# Gromov–Hausdorff convergence of spectral truncations for tori

Malte Leimbach  and  Walter D. van Suijlekom  Institute for Mathematics,
Astrophysics and Particle Physics, Radboud University Nijmegen, Heyendaalseweg
135, 6525 AJ Nijmegen, The Netherlands<EMAIL_ADDRESS><EMAIL_ADDRESS>
(Date: January 17, 2024)

###### Abstract.

We consider operator systems associated to spectral truncations of tori. We
show that their state spaces, when equipped with the Connes distance function,
converge in the Gromov–Hausdorff sense to the space of all Borel probability
measures on the torus equipped with the Monge–Kantorovich distance. A crucial
role will be played by the relationship between Schur and Fourier multipliers.
Along the way, we introduce the spectral Fejér kernel and show that it is a
good kernel. This allows to make the estimates sufficient to prove the desired
convergence of state spaces. We conclude with some structure analysis of the
pertinent operator systems, including the C*-envelope and the propagation
number, and with an observation about the dual operator system.

###### Contents

  1. 1 Introduction
  2. 2 Preliminaries
    1. 2.1 Actions and commutators
    2. 2.2 Good kernels
    3. 2.3 Fourier and Schur multipliers
  3. 3 Convergence of Spectral Truncations of the $d$-Torus
    1. 3.1 A candidate for the $\mathrm{C}^{1}$-approximate order isomorphism
    2. 3.2 A transference result
    3. 3.3 The spectral Fejér kernel
  4. 4 Structure Analysis of the Operator System $\mathrm{C}(\mathbb{T}^{d})^{(\Lambda)}$
    1. 4.1 C*-envelope and propagation number
    2. 4.2 Dual
  5. A Some convex geometry

##  1\. Introduction

In a spectral approach to geometry, such as the one advocated in
noncommutative geometry [7], a natural question that arises is how one may
approximate a geometric space if only part of the spectral data is available
[8]. In fact, for the so-called spectral truncations that are relevant in this
paper one is naturally led to consider quantum metric spaces in the sense of
Rieffel [24].

More precisely, we consider a spectral triple $(A,\mathcal{H},D)$ consisting
of a C*-algebra $A$ acting as bounded operators on a Hilbert space
$\mathcal{H}$, together with a self-adjoint operator $D$ such that $[D,a]$
extends to a bounded operator for $a$ in a dense $*$-subalgebra of $A$, and
such that the resolvent of $D$ is a compact operator. A spectral truncation
for this set of data is given in terms of a spectral projection $P_{\Lambda}$
of $D$ onto the eigenspaces with eigenvalues of modulus $\leq\Lambda$. The
C*-algebra $A$ is then replaced by an operator system, to wit
$P_{\Lambda}AP_{\Lambda}$. It acts on the Hilbert space
$P_{\Lambda}\mathcal{H}$ and also $D$ restricts to a self-adjoint operator
thereon.

The above question may then be sharply formulated in terms of Gromov–Hausdorff
convergence (as $\Lambda\to\infty$) of the state spaces
$\mathcal{S}(P_{\Lambda}AP_{\Lambda})$ when equipped with the Connes distance
function [7]

(1) $\displaystyle d_{P_{\Lambda}AP_{\Lambda}}(\varphi,\psi)$
$\displaystyle=\sup_{a\in
P_{\Lambda}AP_{\Lambda}}\\{|\varphi(a)-\psi(a)|\,:\,\|[P_{\Lambda}DP_{\Lambda},a]\|=1\\}.$
The limit structure one is aiming for is then of course the state space
$\mathcal{S}(A)$, this time equipped with the metric (2) $\displaystyle
d_{A}(\varphi,\psi)$ $\displaystyle=\sup_{a\in
A}\\{|\varphi(a)-\psi(a)|\,:\,\|[D,a]\|=1\\}.$

Note that in the definition of quantum metric spaces there is an additional
(and natural) requirement of compatibility betweeen the above metric structure
and the given weak-$\ast$ topology on the pertinent state spaces (_cf._ [24,
Definition 2.1]). If $A=\mathrm{C}(M)$ and $D=D_{M}$ is the Dirac operator on
a compact Riemannian spin manifold $M$ —indeed our case of interest— this is
the case, as was shown in [6]. In fact, Connes’ distance function then
coincides with the Monge–Kantorovich distance on the state space
$\mathcal{S}(\mathrm{C}(M))$ of all Borel probability measures. In addition,
for finite-dimensional operator systems —such as the above
$P_{\Lambda}AP_{\Lambda}$— Connes’ distance function metrizes the weak-$\ast$
topology as follows from [23, Theorem 1.8] (see also [31, Remark 6]).

In a previous work, a convergence of spectral truncations has been established
for the circle, when equipped with the natural Dirac operator [31]. This was
formulated in the following general framework, which we will also exploit
here.

######  Definition 1.1.

An operator system spectral triple is a triple $(\mathcal{E},\mathcal{H},D)$
where $\mathcal{E}$ is a dense subspace of a (concrete) operator system $E$ in
$\mathcal{B}(\mathcal{H})$, $\mathcal{H}$ is a Hilbert space and $D$ is a
self-adjoint operator in $\mathcal{H}$ with compact resolvent and such that
$[D,T]$ is a bounded operator for all $T\in\mathcal{E}$.

The relation between sequences of operator system spectral triples and a limit
structure is then captured by the following notion of an approximate order
isomorphism.

######  Definition 1.2.

Let $(\mathcal{E},\mathcal{H},D)$ and
$(\mathcal{E}_{\Lambda},\mathcal{H}_{\Lambda},D_{\Lambda})$, for all
$\Lambda\geq 0$, be operator system spectral triples. For all $\Lambda$, let
$\rho_{\Lambda}:\mathcal{E}\rightarrow\mathcal{E}_{\Lambda}$ and
$\sigma_{\Lambda}:\mathcal{E}_{\Lambda}\rightarrow\mathcal{E}$ be linear maps
with the following properties:

  1. (i)

$\rho_{\Lambda}$ and $\sigma_{\Lambda}$ are positive and unital,

  2. (ii)

$\rho_{\Lambda}$ and $\sigma_{\Lambda}$ are _$\mathrm{C}^{1}$ -contractive_,
i.e. they are contractive both with respect to norm $\|\cdot\|$ and with
respect to Lipschitz seminorm $\|[D,\cdot]\|$, respectively
$\|[D_{\Lambda},\cdot]\|$,

  3. (iii)

the compositions $\sigma_{\Lambda}\circ\rho_{\Lambda}$ and
$\rho_{\Lambda}\circ\sigma_{\Lambda}$ approximate the respective identities on
$\mathcal{E}$ and $\mathcal{E}_{\Lambda}$ with respect to Lipschitz seminorm,
i.e.

(3) $\displaystyle\|a-\rho_{\Lambda}\circ\sigma_{\Lambda}(a)\|$
$\displaystyle\leq\gamma_{\Lambda}\|[D,a]\|$ (4)
$\displaystyle\|T-\sigma_{\Lambda}\circ\rho_{\Lambda}(T)\|$
$\displaystyle\leq\gamma_{\Lambda}\|[D_{\Lambda},T]\|,$

for some net $(\gamma_{\Lambda})_{\Lambda\geq 0}$ with
$\gamma_{\Lambda}\rightarrow 0$ as $\Lambda\rightarrow\infty$. Then we call
the pair of maps $(\rho,\sigma)$ (by which we mean the collection of pairs of
maps $(\rho_{\Lambda},\sigma_{\Lambda})$) a _$\mathrm{C}^{1}$ -approximate
order isomorphism_.

It was then shown in [31, Theorem 5] that if the metrics
$d_{\mathcal{E}_{\Lambda}}$, for all $\Lambda\geq 0$, and $d_{\mathcal{E}}$
(defined as in (1) and (2)) metrize the respective weak∗-topologies on the
state spaces $\mathcal{S}(\mathcal{E}_{\Lambda})$ and
$\mathcal{S}(\mathcal{E})$ and if a $\mathrm{C}^{1}$-approximate order
isomorphism exists, then the sequence of metric spaces
$(\mathcal{S}(\mathcal{E}_{\Lambda}),d_{\mathcal{E}_{\Lambda}})$ converges to
$(\mathcal{S}(\mathcal{E}),d_{\mathcal{E}})$ in Gromov–Hausdorff distance. By
exploiting this criterion it was shown in [31] that spectral truncations of
the circle converge.

Other results on Gromov–Hausdorff convergence that can be cast in this general
framework include [2, 1, 25]. Note also the recent developments around the
related notion of propinquity [17, 18, 19], also in the context of spectral
triples, but which, however, mainly focuses on C*-algebras. It should be
mentioned that, even though we are phrasing the question about convergence of
spectral truncations in terms of Gromov–Hausdorff distance, another point of
view would be quantum Gromov–Hausdorff distance [24]. In fact, these notions
are not equivalent [16]. However, as the authors point out, it follows from
[15, Proposition 2.14] that Gromov–Hausdorff convergence still implies quantum
Gromov–Hausdorff convergence in the case which we consider below.

In this article, we show that the conditions in the above Definition can be
met for tori in any dimension $d\geq 1$. Let us spend the remainder of this
introduction by giving an extended overview of our setup and approach.

Consider the spectral triple of the $d$-dimensional torus
$\mathbb{T}^{d}=\mathbb{R}^{d}/2\pi\mathbb{Z}^{d}$:

(5)
$\displaystyle\left(\mathrm{C}^{\infty}(\mathbb{T}^{d}),\mathrm{L}^{2}(S(\mathbb{T}^{d})),D\right)$

This consists of the $*$-algebra of smooth functions acting on the Hilbert
space of $\mathrm{L}^{2}$-sections of the spinor bundle $S(\mathbb{T}^{d})$
(by multiplication) and the Dirac operator $D$ which acts on the dense
subspace of smooth sections of the spinor bundle. We identify
$S(\mathbb{T}^{d})$ with the trivial bundle $\mathbb{T}^{d}\otimes V$, where
$V:=\mathbb{C}^{\lfloor\nicefrac{{d}}{{2}}\rfloor}$, and we write
$\mathcal{H}:=\mathrm{L}^{2}(\mathbb{T}^{d})\otimes
V\cong\mathrm{L}^{2}(S(\mathbb{T}^{d}))$ for the Hilbert space. Recall that
$D=-i\sum_{\mu=1}^{d}\partial_{\mu}\otimes\gamma^{\mu}$ and that the spectrum
of $D$ (which is point-spectrum only) is given by
$\sigma(D)=\\{\pm(n_{1}^{2}+\cdots+n_{d}^{2})^{\nicefrac{{1}}{{2}}}\,:\,n_{i}\in\mathbb{Z}\\}$.
The Dirac operator gives rise to the distance function (2) on the state space
$\mathcal{S}(\mathrm{C}(\mathbb{T}^{d}))$ which metrizes the weak∗-topology on
it and recovers the usual Riemannian distance on $\mathbb{T}^{d}$ when
restricted to pure states.

For any $\Lambda\geq 0$, let $P_{\Lambda}$ be the orthogonal projection to the
subspace of $\mathcal{H}$ spanned by the eigenspinors $e_{\lambda}$ of the
eigenvalues $\lambda$ with $|\lambda|\leq\Lambda$. More concretely, we have
$P_{\Lambda}\mathcal{H}=\mathrm{span}\\{e_{n}\,:\,n\in\mathbb{Z}^{d},\|n\|\leq\Lambda\\}\otimes
V$, with $e_{n}(x):=e^{in\cdot x}$, for all $x\in\mathbb{T}^{d}$. The spectral
projection $P_{\Lambda}$ gives rise to the following _operator system spectral
triple_ :

(6)
$\displaystyle\left(P_{\Lambda}\mathrm{C}^{\infty}(\mathbb{T}^{d})P_{\Lambda},P_{\Lambda}\mathcal{H},P_{\Lambda}DP_{\Lambda}\right)$

We use the notation
$\mathrm{C}(\mathbb{T}^{d})^{(\Lambda)}:=P_{\Lambda}\mathrm{C}^{\infty}(\mathbb{T}^{d})P_{\Lambda}$
and write $D_{\Lambda}:=P_{\Lambda}DP_{\Lambda}$. We also abbreviate
$d_{\Lambda}:=d_{P_{\Lambda}\mathrm{C}^{\infty}(\mathbb{T}^{d})P_{\Lambda}}$
for the distance function defined in (1).

Observe that elements $T$ of the operator system
$\mathrm{C}(\mathbb{T}^{d})^{(\Lambda)}$ are of the form
$T=(t_{k-l})_{k,l\in\mkern
1.5mu\overline{\mkern-1.5mu\mathrm{B}\mkern-1.5mu}\mkern 1.5mu_{\Lambda}{Z}}$,
where $\mkern 1.5mu\overline{\mkern-1.5mu\mathrm{B}\mkern-1.5mu}\mkern
1.5mu_{\Lambda}{Z}:=\mkern
1.5mu\overline{\mkern-1.5mu\mathrm{B}\mkern-1.5mu}\mkern
1.5mu_{\Lambda}\cap\mathbb{Z}^{d}$ is the set of $\mathbb{Z}^{d}$-lattice
points in the closed ball of radius $\Lambda$, and where $t_{k-l}=\langle
e_{k},Te_{l}\rangle$. In particular, $\mathrm{C}(\mathbb{T}^{1})^{(\Lambda)}$
is the operator system of
$(2\lfloor\Lambda\rfloor+1)\times(2\lfloor\Lambda\rfloor+1)$-Toeplitz matrices
which was investigated at length in [8] and [11].

