitary elements are homotopy
equivalences. This implies that these maps induce the following isomorphisms
on homotopy groups for all $i\geq 0$:

$\displaystyle\pi_{i}(\operatorname{GL}_{1}(C^{\ast}_{u}(|\mathbb{Z}^{n}|)))\cong\pi_{i}(\operatorname{U}_{1}(C^{\ast}_{u}(|\mathbb{Z}^{n}|)))\cong\begin{cases}K_{1}(C^{\ast}_{u}(|\mathbb{Z}^{n}|))&\text{$i$
is even,}\\\ K_{0}(C^{\ast}_{u}(|\mathbb{Z}^{n}|))&\text{$i$ is
odd,}\end{cases}$
$\displaystyle\pi_{i}(\operatorname{GL}_{1}(C^{\ast}(|\mathbb{Z}^{n}|)))\cong\pi_{i}(\operatorname{U}_{1}(C^{\ast}(|\mathbb{Z}^{n}|)))\cong\begin{cases}K_{1}(C^{\ast}(|\mathbb{Z}^{n}|))&\text{$i$
is even,}\\\ K_{0}(C^{\ast}(|\mathbb{Z}^{n}|))&\text{$i$ is odd.}\end{cases}$

Let $K(V,i)$ denote the Eilenberg–MacLane space of type $(V,i)$ and
$B\operatorname{U}_{\infty}(\mathbb{C})$ denote the classifying space of the
unitary group $\operatorname{U}_{\infty}(\mathbb{C})$. Also, for based spaces
$X_{i}$ ($i=1,2,\ldots$), define

$\prod_{i\geq 1}^{\circ}X_{i}=\lim_{k\to\infty}(X_{1}\times
X_{2}\times\cdots\times X_{k}).$

For the homotopy type of
$\operatorname{U}_{1}(C^{\ast}_{u}(|\mathbb{Z}^{n}|))$, we show the following
results when $n=1,2$.

######  Theorem 1.2.

There exist homotopy equivalences of infinite loop spaces

$\operatorname{GL}_{1}(C^{\ast}_{u}(|\mathbb{Z}|))\simeq\operatorname{U}_{1}(C^{\ast}_{u}(|\mathbb{Z}|))\simeq\mathbb{Z}\times
B\operatorname{U}_{\infty}(\mathbb{C})\times\prod_{i\geq
1}^{\circ}K(\ell^{\infty}(\mathbb{Z},\mathbb{Z})_{S},2i-1).$

######  Theorem 1.3.

There exist homotopy equivalences of infinite loop spaces

$\operatorname{GL}_{1}(C^{\ast}_{u}(|\mathbb{Z}^{2}|))\simeq\operatorname{U}_{1}(C^{\ast}_{u}(|\mathbb{Z}^{2}|))\simeq
V_{1}\times\operatorname{U}_{\infty}(\mathbb{C})\times\prod_{i\geq
1}^{\circ}(K(V_{0},2i-1)\times K(V_{1},2i)),$

where $V_{0},V_{1}$ are the rational vector spaces given by

$V_{0}=\ker[K_{0}(C^{\ast}_{u}(|\mathbb{Z}^{2}|))\to
K_{0}(C^{\ast}(|\mathbb{Z}^{2}|))],\quad
V_{1}=K_{1}(C^{\ast}_{u}(|\mathbb{Z}^{2}|))$

and the product factor $V_{1}$ is a discrete space.

More detailed descriptions of the vector spaces $V_{0}$ and $V_{1}$ appear in
the proof of Lemma 7.7.

We will see the existence of a homotopy section of the inclusion
$\operatorname{U}_{1}(C^{\ast}_{u}(|\mathbb{Z}|))\to\operatorname{U}_{1}(B^{\mathrm{SW}})$
in Section 6, where $\operatorname{U}_{1}(B^{\mathrm{SW}})$ is the
Segal–Wilson restricted unitary group [SW85] having the homotopy type of
$\mathbb{Z}\times B\operatorname{U}_{\infty}(\mathbb{C})$. This implies
Theorem 1.2. We also show in Section 7 that, for any integer $n\geq 1$, the
inclusion
$\operatorname{U}_{1}(C^{\ast}_{u}(|\mathbb{Z}^{n}|))\to\operatorname{U}_{1}(C^{\ast}(|\mathbb{Z}^{n}|))$
admits a homotopy section if and only if the homomorphism
$K_{\ast}(C^{\ast}_{u}(|\mathbb{Z}^{n}|))\to
K_{\ast}(C^{\ast}(|\mathbb{Z}^{n}|))$ is surjective. Since we can see it is
surjective when $n=1,2$, Theorems 1.2 again and 1.3 follows. If one could show
the surjectivity for $n\geq 3$, then a similar homotopy decomposition will
immediately follow.

This paper is organized as follows. We fix our notation in Section 2. In
Section 3, we recall Rieffel’s results on stability and show Theorem 1.1. In
Section 4, we recall the Bott periodicity realized as a $\ast$-homomorphism.
In Section 5, we recall the Segal–Wilson restricted unitary group and show its
stability. In Section 6, we show Theorem 1.2 using the Segal–Wilson restricted
unitary group. In Section 7, we discuss the homotopy type of
$\operatorname{U}_{d}(C^{\ast}_{u}(|\mathbb{Z}^{n}|))$ for general $n\geq 1$
and show Theorems 1.2 again and 1.3.

##  2\. Notation

The $C^{\ast}$-algebra of bounded operators on a Hilbert space $V$ is denoted
by $\mathcal{B}(V)$ and the subalgebra of compact operators by
$\mathcal{K}(V)$. We write the operator norm of $T\in\mathcal{B}(V)$ as
$\|T\|$. The Hilbert space of square summable sequences indexed by a discrete
group $\Gamma$ will be written as

$\ell^{2}(\Gamma)=\\{(v_{g})_{g}\mid\sum_{g\in\Gamma}|v_{g}|^{2}<\infty\\}.$

We also consider the tensor product Hilbert space
$\ell^{2}(\Gamma)\otimes\mathcal{H}$ with an infinite dimensional separable
Hilbert space $\mathcal{H}$.

A bounded operator $T\in\mathcal{B}(\ell^{2}(\Gamma))$ can be expressed in the
matrix form as

$T=(T_{g,h})_{g,h},\quad T_{g,h}\in\mathbb{C}.$

For $T\in\mathcal{B}(\ell^{2}(\Gamma)\otimes\mathcal{H})$, we also have a
similar expression $T=(T_{g,h})_{g,h}$ with
$T_{g,h}\in\mathcal{B}(\mathcal{H})$.

######  Definition 2.1.

Let $\Gamma$ be a finitely generated group and $d$ denote the word metric with
respect to some finite set of generators. We say that a bounded operator
$T\in\mathcal{B}(\ell^{2}(\Gamma))$ has finite propagation if

$\operatorname{prop}(T)=\sup\\{d(g,h)\mid T_{g,h}\neq 0\\}$

is finite. We define finite propagation for
$T=(T_{g,h})_{g,h}\in\mathcal{B}(\ell^{2}(\Gamma)\otimes\mathcal{H})$
similarly.

######  Example 2.2.

The shift $S_{x}\in\mathcal{B}(\ell^{2}(\Gamma))$ by $x\in\Gamma$ is defined
by

$S_{x}=((S_{x})_{g,h})_{g,h},\quad S_{g,h}=\begin{cases}1&g^{-1}h=x,\\\
0&\text{otherwise.}\end{cases}$

The operator $S_{x}$ is a unitary operator with
$\operatorname{prop}(S_{x})=d(x,1)$.

