69

数理解析研究所講究録
第2023巻 2017年 69-76

Inner and outer continuity of the spectra

for families of magnetic operators on \mathb {Z}^{d}S. Richard*

Graduate school of mathematics, Nagoya University, Chikusa‐ku,
Nagoya 464‐S602, Japan; On leave of absence from Univ. Lyon, Uni‐
versité Claude Bernard Lyon 1, CNRS UMR 5208, Institut Camille
Jordan, 43 blvd. du11 novembre 1918, F‐69622 Villeurbanne cedex,
France.

E‐‐mails: richard@math.nagoya‐u.ac. jp
Abstract
In this note we consider magnetic self‐adjoint operators on \mathb {Z}^{d} whose symbols
and magnetic fields depend on a parameter $\epsilon$.

Sufficient conditions are imposedsuch that the spectrum of these operators varies continuously with respect to $\epsilon$.
The emphasize is put on a construction which is independent of any particular
choice of the magnetic potentials.

2010 Mathematics Subject Classification: 81\mathrm{Q}10, 47\mathrm{L}65

Keywords: Discrete operators, magnetic field, spectrum, twisted crossed productalgebraIntroduction

1

This paper is an extended version of a presentation made at the conference Spectral
and Scattering Theory and Related Topics at Rims in Kyoto in January 2016. The
presentation was based on the paper [11] to which we refer for more details and for the

proofs.
In the Hilbert space \mathcal{H}operators of the form:=l^{2}(\mathbb{Z}^{d}) and for some fixed parameter $\epsilon$ let us consider[H^{ $\epsilon$}u](x):=\displaystyle \sum_{y\in \mathb {Z}^{d} h^{ $\epsilon$}(x;y-x)\mathrm{e}^{i$\phi$^{ $\epsilon$}(x,y)}u(y)(1.1)

with u\in \mathcal{H} of finite support, x\in \mathbb{Z}^{d} and where h^{ $\epsilon$} : \mathbb{Z}^{d}\times \mathbb{Z}^{d}\rightar ow \mathbb{C} and $\phi$^{ $\epsilon$} : \mathbb{Z}^{d}\times \mathbb{Z}^{d}\rightar ow \mathbb{R}satisfy*

Supported by JSPS Grant‐in‐Aid for Young Scientists (A) no. 26707005.

70(i) \displaystyle \sum_{x\in \mathbb{Z}^{d} \sup_{q\in \mathrm{Z}^{d} |h^{ $\epsilon$}(q;x)|<\infty,(ii) h^{ $\epsilon$}(q+x;-x)=h^{ $\epsilon$}(q;x) for any q, x\in \mathbb{Z}^{d},(iii) $\phi$^{ $\epsilon$}(x, y)=-$\phi$^{ $\epsilon$}(y, x) for all x, y\in \mathbb{Z}^{d}.
Such operators are usually called discrete magnetic Schrödinger operators. Note thatcondition (i) ensures that H^{ $\epsilon$} extends continuously to a bounded operator in \mathcal{H} , andbe simply rewritten as

canh^{ $\epsilon$}\in l^{1}(\mathbb{Z}^{d};l^{\infty}(\mathbb{Z}^{d})the operator H^{ $\epsilon$} is self‐adjoint. Note also that

Conditions

.

a

map

(ii) and (iii) imply that$\phi$ : \mathbb{Z}^{d}\times \mathbb{Z}^{d}\rightarrow \mathbb{R} satisfying$\phi$(x, y)=- $\phi$(y, x) for any x, y\in \mathbb{Z}^{d} will be called a magnetic potential.

Our aim is exhibit some continuity properties of the spectrum of these operators

under suitable and natural assumptions. Natural conditions on the family of symbols h^{ $\epsilon$}

imposed below. However, it is well‐known (at least in the continuous setting) that
continuity conditions should not be imposed on the magnetic potentials but rather on
the magnetic fields. This requirement comes from the non‐unicity for the choice of amagnetic potential. Indeed, if $\phi$ : \mathbb{Z}^{d}\times \mathbb{Z}^{d}\rightar ow \mathbb{R} is defined for any x, y\in \mathbb{Z}^{d} byare$\phi$'(x, y)\equiv[ $\phi$+\nabla $\varphi$](x, y) := $\phi$(x, y)+ $\varphi$(y)- $\varphi$(x)(1.2)

for some $\varphi$ : \mathbb{Z}^{d}\rightar ow \mathbb{R} , then the magnetic operators constructed with $\phi$ and $\phi$' are known
to be unitarily equivalent. This property is called the gauge invariance of the magnetic

operators and imposes a slightly more elaborated notion of continuity for the magnetic

contribution, as emphasized below.
Before explaining more in details the necessary construction, let us state a simplified
version of our main theorem in which the $\epsilon$ ‐dependence on $\phi$^{ $\epsilon$} is very simple. A moregeneral setting will be introduced in the subsequent sections. The continuity we shallconsider for the spectrum corresponds to the stability of the spectral gaps as well as thestability of the spectral compounds. In a more precise terminology we shall prove innerand outer continuity for the family of spectra. The following definition is borrowed from
[1] but originally inspired by [3].
Definition 1.1. Let $\Omega$ be a compact Hausdorff space, and let \{$\sigma$_{ $\epsilon$}\}_{ $\epsilon$\in $\Omega$} be a family ofclosed subsets of \mathbb{R}.
1.