The candidate for the map
$\rho_{\Lambda}:\mathrm{C}^{\infty}(\mathbb{T}^{d})\rightarrow\mathrm{C}(\mathbb{T}^{d})^{(\Lambda)}$
in 1.2 is canonically inherent in the problem, namely, it is the compression
map given by $\rho_{\Lambda}(f)=P_{\Lambda}fP_{\Lambda}$. It is easy to see
that this map is positive, unital and $\mathrm{C}^{1}$-contractive (3.1). It
is, however, less obvious what the candidate for the map
$\sigma_{\Lambda}:\mathrm{C}(\mathbb{T}^{d})^{(\Lambda)}\rightarrow\mathrm{C}^{\infty}(\mathbb{T}^{d})$
should be. Inspired by the choice of the map given in [31] in the case of the
circle, we propose the following map:

(7)
$\displaystyle\sigma_{\Lambda}(T):=\frac{1}{\mathcal{N}_{\mathrm{B}}(\Lambda)}\mathrm{Tr}\left(\ket{\psi}\bra{\psi}\alpha(T)\right)$

Here, $\alpha$ is the $\mathbb{T}^{d}$-action (20) on
$\mathrm{C}(\mathbb{T}^{d})^{(\Lambda)}$, the vector $\ket{\psi}$ is given by
$\ket{\psi}=\sum_{n\in\mkern
1.5mu\overline{\mkern-1.5mu\mathrm{B}\mkern-1.5mu}\mkern
1.5mu_{\Lambda}{Z}}e_{n}$, and $\mathcal{N}_{\mathrm{B}}(\Lambda):=\\#\mkern
1.5mu\overline{\mkern-1.5mu\mathrm{B}\mkern-1.5mu}\mkern
1.5mu_{\Lambda}\cap\mathbb{Z}^{d}=\\#\overline{\mathrm{B}}_{\Lambda}{Z}$ is
the number of $\mathbb{Z}^{d}$-lattice points in the closed ball of radius
$\Lambda$. Note that the image of an element $T=(t_{k-l})_{k,l\in\mkern
1.5mu\overline{\mkern-1.5mu\mathrm{B}\mkern-1.5mu}\mkern 1.5mu_{\Lambda}{Z}}$
of the operator system $\mathrm{C}(\mathbb{T}^{d})^{(\Lambda)}$ under this map
is a function on $\mathbb{T}^{d}$ as follows:

(8) $\displaystyle\sigma_{\Lambda}(T)(x)$
$\displaystyle=\frac{1}{\mathcal{N}_{\mathrm{B}}(\Lambda)}\mathrm{Tr}\left((1)_{k,l\in\mkern
1.5mu\overline{\mkern-1.5mu\mathrm{B}\mkern-1.5mu}\mkern
1.5mu_{\Lambda}{Z}}(t_{k-l}e^{i(k-l)\cdot x})_{k,l\in\mkern
1.5mu\overline{\mkern-1.5mu\mathrm{B}\mkern-1.5mu}\mkern
1.5mu_{\Lambda}{Z}}\right)$ (9)
$\displaystyle=\frac{1}{\mathcal{N}_{\mathrm{B}}(\Lambda)}\mathrm{Tr}\left(\left(\sum_{m\in\mkern
1.5mu\overline{\mkern-1.5mu\mathrm{B}\mkern-1.5mu}\mkern
1.5mu_{\Lambda}{Z}}t_{m-l}e^{i(m-l)\cdot x}\right)_{k,l\in\mkern
1.5mu\overline{\mkern-1.5mu\mathrm{B}\mkern-1.5mu}\mkern
1.5mu_{\Lambda}{Z}}\right)$ (10)
$\displaystyle=\frac{1}{\mathcal{N}_{\mathrm{B}}(\Lambda)}\sum_{m,n\in\mkern
1.5mu\overline{\mkern-1.5mu\mathrm{B}\mkern-1.5mu}\mkern
1.5mu_{\Lambda}{Z}}t_{m-n}e^{i(m-n)\cdot x},$

for all $x\in\mathbb{T}^{d}$. One may consider [24, Section 2] as another
instance of inspiration for this choice of map $\sigma_{\Lambda}$ by realizing
that the map $\sigma_{\Lambda}$ is the formal adjoint of the map
$\rho_{\Lambda}$ when the $\ast$-algebra $\mathrm{C}^{\infty}(\mathbb{T}^{d})$
is equipped with the $\mathrm{L}^{2}$-inner product and the operator system
$\mathrm{C}(\mathbb{T}^{d})^{(\Lambda)}$ is equipped with the Hilbert–Schmidt
inner product. Similarly as for $\rho_{\Lambda}$, it is easy to see that the
map $\sigma_{\Lambda}$ is positive, unital and $\mathrm{C}^{1}$-contractive
(3.2).

In order to show that our choice of maps $\rho_{\Lambda}$ and
$\sigma_{\Lambda}$ gives rise to a $\mathrm{C}^{1}$-approximate order
isomorphism it remains to show that their compositions approximate the
respective identities on $\mathrm{C}^{\infty}(\mathbb{T}^{d})$ and
$\mathrm{C}(\mathbb{T}^{d})^{(\Lambda)}$ in Lipschitz seminorm. We show by
direct computations (3.3) that the maps $\sigma_{\Lambda}\circ\rho_{\Lambda}$
and $\rho_{\Lambda}\circ\sigma_{\Lambda}$ act on
$\mathrm{C}^{\infty}(\mathbb{T}^{d})$, respectively
$\mathrm{C}(\mathbb{T}^{d})^{(\Lambda)}$ as follows:

(11) $\displaystyle\sigma_{\Lambda}\circ\rho_{\Lambda}(f)$
$\displaystyle=\left(\mathfrak{m}_{\Lambda}\widehat{f}\right)\widecheck{\vphantom{f}}=:\mathcal{F}_{\mathfrak{m}_{\Lambda}}(f)$
(12) $\displaystyle\rho_{\Lambda}\circ\sigma_{\Lambda}(T)$
$\displaystyle=\left(\mathfrak{m}_{\Lambda}(k-l)t_{k-l}\right)_{k,l\in\mkern
1.5mu\overline{\mkern-1.5mu\mathrm{B}\mkern-1.5mu}\mkern
1.5mu_{\Lambda}{Z}}=:\mathcal{S}_{\mathfrak{m}_{\Lambda}}(T)$

The map $\mathcal{F}_{\mathfrak{m}_{\Lambda}}$ is known as _Fourier
multiplication_ and the map $\mathcal{S}_{\mathfrak{m}_{\Lambda}}$ as _Schur
multiplication_ , respectively with _symbol_

(13)
$\displaystyle\mathfrak{m}_{\Lambda}(n):=\frac{\mathcal{N}_{\mathrm{L}}(\Lambda,n)}{\mathcal{N}_{\mathrm{B}}(\Lambda)},$

where $\mathcal{N}_{\mathrm{L}}(\Lambda,n):=\\#\mkern
1.5mu\overline{\mkern-1.5mu\mathrm{B}\mkern-1.5mu}\mkern
1.5mu_{\Lambda}\cap\mkern
1.5mu\overline{\mkern-1.5mu\mathrm{B}\mkern-1.5mu}\mkern
1.5mu_{\Lambda}(n)\cap\mathbb{Z}^{d}=\\#\overline{\mathrm{L}}_{\Lambda}{Z}(n)$
is the number of $\mathbb{Z}^{d}$-lattice points in the intersection of the
closed ball of radius $\Lambda$ with a copy of itself translated by $n$ (we
call this intersection a _lense_ and denote the set of
$\mathbb{Z}^{d}$-lattice points in it by
$\overline{\mathrm{L}}_{\Lambda}{Z}(n)$).

We denote the compression of the Dirac operator by
$D_{\Lambda}:=P_{\Lambda}DP_{\Lambda}$. We apply an “antiderivative trick”
(3.4) to see that for obtaining estimates of the maps
$\mathrm{id}_{\mathrm{C}^{\infty}(\mathbb{T}^{d})}-\mathcal{F}_{\mathfrak{m}_{\Lambda}}$
and
$\mathrm{id}_{\mathrm{C}(\mathbb{T}^{d})^{(\Lambda)}}-\mathcal{S}_{\mathfrak{m}_{\Lambda}}$
in Lipschitz seminorm, one needs to estimate the following two maps:

(14)
$\displaystyle\mathcal{F}_{\mathfrak{w}_{\Lambda}}:=\frac{i}{2}\sum_{\mu=1}^{d}\mathcal{F}_{\mathfrak{w}^{\mu}_{\Lambda}}\otimes\\{\gamma^{\mu},\cdot\\}$
$\displaystyle:[D,\mathrm{C}^{\infty}(\mathbb{T}^{d})]\to\mathrm{C}^{\infty}(\mathbb{T}^{d});$
(15)
$\displaystyle\mathcal{S}_{\mathfrak{w}_{\Lambda}}:=\frac{i}{2}\sum_{\mu=1}^{d}\mathcal{S}_{\mathfrak{w}^{\mu}_{\Lambda}}\otimes\\{\gamma^{\mu},\cdot\\}$
$\displaystyle:[D,\mathrm{C}(\mathbb{T}^{d})^{(\Lambda)}]\to\mathrm{C}(\mathbb{T}^{d})^{(\Lambda)},$

where $\mathcal{F}_{\mathfrak{w}_{\Lambda}^{\mu}}$ and
$\mathcal{S}_{\mathfrak{w}_{\Lambda}^{\mu}}$ are now respectively Fourier and
Schur multiplication with the symbol

(16) $\displaystyle\mathfrak{w}_{\Lambda}^{\mu}(n)=\begin{cases}0,\text{ if
}n=0\\\ (1-\mathfrak{m}_{\Lambda}(n))\frac{n_{\mu}}{\|n\|^{2}},\text{ if
}n\neq 0.\end{cases}$

A variation of the classical Bożejko-Fendler _transference theorem_ for
Fourier and Schur multipliers (3.5) then shows that the cb-norm of
$\mathcal{S}_{\mathfrak{w}_{\Lambda}}$ is bounded by the cb-norm of
$\mathcal{F}_{\mathfrak{w}_{\Lambda}}$. Since the latter map takes values in a
commutative $\mathrm{C}^{*}$-algebra its cb-norm coincides with its norm, so
this is what is left to estimate.

It is not hard to see that $\mathcal{F}_{\mathfrak{m}_{\Lambda}}$ is an
approximate identity controlled in Lipschitz seminorm, if the convolution
kernel $K_{\mathfrak{m}_{\Lambda}}=\widecheck{\mathfrak{m}}_{\Lambda}$ is a
_good kernel_ (2.2). We show that this is indeed the case by exploiting the
fact that $K_{\mathfrak{m}_{\Lambda}}$ is the square of the _spherical
Dirichlet kernel_ , whence positive. Since this is analogous to the
1-dimensional case, we call $K_{\mathfrak{m}_{\Lambda}}$ the _spectral Fejér
kernel_. As outlined above, our main result that
$(\rho_{\Lambda},\sigma_{\Lambda})$ is a $\mathrm{C}^{1}$-approximate order
isomorphism and hence that spectral truncations of the $d$-torus converge for
all $d\geq 1$ is now simply a corollary of the fact that the _spectral Fejér
kernel_ is _good_.

In the last section, we give a computation of the propagation number of the
operator system $\mathrm{C}(\mathbb{T}^{d})^{(\Lambda)}$. We conclude by
pointing out some obstacles on the road to determining its operator system
dual. In particular, we argue that, for $d\geq 2$, the operator system dual
cannot quite be the one that we would have expected.

### Acknowledgements

We thank Nigel Higson for the comments at an early stage of this project. ML
thanks Gerrit Vos for an introduction to Fourier and Schur multipliers and
Dimitris Gerontogiannis for fruitful discussions. After having published a
first preprint version of this article we were pointed to 3.8, which makes
many technicalities obsolete and at the same time enables us to prove
convergence of spectral truncations of tori in any dimension (also higher than
$3$). We are indepted to Yvann Gaudillot-Estrada for this contribution.
Furthermore, we thank Jens Kaad and Tirthankar Bhattacharyya for their input.
This work was funded by NWO under grant OCENW.KLEIN.376.

##  2\. Preliminaries

###  2.1. Actions and commutators

We spell out a few simple facts used throughout this article. Recall the usual
action of $\mathrm{C}(\mathbb{T}^{d})$ on $\mathcal{H}$:

(17) $\displaystyle f(g\otimes v)=(fg)\otimes
v=\sum_{n\in\mathbb{Z}^{d}}\sum_{m\in\mathbb{Z}^{d}}\widehat{f}(n-m)\widehat{g}(m)e_{n}\otimes
v$

This induces an action of $\mathrm{C}(\mathbb{T}^{d})^{(\Lambda)}$ on
$P_{\Lambda}\mathcal{H}$:

(18) $\displaystyle T\left(\sum_{k\in\mkern
1.5mu\overline{\mkern-1.5mu\mathrm{B}\mkern-1.5mu}\mkern
1.5mu_{\Lambda}{Z}}a_{k}e_{k}\otimes v\right)=\sum_{k\in\mkern
1.5mu\overline{\mkern-1.5mu\mathrm{B}\mkern-1.5mu}\mkern
1.5mu_{\Lambda}{Z}}\sum_{l\in\mkern
1.5mu\overline{\mkern-1.5mu\mathrm{B}\mkern-1.5mu}\mkern
1.5mu_{\Lambda}{Z}}t_{k-l}a_{l}e_{k}\otimes v$

Furthermore, we have the standard $\mathbb{T}^{d}$-action on
$\mathrm{C}(\mathbb{T}^{d})$ (as a subalgebra of $\mathcal{B}(\mathcal{H})$):

(19)
$\displaystyle\alpha_{\theta}(f)=\sum_{n\in\mathbb{Z}^{d}}\widehat{f}(n)e_{n}e^{in\cdot\theta}$

This induces an action of $\mathbb{T}^{d}$ on
$\mathrm{C}(\mathbb{T}^{d})^{(\Lambda)}$ (as an operator subsystem of
$\mathcal{B}(P_{\Lambda}\mathcal{H})$):

(20)
$\displaystyle\alpha_{\theta}(T)=\left(t_{k-l}e^{i(k-l)\cdot\theta}\right)_{k,l\in\mkern
1.5mu\overline{\mkern-1.5mu\mathrm{B}\mkern-1.5mu}\mkern 1.5mu_{\Lambda}{Z}}$

Recall that the following holds:

(21)
$\displaystyle[D,f]=\sum_{\mu=1}^{d}\sum_{n\in\mathbb{Z}^{d}}n_{\mu}\widehat{f}(n)e_{n}\otimes\gamma^{\mu}$

Similarly, we have:

(22)
$\displaystyle[D_{\Lambda},T]=\sum_{\mu=1}^{d}\left((k_{\mu}-l_{\mu})t_{k-l}\right)_{k,l\in\mkern
1.5mu\overline{\mkern-1.5mu\mathrm{B}\mkern-1.5mu}\mkern
1.5mu_{\Lambda}{Z}}\otimes\gamma^{\mu}$

###  2.2. Good kernels

We follow the convention in [30] and call an approximate identity in the
Banach $\ast$-algebra $\mathrm{L}^{1}(\mathbb{T}^{d})$ (with convolution) a
_good kernel_ :

######  Definition 2.1.

For all $\Lambda>0$, let $K_{\Lambda}\in\mathrm{L}^{1}(\mathbb{T}^{d})$. The
family $\\{K_{\Lambda}\\}_{\Lambda>0}$ is called a _good kernel_ , if the
following holds:

  1. (1)

$\int_{\mathbb{T}^{d}}K_{\Lambda}(x)dx=1$ and

  2. (2)

for all $\delta>0$, we have that
$\int_{\mathbb{T}^{d}\setminus\mathrm{B}_{\delta}(0)}K_{\Lambda}(x)dx\rightarrow
0$, as $\Lambda\rightarrow\infty$.

Good kernels provide a way to approximate the identity on
$\mathrm{C}^{\infty}(\mathbb{T}^{d})$ not only in $\mathrm{L}^{1}$\- and
$\sup$-norm, but also in Lipschitz seminorm:

######  Lemma 2.2.

If $K_{\Lambda}$ is a _good kernel_ , then, for all
$f\in\mathrm{C}^{\infty}(\mathbb{T}^{d})$, the following holds:

(23) $\displaystyle\|f-K_{\Lambda}\ast f\|\leq\gamma_{\Lambda}\|[D,f]\|,$

where $\gamma_{\Lambda}\rightarrow 0$ as $\Lambda\rightarrow\infty$.