It is easy to see that the definition of having finite propagation is
independent of the choice of generators while the value of
$\operatorname{prop}(T)$ depends on the word metric. Since we have

$\operatorname{prop}(ST)\leq\operatorname{prop}(S)+\operatorname{prop}(T),\quad\operatorname{prop}(T^{\ast})=\operatorname{prop}(T),\quad\operatorname{prop}(1)=0$

for any finite propagation operators $S,T\in\mathcal{B}(\ell^{2}(\Gamma))$,
the subset of finite propagation operators becomes a unital $\ast$-subalgebra
of $\mathcal{B}(\ell^{2}(\Gamma))$. Similar properties hold for finite
propagation operators $S,T\in\mathcal{B}(\ell^{2}(\Gamma)\otimes\mathcal{H})$
such that the components $T_{g,h}$ and $S_{g,h}$ are compact operators.

######  Definition 2.3.

The uniform Roe algebra $C^{\ast}_{u}(|\Gamma|)$ of $\Gamma$ is the norm
closure of the algebra of finite propagation operators in
$\mathcal{B}(\ell^{2}(\Gamma))$.

######  Definition 2.4.

The Roe algebra $C^{\ast}(|\Gamma|)$ of $\Gamma$ is the norm closure of the
algebra of finite propagation operators
$T\in\mathcal{B}(\ell^{2}(\Gamma)\otimes\mathcal{H})$ such that each component
$T_{g,h}$ is a compact operator.

######  Remark 2.5.

We follow the usual notation $C^{\ast}(|\Gamma|)$ for the Roe algebra of
$\Gamma$ to distinguish it from the group $C^{\ast}$-algebra of $\Gamma$
though we do not consider the latter here.

We will consider the uniform Roe algebra $C^{\ast}_{u}(|\Gamma|)$ is a
subalgebra of the Roe algebra $C^{\ast}(|\Gamma|)$ with respect to some
inclusion $\mathbb{C}\subset\mathcal{H}$.

We use the symbol $\ell^{\infty}(\Gamma,\mathbb{C})$ to express the Banach
algebra of $\mathbb{C}$-valued bounded sequences indexed by $\Gamma$ rather
than the simpler symbol $\ell^{\infty}(\Gamma)$ since we also consider the
abelian group of $\mathbb{Z}$-valued bounded sequences
$\ell^{\infty}(\Gamma,\mathbb{Z})$.

The group $\Gamma$ acts on the algebras $\ell^{\infty}(\Gamma,\mathbb{C})$ and
$\ell^{\infty}(\Gamma,\mathcal{K}(\mathcal{H}))$ by right translation. The
action by $x\in\Gamma$ is compatible with the conjugation by $S_{x}$ through
the diagonal inclusion $\ell^{\infty}(\Gamma,\mathbb{C})\to
C^{\ast}_{u}(|\Gamma|)$ or $\ell^{\infty}(\Gamma,\mathcal{K}(\mathcal{H}))\to
C^{\ast}(|\Gamma|)$ given by

$(t_{g})_{g}\mapsto(T_{g,h})_{g,h},\quad T_{g,h}=\begin{cases}t_{g}&g=h,\\\
0&\text{otherwise.}\end{cases}$

Moreover, this inclusion extends to the well-known isomorphisms

$\ell^{\infty}(\Gamma,\mathbb{C})\rtimes\Gamma\cong
C^{\ast}_{u}(|\Gamma|),\quad\ell^{\infty}(\Gamma,\mathcal{K}(\mathcal{H}))\rtimes\Gamma\cong
C^{\ast}(|\Gamma|)$

from the reduced crossed products of $C^{\ast}$-algebras. For example, see
[Roe03, Theorem 4.28].

The $d\times d$-matrix algebra $M_{d}(A)$ of a $C^{\ast}$-algebra $A$ is again
a $C^{\ast}$-algebra. The spaces of invertible elements and unitary elements
in $M_{d}(A)$ will be denoted as $\operatorname{GL}_{d}(A)$ and
$\operatorname{U}_{d}(A)$. The stabilizing maps are given as

$\operatorname{GL}_{1}(A)\to\operatorname{GL}_{\infty}(A)=\lim_{d\to\infty}\operatorname{GL}_{d}(A),\quad\operatorname{U}_{1}(A)\to\operatorname{U}_{\infty}(A)=\lim_{d\to\infty}\operatorname{U}_{d}(A),$

where the inductive limits are taken along the inclusions
$\operatorname{GL}_{d}(A)\subset\operatorname{GL}_{d+1}(A)$ and
$\operatorname{U}_{d}(A)\subset\operatorname{U}_{d+1}(A)$. The inductive limit
spaces $\operatorname{GL}_{\infty}(A)$ and $\operatorname{U}_{\infty}(A)$ are
well-known to be homotopy equivalent to the spaces
$\operatorname{GL}_{1}(A\otimes\mathcal{K}(\mathcal{H}))$ and
$\operatorname{U}_{1}(A\otimes\mathcal{K}(\mathcal{H}))$.

##  3\. Stability

The aim of this section is to prove Theorem 1.1. Once the assumption of the
following result by Rieffel [Rie87] is verified, the theorem will immediately
follow.

######  Theorem 3.1 (Rieffel).

Let $A$ be a unital $C^{\ast}$-algebra. If $A$ is tsr-boundedly divisible,
then the stabilizing maps

$\operatorname{GL}_{1}(A)\to\operatorname{GL}_{\infty}(A)\quad\text{and}\quad\operatorname{U}_{1}(A)\to\operatorname{U}_{\infty}(A)$

are homotopy equivalences.

######  Remark 3.2.

The original statement of Theorem 4.13 in [Rie87] is involved only with
homotopy groups. But what is actually proved there is slightly stronger as
above.

For the definitions of the topological stable rank
$\operatorname{tsr}(A)\in\mathbb{Z}_{\geq 1}$, see [Rie83]. A
$C^{\ast}$-algebra $A$ is said to be tsr-boundedly divisible [Rie87] if there
is a constant $K$ such that for any integer $m$, there exists an integer
$d\geq m$ such that $A$ is isomorphic to $M_{d}(B)$ for some
$C^{\ast}$-algebra $B$ with $\operatorname{tsr}(B)\leq K$. To verify the
assumption, we need the following two lemmas.

######  Lemma 3.3.

The topological stable ranks of $C^{\ast}_{u}(|\mathbb{Z}^{n}|)$ and
$C^{\ast}(|\mathbb{Z}^{n}|)$ are estimated as

$\operatorname{tsr}(C^{\ast}_{u}(|\mathbb{Z}^{n}|))\leq
n+1\quad\text{and}\quad\operatorname{tsr}(C^{\ast}(|\mathbb{Z}^{n}|))\leq
n+1.$

######  Remark 3.4.

We will see that both $C^{\ast}_{u}(|\mathbb{Z}^{n}|)$ and
$C^{\ast}(|\mathbb{Z}^{n}|)$ are tsr-boundedly divisible using this lemma.
Thus we will actually obtain the estimates
$\operatorname{tsr}(C^{\ast}_{u}(|\mathbb{Z}^{n}|))\leq 2$ and
$\operatorname{tsr}(C^{\ast}(|\mathbb{Z}^{n}|))\leq 2$ by [Rie87, Proposition
4.6].