The family \{$\sigma$_{ $\epsilon$}\}_{ $\epsilon$\in $\Omega$} is outer continuous at $\epsilon$_{0}\in $\Omega$ if for any compact subset \mathcal{K} of\mathbb{R} such that \mathcal{K}\cap$\sigma$_{ $\epsilon$ 0}=\emptyset there exists a neighbourhood \mathcal{N}=\mathcal{N}(\mathcal{K}, $\epsilon$_{0})such that \mathcal{K}\cap$\sigma$_{ $\epsilon$}=\emptyset for any $\epsilon$\in \mathcal{N},2.

of $\epsilon$_{0} in $\Omega$

The family \{$\sigma$_{ $\epsilon$}\}_{ $\epsilon$\in $\Omega$} is inner continuous at $\epsilon$_{0}\in $\Omega$ if for any open subset \mathcal{O} of \mathbb{R}such that \mathcal{O}\cap$\sigma$_{ $\epsilon$ 0}\neq\emptyset there exists a neighbourhood \mathcal{N}=\mathcal{N}(\mathcal{O}, $\epsilon$_{0})that \mathcal{O}\cap$\sigma$_{ $\epsilon$}\neq\emptyset for any $\epsilon$\in \mathcal{N}.

With this definition at hand we can now choose $\Omega$of $\epsilon$_{0} in $\Omega$ such:=[0 1 ] and state a special,

instance of our main result which is presented in Theorem 3.1. Note that the following
statement is

inspired from [8] and a comparison with the existing literature will beestablished just afterwards.

71

Theorem 1.2. For each $\epsilon$\in $\Omega$ :=[0 , 1 ] let h^{ $\epsilon$} : \mathbb{Z}^{d}\times \mathbb{Z}^{d}\rightar ow \mathbb{C} satisfy the above conditions(i) and (ii), and

1) \displaystyle \sum_{x\in \mathrm{Z}^{d} \sup_{q\in \mathbb{Z}^{d} \sup_{ $\epsilon$\in[0,1]}|h^{ $\epsilon$}(q;x)|<\infty,2) For any fixed x\in \mathbb{Z}^{d},\displaystyle \lim_{ $\epsilon$\rightar ow $\epsilon$}\sup_{q\in \mathb {Z}^{d} |h^{$\epsilon$'}(q;x)-h^{ $\epsilon$}(q;x)|=0.
Let also $\phi$ be a magnetic potential which satisfies| $\phi$(x, y)+ $\phi$(y, z)+ $\phi$(z, x)|\leq
where \triangle(x, y, z)

means

area

\triangle(x, y, z),

the triangle in \mathbb{R}^{d} determined by the three points x, y, z\in \mathbb{Z}^{d}.

Then for H^{ $\epsilon$} defined on u\in \mathcal{H} by[H^{ $\epsilon$}u](x):=\displaystyle \sum_{y\in \mathrm{Z}^{d} h^{ $\epsilon$}(x;y-x)\mathrm{e}^{i $\epsilon \phi$(x,y)}u(y)the family of spectra $\sigma$(H^{ $\epsilon$}) forms an outer and an inner continuous family at every

points $\epsilon$\in $\Omega$.

It is certainly impossible to mention all papers dealing with continuity propertiesof families of such operators, but let us cite a few of them which are relevant for ourinvestigations. First of all, let us mention the seminal paper [3] in which the author
proves the Lipschitz continuity of gap boundaries with respect to the variation of aconstant magnetic field for a family of pseudodifferential operators acting on \mathb {Z}^{2} In [6]and based on the framework introduced in [13], similar Lipschitz continuity is proved forself‐adjoint operators acting on a crystal lattice, a natural generalization of \mathb {Z}^{d} Papers[8] and [5] also deal with families of magnetic pseudodifferential operators on \mathb {Z}^{2} and the.

.

,results contained in [8] partially motivated our work. Let us still mention two additional
papers which are at the root of our work: [7] in which a general framework for magnetic

systems, involving twisted crossed product C^{*} ‐algebras, is introduced, and the reference[1] which contains results similar to ours but in a continuous setting.