###### Proof.

The proof is as in [2, Lemma 5.13]. For all $x\in\mathbb{T}^{d}$, we have:

(24) $\displaystyle|K_{\Lambda}\ast f(x)-f(x)|$
$\displaystyle\leq\int_{\mathbb{T}^{d}}|K_{\Lambda}(y)(f(x-y)-f(x))|dy$ (25)
$\displaystyle\leq\int_{\mathbb{T}^{d}}|K_{\Lambda}(y)|\|y\|\|f\|_{\text{Lip}}dy$
(26) $\displaystyle=\int_{\mathbb{T}^{d}}|K_{\Lambda}(y)|\|y\|dy\|[D,f]\|$

We set $\gamma_{\Lambda}:=\int_{\mathbb{T}^{d}}|K_{\Lambda}(y)|\|y\|dy$. Let
$\varepsilon>0$. Let $\Lambda_{0}$ be large enough such that, for all
$\Lambda\geq\Lambda_{0}$,
$\int_{\|y\|\geq\varepsilon}|K_{\Lambda}(y)|dy<\varepsilon$. Then, for
$\Lambda\geq\Lambda_{0}$, we obtain:

(27) $\displaystyle\gamma_{\Lambda}$
$\displaystyle=\int_{\|y\|\geq\varepsilon}|K_{\Lambda}(y)|\|y\|dy+\int_{\|y\|<\varepsilon}|K_{\Lambda}(y)|\|y\|dy$
(28) $\displaystyle\leq
d\left(\int_{\|y\|\geq\varepsilon}|K_{\Lambda}(y)|dy+\varepsilon\int_{\|y\|<\varepsilon}|K_{\Lambda}(y)|dy\right)$
(29) $\displaystyle\leq d\left(\varepsilon+\varepsilon C\right),$

where $C=\sup_{\Lambda>0}\int_{\mathbb{T}^{d}}|K_{\Lambda}(y)|dy<\infty$. ∎

###  2.3. Fourier and Schur multipliers

Let $\Gamma$ be a discrete group and let
$\lambda:\Gamma\rightarrow\mathcal{B}(\ell^{2}(\Gamma))$ be its left-regular
representation given by $\lambda_{g}f(h)=f(gh)$. We denote by
$\mathrm{C}_{\lambda}^{\ast}(\Gamma)$ the reduced group
$\mathrm{C}^{\ast}$-algebra, i.e. the completion of the group ring
$\mathbb{C}[\Gamma]$ in $\mathcal{B}(\ell^{2}(\Gamma))$ with respect to the
norm $\|x\|_{\mathrm{red}}:=\|\lambda(x)\|_{\mathcal{B}(\ell^{2}(\Gamma))}$. A
function $\varphi:\Gamma\rightarrow\mathbb{C}$ gives rise to a _multiplier_ on
the group ring as follows:

(30) $\displaystyle\mathbb{C}[\Gamma]$
$\displaystyle\rightarrow\mathbb{C}[\Gamma]$ (31)
$\displaystyle\sum_{g\in\Gamma}a_{g}g$
$\displaystyle\mapsto\sum_{g\in\Gamma}\varphi(g)a_{g}g$

If this map extends to a bounded linear map
$\mathcal{M}_{\varphi}:\mathrm{C}_{\lambda}^{\ast}(\Gamma)\rightarrow\mathrm{C}_{\lambda}^{\ast}(\Gamma)$
we call this extension the _multiplier on
$\mathrm{C}_{\lambda}^{\ast}(\Gamma)$ with symbol $\varphi$_. We record the
obvious fact that if $\varphi$ is finitely supported it always induces a
multiplier on $\mathrm{C}_{\lambda}^{\ast}(\Gamma)$.

Recall that if $\Gamma$ is abelian, then
$\mathrm{C}_{\lambda}^{\ast}(\Gamma)=\mathrm{C}^{\ast}(\Gamma)\cong\mathrm{C}(\widehat{\Gamma})$,
where $\widehat{\Gamma}$ is the Pontryagin dual of $\Gamma$. In this case, we
call the multiplier on $\mathrm{C}(\widehat{\Gamma})$ the _Fourier multiplier
with symbol $\varphi$_ and denote it by $\mathcal{F}_{\varphi}$. The Fourier
multiplier takes on the following form:

(32)
$\displaystyle\mathcal{F}_{\varphi}(f)=\left(g\mapsto\varphi(g)\widehat{f}(g)\right)\widecheck{\vphantom{f}}=\widecheck{\varphi}\ast
f$

See e.g. [14, Chapter 6] for the relevant Fourier theory of locally compact
abelian groups.

Let $k:\Gamma\times\Gamma\rightarrow\mathbb{C}$ be a function, also called a
_kernel_. A kernel $k$ gives rise to a linear map with domain
$\mathcal{B}(\ell^{2}(\Gamma))$ given by
$\mathcal{S}_{k}:(t_{g,h})_{g,h\in\Gamma}\mapsto(k(g,h)t_{g,h})_{g,h\in\Gamma}$,
where $t_{g,h}=\langle\delta_{g},T\delta_{h}\rangle$, for
$T\in\mathcal{B}(\ell^{2}(\Gamma))$. If this map is bounded with range in
$\mathcal{B}(\ell^{2}(\Gamma))$, we call it a _Schur multiplier_. See e.g.
[32] for a survey and [22, Chapter 5] as a standard reference which includes a
discussion of the connection with Grothendieck’s theorem. We collect some
well-known facts about Schur multipliers.

######  Proposition 2.3.

Let $k:\Gamma\times\Gamma\rightarrow\mathbb{C}$ be a kernel. Then the
following are equivalent:

  1. (i)

$\mathcal{S}_{k}$ is a Schur multiplier of norm $\|\mathcal{S}_{k}\|\leq 1$.

  2. (ii)

$\mathcal{S}_{k}$ is a completely bounded Schur multiplier of
$\mathrm{cb}$-norm $\|\mathcal{S}_{k}\|_{\mathrm{cb}}\leq 1$.

  3. (iii)

There exists a Hilbert space $\mathcal{H}$ and families of vectors
$\\{\xi_{g}\\}_{g\in\Gamma},\\{\eta_{h}\\}_{h\in\Gamma}\subset\mathcal{H}$
with $\|\xi_{g}\|,\|\eta_{h}\|\leq 1$ such that
$k(g,h)=\langle\xi_{g},\eta_{h}\rangle$, for all $g,h\in\Gamma$.

For an elementary proof of the equivalence of (i) and (ii), we refer to [21,
Theorem 8.7 and Corollary 8.8]. A proof of the equivalence of (ii) and (iii)
can be found e.g. in [5, Theorem D.4], which we now sketch: Assuming that
$\|\mathcal{S}_{k}\|_{\mathrm{cb}}\leq 1$, Wittstock’s factorization theorem
gives a factorization of $\mathcal{S}_{k}$ through $\mathcal{B}(\mathcal{H})$,
for some Hilbert space $\mathcal{H}$, which allows to construct appropriate
$\xi_{g}$ and $\eta_{h}$. For the converse implication, the map
$\mathcal{S}_{k}$ is factorized through
$\mathcal{B}(\ell^{2}(\Gamma)\otimes\mathcal{H})$ as
$\mathcal{S}_{k}(T)=V^{\ast}(T\otimes\mathbf{1}_{\mathcal{H}})W$, for the
contractions $V\delta_{g}:=\delta_{g}\otimes\xi_{g}$ and
$W\delta_{h}:=\delta_{h}\otimes\eta_{h}$. The same works when tensoring with
$\mathbf{1}_{M_{n}}$, for arbitrary $n\in\mathbb{N}$, which shows complete
contractivity of $\mathcal{S}_{k}$.

We are mainly interested in Schur multipliers $\mathcal{S}_{k}$ induced by a
function $\varphi:\Gamma\rightarrow\mathbb{C}$, i.e.
$k(g,h):=\varphi(gh^{-1})$. We call such a Schur multiplier a _Schur
multiplier with symbol $\varphi$_ and slightly abuse notation to denote it by
$\mathcal{S}_{\varphi}$. It is easy to see that
$\mathcal{S}_{\varphi}\big{|}_{\mathrm{C}_{\lambda}^{\ast}(\Gamma)}=\mathcal{M}_{\varphi}$.
Indeed, let $f\in\mathrm{C}_{\lambda}^{\ast}(\Gamma)$ and
$\\{\delta_{g}\\}_{g\in\Gamma}$ be an orthonormal basis for
$\ell^{2}(\Gamma)$. Then the matrix associated to $f$ (viewed as an element of
$\mathcal{B}(\ell^{2}(\Gamma))$) is a Toeplitz matrix in the following sense:

(33)
$\displaystyle\langle\delta_{g},f\delta_{h}\rangle=\langle\delta_{g},\sum_{\gamma\in\Gamma}f_{\gamma}\lambda_{\gamma}(\delta_{h})\rangle=\sum_{\gamma\in\Gamma}\langle\delta_{g},f_{\gamma}\delta_{\gamma
h}\rangle=f_{gh^{-1}}$

It follows that the matrix associated to $\mathcal{M}_{\varphi}(f)$ is the
following Toeplitz matrix:

(34)
$\displaystyle\langle\delta_{g},\mathcal{M}_{\varphi}(f)\delta_{h}\rangle=\varphi(gh^{-1})f_{gh^{-1}},$

which shows that
$\mathcal{S}_{\varphi}((f_{gh^{-1}})_{g,h\in\Gamma})=((\mathcal{M}_{\varphi}(f))_{g,h})_{g,h\in\Gamma}$.

##  3\. Convergence of Spectral Truncations of the $d$-Torus

The goal of this section is to prove that the maps
$\rho_{\Lambda}:\mathrm{C}^{\infty}(\mathbb{T}^{d})\rightarrow\mathrm{C}(\mathbb{T}^{d})^{(\Lambda)}$,
given by the compression $\rho_{\Lambda}(f):=P_{\Lambda}fP_{\Lambda}$, and
$\sigma_{\Lambda}:\mathrm{C}(\mathbb{T}^{d})^{(\Lambda)}\rightarrow\mathrm{C}^{\infty}(\mathbb{T}^{d})$,
given by
$\sigma_{\Lambda}(T):=\frac{1}{\mathcal{N}_{\mathrm{B}}(\Lambda)}\mathrm{Tr}(\ket{\psi}\bra{\psi}\alpha(T))$,
form a $\mathrm{C}^{1}$-approximate order isomorphism in the sense of
Definition 1.2.

###  3.1. A candidate for the $\mathrm{C}^{1}$-approximate order isomorphism

We begin by checking unitality, positivity and $\mathrm{C}^{1}$-contractivity
for the maps $\rho_{\Lambda}$ and $\sigma_{\Lambda}$.

######  Lemma 3.1.

The map
$\rho_{\Lambda}:\mathrm{C}^{\infty}(\mathbb{T}^{d})\rightarrow\mathrm{C}(\mathbb{T}^{d})^{(\Lambda)}$
is unital, positive and $\mathrm{C}^{1}$-contractive.

###### Proof.

Unitality is clear as
$P_{\Lambda}\mathbf{1}P_{\Lambda}\left(\sum_{n\in\overline{\mathrm{B}}_{\Lambda}{Z}}a_{n}e_{n}\otimes
s_{n}\right)=\sum_{n\in\overline{\mathrm{B}}_{\Lambda}{Z}}a_{n}e_{n}\otimes
s_{n}$, for all
$\sum_{n\in\overline{\mathrm{B}}_{\Lambda}{Z}}a_{n}e_{n}\otimes s_{n}\in
P_{\Lambda}\mathcal{H}$. Also positivity is obvious since $\langle
PaP\varphi,P\varphi\rangle_{P\mathcal{K}}=\langle
aP\varphi,P\varphi\rangle_{\mathcal{K}}\geq 0$, for any Hilbert space
$\mathcal{K}$, any projection $P\in\mathcal{B}(\mathcal{K})$ and any positive
operator $a\in\mathcal{B}(\mathcal{K})_{+}$. Contractivity in norm follows
from Plancherel’s theorem:

$\|P_{\Lambda}fP_{\Lambda}\|^{2}=\sup_{\underset{\|\varphi\|\leq 1}{\varphi\in
P_{\Lambda}\mathcal{H}}}\|P_{\Lambda}fP_{\Lambda}\varphi\|^{2}\leq\displaystyle{\sum_{n\in\overline{\mathrm{B}}_{\Lambda}{Z}}}\left|\widehat{f}(n)\right|^{2}\leq\sum_{n\in\mathbb{Z}^{d}}\left|\widehat{f}(n)\right|^{2}=\|f\|^{2}$

For contractivity in Lipschitz seminorm, we first observe that
$\rho_{\Lambda}$ commutes with $[D,\cdot]$ in the following sense, which is an
immediate consequence of the fact that $D$ commutes with $P_{\Lambda}$:

(35)
$\displaystyle[D_{\Lambda},\rho_{\Lambda}(f)]=P_{\Lambda}[D,f]P_{\Lambda}$

This immediately gives $\|[D,\cdot]\|$-contractivity:

(36)
$\displaystyle\|[D_{\Lambda},\rho_{\Lambda}(f)]\|=\|P_{\Lambda}[D,f]P_{\Lambda}\|\leq\|[D,f]\|$

∎

######  Lemma 3.2.

The map
$\sigma_{\Lambda}:\mathrm{C}(\mathbb{T}^{d})^{(\Lambda)}\rightarrow\mathrm{C}^{\infty}(\mathbb{T}^{d})$
is unital, positive and $\mathrm{C}^{1}$-contractive.