###### Proof.

Let $A=\mathbb{C}$ or $\mathcal{K}(\mathcal{H})$. Since the invertible
elements in $\ell^{\infty}(\mathbb{Z}^{n},\mathbb{C})$ and
$\mathbb{C}\oplus\ell^{\infty}(\mathbb{Z}^{n},\mathcal{K}(\mathcal{H}))$ are
dense, we have

$\operatorname{tsr}(\ell^{\infty}(\mathbb{Z}^{n},A))=1$

by [Rie83, Proposition 3.1]. Considering the restricted action of
$\mathbb{Z}^{m}\subset\mathbb{Z}^{n}$ on the first $m$ factors of
$\mathbb{Z}^{n}$, we obtain the isomorphism

$\ell^{\infty}(\mathbb{Z}^{n},A)\rtimes\mathbb{Z}^{m+1}\cong(\ell^{\infty}(\mathbb{Z}^{n},A)\rtimes\mathbb{Z}^{m})\rtimes\mathbb{Z}.$

Thus, by [Rie83, Theorem 7.1], we get the desired estimates on
$\operatorname{tsr}(C^{\ast}_{u}(|\mathbb{Z}|))$ and
$\operatorname{tsr}(C^{\ast}(|\mathbb{Z}^{n}|))$. ∎

######  Lemma 3.5.

For any integer $d\geq 1$, there exist isomorphisms

$\phi\colon C^{\ast}_{u}(|\mathbb{Z}^{n}|)\cong
M_{d}(C^{\ast}_{u}(|\mathbb{Z}^{n}|))\quad\text{and}\quad\phi\colon
C^{\ast}(|\mathbb{Z}^{n}|)\cong M_{d}(C^{\ast}(|\mathbb{Z}^{n}|)).$

###### Proof.

Let $V=\mathbb{C}$ or $\mathcal{H}$. According to the decomposition

$\ell^{2}(\mathbb{Z}^{n})\otimes
V=\bigoplus_{(i_{1},\ldots,i_{n})\in\mathbb{Z}^{n}}V_{(i_{1},\ldots,i_{n})},\quad
V_{(i_{1},\ldots,i_{n})}\cong V,$

we have the matrix expression for
$T\in\mathcal{B}(\ell^{2}(\mathbb{Z}^{n})\otimes V)$

$T=(T_{(i_{1},\ldots,i_{n})(j_{1},\ldots,j_{n})})_{(i_{1},\ldots,i_{n})(j_{1},\ldots,j_{n})},\quad
T_{(i_{1},\ldots,i_{n})(j_{1},\ldots,j_{n})}\colon V_{(j_{1},\ldots,j_{n})}\to
V_{(i_{1},\ldots,i_{n})}.$

Consider the map $\phi\colon\mathcal{B}(\ell^{2}(\mathbb{Z}^{n})\otimes V)\to
M_{d}(\mathcal{B}(\ell^{2}(\mathbb{Z}^{n})\otimes V))$ given by

$\phi(T)_{(i_{1},\ldots,i_{n})(j_{1},\ldots,j_{n})}=\begin{pmatrix}T_{(di_{1},\ldots,i_{n})(dj_{1},\ldots,j_{n})}&\cdots&T_{(di_{1},\ldots,i_{n})(dj_{1}+d-1,\ldots,j_{n})}\\\
\vdots&\ddots&\vdots\\\
T_{(di_{1}+d-1,\ldots,i_{n})(dj_{1},\ldots,j_{n})}&\cdots&T_{(di_{1}+d-1,\ldots,i_{n})(dj_{1}+d-1,\ldots,j_{n})}\end{pmatrix}\in
M_{d}(\mathcal{B}(V)).$

The restrictions to
$C^{\ast}_{u}(|\mathbb{Z}^{n}|)\subset\mathcal{B}(\ell^{2}(\mathbb{Z}^{n}))$
and
$C^{\ast}(|\mathbb{Z}^{n}|)\subset\mathcal{B}(\ell^{2}(\mathbb{Z}^{n})\otimes\mathcal{H})$
are desired isomorphisms. ∎

######  Remark 3.6.

When $n=1$, the map $\phi$ is just taking the block matrix of which each block
is a $d\times d$-matrix.

###### Proof of Theorem 1.1.

By Lemmas 3.3 and 3.5, we can apply Theorem 3.1 to
$C^{\ast}_{u}(|\mathbb{Z}^{n}|)$ and $C^{\ast}(|\mathbb{Z}^{n}|)$. This
completes the proof of the theorem. ∎

##  4\. Bott periodicity

Let us recall the Bott periodicity of $C^{\ast}$-algebras here. Let $A$ be a
$C^{\ast}$-algebra, which might be non-unital. The direct sum
$\mathbb{C}\oplus A$ is considered as the unitization with unit
$(1,0)\in\mathbb{C}\oplus A$. Define the unitary group
$\operatorname{U}^{\prime}_{d}(A)$ by

$\operatorname{U}^{\prime}_{d}(A)=\\{U\in\operatorname{U}_{n}(\mathbb{C}\oplus
A)\mid U-(I_{d},0)\in M_{d}(A)\\}.$

If $A$ is already unital, we have a canonical isomorphism
$\operatorname{U}^{\prime}_{d}(A)\cong\operatorname{U}_{d}(A)$. So we use the
same symbol $\operatorname{U}_{d}(A)$ for $\operatorname{U}^{\prime}_{d}(A)$
even if $A$ is not unital.

Consider the following space of continuous functions:

$C_{0}(\mathbb{R}^{m},A)=\\{T\colon\mathbb{R}^{m}\to A\mid T\text{ is
continuous and }\lim_{|z|\to\infty}T(z)=0\\}.$

This is a $C^{\ast}$-algebra without unit. Notice that
$C_{0}(\mathbb{R}^{m},A)$ is isomorphic to the space $\Omega^{m}A$ of based
maps from the $m$-sphere $S^{m}$ to $A$ where the basepoint $\ast\in S^{m}$ is
mapped to $0\in A$.

Set the element

$p_{B}(z)=\frac{1}{1+|z|^{2}}\begin{pmatrix}|z|^{2}&z\\\
\bar{z}&1\end{pmatrix}\in M_{2}(\mathbb{C}\oplus
C_{0}(\mathbb{R}^{2},\mathbb{C}))\quad(z\in\mathbb{R}^{2}),$

where we identify $\mathbb{R}^{2}\cong\mathbb{C}$ in the matrix entries. The
Bott map $\beta\colon A\to M_{2}(A\oplus C_{0}(\mathbb{R}^{2},A))$ is a
$\ast$-homomorphism defined by

$\beta(a)=p_{B}\begin{pmatrix}a&0\\\ 0&a\end{pmatrix}=\begin{pmatrix}a&0\\\
0&a\end{pmatrix}p_{B}.$

Then we have the commutative square of unital $C^{\ast}$-algebras

$\textstyle{\mathbb{C}\oplus
A\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\epsilon}$$\scriptstyle{\beta}$$\textstyle{\mathbb{C}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\eta}$$\textstyle{M_{2}(\mathbb{C}\oplus
A\oplus
C_{0}(\mathbb{R}^{2},A))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\epsilon}$$\textstyle{M_{2}(\mathbb{C}\oplus
A)}$

where $\epsilon\colon\mathbb{C}\oplus A\to\mathbb{C}$ and $\epsilon\colon
M_{2}(\mathbb{C}\oplus A\oplus C_{0}(\mathbb{R}^{2},A))\to
M_{2}(\mathbb{C}\oplus A)$ are the projections and $\eta\colon\mathbb{C}\to
M_{2}(\mathbb{C}\oplus A)$ is the unit map. This square induces the
$\ast$-homomorphism between the kernels of $\epsilon$:

$\beta\colon A\to M_{2}(C_{0}(\mathbb{R}^{2},A)).$

We call this $\beta$ the Bott map as well. It is natural in the following
sense: if $f\colon A\to B$ is a $\ast$-homomorphism between
$C^{\ast}$-algebras, then the following square commutes:

$\textstyle{A\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f}$$\scriptstyle{\beta}$$\textstyle{B\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\beta}$$\textstyle{M_{2}(C_{0}(\mathbb{R}^{2},A))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{f_{\ast}}$$\textstyle{M_{2}(C_{0}(\mathbb{R}^{2},B))}$

######  Proposition 4.1.

The Bott map $\beta\colon A\to M_{2}(C_{0}(\mathbb{R}^{2},A))$ induces an
isomorphism on $K$-theory.