###### Proof.

Unitality is clear as, for all $x\in\mathbb{T}^{d}$, we have:

(37)
$\displaystyle\sigma_{\Lambda}(\mathbf{1})(x)=\frac{1}{\mathcal{N}_{\mathrm{B}}(\Lambda)}\mathrm{Tr}\left(\ket{\psi}\bra{\psi}\alpha_{x}(\mathbf{1})\right)=\frac{1}{\mathcal{N}_{\mathrm{B}}(\Lambda)}\mathrm{Tr}\left(\ket{\psi}\bra{\psi}\mathbf{1}\right)=1$

Positivity is also immediate from the definition. Namely, let
$T\in\mathrm{C}(\mathbb{T}^{d})^{(\Lambda)}_{+}$ be a positive operator on
$P_{\Lambda}\mathcal{H}$ and let $P_{\Lambda}\mathcal{H}\ni\zeta\mapsto
Q_{T}(\zeta):=\langle\zeta,T\zeta\rangle$ be its associated quadratic form.
For $\zeta=\sum_{n\in\mkern
1.5mu\overline{\mkern-1.5mu\mathrm{B}\mkern-1.5mu}\mkern
1.5mu_{\Lambda}{Z}}e_{n}$, we obtain:

(38) $\displaystyle
0\leq\frac{1}{\mathcal{N}_{\mathrm{B}}(\Lambda)}Q_{T}(\zeta)(x)=\frac{1}{\mathcal{N}_{\mathrm{B}}(\Lambda)}\mathrm{Tr}\left(\ket{\psi}\bra{\psi}\left(e_{-(m-n)}t_{m-n}\right)_{m,n}\right)=\sigma_{\Lambda}(T)(x),$

for all $x\in\mathbb{T}^{d}$. For contractivity, we compute:

(39) $\displaystyle|\sigma_{\Lambda}(T)(x)|$
$\displaystyle\leq\frac{1}{\mathcal{N}_{\mathrm{B}}(\Lambda)}\operatorname{Tr}(\ket{\psi}\bra{\psi})\|\alpha_{x}(T)\|=\|T\|$

For contractivity in Lipschitz seminorm, we first observe that
$\sigma_{\Lambda}$ commutes with $[D,\cdot]$ in the following sense, which is
an easy consequence of (22):

(40) $\displaystyle[D,\sigma_{\Lambda}(T)]$
$\displaystyle=-i\sum_{\mu=1}^{d}\frac{1}{\mathcal{N}_{\mathrm{B}}(\Lambda)}\sum_{n\in\mkern
1.5mu\overline{\mkern-1.5mu\mathrm{B}\mkern-1.5mu}\mkern
1.5mu_{\Lambda}{Z}}in_{\mu}t_{n}e_{n}\otimes\gamma^{\mu}$ (41)
$\displaystyle=\sum_{\mu=1}^{d}\frac{1}{\mathcal{N}_{\mathrm{B}}(\Lambda)}\mathrm{Tr}\left(\ket{\psi}\bra{\psi}\alpha\left(((n_{\mu}-m_{\mu})t_{n-m})_{n,m\in\mkern
1.5mu\overline{\mkern-1.5mu\mathrm{B}\mkern-1.5mu}\mkern
1.5mu_{\Lambda}{Z}}\right)\right)\otimes\gamma^{\mu}$ (42)
$\displaystyle=\sigma_{\Lambda}\otimes\mathbf{1}([D_{\Lambda},T])$

This immediately gives $\|[D,\cdot]\|$-contractivity:

(43)
$\displaystyle\|[D,\sigma_{\Lambda}(T)]\|=\|\sigma_{\Lambda}\otimes\mathbf{1}\left([D_{\Lambda},T]\right)\|\leq\|[D_{\Lambda},T]\|$

∎

We now compute the compositions $\sigma_{\Lambda}\circ\rho_{\Lambda}$ and
$\rho_{\Lambda}\circ\sigma_{\Lambda}$:

######  Lemma 3.3.

The two compositions
$\sigma_{\Lambda}\circ\rho_{\Lambda}:\mathrm{C}^{\infty}(\mathbb{T}^{d})\rightarrow\mathrm{C}^{\infty}(\mathbb{T}^{d})$
and
$\rho_{\Lambda}\circ\sigma_{\Lambda}:\mathrm{C}(\mathbb{T}^{d})^{(\Lambda)}\rightarrow\mathrm{C}(\mathbb{T}^{d})^{(\Lambda)}$
are given respectively by the Fourier multiplier and by the Schur multiplier
with the symbol $\mathfrak{m}_{\Lambda}$:

(44) $\displaystyle\sigma_{\Lambda}\circ\rho_{\Lambda}$
$\displaystyle=\mathcal{F}_{\mathfrak{m}_{\Lambda}}$ (45)
$\displaystyle\rho_{\Lambda}\circ\sigma_{\Lambda}$
$\displaystyle=\mathcal{S}_{\mathfrak{m}_{\Lambda}}$

###### Proof.

Both identities are just simple computations:

(46) $\displaystyle\sigma_{\Lambda}\circ\rho_{\Lambda}(f)(x)$
$\displaystyle=\frac{1}{\mathcal{N}_{\mathrm{B}}(\Lambda)}\mathrm{Tr}\left(\ket{\psi}\bra{\psi}\alpha_{x}(P_{\Lambda}fP_{\Lambda})\right)$
(47)
$\displaystyle=\frac{1}{\mathcal{N}_{\mathrm{B}}(\Lambda)}\mathrm{Tr}\left(\ket{\psi}\bra{\psi}\alpha_{x}\left(\left(\widehat{f}({k-l})\right)_{k,l\in\overline{\mathrm{B}}_{\Lambda}{Z}}\right)\right)$
(48)
$\displaystyle=\frac{1}{\mathcal{N}_{\mathrm{B}}(\Lambda)}\sum_{k,l\in\overline{\mathrm{B}}_{\Lambda}{Z}}\widehat{f}({k-l})e^{i(k-l)\cdot
x}$ (49)
$\displaystyle=\sum_{n\in\overline{\mathrm{B}}_{\Lambda}{Z}-\overline{\mathrm{B}}_{\Lambda}{Z}}\frac{\mathcal{N}_{\mathrm{L}}(\Lambda,n)}{\mathcal{N}_{\mathrm{B}}(\Lambda)}\widehat{f}(n)e_{n}(x)$
(51) $\displaystyle\rho_{\Lambda}\circ\sigma_{\Lambda}(T)$
$\displaystyle=\rho_{\Lambda}\left(\frac{1}{\mathcal{N}_{\mathrm{B}}(\Lambda)}\mathrm{Tr}\left(\ket{\psi}\bra{\psi}\alpha_{\bullet}(T)\right)\right)$
(52)
$\displaystyle=\rho_{\Lambda}\left(\frac{1}{\mathcal{N}_{\mathrm{B}}(\Lambda)}\sum_{k,l\in\overline{\mathrm{B}}_{\Lambda}^{\mathbb{Z}}}t_{k-l}e_{k-l}\right)$
(53)
$\displaystyle=\rho_{\Lambda}\left(\sum_{n\in\overline{\mathrm{B}}_{\Lambda}^{\mathbb{Z}}-\overline{\mathrm{B}}_{\Lambda}^{\mathbb{Z}}}\frac{\mathcal{N}_{\mathrm{L}}(\Lambda,n)}{\mathcal{N}_{\mathrm{B}}(\Lambda)}t_{n}e_{n}\right)$
(54)
$\displaystyle=\left(\frac{\mathcal{N}_{\mathrm{L}}(\Lambda,m-n)}{\mathcal{N}_{\mathrm{B}}(\Lambda)}t_{m-n}\right)_{m,n\in\overline{\mathrm{B}}_{\Lambda}^{\mathbb{Z}}}$

∎

###  3.2. A transference result

In order to show that the pair $(\rho_{\Lambda},\sigma_{\Lambda})$ is a
$\mathrm{C}^{1}$-approximate order isomorphism it remains to check that the
compositions $\sigma_{\Lambda}\circ\nolinebreak\rho_{\Lambda}$ and
$\rho_{\Lambda}\circ\sigma_{\Lambda}$ approximate the identity respectively on
$\mathrm{C}(\mathbb{T}^{d})$ and $\mathrm{C}(\mathbb{T}^{d})^{(\Lambda)}$ in
Lipschitz seminorm.

Recall the definition of $\mathfrak{w}_{\Lambda}^{\mu}(n)$, for
$\mu=1,\dots,d$ and $n\in\mathbb{Z}^{d}$ from (16).

######  Lemma 3.4.

For every $f\in\mathrm{C}^{\infty}(\mathbb{T}^{d})$ we have the following
equality of bounded operators on the Hilbert space
$\mathrm{L}^{2}(\mathbb{T}^{d})\otimes V$:

(55)
$\displaystyle\left(f-\sigma_{\Lambda}\circ\rho_{\Lambda}(f)\right)\otimes\mathbf{1}=\frac{i}{2}\left(\sum_{\mu=1}^{d}\mathcal{F}_{\mathfrak{w}_{\Lambda}^{\mu}}\otimes\\{\gamma^{\mu},\cdot\\}\right)([D,f]),$

where $\mathcal{F}_{\mathfrak{w}_{\Lambda}^{\mu}}$ is the Fourier multiplier
on the $\ast$-algebra $\mathrm{C}^{\infty}(\mathbb{T}^{d})$ with symbol
$\mathfrak{w}_{\Lambda}^{\mu}$.

Similarly, for every $T\in\mathrm{C}(\mathbb{T}^{d})^{(\Lambda)}$, we have the
following equality of bounded operators on the Hilbert space
$P_{\Lambda}\mathrm{L}^{2}(\mathbb T^{d})\otimes V$:

(56)
$\displaystyle\left(T-\rho_{\Lambda}\circ\sigma_{\Lambda}(T)\right)\otimes\mathbf{1}=\frac{i}{2}\left(\sum_{\mu=1}^{d}\mathcal{S}_{\mathfrak{w}_{\Lambda}^{\mu}}\otimes\\{\gamma^{\mu},\cdot\\}\right)([D_{\Lambda},T]),$

where $\mathcal{S}_{\mathfrak{w}_{\Lambda}^{\mu}}$ denotes Schur (i.e.
entrywise) multiplication on the operator system
$\mathrm{C}(\mathbb{T}^{d})^{(\Lambda)}$ with symbol
$\mathfrak{w}_{\Lambda}^{\mu}$.

###### Proof.

For the first claim, we have:

(57) $\displaystyle f-\sigma_{\Lambda}\circ\rho_{\Lambda}(f)$
$\displaystyle=f-\mathcal{F}_{\mathfrak{m}_{\Lambda}}(f)=\sum_{n\in\mathbb{Z}^{d}}(1-\mathfrak{m}_{\Lambda}(n))\widehat{f}(n)e_{n}$
(58)
$\displaystyle=\sum_{\mu=1}^{d}\sum_{n\in\mathbb{Z}^{d}}(1-\mathfrak{m}_{\Lambda}(n))\frac{n_{\mu}}{\|n\|^{2}}\cdot
n_{\mu}\widehat{f}(n)e_{n}$ (59)
$\displaystyle=\sum_{\mu,\nu=1}^{d}\mathcal{F}_{\mathfrak{w}^{\mu}_{\Lambda}}(-i\partial_{\nu}f)\cdot\delta^{\mu\nu}.$

The first claimed result then follows by writing
$2\delta^{\mu\nu}\cdot\mathbf{1}=\\{\gamma^{\mu},\gamma^{\nu}\\}$.

The computation for the second claim is largely analogous:

(60) $\displaystyle T-\rho_{\Lambda}\circ\sigma_{\Lambda}(T)$
$\displaystyle=T-\mathcal{S}_{\mathfrak{m}_{\Lambda}}(T)$ (61)
$\displaystyle=\sum_{\mu=1}^{d}\left((1-\mathfrak{m}_{\Lambda}(k-l))\frac{k_{\mu}-l_{\mu}}{\|k-l\|^{2}}\cdot(k_{\mu}-l_{\mu})t_{k-l}\right)_{k,l\in\mkern
1.5mu\overline{\mkern-1.5mu\mathrm{B}\mkern-1.5mu}\mkern
1.5mu_{\Lambda}^{\mathbb{Z}}}$ (62)
$\displaystyle=i\sum_{\mu,\nu=1}^{d}\mathcal{S}_{\mathfrak{w}^{\mu}_{\Lambda}}\left(\left((k_{\nu}-l_{\nu})t_{k-l}\right)_{k,l\in\mkern
1.5mu\overline{\mkern-1.5mu\mathrm{B}\mkern-1.5mu}\mkern
1.5mu_{\Lambda}^{\mathbb{Z}}}\right)\cdot\delta^{\mu\nu}.$

The result now follows by combining the defining relations for the gamma-
matrices as before with the expression (22) for the operator
$[D_{\Lambda},T]\in\mathcal{B}(P_{\Lambda}\mathcal{H})$. ∎

It is a classical result of Bożejko and Fendler [4] that for any discrete
group $\Gamma$ and function $\varphi:\Gamma\to\mathbb{C}$ the cb-norm of Schur
multiplication $S_{\varphi}$ on $\mathcal{B}(\ell^{2}(\Gamma))$ coincides with
the cb-norm of Fourier multiplication $\mathcal{F}_{\varphi}$ on
$\mathrm{C}^{*}_{\lambda}(\Gamma)$ (see also [22, Theorem 6.4] and [5,
Proposition D.6]). However, the two linear maps obtained in 3.4 act on the
operator subsystems of differential forms on $\mathcal{H}$ and
$P_{\Lambda}\mathcal{H}$, respectively, so the result of Bożejko and Fendler
does not apply directly. We prove a variation on it which relates the cb-norms
of the two linear maps which appear in 3.4:

(63)
$\displaystyle\mathcal{F}_{\mathfrak{w}_{\Lambda}}:=\frac{i}{2}\sum_{\mu=1}^{d}\mathcal{F}_{\mathfrak{w}^{\mu}_{\Lambda}}\otimes\\{\gamma^{\mu},\cdot\\}$
$\displaystyle:[D,\mathrm{C}^{\infty}(\mathbb{T}^{d})]\to\mathrm{C}^{\infty}(\mathbb{T}^{d});$
(64)
$\displaystyle\mathcal{S}_{\mathfrak{w}_{\Lambda}}:=\frac{i}{2}\sum_{\mu=1}^{d}\mathcal{S}_{\mathfrak{w}^{\mu}_{\Lambda}}\otimes\\{\gamma^{\mu},\cdot\\}$
$\displaystyle:[D_{\Lambda},\mathrm{C}(\mathbb{T}^{d})^{(\Lambda)}]\to\mathrm{C}(\mathbb{T}^{d})^{(\Lambda)},$

Here we consider $[D,\mathrm{C}^{\infty}(\mathbb{T}^{d})]$ and
$[D_{\Lambda},\mathrm{C}(\mathbb{T}^{d})^{(\Lambda)}]$ as (dense subsets of)
operator systems in $\mathcal{B}(\mathrm{L}^{2}(\mathbb{T}^{d})\otimes V)$.

######  Lemma 3.5.

For the above two linear maps we have the following norm inequality:

$\|\mathcal{S}_{\mathfrak{w}_{\Lambda}}\|\leq\|\mathcal{F}_{\mathfrak{w}_{\Lambda}}\|$

###### Proof.

We vary on the proof given in [22, Theorem 6.4]. First identify
$\mathrm{L}^{2}(\mathbb{T}^{d})\otimes V\cong\ell^{2}(\mathbb{Z}^{d})\otimes
V$ using the Fourier basis $\\{e_{n}\\}_{n\in\mathbb{Z}^{d}}$, and write
$\mathcal{H}:=\ell^{2}(\mathbb{Z}^{d})\otimes V$. Consider the unitary
operator $U$ defined on $\mathcal{H}\otimes\mathcal{H}$ by a combination of a
shift in Fourier space and a tensor flip in spinor space:

$U(e_{n}\otimes v\otimes e_{m}\otimes v^{\prime})=e_{n}\otimes
v^{\prime}\otimes e_{n+m}\otimes v$

Recall that an elementary matrix $E_{kl}$ ($k,l\in\mathbb{Z}^{d}$) acts on
$\mathcal{H}$ as:

(65) $\displaystyle E_{kl}(e_{n}\otimes v)$
$\displaystyle=\delta_{ln}e_{k}\otimes v,$ in contrast to a generator $e_{k}$
in the group $\mathrm{C}^{*}$-algebra
$\mathrm{C}^{*}(\mathbb{Z}^{d})\cong\mathrm{C}(\mathbb{T}^{d})$, which acts as
(66) $\displaystyle e_{k}(e_{n}\otimes v)$ $\displaystyle=e_{n+k}\otimes v.$

Note furthermore that under the identification
$\mathrm{L}^{2}(\mathbb{T}^{d})\cong\ell^{2}(\mathbb{Z}^{d})$ we have

(67) $D(e_{n}\otimes
v)=ne_{n}\otimes\gamma^{\mu}v,\qquad(n\in\mathbb{Z}^{d},v\in V).$

We then find that

(68) $\displaystyle U(E_{kl}\otimes\mathbf{1}_{\mathcal{H}})U^{*}$
$\displaystyle=E_{kl}\otimes e_{k-l}$ (69) $\displaystyle
U(E_{kl}\gamma^{\mu}\otimes\mathbf{1}_{\mathcal{H}})U^{*}$
$\displaystyle=E_{kl}\otimes e_{k-l}\gamma^{\mu}$

where $\gamma^{\mu}$ acts of course on the spinor space $V$. Note that in view
of Equation (67) we also have
$U([D,E_{kl}]\otimes\mathbf{1}_{\mathcal{H}})U^{*}=E_{kl}\otimes[D,e_{k-l}]\in\mathcal{K}(\mathcal{H})\otimes[D,\mathrm{C}^{\infty}(\mathbb{T}^{d})]$.

Using this we may now show

(70) $\displaystyle
U\left((\mathcal{S}_{\mathfrak{w}_{\Lambda}}\otimes\mathrm{id}_{\mathcal{B}(\mathcal{H})})\big{(}[D,E_{kl}]\otimes\mathbf{1}_{\mathcal{H}}\big{)}\right)U^{*}$
$\displaystyle=\sum_{\mu}U\left(\mathcal{S}_{\mathfrak{w}_{\Lambda}}(\gamma^{\mu}(k-l)_{\mu}E_{kl})\otimes\mathbf{1}_{\mathcal{H}}\right)U^{*}$
(71)
$\displaystyle=i\sum_{\mu}U\left({\mathfrak{w}^{\mu}_{\Lambda}}(k-l)(k-l)_{\mu}E_{kl}\otimes\mathbf{1}_{\mathcal{H}}\right)U^{*}$
(72)
$\displaystyle=i\sum_{\mu}E_{kl}\otimes{\mathfrak{w}^{\mu}_{\Lambda}}(k-l)(k-l)_{\mu}e_{k-l}$
(73)
$\displaystyle=E_{kl}\otimes\mathcal{F}_{\mathfrak{w}_{\Lambda}}([D,e_{k-l}])$
(74)
$\displaystyle=(\mathrm{id}_{\mathcal{B}(\mathcal{H})}\otimes\mathcal{F}_{\mathfrak{w}_{\Lambda}})\left(U([D,E_{kl}]\otimes\mathbf{1}_{\mathcal{H}})U^{*}\right).$

This extends by linearity to arbitrary $x=\sum_{k,l\in\mkern
1.5mu\overline{\mkern-1.5mu\mathrm{B}\mkern-1.5mu}\mkern
1.5mu_{\Lambda}{Z}}t_{k-l}E_{kl}\in\mathrm{C}(\mathbb{T}^{d})^{(\Lambda)}$ to
yield

$U\left((\mathcal{S}_{\mathfrak{w}_{\Lambda}}\otimes\text{id}_{\mathcal{B}(\mathcal{H})})\big{(}[D,x]\otimes\mathbf{1}_{\mathcal{H}}\big{)}\right)U^{*}=(\text{id}_{\mathcal{B}(\mathcal{H})}\otimes\mathcal{F}_{\mathfrak{w}_{\Lambda}})\left(U([D,x]\otimes\mathbf{1}_{\mathcal{H}})U^{*}\right).$

From this we obtain the following estimate:

(75) $\displaystyle\|\mathcal{S}_{\mathfrak{w}_{\Lambda}}([D,x])\|$
$\displaystyle=\|\mathcal{S}_{\mathfrak{w}_{\Lambda}}([D,x])\otimes\mathbf{1}_{\mathcal{H}}\|$
(76)
$\displaystyle=\|(\mathcal{S}_{\mathfrak{w}_{\Lambda}}\otimes\mathrm{id}_{\mathcal{B}(\mathcal{H})})([D,x]\otimes\mathbf{1}_{\mathcal{H}})\|$
(77)
$\displaystyle=\|U\left((\mathcal{S}_{\mathfrak{w}_{\Lambda}}\otimes\mathrm{id}_{\mathcal{B}(\mathcal{H})})\big{(}[D,x]\otimes\mathbf{1}_{\mathcal{H}}\big{)}\right)U^{*}\|$
(78)
$\displaystyle=\|(\mathrm{id}_{\mathcal{B}(\mathcal{H})}\otimes\mathcal{F}_{\mathfrak{w}_{\Lambda}})\left(U([D,x]\otimes\mathbf{1}_{\mathcal{H}})U^{*}\right)\|$
(79)
$\displaystyle\leq\|\mathrm{id}_{\mathcal{K}(\mathcal{H})}\otimes\mathcal{F}_{\mathfrak{w}_{\Lambda}}\|\|U([D,x]\otimes\mathbf{1}_{\mathcal{H}})U^{*}\|$
(80)
$\displaystyle=\|\mathcal{F}_{\mathfrak{w}_{\Lambda}}\|_{\textup{cb}}\|[D,x]\|,$

where the penultimate step follows from the fact that
$U([D,x]\otimes\mathbf{1}_{\mathcal{H}})U^{*}\in\mathcal{K}(\mathcal{H})\otimes[D,\mathrm{C}^{\infty}(\mathbb{T}^{d})]$.
This implies that
$\|\mathcal{S}_{\mathfrak{w}_{\Lambda}}\|\leq\|\mathcal{F}_{\mathfrak{w}_{\Lambda}}\|_{\textup{cb}}$.

Finally, supposing that $\mathcal{F}_{{\mathfrak{w}_{\Lambda}}}$ is a bounded
linear map its norm and $\mathrm{cb}$-norm coincide because its range is a
subset of a commutative $\mathrm{C}^{\ast}$-algebra, namely
$\mathrm{C}(\mathbb{T}^{d})$ (cf. [21, Theorem 3.9]). ∎

######  Remark 3.6.

Note that the classical transference theorem is stated as an equality of the
cb-norms of a Schur multiplier $\mathcal{S}_{\varphi}$ and a Fourier
multiplier $\mathcal{F}_{\varphi}$. However, for this it is crucial that
$\mathcal{F}_{\varphi}$ and $\mathcal{S}_{\varphi}$ are defined on the
$\mathrm{C}^{*}$-algebras $\mathrm{C}_{\lambda}^{*}(\Gamma)$ and
$\mathcal{B}(\ell^{2}(\Gamma))$ which is not the situation we find in the
above lemma. In fact, the maps (63) and (64) are only defined on operator
subsystems and do not extend to the $\mathrm{C}^{*}$-algebras
$\mathrm{C}(\mathbb{T}^{d})$ and $\mathcal{B}(\mathcal{H})$ respectively, so
equality in 3.5 is not to be expected.

Our task is thus reduced to the computation of the norm of the map
$\mathcal{F}_{\mathfrak{w}_{\Lambda}}:[D,\mathrm{C}^{\infty}(\mathbb{T}^{d})]\to\mathrm{C}^{\infty}(\mathbb{T}^{d})$
given in Equation (63).

###  3.3. The spectral Fejér kernel

We define $K_{\mathfrak{m}_{\Lambda}}$ as the convolution kernel corresponding
to the Fourier multiplier $\mathcal{F}_{\mathfrak{m}_{\Lambda}}$ as in (32),
i.e.

(81) $\displaystyle
K_{\mathfrak{m}_{\Lambda}}:=\widecheck{\mathfrak{m}}_{\Lambda}.$

In view of 2.2 it would be desirable to see that $K_{\mathfrak{m}_{\Lambda}}$
is a _good kernel_. Indeed, by 3.4 this would give precisely the estimate of
$\mathcal{F}_{\mathfrak{w}_{\Lambda}}$ which remains to show.

######  Lemma 3.7.

For every $n\in\mathbb{Z}^{d}$, we have that
$\mathfrak{m}_{\Lambda}(n)\rightarrow 1$, as $\Lambda\rightarrow\infty$.

###### Proof.

This follows from two simple geometric observations. One is that
$|\mathcal{N}_{\mathrm{B}}(\Lambda)-\mathcal{V}_{\mathrm{B}}(\Lambda)|\leq\sqrt{d}\mathcal{A}_{\mathrm{B}}(\Lambda)$,
where $\mathcal{V}_{\mathrm{B}}(\Lambda)$ is the volume of the $d$-dimensional
ball of radius $\Lambda$ and $\mathcal{A}_{\mathrm{B}}(\Lambda)$ is its
surface area. The other is that a lense (i.e. the intersection of two
$d$-dimensional balls of radius $\Lambda$ one of them shifted by a parameter
$n\in\mathbb{Z}^{d}$) contains the ball of radius $\Lambda-\|n\|$, if this
number is non-negative. Together, these observations yield the following
estimates:

(82) $\displaystyle|1-\mathfrak{m}_{\Lambda}(n)|$
$\displaystyle=\frac{1}{\mathcal{N}_{\mathrm{B}}(\Lambda)}|\mathcal{N}_{\mathrm{B}}(\Lambda)-\mathcal{N}_{\mathrm{L}}(\Lambda,n)|$
(83)
$\displaystyle\leq\frac{1}{\mathcal{V}_{\mathrm{B}}(\Lambda-\sqrt{d})}|\mathcal{V}_{\mathrm{B}}(\Lambda+\sqrt{d})-\mathcal{V}_{\mathrm{B}}((\Lambda-\sqrt{d}-\|n\|)_{+})|$
(84)
$\displaystyle=\frac{1}{(\Lambda-\sqrt{d})^{d}}\underbrace{|(\Lambda+\sqrt{d})^{d}-((\Lambda-\sqrt{d}-\|n\|)_{+})^{d}|}_{=\mathcal{O}(\Lambda^{d-1})}$
(85) $\displaystyle\rightarrow 0,$

as $\Lambda\rightarrow\infty$. (Here $t_{+}:=t$, if $t\geq 0$, and $t_{+}:=0$,
if $t<0$, for $t\in\mathbb{R}$.) ∎

######  Proposition 3.8.

The function $K_{\mathfrak{m}_{\Lambda}}$ is positive and is a _good kernel_.

###### Proof.

We begin by showing positivity of $K_{\mathfrak{m}_{\Lambda}}$. Indeed,
$K_{\mathfrak{m}_{\Lambda}}=\widecheck{\mathfrak{m}}_{\Lambda}$ is the
(inverse) Fourier transform of a convolution-square:

(86)
$\displaystyle\begin{split}\mathfrak{m}_{\Lambda}(n)&=\frac{\mathcal{N}_{\mathrm{L}}(\Lambda,n)}{\mathcal{N}_{\mathrm{B}}}\\\
&=\frac{1}{\mathcal{N}_{\mathrm{B}}(\Lambda)}\sum_{k\in\mathbb{Z}^{d}}\chi_{\mkern
1.5mu\overline{\mkern-1.5mu{\mathrm{B}}\mkern-1.5mu}\mkern
1.5mu_{\Lambda}{Z}}(k)\chi_{\mkern
1.5mu\overline{\mkern-1.5mu{\mathrm{B}}\mkern-1.5mu}\mkern
1.5mu_{\Lambda}{Z}(n)}(k)\\\
&=\frac{1}{\mathcal{N}_{\mathrm{B}}(\Lambda)}\left(\chi_{\mkern
1.5mu\overline{\mkern-1.5mu{\mathrm{B}}\mkern-1.5mu}\mkern
1.5mu_{\Lambda}{Z}}\ast\chi_{\mkern
1.5mu\overline{\mkern-1.5mu{\mathrm{B}}\mkern-1.5mu}\mkern
1.5mu_{\Lambda}{Z}}\right)(n)\end{split}$

Next, we check the total mass of $K_{\mathfrak{m}_{\Lambda}}$:

(87)
$\displaystyle\int_{\mathbb{T}^{d}}K_{\mathfrak{m}_{\Lambda}}(x)\,\mathrm{d}x$
$\displaystyle=\frac{1}{\mathcal{N}_{\mathrm{B}}(\Lambda)}\left(\chi_{\mkern
1.5mu\overline{\mkern-1.5mu{\mathrm{B}}\mkern-1.5mu}\mkern
1.5mu_{\Lambda}{Z}}\ast\chi_{\mkern
1.5mu\overline{\mkern-1.5mu{\mathrm{B}}\mkern-1.5mu}\mkern
1.5mu_{\Lambda}{Z}}\right)(0)$ (88)
$\displaystyle=\frac{\mathcal{N}_{\mathrm{L}}(\Lambda,0)}{\mathcal{N}_{\mathrm{B}}(\Lambda)}=1$

Last, we argue that the mass of $K_{\mathfrak{m}_{\Lambda}}$ becomes
concentrated around $0$ as $\Lambda\rightarrow\infty$. Fix some $\delta>0$ and
let $\varepsilon>0$ be arbitrary. Let $\varphi$ be a non-negative
trigonometric polynomial such that the following holds:

(89) $\displaystyle 1-\chi_{\mathrm{B}_{\delta}}\leq\varphi\leq
1-\chi_{\mathrm{B}_{\nicefrac{{\delta}}{{2}}}}+\frac{\varepsilon}{2}$

The existence of such a trigonometric polynomial $\varphi$ follows from
Weierstraß’s approximation theorem for trigonometric polynomials since the
$\frac{\varepsilon}{4}$-neighborhood of the continuous function $\psi$, given
by

(90) $\displaystyle\psi(x)=\begin{cases}\frac{\varepsilon}{4},\text{ if
}x\in\mathrm{B}_{\nicefrac{{\delta}}{{2}}},\\\
\frac{2}{\delta}\|x\|+\frac{\varepsilon}{4}-1,\text{ if
}x\in\mathrm{B}_{\delta}\setminus\mathrm{B}_{\nicefrac{{\delta}}{{2}}},\\\
1+\frac{\varepsilon}{4},\text{ if
}\mathrm{T}^{d}\setminus\mathrm{B}_{\delta},\end{cases}$

lies between $1-\chi_{\mathrm{B}_{\delta}}$ and
$1-\chi_{\mathrm{B}_{\nicefrac{{\delta}}{{2}}}}+\frac{\varepsilon}{2}$. Let
$\Lambda_{0}>0$ be large enough such that, for all $\Lambda\geq\Lambda_{0}$,
we have that
$\displaystyle{\max_{n\in\mathrm{supp}(\widehat{\varphi})}}|1-\mathfrak{m}_{\Lambda}(n)|\leq\frac{\varepsilon}{2\|\widehat{\varphi}\|_{\ell^{1}(\mathbb{Z}^{d})}}$.
This is of course possible since the pointwise convergence from 3.7 implies
uniform convergence to $1$ of the restriction of $\mathfrak{m}_{\Lambda}$ to
the finite set $\mathrm{supp}(\widehat{\varphi})$. Then the following holds:

(91)
$\displaystyle\int_{\mathbb{T}^{d}\setminus\mathrm{B}_{\delta}}K_{\mathfrak{m}_{\Lambda}}(x)\,\mathrm{d}x$
$\displaystyle=\int_{\mathbb{T}^{d}}K_{\mathfrak{m}_{\Lambda}}(x)(1-\chi_{\mathrm{B}_{\delta}}(x))\,\mathrm{d}x$
(92)
$\displaystyle\leq\int_{\mathbb{T}^{d}}K_{\mathfrak{m}_{\Lambda}}(x)\varphi(x)\,\mathrm{d}x$
(93)
$\displaystyle\leq\left|\int_{\mathbb{T}^{d}}K_{\mathfrak{m}_{\Lambda}}(x)\varphi(x)\,\mathrm{d}x-\varphi(0)\right|+\frac{\varepsilon}{2}$
(94)
$\displaystyle=\left|\sum_{n\in\mathbb{Z}^{d}}\mathfrak{m}_{\Lambda}(n)\widehat{\varphi}(n)-\sum_{n\in\mathbb{Z}^{d}}\widehat{\varphi}(n)\right|+\frac{\varepsilon}{2}$
(95)
$\displaystyle\leq\|\left(\mathfrak{m}_{\Lambda}-1\right)|_{\mathrm{supp}(\widehat{\varphi})}\|_{\ell^{\infty}(\mathbb{Z}^{d})}\cdot\|\widehat{\varphi}\|_{\ell^{1}(\mathbb{Z}^{d})}+\frac{\varepsilon}{2}$
(96) $\displaystyle\leq\frac{\varepsilon}{2}+\frac{\varepsilon}{2},$

for all $\Lambda\geq\Lambda_{0}$. In the fourth step we applied the Plancherel
formula and in the fifth step the Hölder inequality. ∎

######  Remark 3.9.

Note that (86) shows that the function $K_{\mathfrak{m}_{\Lambda}}$ is
precisely the square of the well-known spherical Dirichlet kernel (_cf._
Lipschitz seminorminition 3.1.6]Gra14). However, by a classical result by du
Bois-Reymond the latter is not a _good kernel_ (_cf._ [12, Proposition
3.3.5]). The feature of $K_{\mathfrak{m}_{\Lambda}}$ which is crucial for its
good behavior is positivity. We emphasize that our _spectral Fejér kernel_
does not coincide with the so-called _circular Fejér kernel_ which is
investigated in [12, Chapter 3]. Note in particular, that the _circular_ Fejér
kernel fails to be _good_ in dimensions $d\geq 3$ which motivates the
introduction of Bochner–Riesz summability methods.