######  Remark 4.2.

This can be seen as a formulation of the Bott periodicity. If you wish to
deduce this proposition from the results appearing in [Bla86], it follows from
the observation 9.2.10 on the generator of
$K_{0}(C_{0}(\mathbb{R}^{2},\mathbb{C}))$ and the Künneth theorem for tensor
products (Theorem 23.1.3).

The Bott periodicity provides the natural homotopy equivalence

$\operatorname{U}_{\infty}(A)\xrightarrow{\beta}\operatorname{U}_{\infty}(M_{2}(C_{0}(\mathbb{R}^{2},A)))\simeq\Omega^{2}\operatorname{U}_{\infty}(A),$

which is a group homomorphism. Thus we obtain the following proposition on
infinite loop structure.

######  Proposition 4.3.

The unitary group $\operatorname{U}_{\infty}(A)$ of a $C^{\ast}$-algebra $A$
is equipped with a canonical infinite loop space structure such that the map
$\operatorname{U}_{\infty}(A)\to\operatorname{U}_{\infty}(B)$ induced from a
$\ast$-homomorphism $A\to B$ is an infinite loop map. Moreover, the underlying
loop structure of $\operatorname{U}_{\infty}(A)$ coincides with the group
structure of $\operatorname{U}_{\infty}(A)$.

######  Remark 4.4.

The last sentence in the proposition means that there exists a homotopy
equivalence
$B\operatorname{U}_{\infty}(A)\simeq\Omega\operatorname{U}_{\infty}(A)$ from
the classifying space $B\operatorname{U}_{\infty}(A)$ of the topological group
$\operatorname{U}_{\infty}(A)$.

##  5\. Segal–Wilson restricted unitary group

To study the homotopy type of
$\operatorname{U}_{1}(C^{\ast}_{u}(|\mathbb{Z}|))$, we will relate it with
other spaces. One is the Segal–Wilson restricted unitary group
$\operatorname{U}_{1}(B^{\mathrm{SW}})$ and the other is the unitary group of
the Roe algebra $\operatorname{U}_{1}(C^{\ast}(|\mathbb{Z}|))$. We recall the
former in this section.

We have another matrix expression for $T\in\mathcal{B}(\ell^{2}(\mathbb{Z}))$
as

$T=\begin{pmatrix}T_{--}&T_{-+}\\\ T_{+-}&T_{++}\end{pmatrix},$

where

$\displaystyle
T_{--}\colon\ell^{2}(\mathbb{Z}_{<0})\to\ell^{2}(\mathbb{Z}_{<0}),\quad
T_{-+}\colon\ell^{2}(\mathbb{Z}_{\geq 0})\to\ell^{2}(\mathbb{Z}_{<0}),$
$\displaystyle
T_{+-}\colon\ell^{2}(\mathbb{Z}_{<0})\to\ell^{2}(\mathbb{Z}_{\geq 0}),\quad
T_{++}\colon\ell^{2}(\mathbb{Z}_{\geq 0})\to\ell^{2}(\mathbb{Z}_{\geq 0}).$

######  Definition 5.1.

We define the $C^{\ast}$-algebra $B^{\mathrm{SW}}$ by

$\displaystyle
B^{\mathrm{SW}}:=\\{T\in\mathcal{B}(\mathcal{H})\mid\text{$T_{-+},T_{+-}$ are
compact}\\}.$

The symbol “SW” stands for Segal–Wilson. The unitary group
$\operatorname{U}_{1}(B^{\mathrm{SW}})$ is called the restricted unitary group
in the work of Segal and Wilson [SW85]. They used it as a model of the
infinite Grassmannian.

######  Lemma 5.2 (Segal–Wilson).

The space $\operatorname{U}_{1}(B^{\mathrm{SW}})$ has the homotopy type of
$\mathbb{Z}\times B\operatorname{U}_{\infty}(\mathbb{C})$. Moreover, the map

$\pi_{0}(\operatorname{U}_{1}(B^{\mathrm{SW}}))\to\mathbb{Z},\quad[U]\mapsto\operatorname{ind}(U_{++}),$

is bijective, where $\operatorname{ind}(U_{++})$ denotes the Fredholm index of
the Fredholm operator $U_{++}$.

Let $S=S_{+1}\in B^{\mathrm{SW}}$ the shift operator as in Example 2.2. We
have $\operatorname{ind}S^{n}=n$.

The goal of this section is to see the following.

######  Proposition 5.3.

The stabilizing maps

$\operatorname{GL}_{1}(B^{\mathrm{SW}})\to\operatorname{GL}_{\infty}(B^{\mathrm{SW}})\quad\text{and}\quad\operatorname{U}_{1}(B^{\mathrm{SW}})\to\operatorname{U}_{\infty}(B^{\mathrm{SW}})$

are homotopy equivalences.

To show this, we do not use a kind of stability as in Section 3.

######  Lemma 5.4.

For any integer $d\geq 1$, the inclusion

$\operatorname{U}_{1}(B^{\mathrm{SW}})\to\operatorname{U}_{d}(B^{\mathrm{SW}})$

induces an isomorphism on $\pi_{0}$.

###### Proof.

Consider the composite of the inclusion and the isomorphism $\phi\colon
B^{\mathrm{SW}}\to M_{d}(B^{\mathrm{SW}})$ similar to the one in the proof of
Lemma 3.5:

$\operatorname{U}_{1}(B^{\mathrm{SW}})\to\operatorname{U}_{d}(B^{\mathrm{SW}})\xrightarrow{\phi^{-1}}\operatorname{U}_{1}(B^{\mathrm{SW}}).$

It is easy to see that the image of the shift
$S\in\operatorname{U}_{1}(B^{\mathrm{SW}})$ under this composite again has
index $1$. This implies the lemma. ∎

######  Lemma 5.5.

The $K$-theory of $B^{\mathrm{SW}}$ is computed as

$K_{i}(B^{\mathrm{SW}})\cong\begin{cases}0&i=0,\\\ \mathbb{Z}&i=1,\end{cases}$

where $K_{1}(B^{\mathrm{SW}})$ is generated by the shift
$S\in\operatorname{U}_{1}(B^{\mathrm{SW}})$.

###### Proof.

This follows from the isomorphisms

$K_{0}(B^{\mathrm{SW}})\cong\lim_{d\to\infty}\pi_{1}(\operatorname{U}_{d}(B^{\mathrm{SW}}))\quad\text{and}\quad
K_{1}(B^{\mathrm{SW}})\cong\lim_{d\to\infty}\pi_{0}(\operatorname{U}_{d}(B^{\mathrm{SW}}))$

and Lemmas 5.2 and 5.4. ∎

######  Lemma 5.6.

For any $i\geq 0$, there exists an integer $m\geq 1$ such that the iterated
Bott map

$\beta^{i}\colon\operatorname{U}_{d}(B^{\mathrm{SW}})\to\operatorname{U}_{d}(M_{2^{i}}(C_{0}(\mathbb{R}^{2i},B^{\mathrm{SW}})))$

induces an isomorphism on $\pi_{0}$ if $d\geq m$.