######  Theorem 3.10.

Spectral truncations of $\mathbb{T}^{d}$ converge, for all $d\geq 1$.

###### Proof.

From
$(T-\rho_{\Lambda}\circ\sigma_{\Lambda}(T))\otimes\mathbf{1}=\mathcal{S}_{\mathfrak{w}_{\Lambda}}(T)$
(3.4) and
$\|\mathcal{S}_{\mathfrak{w}_{\Lambda}}\|\leq\|\mathcal{F}_{\mathfrak{w}_{\Lambda}}\|$
(3.5), together with
$\mathcal{F}_{\mathfrak{w}_{\Lambda}}(f)=(f-\sigma_{\Lambda}\circ\rho_{\Lambda}(f))\otimes\mathbf{1}=(f-K_{\mathfrak{w}_{\Lambda}}\ast
f)\otimes\mathbf{1}$ (again 3.4), we obtain from 2.2 a sequence
$\gamma_{\Lambda}\rightarrow 0$, as $\Lambda\rightarrow\infty$, such that

(97) $\displaystyle\|T-\rho_{\Lambda}\circ\sigma_{\Lambda}(T)\|$
$\displaystyle\leq\gamma_{\Lambda}\|[D_{\Lambda},T]\|\quad\text{and}$ (98)
$\displaystyle\|f-\sigma_{\Lambda}\circ\rho_{\Lambda}(f)\|$
$\displaystyle\leq\gamma_{\Lambda}\|[D,f]\|,$

since the spectral Fejér kernel $K_{\mathfrak{m}_{\Lambda}}$ is a good kernel
(3.8). Together with 3.1 and 3.2 this shows that
$(\rho_{\Lambda},\sigma_{\Lambda})$ is a $\mathrm{C}^{1}$-approximate order
isomorphism which by [31, Theorem 5] implies our result. ∎

######  Remark 3.11.

We point out that similarly convergence of other kinds of truncations of
$\mathbb{T}^{d}$ can be shown. In particular, by replacing our projections
$P_{\Lambda}$ with the projections $P_{N}^{\square}$ with
$\mathrm{ran}(P_{N}^{\square})=\\{e_{n}\,:\,n_{i}=-N,\dots,N,\text{ for
}i=1,\dots,d\\}$, analogous arguments to the ones presented in this chapter
yield a new proof that the “box-truncations” of $\mathbb{T}^{d}$, which were
considered in [2], converge.

##  4\. Structure Analysis of the Operator System
$\mathrm{C}(\mathbb{T}^{d})^{(\Lambda)}$

###  4.1. C*-envelope and propagation number

Recall [13] (see also [21, Chapter 15]) that a _$\mathrm{C}^{*}$ -extension_
of a unital operator system ${E}$ is a unital $\mathrm{C}^{*}$-algebra ${A}$
together with an injective completely positive map $\iota:{E}\rightarrow{A}$
such that $\mathrm{C}^{*}(\iota({E}))={A}$. A $\mathrm{C}^{*}$-extension ${A}$
of ${E}$ is called the _$\mathrm{C}^{*}$ -envelope_ and denoted by
$\mathrm{C}_{\mathrm{env}}^{*}({E})$ if, for every unital
$\mathrm{C}^{*}$-algebra ${B}$ and every unital completely positive map
$\phi:{A}\rightarrow{B}$, the map $\phi$ is a complete order injection if the
composition $\phi\circ\iota$ is. Recall furthermore from [8] that the
_propagation number_ $\mathrm{prop}({E})$ is the smallest positive integer $n$
such that $(\iota({E}))^{\circ n}\subseteq\mathrm{C}_{\mathrm{env}}^{*}({E})$
is a $\mathrm{C}^{*}$-algebra, where ${F}^{\circ
n}=\mathrm{span}\\{f_{1}\cdots f_{n}\,:\,f_{i}\in{F}\\}$, for a unital
operator subsystem ${F}$ of a unital $\mathrm{C}^{\ast}$-algebra ${A}$.

For $p\in\mkern 1.5mu\overline{\mkern-1.5mu\mathrm{B}\mkern-1.5mu}\mkern
1.5mu_{\Lambda}{Z}+\mkern
1.5mu\overline{\mkern-1.5mu\mathrm{B}\mkern-1.5mu}\mkern 1.5mu_{\Lambda}{Z}$,
we define the following operator in
$\mathrm{C}(\mathbb{T}^{d})^{(\Lambda)}\subset\mathcal{B}(P_{\Lambda}\mathcal{H})$:

(99) $\displaystyle T_{p}:=\sum_{n\in\mkern
1.5mu\overline{\mkern-1.5muL\mkern-1.5mu}\mkern
1.5mu_{\Lambda}^{\mathbb{Z}}(p)}E_{n-p,n}=\sum_{n\in\mkern
1.5mu\overline{\mkern-1.5muL\mkern-1.5mu}\mkern
1.5mu_{\Lambda}^{\mathbb{Z}}(-p)}E_{n,n+p},$

where $E_{k,l}\in\mathcal{B}(P_{\Lambda}\mathcal{H})$ is the _matrix unit_
given by $\langle e_{m},E_{k,l}e_{n}\rangle=\delta_{mk}\delta_{ln}$, for
$k,l,m,n\in\mkern 1.5mu\overline{\mkern-1.5mu\mathrm{B}\mkern-1.5mu}\mkern
1.5mu_{\Lambda}{Z}$. It is not hard to check that $\\{T_{p}\\}_{p\in\mkern
1.5mu\overline{\mkern-1.5mu\mathrm{B}\mkern-1.5mu}\mkern
1.5mu_{\Lambda}{Z}+\mkern
1.5mu\overline{\mkern-1.5mu\mathrm{B}\mkern-1.5mu}\mkern 1.5mu_{\Lambda}{Z}}$
is a basis for the operator system $\mathrm{C}(\mathbb{T}^{d})^{(\Lambda)}$.

With the preparations of Appendix A, we are in position to treat the
$\mathrm{C}^{*}$-envelope and propagation number of the operator system
$\mathrm{C}(\mathbb{T}^{d})^{(\Lambda)}$.

######  Proposition 4.1.

The $\mathrm{C}^{\ast}$-envelope and the propagation number of
$\mathrm{C}(\mathbb{T}^{d})^{(\Lambda)}$ are given by
$\mathrm{C}^{\ast}_{\text{env}}(\mathrm{C}(\mathbb{T}^{d})^{(\Lambda)})=\mathcal{B}(P_{\Lambda}\mathcal{H})$
and $\operatorname{prop}(\mathrm{C}(\mathbb{T}^{d})^{(\Lambda)})=2$.

###### Proof.

The matrix order structure on $\mathrm{C}(\mathbb{T}^{d})^{(\Lambda)}$ is the
one inherited from the inclusion into $\mathcal{B}(P_{\Lambda}\mathcal{H})$.
It remains to show that the inclusion
$\mathrm{C}(\mathbb{T}^{d})^{(\Lambda)}\hookrightarrow\mathcal{B}(P_{\Lambda}\mathcal{H})$
is a $\mathrm{C}^{\ast}$-extension, i.e. that it generates
$\mathcal{B}(P_{\Lambda}\mathcal{H})$. Indeed, if this is the case, it is
clear that
$\mathcal{B}(P_{\Lambda}\mathcal{H})\cong\mathrm{C}^{\ast}_{\text{env}}(\mathrm{C}(\mathbb{T}^{d})^{(\Lambda)})$
since $\mathcal{B}(P_{\Lambda}\mathcal{H})$ is simple.

We will see that $\mathcal{B}(P_{\Lambda}\mathcal{H})$ is in fact spanned by
respective products of two basic operators (99). To this end, let
$p,q\in\mkern 1.5mu\overline{\mkern-1.5mu\mathrm{B}\mkern-1.5mu}\mkern
1.5mu_{\Lambda}{Z}+\mkern
1.5mu\overline{\mkern-1.5mu\mathrm{B}\mkern-1.5mu}\mkern 1.5mu_{\Lambda}{Z}$.
Then, the following holds:

(100) $\displaystyle\begin{split}T_{p}T_{q}&=\left(\sum_{n\in\mkern
1.5mu\overline{\mkern-1.5muL\mkern-1.5mu}\mkern
1.5mu_{\Lambda}^{\mathbb{Z}}(p)}E_{n-p,n}\right)\left(\sum_{n\in\mkern
1.5mu\overline{\mkern-1.5muL\mkern-1.5mu}\mkern
1.5mu_{\Lambda}^{\mathbb{Z}}(-q)}E_{n,n+q}\right)\\\ &=\sum_{n\in\mkern
1.5mu\overline{\mkern-1.5muL\mkern-1.5mu}\mkern
1.5mu_{\Lambda}^{\mathbb{Z}}(p)\cap\mkern
1.5mu\overline{\mkern-1.5muL\mkern-1.5mu}\mkern
1.5mu_{\Lambda}^{\mathbb{Z}}(-q)}E_{n-p,n+q}\\\ &=\sum_{n\in\mkern
1.5mu\overline{\mkern-1.5muL\mkern-1.5mu}\mkern
1.5mu_{\Lambda}^{\mathbb{Z}}(p+q)\cap\mkern
1.5mu\overline{\mkern-1.5muL\mkern-1.5mu}\mkern
1.5mu_{\Lambda}^{\mathbb{Z}}(q)}E_{n-p-q,n},\end{split}$

where we used the fact that $\left(\mkern
1.5mu\overline{\mkern-1.5muL\mkern-1.5mu}\mkern 1.5mu_{\Lambda}(p)\cap\mkern
1.5mu\overline{\mkern-1.5muL\mkern-1.5mu}\mkern
1.5mu_{\Lambda}(-q)\right)+q=\mkern
1.5mu\overline{\mkern-1.5muL\mkern-1.5mu}\mkern 1.5mu_{\Lambda}(p+q)\cap\mkern
1.5mu\overline{\mkern-1.5muL\mkern-1.5mu}\mkern 1.5mu_{\Lambda}(q)$ which can
be easily checked. As a special case, for $l,k\in\mkern
1.5mu\overline{\mkern-1.5mu\mathrm{B}\mkern-1.5mu}\mkern
1.5mu_{\Lambda}{Z}+\mkern
1.5mu\overline{\mkern-1.5mu\mathrm{B}\mkern-1.5mu}\mkern 1.5mu_{\Lambda}{Z}$
such that $l+k\in\mkern
1.5mu\overline{\mkern-1.5mu\mathrm{B}\mkern-1.5mu}\mkern
1.5mu_{\Lambda}{Z}+\mkern
1.5mu\overline{\mkern-1.5mu\mathrm{B}\mkern-1.5mu}\mkern 1.5mu_{\Lambda}{Z}$,
we obtain:

(101) $\displaystyle T_{-k}T_{l+k}=\sum_{n\in\mkern
1.5mu\overline{\mkern-1.5muL\mkern-1.5mu}\mkern
1.5mu_{\Lambda}^{\mathbb{Z}}(l)\cap\mkern
1.5mu\overline{\mkern-1.5muL\mkern-1.5mu}\mkern
1.5mu_{\Lambda}^{\mathbb{Z}}(l+k)}E_{n-l,n}$

Note that this generalizes the formula given in the proof of [8, Proposition
4.2] where $d$ was equal to $1$. 111 Moreover, this formula may be interpreted
in a similar way: The operator $T_{-k}T_{l+k}$ can be regarded as “matrix”
(with multi-indexed entries) which has $0$-entries everywhere except for the
$l$-th “diagonal” where its entries are either $1$ or $0$ depending on the
parameter $k$.

We need some elementary geometric observations. For $\Lambda^{\prime}\geq 0$,
let
$K_{\Lambda^{\prime}}:=\operatorname{co}\left(\overline{\mathrm{B}}_{\Lambda^{\prime}}^{\mathbb{Z}}\right)$
denote the convex hull of the set of $\mathbb{Z}^{d}$-lattice points in the
closed ball of radius $\Lambda^{\prime}$. Note that
$K_{\Lambda^{\prime}}^{\mathbb{Z}}:=K_{\Lambda^{\prime}}\cap\mathbb{Z}^{d}=\overline{\mathrm{B}}_{\Lambda^{\prime}}^{\mathbb{Z}}$.
Furthermore, $K_{\Lambda^{\prime}}$ is a polytope which is symmetric under
reflections along coordinate axes and diagonals (i.e. under changing signs of
coordinates and exchanging coordinates). Clearly, all the extreme points of
$K_{\Lambda^{\prime}}$, the set of which is denoted by
$\operatorname{ex}(K_{\Lambda^{\prime}})$, have integer coordinates. Moreover,
if $x\in K_{\Lambda^{\prime}}$ is of norm $\|x\|=\Lambda^{\prime}$ it is an
extreme point, but not necessarily all extreme points of
$K_{\Lambda^{\prime}}$ are of norm $\Lambda^{\prime}$ as can be seen in the
case $d=2$, $\Lambda^{\prime}=3$ (_cf._ Figure 1).

Figure 1. A plot of the boundaries of the balls $\mathrm{B}_{\Lambda}$ and the
polytopes $K_{\Lambda}$, for $\Lambda=0,1,\sqrt{2},2,\sqrt{5},2\sqrt{2},3$.

In order to prove the claim it is enough to write every rank-one operator
$E_{p,q}\in\mathcal{B}(P_{\Lambda}\mathcal{H})$ as a linear combination of
products of the form (101), where $p,q\in\mkern
1.5mu\overline{\mkern-1.5mu\mathrm{B}\mkern-1.5mu}\mkern 1.5mu_{\Lambda}{Z}$
and $l,k\in\mkern 1.5mu\overline{\mkern-1.5mu\mathrm{B}\mkern-1.5mu}\mkern
1.5mu_{\Lambda}{Z}+\mkern
1.5mu\overline{\mkern-1.5mu\mathrm{B}\mkern-1.5mu}\mkern 1.5mu_{\Lambda}{Z}$
such that $l+k\in\mkern
1.5mu\overline{\mkern-1.5mu\mathrm{B}\mkern-1.5mu}\mkern
1.5mu_{\Lambda}{Z}+\mkern
1.5mu\overline{\mkern-1.5mu\mathrm{B}\mkern-1.5mu}\mkern 1.5mu_{\Lambda}{Z}$.
To this end, fix $p,q\in\mkern
1.5mu\overline{\mkern-1.5mu\mathrm{B}\mkern-1.5mu}\mkern 1.5mu_{\Lambda}{Z}$
and set $l:=q-p\in\mkern
1.5mu\overline{\mkern-1.5mu\mathrm{B}\mkern-1.5mu}\mkern
1.5mu_{\Lambda}{Z}+\mkern
1.5mu\overline{\mkern-1.5mu\mathrm{B}\mkern-1.5mu}\mkern 1.5mu_{\Lambda}{Z}$.
Set $\Lambda^{\prime}:=\|q\|$. We claim that we can find an extreme point
$m\in\operatorname{ex}(K_{\Lambda})$ such that the following holds:

(102) $\displaystyle K_{\Lambda^{\prime}}\cap(K_{\Lambda}-m+q)=\\{q\\}.$

To see this, note that $q$ is an extreme point of $K_{\Lambda^{\prime}}$ and
that the smallest cone $[0,\infty)\cdot(K_{\Lambda^{\prime}}-q)+q$ which
contains $K_{\Lambda^{\prime}}$ is locally compact (trivially in this finite
dimensional case), closed, convex, proper and has vertex $q$. By A.3 we can
find an extreme point $m\in\operatorname{ex}(K_{\Lambda})$ such that
$(K_{\Lambda}-m)\cap(K_{\Lambda^{\prime}}-q)=(K_{\Lambda}-m)\cap\left([0,\infty)\cdot(K_{\Lambda^{\prime}}-q)+q\right)=\\{0\\}$,
which is equivalent to (102).

Now, fix an extreme point $m\in\operatorname{ex}(K_{\Lambda})$ such that (102)
is satisfied and set $k:=q-l-m=p-m$. Note that for this $l$ and $k$ the
product (101) makes sense, i.e. $k\in\mkern
1.5mu\overline{\mkern-1.5mu\mathrm{B}\mkern-1.5mu}\mkern
1.5mu_{\Lambda}{Z}+\mkern
1.5mu\overline{\mkern-1.5mu\mathrm{B}\mkern-1.5mu}\mkern 1.5mu_{\Lambda}{Z}$
and $l+k\in\mkern 1.5mu\overline{\mkern-1.5mu\mathrm{B}\mkern-1.5mu}\mkern
1.5mu_{\Lambda}{Z}+\mkern
1.5mu\overline{\mkern-1.5mu\mathrm{B}\mkern-1.5mu}\mkern 1.5mu_{\Lambda}{Z}$.
Furthermore, the following holds:

(103) $\displaystyle\left(\mkern
1.5mu\overline{\mkern-1.5mu\mathrm{L}\mkern-1.5mu}\mkern
1.5mu_{\Lambda}{Z}(l)\cap\mkern
1.5mu\overline{\mkern-1.5mu\mathrm{L}\mkern-1.5mu}\mkern
1.5mu_{\Lambda}{Z}(l+k)\right)\cap K_{\Lambda^{\prime}}=\\{q\\}$

The converse inclusion is clear from (102) together with $\mkern
1.5mu\overline{\mkern-1.5mu\mathrm{L}\mkern-1.5mu}\mkern
1.5mu_{\Lambda}{Z}(l+k)\subset\mkern
1.5mu\overline{\mkern-1.5mu\mathrm{B}\mkern-1.5mu}\mkern
1.5mu_{\Lambda}{Z}(l+k)=K_{\Lambda}{Z}+q-m$, where $q-m=l+k$ was used.

This shows that, for $l=q-p$ and $k=p-m$, the rank-one operator $E_{p,q}$ is a
summand of $T_{-k}T_{l+k}$ as in (101) where, for all the other rank-one
operators $E_{n-l,n}$ appearing in the sum, we have that $\|n\|>\|q\|$, i.e.

(104) $\displaystyle
E_{p,q}=T_{-k}T_{l+k}-\sum_{\underset{\|n\|>\|q\|}{n\in\mkern
1.5mu\overline{\mkern-1.5mu\mathrm{L}\mkern-1.5mu}\mkern
1.5mu_{\Lambda}{Z}(l+k)\cap\mkern
1.5mu\overline{\mkern-1.5mu\mathrm{L}\mkern-1.5mu}\mkern
1.5mu_{\Lambda}{Z}(l)}}E_{n-l,n}.$

Moreover, for each $E_{n-l,n}$ in the above sum, a similar expression can be
obtained, and so forth. Hence, after finitely many steps this gives a finite
linear combination of products of the form (101) for $E_{p,q}$.

Altogether, this proves that
$\mathcal{B}(P_{\Lambda}\mathcal{H})\subseteq\operatorname{span}\left\\{T_{-k}T_{l+k}\,\big{|}\,l,k,l+k\in\mkern
1.5mu\overline{\mkern-1.5mu\mathrm{B}\mkern-1.5mu}\mkern
1.5mu_{\Lambda}{Z}+\mkern
1.5mu\overline{\mkern-1.5mu\mathrm{B}\mkern-1.5mu}\mkern
1.5mu_{\Lambda}{Z}\right\\}$ which shows that
$\mathrm{C}^{\ast}_{\text{env}}(\mathrm{C}(\mathbb{T}^{d})^{(\Lambda)})\cong\mathcal{B}(P_{\Lambda}\mathcal{H})$
and $\operatorname{prop}(\mathrm{C}(\mathbb{T}^{d})^{(\Lambda)})\leq 2$.
Realizing that $E_{0,0}\notin\mathrm{C}(\mathbb{T}^{d})^{(\Lambda)}$ it is
clear that $\operatorname{prop}(\mathrm{C}(\mathbb{T}^{d})^{(\Lambda)})>1$.
This finishes the proof. ∎

We illustrate the procedure of expressing elementary matrices
$E_{p,q}\in\mathcal{B}(P_{\Lambda}\mathcal{H})$ in terms of products of basic
operators of the form (101) as described in the above proof in the following
two examples.

######  Example 4.2.

Let $p,q\in\mkern 1.5mu\overline{\mkern-1.5mu\mathrm{B}\mkern-1.5mu}\mkern
1.5mu_{\Lambda}{Z}$ such that $\|q\|=\Lambda$. Finding an
$m\in\operatorname{ex}(K_{\Lambda})$ such that (102) holds is particularly
easy in this case, namely, set $m:=-q$. Then set $k:=p-m=p+q$ and, with
$l=q-p$, we obtain:

(105) $\displaystyle
E_{p,q}=T_{-k}T_{l+k}=T_{-p-q}T_{q-p+p+q}=T_{-p-q}T_{2q},$

according to (104), since there are no $n\in\mkern
1.5mu\overline{\mkern-1.5mu\mathrm{B}\mkern-1.5mu}\mkern 1.5mu_{\Lambda}{Z}$
with $\|n\|>\|q\|=\Lambda$.

######  Example 4.3.

Let $d=2$ and $\Lambda=\sqrt{2}$, i.e. $\mkern
1.5mu\overline{\mkern-1.5mu\mathrm{B}\mkern-1.5mu}\mkern 1.5mu_{\Lambda}{Z}$
consists of $9$ points and $K_{\Lambda}=\operatorname{co}\left(\mkern
1.5mu\overline{\mkern-1.5mu\mathrm{B}\mkern-1.5mu}\mkern
1.5mu_{\Lambda}{Z}\right)$ is the square with side length $2$. For $p=(0,0)$
and $q=(1,0)$, we want to express the matrix unit $E_{p,q}$ as a linear
combination of products of basic operators (101) with $l=q-p=(1,0)$. Set
$\Lambda^{\prime}:=\|q\|=1$. Now, find an extreme point
$m\in\operatorname{ex}(K_{\Lambda})$ such that (102) holds. A valid choice is
e.g. $m:=(-1,-1)$. Set $k:=p-m=(1,1)$. Then we have:

$\mkern 1.5mu\overline{\mkern-1.5mu\mathrm{L}\mkern-1.5mu}\mkern
1.5mu_{\Lambda}{Z}(l)\cap\mkern
1.5mu\overline{\mkern-1.5mu\mathrm{L}\mkern-1.5mu}\mkern
1.5mu_{\Lambda}{Z}(l+k)=\mkern
1.5mu\overline{\mkern-1.5mu\mathrm{L}\mkern-1.5mu}\mkern
1.5mu_{\sqrt{2}}{Z}(1,0)\cap\mkern
1.5mu\overline{\mkern-1.5mu\mathrm{L}\mkern-1.5mu}\mkern
1.5mu_{\sqrt{2}}{Z}(2,1)=\\{(1,0),(1,1)\\}$

Therefore:

$T_{-k}T_{l+k}=T_{(-1,-1)}T_{(2,1)}=\sum_{n\in\mkern
1.5mu\overline{\mkern-1.5mu\mathrm{L}\mkern-1.5mu}\mkern
1.5mu_{\sqrt{2}}{Z}(1,0)\cap\mkern
1.5mu\overline{\mkern-1.5mu\mathrm{L}\mkern-1.5mu}\mkern
1.5mu_{\sqrt{2}}{Z}(2,1)}E_{n-(1,0),n}=\underbrace{E_{(0,0),(1,0)}}_{=E_{p,q}}+E_{(0,1),(1,1)}$

Note that $\|(1,1)\|=\sqrt{2}>\|q\|=1$.

By 4.2, we have $E_{(0,1),(1,1)}=T_{(-1,-2)}T_{(2,2)}$. Altogether, we obtain:

$E_{(0,0),(1,0)}=T_{(-1,-1)}T_{(2,1)}-T_{(-1,-2)}T_{(2,2)}$

######  Remark 4.4.

We point out that analogously to the proof of 4.1 one can show that the
$\mathrm{C}^{*}_{\mathrm{env}}(P_{\Lambda}^{K}\mathrm{C}^{\infty}(\mathbb{T}^{d})P_{\Lambda}^{K})=\mathcal{B}(P_{\Lambda}^{K}\mathcal{H})$
and
$\mathrm{prop}(P_{\Lambda}^{K}\mathrm{C}^{\infty}(\mathbb{T}^{d})P_{\Lambda}^{K})=2$,
where $K$ is a convex compact subset of $\mathbb{R}^{d}$ which is symmetric
with respect to reflections along the coordinate axes and diagonals and where
$P_{\Lambda}^{K}\in\mathcal{B}(\mathcal{H})$ is the orthogonal projection with
$\mathrm{ran}(P_{\Lambda}^{K})=\\{e_{n}\,:\,n\in(\Lambda\cdot
K)\cap\mathbb{Z}^{d}\\}$. In particular, this shows that the propagation
number of the operator system obtained from “box-truncations”, which were
considered in [2], is also $2$.

###  4.2. Dual

In [11, Theorem 3.1] it was shown that the operator system of
$(N\times\nolinebreak N)$-Toeplitz matrices $\mathrm{C}(S^{1})^{(N)}$ is dual
to the Fejér–Riesz system $\mathrm{C}(S^{1})_{(N)}$ which consists of
trigonometric polynomials of the form $\sum_{n=-N+1}^{N-1}a_{n}e_{n}$. In
fact, this was already stated in [8] but it was only shown in [11] that
$\mathrm{C}(S^{1})^{(N)}\cong\left(\mathrm{C}(S^{1})_{(N)}\right)^{\mathrm{d}}$
in the sense that there is a unital complete order isomorphism. For the proof,
the (operator valued) Fejér–Riesz theorem plays an essential role. Recall that
the Fejér–Riesz theorem states that every non-negative Laurent polynomial
$P=\sum_{k=-N}^{N}a_{k}e_{k}\geq 0$ can be expressed as a hermitian square of
an analytic polynomial $Q=\sum_{k=0}^{N}b_{k}e_{k}$, i.e.

$P=Q^{*}Q.$

A generalization to the case where the coefficients $a_{k}$ and $b_{k}$ are
operators on a Hilbert space is due to Rosenblum. See [10] for a survey on the
operator-valued Fejér–Riesz theorem.

In view of the duality result for $S^{1}$, it would be natural to expect that
the operator system $\mathrm{C}(\mathbb{T}^{d})^{(\Lambda)}$ is dual to the
operator system $\mathrm{C}(\mathbb{T}^{d})_{(2\Lambda)}$ which consists of
trigonometric polynomials on the $d$-torus of the form $\sum_{n\in\mkern
1.5mu\overline{\mkern-1.5mu\mathrm{B}\mkern-1.5mu}\mkern
1.5mu_{2\Lambda}{Z}}a_{n}e_{n}$. However, this duality must fail even
algebraically as soon as $d\geq 2$. Indeed, it is clear that
$\dim\left(\mathrm{C}(\mathbb{T}^{d})_{(2\Lambda)}\right)=\\#\mkern
1.5mu\overline{\mkern-1.5mu\mathrm{B}\mkern-1.5mu}\mkern
1.5mu_{2\Lambda}{Z}=\mathcal{N}_{\mathrm{B}}(2\Lambda)$ and from the above
considerations about a basis we conclude that
$\dim\left(\mathrm{C}(\mathbb{T}^{d})^{(\Lambda)}\right)=\\#\left(\mkern
1.5mu\overline{\mkern-1.5mu\mathrm{B}\mkern-1.5mu}\mkern
1.5mu_{\Lambda}{Z}-\mkern
1.5mu\overline{\mkern-1.5mu\mathrm{B}\mkern-1.5mu}\mkern
1.5mu_{\Lambda}{Z}\right)=\\#\left(\mkern
1.5mu\overline{\mkern-1.5mu\mathrm{B}\mkern-1.5mu}\mkern
1.5mu_{\Lambda}{Z}+\mkern
1.5mu\overline{\mkern-1.5mu\mathrm{B}\mkern-1.5mu}\mkern
1.5mu_{\Lambda}{Z}\right)$. In general, however, the inclusion $\mkern
1.5mu\overline{\mkern-1.5mu\mathrm{B}\mkern-1.5mu}\mkern
1.5mu_{\Lambda}{Z}+\mkern
1.5mu\overline{\mkern-1.5mu\mathrm{B}\mkern-1.5mu}\mkern
1.5mu_{\Lambda}{Z}\subset\mkern
1.5mu\overline{\mkern-1.5mu\mathrm{B}\mkern-1.5mu}\mkern 1.5mu_{2\Lambda}{Z}$
is strict, as the following examples demonstrate:

######  Example 4.5.

If $d=2$, we have that $(3,2)\in\mkern
1.5mu\overline{\mkern-1.5mu\mathrm{B}\mkern-1.5mu}\mkern
1.5mu_{4}{Z}\setminus\left(\mkern
1.5mu\overline{\mkern-1.5mu\mathrm{B}\mkern-1.5mu}\mkern 1.5mu_{2}{Z}+\mkern
1.5mu\overline{\mkern-1.5mu\mathrm{B}\mkern-1.5mu}\mkern 1.5mu_{2}{Z}\right)$.
If $d\geq 3$, we have that $(1,\cdots,1)\in\mkern
1.5mu\overline{\mkern-1.5mu\mathrm{B}\mkern-1.5mu}\mkern
1.5mu_{2}{Z}\setminus\left(\mkern
1.5mu\overline{\mkern-1.5mu\mathrm{B}\mkern-1.5mu}\mkern 1.5mu_{1}{Z}+\mkern
1.5mu\overline{\mkern-1.5mu\mathrm{B}\mkern-1.5mu}\mkern 1.5mu_{1}{Z}\right)$.