###### Proof.

From the isomorphisms

$\pi_{0}(\operatorname{U}_{d}(M_{2^{i}}(C_{0}(\mathbb{R}^{2i},B^{\mathrm{SW}})))\cong\pi_{2i}(\operatorname{U}_{2^{i}d}(B^{\mathrm{SW}}))\cong\pi_{2i}(\operatorname{U}_{1}(B^{\mathrm{SW}}))\cong\mathbb{Z}$

and

$K_{1}(M_{2^{i}}(C_{0}(\mathbb{R}^{2i},B^{\mathrm{SW}})))\cong\lim_{d\to\infty}\pi_{0}(\operatorname{U}_{d}(M_{2^{i}}(C_{0}(\mathbb{R}^{2i},B^{\mathrm{SW}})))\cong
K_{1}(B^{\mathrm{SW}})\cong\mathbb{Z},$

we can find an integer $m\geq 1$ such that the stabilizing map

$\pi_{0}(\operatorname{U}_{d}(M_{2^{i}}(C_{0}(\mathbb{R}^{2i},B^{\mathrm{SW}})))\to
K_{1}(C_{0}(M_{2^{i}}(\mathbb{R}^{2i},B^{\mathrm{SW}})))$

is an isomorphism if $d\geq m$. Consider the commutative diagram

$\textstyle{\pi_{0}(\operatorname{U}_{d}(B^{\mathrm{SW}}))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\scriptstyle{(\beta^{i})_{\ast}}$$\textstyle{K_{1}(B^{\mathrm{SW}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{(\beta^{i})_{\ast}}$$\textstyle{\pi_{0}(\operatorname{U}_{d}(M_{2^{i}}(C_{0}(\mathbb{R}^{2i},B^{\mathrm{SW}})))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\textstyle{K_{1}(M_{2^{i}}(C_{0}(\mathbb{R}^{2i},B^{\mathrm{SW}}))}$

where the top arrow is an isomorphism by Lemma 5.4 and the right Bott map
$\beta^{i}$ is an isomorphism by Proposition 4.1. Then the lemma follows. ∎

###### Proof of Proposition 5.3.

Take an integer $i\geq 0$. We can find an integer $m\geq 1$ as in Proposition
5.6 and

$\pi_{2i}(\operatorname{U}_{d}(B^{\mathrm{SW}}))\to\pi_{2i}(\operatorname{U}_{\infty}(B^{\mathrm{SW}}))\cong\mathbb{Z}$

is an isomorphism if $d\geq m$. Consider the following commutative diagram:

$\textstyle{\operatorname{U}_{1}(C_{0}(\mathbb{R}^{2i},B^{\mathrm{SW}}))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\textstyle{\operatorname{U}_{1}(M_{2^{i}}(C_{0}(\mathbb{R}^{2i},B^{\mathrm{SW}})))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{U}_{1}(B^{\mathrm{SW}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\beta^{i}}$isom.
on
$\pi_{0}$$\textstyle{\operatorname{U}_{d}(C_{0}(\mathbb{R}^{2i},B^{\mathrm{SW}}))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\textstyle{\operatorname{U}_{d}(M_{2^{i}}(C_{0}(\mathbb{R}^{2i},B^{\mathrm{SW}})))}$$\textstyle{\operatorname{U}_{d}(B^{\mathrm{SW}})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\beta^{i}}$

where the left horizontal arrows are the isomorphisms similar to the one in
Lemma 3.5 and the vertical arrows are the inclusions. Since the composite

$\operatorname{U}_{1}(B^{\mathrm{SW}})\to\operatorname{U}_{d}(B^{\mathrm{SW}})\xrightarrow{\beta^{i}}\operatorname{U}_{d}(M_{2^{i}}(C_{0}(\mathbb{R}^{2i},B^{\mathrm{SW}})))$

induces an isomorphism on $\pi_{0}$, the middle vertical arrow

$\operatorname{U}_{1}(M_{2^{i}}(C_{0}(\mathbb{R}^{2i},B^{\mathrm{SW}})))\to\operatorname{U}_{d}(M_{2^{i}}(C_{0}(\mathbb{R}^{2i},B^{\mathrm{SW}})))$

induces a surjection on $\pi_{0}$. But it is indeed an isomorphism as their
$\pi_{0}$ are isomorphic to $\mathbb{Z}$. Then the map

$\operatorname{U}_{1}(C_{0}(\mathbb{R}^{2i},B^{\mathrm{SW}}))\to\operatorname{U}_{d}(C_{0}(\mathbb{R}^{2n},B^{\mathrm{SW}}))$

induces an isomorphism on $\pi_{0}$. This implies that the map

$\operatorname{U}_{1}(B^{\mathrm{SW}})\to\operatorname{U}_{d}(B^{\mathrm{SW}})$

induces an isomorphism on $\pi_{2i}$. Thus the map

$\operatorname{U}_{1}(B^{\mathrm{SW}})\to\operatorname{U}_{\infty}(B^{\mathrm{SW}})$

induces an isomorphism on $\pi_{2i}$. This completes the proof. ∎

##  6\. Homotopy type of $\operatorname{U}_{1}(C^{\ast}_{u}(|\mathbb{Z}|))$

The goal of this section is to prove Theorem 1.2. The components $T_{-+}$ and
$T_{+-}$ of a finite propagation operator $T\in\mathcal{B}(H)$ are finite rank
operators. This implies the inclusion

$C^{\ast}_{u}(|\mathbb{Z}|)\subset B^{\mathrm{SW}}.$

This map is a key to the proof of Theorem 1.2.

We begin with computing the $K$-theory.

######  Proposition 6.1.

The following isomorphism holds:

$K_{\ast}(C^{\ast}_{u}(|\mathbb{Z}|))\cong\begin{cases}\ell^{\infty}(\mathbb{Z},\mathbb{Z})_{S}&i=0,\\\
\mathbb{Z}&i=1,\end{cases}$

where

$\ell^{\infty}(\mathbb{Z},\mathbb{Z})_{S}=\ell^{\infty}(\mathbb{Z},\mathbb{Z})/\\{a-Sa\mid
a\in\ell^{\infty}(\mathbb{Z},\mathbb{Z})\\}$

is the coinvariant by the shift
$S\colon\ell^{\infty}(\mathbb{Z},\mathbb{Z})\to\ell^{\infty}(\mathbb{Z},\mathbb{Z})$.

###### Proof.

Applying the Pimsner–Voiculescu exact sequence [PV80] to the crossed product

$C^{\ast}_{u}(|\mathbb{Z}|)\cong\ell^{\infty}(\mathbb{Z},\mathbb{C})\rtimes\mathbb{Z},$

we get the six-term cyclic exact sequence:

$\textstyle{K_{0}(\ell^{\infty}(\mathbb{Z},\mathbb{C}))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{1-S}$$\textstyle{K_{0}(\ell^{\infty}(\mathbb{Z},\mathbb{C}))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{K_{0}(C^{\ast}_{u}(|\mathbb{Z}|))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{K_{1}(C^{\ast}_{u}(|\mathbb{Z}|))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{K_{1}(\ell^{\infty}(\mathbb{Z},\mathbb{C}))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{K_{1}(\ell^{\infty}(\mathbb{Z},\mathbb{C}))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{1-S}$

As is well-known, we have

$K_{i}(\ell^{\infty}(\mathbb{Z},\mathbb{C}))\cong\begin{cases}\ell^{\infty}(\mathbb{Z},\mathbb{Z})&i=0,\\\
0&i=1,\end{cases}$

where the induced homomorphism
$S\colon\ell^{\infty}(\mathbb{Z},\mathbb{Z})\to\ell^{\infty}(\mathbb{Z},\mathbb{Z})$
is the shift as well. Thus we can compute
$K_{\ast}(C^{\ast}_{u}(|\mathbb{Z}|))$ by the previous exact sequence. ∎

We saw the homotopy stabilities as in Theorem 1.1 and Proposition 5.3. Then it
is sufficient to investigate the inclusion
$\operatorname{U}_{\infty}(C^{\ast}_{u}(|\mathbb{Z}|))\to\operatorname{U}_{\infty}(B^{\mathrm{SW}})$.

######  Lemma 6.2.

The inclusion
$\operatorname{U}_{\infty}(C^{\ast}_{u}(|\mathbb{Z}|))\to\operatorname{U}_{\infty}(B^{\mathrm{SW}})$
induces isomorphisms on $\pi_{2i}$ for $i\geq 0$.

###### Proof.

By Lemma 5.5, $K_{1}(B^{\mathrm{SW}})$ is isomorphic to $\mathbb{Z}$ and
generated by the shift $S\in B^{\mathrm{SW}}$. Since $S\in
C^{\ast}_{u}(|\mathbb{Z}|)$ and
$K_{1}(C^{\ast}_{u}(|\mathbb{Z}|))\cong\mathbb{Z}$, the map
$K_{1}(C^{\ast}_{u}(|\mathbb{Z}|))\to K_{1}(B^{\mathrm{SW}})$ is an
isomorphism. Thus the map
$\pi_{2i}(\operatorname{U}_{\infty}(C^{\ast}_{u}(|\mathbb{Z}|)))\to\pi_{2i}(\operatorname{U}_{\infty}(B^{\mathrm{SW}}))$
is also an isomorphism. ∎

Let $F_{1}$ be the homotopy fiber of the inclusion
$\operatorname{U}_{\infty}(C^{\ast}_{u}(|\mathbb{Z}|))\to\operatorname{U}_{\infty}(B^{\mathrm{SW}})$.

######  Proposition 6.3.

The space $F_{1}$ has the homotopy type of the product of Eilenberg–MacLane
spaces

$\prod_{i\geq 1}^{\circ}K(\ell^{\infty}(\mathbb{Z},\mathbb{Z})_{S},2i-1).$

where $\ell^{\infty}(\mathbb{Z},\mathbb{Z})_{S}$ is a rational vector space of
uncountable dimension.

###### Proof.

Observing the homotopy exact sequence

$\cdots\to\pi_{i}(F_{1})\to\pi_{i}(\operatorname{U}_{\infty}(C^{\ast}_{u}(|\mathbb{Z}|)))\to\pi_{i}(\operatorname{U}_{\infty}(B^{\mathrm{SW}}))\to\pi_{i-1}(F_{1})\to\cdots,$

we can see that the homotopy fiber inclusion
$F_{1}\to\operatorname{U}_{\infty}(C^{\ast}_{u}(|\mathbb{Z}|))$ induces an
isomorpshim on $\pi_{2i-1}$ and $\pi_{2i}(F_{1})=0$ by Lemma 6.2 and the fact
that $\pi_{2i-1}(\operatorname{U}_{1}(B^{\mathrm{SW}}))=0$. By Proposition
6.1, we have $\pi_{2i-1}(F_{1})\cong\ell^{\infty}(\mathbb{Z},\mathbb{Z})_{S}$.
The abelian group $\ell^{\infty}(\mathbb{Z},\mathbb{Z})_{S}$ is a rational
vector space of uncountable dimension as seen in [KKT, Section 5]. By [KKT,
Lemma 5.4], $F_{1}$ has the homotopy type of the product of Eilenberg–MacLane
spaces as above. ∎

The following easy lemma is useful to study the homotopy type of the unitary
group of a $C^{\ast}$-algebra. Let $p_{r}\in M_{d}(\mathbb{C})$ denote the
projection of rank $r$.

######  Lemma 6.4.

Let $A$ be a $C^{\ast}$-algebra, where we do not require the existence of
unit. For any element $u\in K_{0}(A)$, there exists a (non-unital in general)
$\ast$-homomorphism $f\colon\mathbb{C}\to M_{d}(A)$ such that
$f_{\ast}[p_{1}]\in K_{0}(M_{d}(A))\cong K_{0}(A)$ equals to $u$.

###### Proof.

We can find a projection $p\in M_{d}(\mathbb{C}\oplus A)$ and $r\geq 0$ such
that $u=[p]-[p_{r}]$ in $K_{0}(A)$. Define a $\ast$-homomorphism
$f\colon\mathbb{C}\to M_{d}(A)$ by $f(1)=p$. This is the desired map. ∎

######  Proposition 6.5.

The inclusion
$\operatorname{U}_{\infty}(C^{\ast}_{u}(|\mathbb{Z}|))\to\operatorname{U}_{\infty}(B^{\mathrm{SW}})$
admits a homotopy section, which is an infinite loop map.

###### Proof.

Consider the inclusion of based loop spaces
$\operatorname{U}_{\infty}(C_{0}(\mathbb{R},C^{\ast}_{u}(|\mathbb{Z}|)))\to\operatorname{U}_{\infty}(C_{0}(\mathbb{R},B^{\mathrm{SW}}))$.
By Proposition 4.3, Lemma 6.4 and
$K_{0}(C_{0}(\mathbb{R},C^{\ast}_{u}(|\mathbb{Z}|)))\cong\mathbb{Z}$, there
exists an infinite loop map
$f\colon\operatorname{U}_{\infty}(\mathbb{C})\to\operatorname{U}_{\infty}(C_{0}(\mathbb{R},C^{\ast}_{u}(|\mathbb{Z}|)))$
which induces an isomorphism on $\pi_{2i-1}$ for any $i\geq 1$. It follows
from this and Lemma 6.2 that the composite

$\operatorname{U}_{\infty}(\mathbb{C})\xrightarrow{f}\operatorname{U}_{\infty}(C_{0}(\mathbb{R},C^{\ast}_{u}(|\mathbb{Z}|)))\to\operatorname{U}_{\infty}(C_{0}(\mathbb{R},B^{\mathrm{SW}}))$

is a homotopy equivalence. Then the inclusion of based loop spaces
$\operatorname{U}_{\infty}(C_{0}(\mathbb{R},C^{\ast}_{u}(|\mathbb{Z}|)))\to\operatorname{U}_{\infty}(C_{0}(\mathbb{R},B^{\mathrm{SW}}))$
admits a homotopy section. This implies that the inclusion of the double loop
space
$\operatorname{U}_{\infty}(C_{0}(\mathbb{R}^{2},C^{\ast}_{u}(|\mathbb{Z}|)))\to\operatorname{U}_{\infty}(C_{0}(\mathbb{R}^{2},B^{\mathrm{SW}}))$
also admits a homotopy section. Thus the inclusion
$\operatorname{U}_{\infty}(C^{\ast}_{u}(|\mathbb{Z}|))\to\operatorname{U}_{\infty}(B^{\mathrm{SW}})$
admits a homotopy section by Bott periodicity, which is again an infinite loop
map. ∎

###### Proof of Theorem 1.2.

By Proposition 6.5, we have a homotopy equivalence

$\operatorname{U}_{\infty}(C^{\ast}_{u}(|\mathbb{Z}|))\simeq\operatorname{U}_{\infty}(B^{\mathrm{SW}})\times
F_{1}$

as infinite loop spaces. The homotopy types of the spaces
$\operatorname{U}_{\infty}(B^{\mathrm{SW}})$ and $F_{1}$ are determined in
Lemma 5.2 and Proposition 6.3, respectively. Together with the homotopy
stability in Theorem 1.1, this completes the proof of the theorem. ∎

##  7\. Generalization

In this section, we study the relation between the homotopy type of
$\operatorname{U}_{1}(C^{\ast}_{u}(|\mathbb{Z}^{n}|))$ and the inclusion
$\operatorname{U}_{1}(C^{\ast}_{u}(|\mathbb{Z}^{n}|))\subset\operatorname{U}_{1}(C^{\ast}(|\mathbb{Z}^{n}|))$
for general $n\geq 2$. In view of Theorem 1.2, we propose the following
question.