In view of these remarks, a more promising candidate for the dual operator
system of $\mathrm{C}(\mathbb{T}^{d})^{(\Lambda)}$ might be the operator
subsystem $\mathrm{C}(\mathbb{T}^{d})_{(\Lambda)+(\Lambda)}$ of
$\mathrm{C}(\mathbb{T}^{d})$ which consists of trigonometric polynomials on
the $d$-torus of the form $\sum_{n\in\mkern
1.5mu\overline{\mkern-1.5mu\mathrm{B}\mkern-1.5mu}\mkern
1.5mu_{\Lambda}{Z}+\mkern
1.5mu\overline{\mkern-1.5mu\mathrm{B}\mkern-1.5mu}\mkern
1.5mu_{\Lambda}{Z}}a_{n}e_{n}$.

In order to generalize the proof of [11] to this setting, one would need to
show that the following map is a complete order isomorphism:

(106) $\displaystyle\phi:\mathrm{C}(\mathbb{T}^{d})^{(\Lambda)}$
$\displaystyle\rightarrow(\mathrm{C}(\mathbb{T}^{d})_{(\Lambda)+(\Lambda)})^{\mathrm{d}}$
(107) $\displaystyle\mathbf{t}$ $\displaystyle\mapsto\varphi_{\mathbf{t}},$

where

(108) $\displaystyle\varphi_{\mathbf{t}}(f)=\sum_{n\in\mkern
1.5mu\overline{\mkern-1.5mu\mathrm{B}\mkern-1.5mu}\mkern
1.5mu_{\Lambda}{Z}+\mkern
1.5mu\overline{\mkern-1.5mu\mathrm{B}\mkern-1.5mu}\mkern
1.5mu_{\Lambda}{Z}}t_{-n}a_{n},$

for $\mathbf{t}=(t_{k-l})_{k,l\in\mkern
1.5mu\overline{\mkern-1.5mu\mathrm{B}\mkern-1.5mu}\mkern
1.5mu_{\Lambda}{Z}+\mkern
1.5mu\overline{\mkern-1.5mu\mathrm{B}\mkern-1.5mu}\mkern 1.5mu_{\Lambda}{Z}}$
and $f=\sum_{n\in\mkern
1.5mu\overline{\mkern-1.5mu\mathrm{B}\mkern-1.5mu}\mkern
1.5mu_{\Lambda}{Z}+\mkern
1.5mu\overline{\mkern-1.5mu\mathrm{B}\mkern-1.5mu}\mkern
1.5mu_{\Lambda}{Z}}a_{n}e_{n}\in\mathrm{C}(\mathbb{T}^{d})_{(\Lambda)+(\Lambda)}$.
It is easy to see that $\phi$ is unital and an isomorphism of vector spaces,
so it remains to show that $\phi$ and $\phi^{-1}$ are completely positive.

While the proof of complete positivity of $\phi^{-1}$ is largely analogous to
the one in [11], for the proof of complete positivity of $\phi$ a multivariate
version of the (operator-valued) Fejér–Riesz theorem would be needed. In fact,
it would be sufficient to be able to express a (dense subset of the cone of)
non-negative Laurent polynomials in $d$ variables $P=\sum_{n\in\mkern
1.5mu\overline{\mkern-1.5mu\mathrm{B}\mkern-1.5mu}\mkern
1.5mu_{\Lambda}{Z}}a_{n}e_{n}$ as a finite sum of squares of analytic
polynomials $Q_{i}=\sum_{n\in\left(\mkern
1.5mu\overline{\mkern-1.5mu\mathrm{B}\mkern-1.5mu}\mkern
1.5mu_{\Lambda}{Z}\right)_{+}}b_{n}^{i}e_{n}$, where $\left(\mkern
1.5mu\overline{\mkern-1.5mu\mathrm{B}\mkern-1.5mu}\mkern
1.5mu_{\Lambda}{Z}\right)_{+}=\mkern
1.5mu\overline{\mkern-1.5mu\mathrm{B}\mkern-1.5mu}\mkern
1.5mu_{\Lambda}\cap(\mathbb{Z}_{\geq 0})^{d}$. Although it is not known to the
authors if such a result holds for a dense subset of the positive cone , at
least for $d\geq 3$ it must fail for some non-negative Laurent polynomials
even if higher degrees of the analytic polynomials $Q_{i}$ are admitted
([29]). For $d=2$, Dritschel proved ([9, Theorem 4.1]) that if $P$ is a non-
negative Laurent polynomial of degree $(d_{1},d_{2})$ (with Hilbert space
operators as coefficients), then $P$ is a sum of at most $2d_{2}$ squares of
analytic polynomials each of degree at most $(d_{1},d_{2}-1)$. However, as it
is stated in the conclusion of that article it is not clear by how much the
number of summands and in particular the bound on the degree can be improved.
Yet, there are examples of non-negative polynomials of degree $(d_{1},d_{2})$
which are not a finite sum of squares of analytic polynomials of degree
$(d_{1},d_{2})$ ([20], [27], [28, Section 3.6]). Note that it is in fact true
that a strictly positive trigonometric polynomial $P>0$ in $d$ variables is a
finite sum of squares of analytic polynomials ([10, Theorem 5.1]), but the
degrees might get out of control.

We see that it is apparently not as straightforward to determine the operator
system dual of $\mathrm{C}(\mathbb{T}^{d})^{(\Lambda)}$ as one might expect at
first thought and we have to leave this to further research to be conducted.

##  Appendix A Some convex geometry

In order to compute the propagation number of the operator system
$\mathrm{C}(\mathbb{T}^{d})^{(\Lambda)}$, some facts from convex geometry are
required. Since the natural setting for these is locally convex spaces we
formulate all the required results in this abstract language even though we
only make use of them in the finite dimensional case. See e.g. [26, Section
11] and [3, Chapter II] for much of the standard terminology.

Throughout this section, let $X$ be a Hausdorff locally convex space over
$\mathbb{R}$ with continuous dual $X^{\prime}$. Every linear functional $l$ on
$X$ and every real number $\alpha\in\mathbb{R}$ give rise to a hyperplane in
$X$ given by $H_{l=\alpha}=\left\\{x\in X\,|\,l(x)=\alpha\right\\}$. Clearly,
$H_{l=\alpha}=\ker(l-\alpha)$ and the hyperplane $H_{l=\alpha}$ is closed if
and only if $l$ is continuous which is the only case we consider. Every
hyperplane $H_{l=\alpha}$ gives rise to an open positive and an open negative
half-space denoted by $H_{l>\alpha}$ and $H_{l<\alpha}$ respectively and
defined by $H_{l>\alpha}:=\left\\{x\in X\,|\,l(x)>\alpha\right\\}$ and
similarly for $H_{l<\alpha}$. Their respective closures are called the closed
positive and the closed negative half-space associated to $H_{l=\alpha}$ and
denoted by $H_{l\geq\alpha}$ and $H_{l\leq\alpha}$ respectively. Of course
$H_{l\geq\alpha}=\left\\{x\in X\,|\,l(x)\geq\alpha\right\\}$ and similarly for
$H_{l\leq\alpha}$.

If $K,L\subseteq X$ are two convex sets, it is said that they are _separated_
by the hyperplane $H_{l=\alpha}$ if $K\subseteq H_{l\geq\alpha}$ and
$L\subseteq H_{l\leq\alpha}$. The sets $K$ and $L$ are called _properly
separated_ if additionally $K\nsubset H_{l=\alpha}$ or $L\nsubset
H_{l=\alpha}$. The sets $K$ and $L$ are called _strictly separated_ by
$H_{l=\alpha}$ if $K\subseteq H_{l>\alpha}$ and $L\subseteq H_{l<\alpha}$. A
hyperplane $H_{l=\alpha}$ is called a _supporting hyperplane_ for a non-empty
convex set $K\subset X$, if $K\subseteq H_{l\geq\alpha}$ and $x\in
H_{l=\alpha}$, for at least one $x\in K$. A supporting hyperplane
$H_{l=\alpha}$ for $K$ is called _non-trivial_ if $K\nsubseteq H_{l=\alpha}$.

Recall that a cone $C\subseteq X$ is called _pointed_ if $0\in C$, _salient_
if it does not contain any $1$-dimensional subspaces of $X$ and _proper_ if
$C\cap(-C)=\\{0\\}$. We only consider convex cones. A convex pointed cone is
salient if and only if it is proper. For $x\in X$, we call a set $C+x$ a _cone
with vertex $x$_ if $C$ is a pointed cone. A cone with vertex $x$ is called
_proper_ if $C-x$ is a proper cone.

Let $K\subset X$ be convex set with $0\notin K$. Set $C:=[0,\infty)\cdot K$.
Clearly, $C$ is a convex pointed cone. Moreover, $C$ is the smallest convex
pointed cone which which contains $K$ in the sense that every convex pointed
cone which contains $K$ must contain $C$.

If $C$ is a convex pointed cone, a subset $B\subset C$ is called a _base_ (or
_sole_ in [3, II, §8.3]) if there exists a closed hyperplane $H\notowner 0$
such that $B=H\cap C$ and such that $C$ is the smallest convex pointed cone
which contains $B$. It is well-known that a convex subset $B$ of a convex
pointed cone $C$ is a base if and only if, for every $x\in C\setminus\\{0\\}$,
there exists a unique pair $(\lambda,y)\in(0,\infty)\times B$ such that
$x=\lambda y$.

The statement of the following lemma can be found in [3, II, §7.2, Exercise
21a].

######  Lemma A.1.

Let $C\subset X$ be a locally compact, closed, convex, proper cone with vertex
$x$. Then there exists a closed supporting hyperplane $H$ of $C$ such that
$H\cap C=\\{x\\}$.

###### Proof.

To simplify notation we assume, without loss of generality, that $x=0$. Let
$U\subset X$ be a convex open neighborhood of $0$ such that $K:=\mkern
1.5mu\overline{\mkern-1.5muU\mkern-1.5mu}\mkern 1.5mu\cap C$ is compact. We
claim that $C=[0,\infty)\cdot K$, i.e. $C$ is the smallest convex pointed cone
which contains $K$. In fact, the inclusion $C\supseteq[0,\infty)\cdot K$ is
clear from the cone property. To see that $C\subseteq[0,\infty)\cdot K$, let
$y\in C$. Since every $0$-neighborhood in a locally convex space is absorbent,
there exists a positive scalar $\lambda>0$ such that $y\in\lambda U$. Hence,
$\frac{1}{\lambda}y\in U\cap\frac{1}{\lambda}C=U\cap C\subset K$ and therefore
$y=\lambda\frac{1}{\lambda}y\in\lambda K\subset[0,\infty)\cdot K$.

By [3, II, §7.1, Proposition 2], there is an open half-space
$H_{l<\alpha}\subset X$ such that $0\in H_{l<\alpha}\cap K\subset U\cap C$. We
may assume that the scalar $\alpha$ is positive, otherwise pass to the
functional $-l$ instead. The boundary $\partial H_{l<\alpha}=H_{l=\alpha}$ of
this half-space is a closed hyperplane of $X$ which does not contain $0$.

We claim that the cone $C$ is the smallest closed convex pointed cone which
contains $H_{l=\alpha}\cap C$, i.e. $C=[0,\infty)\cdot(H_{l=\alpha}\cap C)$.
To see this, observe that the following inclusion holds:

(109) $\displaystyle[0,\infty)\cdot\left(H_{l=\alpha}\cap
C\right)=[0,\infty)\cdot\left(H_{l\leq\alpha}\cap
C\right)\subseteq[0,\infty)\cdot K=C$

The converse inclusion $[0,\infty)\cdot(H_{l\leq\alpha}\cap
C)\supseteq[0,\infty)\cdot K$ follows from the continuity and hence
boundedness on the compact set $K$ of the functional $l$ by observing that
there exists a positive real number $\lambda>0$ such that $\lambda
H_{l\leq\alpha}=H_{l\leq\lambda\alpha}\supseteq K$.

Now, set $H:=H_{l=0}$. Then we have:

(110) $\displaystyle H\cap C=H\cap[0,\infty)\cdot(H_{l=\alpha}\cap
C)=(H\cap[0,\infty)\cdot H_{l=\alpha})\cap C=\\{0\\}$

So $H$ is the desired closed supporting hyperplane for $C$. ∎

######  Lemma A.2.

Let $K\subset X$ be a non-empty compact convex subset. Let $l\in X^{\prime}$
be a continuous linear functional and $H_{l=0}$ be the associated closed
hyperplane through $0$. Then there exists an extreme point
$x\in\operatorname{ex}(K)$ such that $K-x\subset H_{l\geq 0}$.

###### Proof.

By continuity of $l$, the image $l(K)\subset\mathbb{R}$ is bounded. Set
$\alpha:=\inf(l(K))$. In other words, $\alpha$ is the largest real number such
that $K\subset H_{l\geq\alpha}$. In particular, $H_{l=\alpha}$ is a closed
supporting hyperplane of $K$. By [3, II, §7.1, Corollary to Proposition 1],
the hyperplane $H_{l=\alpha}$ contains an extreme point $x$ of $K$. Moreover,
$K-x$ is contained in the positive half-space $H_{l\geq 0}$. ∎

Combining the previous two lemmas, we obtain the following immediate
consequence:

######  Corollary A.3.

Let $C\subset X$ be a locally compact, closed, convex, proper cone with vertex
$x$ and let $K\subset X$ be a non-empty compact convex set. Then there exists
an extreme point $y\in\operatorname{ex}(K)$ and a closed hyperplane $H$ which
separates $C-x$ and $K-y$. Moreover, these two sets only intersect in $0$,
i.e. $(C-x)\cap(K-y)=\\{0\\}$.

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