######  Question 7.1.

Does the inclusion
$\operatorname{U}_{d}(C^{\ast}_{u}(|\Gamma|))\to\operatorname{U}_{d}(C^{\ast}(|\Gamma|))$
admits a homotopy section? Are the homotopy groups of its homotopy fiber are
rational vector spaces?

Let us see the case when $\Gamma=\mathbb{Z}^{n}$ in view of this question.

######  Lemma 7.2.

The $K$-theory of the Roe algebra $C^{\ast}(|\mathbb{Z}^{n}|)$ is computed as

$K_{i}(C^{\ast}(|\mathbb{Z}^{n}|))\cong\begin{cases}\mathbb{Z}&\text{$i\equiv
n$ mod $2$,}\\\ 0&\text{$i\not\equiv n$ mod $2$.}\end{cases}$

###### Proof.

Let

$A_{m}=\ell^{\infty}(\mathbb{Z}^{n},\mathcal{K}(\mathcal{H}))\rtimes\mathbb{Z}^{m}$

with respect to the action of $\mathbb{Z}^{m}$ ($m\leq n$) on the first $m$
factors of $\mathbb{Z}^{n}$. Let $S_{j}$ denote the shift on the $j$-th
factor. Then by the Pimsner–Voiculescu exact sequence

$\textstyle{K_{0}(A_{m-1})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{1-S_{m}}$$\textstyle{K_{0}(A_{m-1})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{K_{0}(A_{m})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{K_{1}(A_{m})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{K_{1}(A_{m-1})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{K_{1}(A_{m-1})\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{1-S_{m}}$

for $A_{m}=A_{m-1}\rtimes_{S_{m}}\mathbb{Z}$, we obtain the short exact
sequence

(1) $\displaystyle 0\to K_{i}(A_{m-1})_{S_{m}}\to$ $\displaystyle
K_{i}(A_{m})\to K_{1-i}(A_{m-1})^{S_{m}}\to 0$

for $i=0,1$, where $K_{i}(A_{m-1})_{S_{m}}$ and $K_{i}(A_{m-1})^{S_{m}}$
denote the coinvariant and the invariant by $S_{m}$, respectively. Since
$A_{0}=\ell^{\infty}(\mathbb{Z}^{n},\mathcal{K}(\mathcal{H}))$ and we have the
well-known isomorphism

$K_{i}(\ell^{\infty}(\mathbb{Z}^{n},\mathcal{K}))\cong\begin{cases}\mathbb{Z}^{\mathbb{Z}^{n}}&i=0,\\\
0&i=1,\end{cases}$

where $\mathbb{Z}^{\mathbb{Z}^{n}}$ is the group of all $\mathbb{Z}$-valued
sequences over $\mathbb{Z}^{n}$, we obtain

$K_{i}(A_{m})\cong\begin{cases}\mathbb{Z}^{\mathbb{Z}^{n-m}}&\text{$i\equiv m$
mod $2$,}\\\ 0&\text{$i\not\equiv m$ mod $2$,}\end{cases}$

by induction on $m$. The lemma is just the case when $m=n$. ∎

Together with the previous lemma, the homotopy type of
$\operatorname{U}_{\infty}(C^{\ast}(|\mathbb{Z}^{n}|))$ is determined by the
following lemma.

######  Lemma 7.3.

Let $A$ be a $C^{\ast}$-algebra, where we do not require the existence of
unit. Consider the following two conditions on $K$-theory:

$\text{(i)}\quad K_{i}(A)\cong\begin{cases}\mathbb{Z}&i=0,\\\
0&i=1,\end{cases}\qquad\qquad\text{(ii)}\quad
K_{i}(A)\cong\begin{cases}0&i=0,\\\ \mathbb{Z}&i=1.\end{cases}$

If (i) holds, then $\operatorname{U}_{\infty}(A)$ has the homotopy type of
$\operatorname{U}_{\infty}(\mathbb{C})$ as an infinite loop space. If (ii)
holds, then $\operatorname{U}_{\infty}(A)$ has the homotopy type of
$\mathbb{Z}\times B\operatorname{U}_{\infty}(\mathbb{C})$ as an infinite loop
space.

###### Proof.

Suppose the condition (i). By Lemma 6.4, there exists a homotopy equivalence
$f\colon\operatorname{U}_{\infty}(\mathbb{C})\to\operatorname{U}_{\infty}(A)$,
which is an infinite loop map. When the condition (ii) holds, apply the result
for the condition (i) to the algebra $C_{0}(\mathbb{R},A)$. This implies that
$\operatorname{U}_{\infty}(C_{0}(\mathbb{R},A))$ is homotopy equivalent to
$\operatorname{U}_{\infty}(\mathbb{C})$. By the Bott periodicity,
$\operatorname{U}_{\infty}(A)$ is homotopy equivalent to
$\Omega\operatorname{U}_{\infty}(\mathbb{C})\simeq\mathbb{Z}\times
B\operatorname{U}_{\infty}(\mathbb{C})$. ∎

######  Proposition 7.4.

The inclusion
$\operatorname{U}_{\infty}(C^{\ast}_{u}(|\mathbb{Z}^{n}|))\to\operatorname{U}_{\infty}(C^{\ast}(|\mathbb{Z}^{n}|))$
admits a homotopy section as an infinite loop map if and only if the
homomorphism $K_{\ast}(C^{\ast}_{u}(|\mathbb{Z}^{n}|))\to
K_{\ast}(C^{\ast}(|\mathbb{Z}^{n}|))$ is surjective.

###### Proof.

The only if part is obvious. For the if part, when $n$ is odd, this follows
from Lemma 7.2 and the same argument as in the proof of Proposition 6.5. When
$n$ is even, apply the same argument to the map on the based loop spaces
$\operatorname{U}_{\infty}(C_{0}(\mathbb{R},C^{\ast}_{u}(|\mathbb{Z}^{n}|)))\to\operatorname{U}_{\infty}(C_{0}(\mathbb{R},C^{\ast}(|\mathbb{Z}^{n}|)))$.
Then the proposition follows from the existence of the homotopy section of the
map on the double loop spaces
$\operatorname{U}_{\infty}(C_{0}(\mathbb{R}^{2},C^{\ast}_{u}(|\mathbb{Z}^{n}|)))\to\operatorname{U}_{\infty}(C_{0}(\mathbb{R}^{2},C^{\ast}(|\mathbb{Z}^{n}|)))$
and the Bott periodicity. ∎

Now all we have to do is to see that the homomorphism
$K_{\ast}(C^{\ast}_{u}(|\mathbb{Z}^{n}|))\to
K_{\ast}(C^{\ast}(|\mathbb{Z}^{n}|))$ is surjective. Let

$B_{m}=\ell^{\infty}(\mathbb{Z}^{n},\mathbb{C})\rtimes\mathbb{Z}^{m}$

with respect to the action $\mathbb{Z}^{m}$ ($m\leq n$) on the first $m$
factors of $\mathbb{Z}^{n}$ and $S_{j}$ denote the shift on the $j$-th factor.
We obtain the short exact sequences similar to (1)

(2) $\displaystyle 0\to K_{i}(B_{m-1})_{S_{m}}\to$ $\displaystyle
K_{i}(B_{m})\to K_{1-i}(B_{m-1})^{S_{m}}\to 0$

for $i=0,1$. For $n=1,2$, we can see the surjectivity as follows.

######  Lemma 7.5.

The homomorphism $K_{1}(C^{\ast}_{u}(|\mathbb{Z}|))\to
K_{1}(C^{\ast}(|\mathbb{Z}|))$ is an isomorphism.

###### Proof.

Consider the commutative square

$\textstyle{K_{1}(C^{\ast}_{u}(|\mathbb{Z}|))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\textstyle{\ell^{\infty}(\mathbb{Z},\mathbb{Z})^{S}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\textstyle{K_{1}(C^{\ast}(|\mathbb{Z}|))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\textstyle{(\mathbb{Z}^{\mathbb{Z}})^{S}}$

obtained from the exact sequences (1) and (2). Thus the lemma follows. ∎

######  Lemma 7.6.

The homomorphism $K_{0}(C^{\ast}_{u}(|\mathbb{Z}^{2}|))\to
K_{0}(C^{\ast}(|\mathbb{Z}^{2}|))$ is surjective.

###### Proof.

When $n=2$, we can compute $K_{\ast}(B_{1})$ by the exact sequence (2) as
follows:

$K_{i}(B_{1})\cong\begin{cases}\ell^{\infty}(\mathbb{Z}^{2},\mathbb{Z})_{S_{1}}&i=0,\\\
\ell^{\infty}(\mathbb{Z}^{2},\mathbb{Z})^{S_{1}}&i=1.\end{cases}$

Again by the exact sequences (1) and (2) for $m=2$, we have the commutative
diagram

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\ell^{\infty}(\mathbb{Z}^{2},\mathbb{Z})_{S_{1}S_{2}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{K_{0}(C^{\ast}_{u}(|\mathbb{Z}^{2}|))\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\ell^{\infty}(\mathbb{Z}^{2},\mathbb{Z})^{S_{1}S_{2}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{K_{0}(C^{\ast}(|\mathbb{Z}^{2}|))\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\cong}$$\textstyle{(\mathbb{Z}^{\mathbb{Z}^{2}})^{S_{1}S_{2}}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0}$

Thus the homomorphism $K_{0}(C^{\ast}_{u}(|\mathbb{Z}^{2}|))\to
K_{0}(C^{\ast}(|\mathbb{Z}^{2}|))$ is surjective by the right square. ∎

To determine the homotopy type of $C^{\ast}_{u}(|\mathbb{Z}^{2}|)$, we also
need its $K$-theory.

######  Lemma 7.7.

The $K$-theory $K_{1}(C^{\ast}_{u}(|\mathbb{Z}^{2}|))$ and the kernel of the
homomorphism $K_{0}(C^{\ast}_{u}(|\mathbb{Z}^{2}|))\to
K_{0}(C^{\ast}(|\mathbb{Z}^{2}|))$ are rational vector spaces of uncountable
dimension.

###### Proof.

As seen in the proof of Lemma 7.6, the latter group is isomorphic to
$\ell^{\infty}(\mathbb{Z}^{2},\mathbb{Z})_{S_{1}S_{2}}$. The coinvariant
$\ell^{\infty}(\mathbb{Z}^{2},\mathbb{Z})_{S_{1}}$ can be seen to be a
rational vector space of uncountable dimension by the same argument as in
[KKT, Section 5]. Then, since
$S_{2}\colon\ell^{\infty}(\mathbb{Z}^{2},\mathbb{Z})_{S_{1}}\to\ell^{\infty}(\mathbb{Z}^{2},\mathbb{Z})_{S_{1}}$
is a linear map on a rational vector space, the coinvariant
$\ell^{\infty}(\mathbb{Z}^{2},\mathbb{Z})_{S_{1}S_{2}}$ is a rational vector
space of uncountable dimension. For $K_{1}(C^{\ast}_{u}(|\mathbb{Z}^{2}|))$,
we obtain the exact sequence

$0\to(\ell^{\infty}(\mathbb{Z}^{2},\mathbb{Z})^{S_{1}})_{S_{2}}\to
K_{1}(C^{\ast}_{u}(|\mathbb{Z}^{2}|))\to(\ell^{\infty}(\mathbb{Z}^{2},\mathbb{Z})_{S_{1}})^{S_{2}}\to
0$

from (2). Since
$(\ell^{\infty}(\mathbb{Z}^{2},\mathbb{Z})^{S_{1}})_{S_{2}}\cong\ell^{\infty}(\mathbb{Z},\mathbb{Z})_{S}$
and $\ell^{\infty}(\mathbb{Z}^{2},\mathbb{Z})_{S_{1}}$ are rational vector
spaces, $K_{1}(C^{\ast}_{u}(|\mathbb{Z}^{2}|))$ is also a rational vector
space of uncountable dimension. ∎

###### Proof of Theorem 1.3.

By Proposition 7.4 and Lemma 7.6, the inclusion
$\operatorname{U}_{\infty}(C^{\ast}_{u}(|\mathbb{Z}^{2}|))\to\operatorname{U}_{\infty}(C^{\ast}(|\mathbb{Z}^{2}|))$
admits a homotopy section as an infinite loop map. Let $F_{2}$ be the homotopy
fiber of the inclusion. Then we have a homotopy equivalence

$\operatorname{U}_{\infty}(C^{\ast}_{u}(|\mathbb{Z}^{2}|))\simeq\operatorname{U}_{\infty}(C^{\ast}(|\mathbb{Z}^{2}|))\times
F_{2}$

as infinite loop spaces. By Lemmas 7.2 and 7.3,
$\operatorname{U}_{\infty}(C^{\ast}(|\mathbb{Z}^{2}|))$ is homotopy equivalent
to $\operatorname{U}_{\infty}(\mathbb{C})$ as an infinite loop space. By the
naturality of the Bott maps

$\beta\colon\operatorname{U}_{\infty}(C^{\ast}_{u}(|\mathbb{Z}^{2}|))\xrightarrow{\simeq}\operatorname{U}_{\infty}(C_{0}(\mathbb{R}^{2},C^{\ast}_{u}(|\mathbb{Z}^{2}|)))\quad\text{and}\quad\beta\colon\operatorname{U}_{\infty}(C^{\ast}(|\mathbb{Z}^{2}|))\xrightarrow{\simeq}\operatorname{U}_{\infty}(C_{0}(\mathbb{R}^{2},C^{\ast}(|\mathbb{Z}^{2}|))),$

we have the homotopy equivalence

$\tilde{\beta}\colon F_{2}\xrightarrow{\simeq}\Omega^{2}F_{2}$

as well. The homotopy group of $F_{2}$ can be computed by Lemma 7.6:

$\pi_{i}(F_{2})\cong\begin{cases}V_{1}&\text{$i$ is even,}\\\ V_{0}&\text{$i$
is odd,}\end{cases}$

where

$V_{0}=\ker[K_{0}(C^{\ast}_{u}(|\mathbb{Z}^{2}|))\to
K_{0}(C^{\ast}(|\mathbb{Z}^{2}|))],\quad
V_{1}=K_{1}(C^{\ast}_{u}(|\mathbb{Z}^{2}|))$

are rational vector spaces by Lemma 7.7. Again as in the proof of [KKT, Lemma
5.4], we can find maps

$\prod_{i\geq 1}^{\circ}K(V_{0},2i-1)\to F_{2}\quad\text{and}\quad\prod_{i\geq
1}^{\circ}K(V_{1},2i-1)\to\Omega F_{2}$

inducing isomorphisms on the odd degree homotopy groups. Then, using the
homotopy equivalence $\tilde{\beta}$, we obtain the homotopy equivalence

$\prod_{i\geq 1}^{\circ}(K(V_{0},2i-1)\times K(V_{1},2i))\to F_{2}.$

This completes the proof of the theorem. ∎

Moreover, Lemma 7.5 provides another proof of Theorem 1.2 in a similar manner.

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