Let us now emphasize that the framework presented in Section 3 does not allow
us

to get any quantitative estimate, as emphasized in the recent paper [2]. Indeed, thevery weak continuity requirement we impose on the $\epsilon$ ‐dependence on our objets can notlead to any Lipschitz or Hölder continuity. More stringent assumptions are necessary
for that purpose, and such estimates certainly deserve further investigations.

Our approach relies on the concepts of twisted crossed product C^{*} ‐algebras andfield of such algebras,

mainly borrowed from [12, 14]. In the discrete setting,[3, 6, 13]. However, instead ofawithscalarwhich
issufficientfor the case of a constantconsidering 2‐cocycle
values,ourtakevaluesintheofmagnetic field,
2‐cocycles
group
unitary elements of l^{\infty}(\mathbb{Z}^{d})on asuch algebras have already been used, for example in.

72This allows us to consider arbitrary magnetic potential on \mathb {Z}^{d} and to encompass all thecorresponding operators in a single algebra.
Let us finally describe the content of this paper. In Section 2 we introduce the frame‐
work for a single magnetic system, i.e for a fixed $\epsilon$ For that reason, no $\epsilon$ ‐dependence isindicated in this section. In Section 3 the -$\epsilon$ ‐dependence is introduced and the continuous
dependence on this parameter is studied. Our main result is presented in Theorem 3.1.
.

2.

Discrete magnetic systems

In this section we introduce a quantity which does not depend on the choice of a par‐ticular magnetic potential. In that respect, this function depends only on the magnetic

field, as emphasized below. Note that no $\epsilon$ ‐dependence is written in this section.
We start by recalling that a magnetic potential consists in a map $\phi$ : \mathbb{Z}^{d}\times \mathbb{Z}^{d}\rightar ow \mathbb{R}satisfying for any x, y\in \mathbb{Z}^{d} the relation$\phi$(x, y)=- $\phi$(y, x).

Then, given such a magnetic potential $\phi$ let us introduce and study a new map$\omega$:\mathbb{Z}^{d}\times \mathbb{Z}^{d}\rightar ow l^{\infty}(\mathbb{Z}^{d})defined for q, x, y\in \mathbb{Z}^{d} by[ $\omega$(x, y)](q)\equiv $\omega$(q;x, y) :=\exp\{i[ $\phi$(q, q+x)+ $\phi$(q+x, q+x+y)+ $\phi$(q+x+y, q)]\}. (2.1)
Note that $\omega$(x, y) is unitary‐valued since | $\omega$(q, ; x, y)|=1 for any q\in \mathbb{Z}^{d}. \ln fact, $\omega$(x, y)takes values in the unitary group of the algebra l^{\infty}(\mathbb{Z}^{d}) which shall simply denote by,\mathscr{U}(\mathbb{Z}^{d}) i.e. %(Zd) =\{f:\mathbb{Z}^{d}\rightarrow \mathrm{T}\}.
,

Let us also introduce the action $\theta$ of \mathb {Z}^{d} by translations, namely on any f\in l^{\infty}(\mathbb{Z}^{d})

one

sets

$\theta$_{x}f(y)=f(x+y).

In particular, since $\omega$(x, y)\in l^{\infty}(\mathbb{Z}^{d}) we have[$\theta$_{z} $\omega$(x, y)](q):=[ $\omega$(x, y)](q+z)= $\omega$(q+z;x, y).

Based on these definitions, the following properties for $\omega$ can be obtained by straight‐
forward computations. Recall that the notation $\phi$+\nabla $\varphi$ has been introduced in (1.2).
Lemma 2.1. Let $\phi$ be a magnetic potential and let $\omega$ be defined by (2.1). Then for any
x, y, z\in \mathbb{Z}^{d} the following properties hold:(i) $\omega$(x+y, z) $\omega$(x, y)=$\theta$_{x} $\omega$(y, z) $\omega$(x, y+z)(ii) $\omega$(x, 0)= $\omega$(0, x)=1

(normalization)(2‐cocycle property)

73(iii) $\omega$(x, -x)=1(additional property)(iv) For any $\varphi$ : \mathbb{Z}^{d}\rightar ow \mathbb{R} the magnetic potentials $\phi$ and $\phi$+\nabla $\varphi$ define the samefunction $\omega$(independence property).
,With the above lemma, one directly infers that $\omega$ is a normalized 2‐cocycle on \mathbb{Z}^{d}with values in \mathscr{U}(\mathbb{Z}^{d}) and which satisfies the additional property (iii). In addition, thismap depends only on equivalent classes of magnetic potentials, as emphasized in (iv).
One could argue that the 2‐cocycle $\omega$ depends only on the magnetic field as introduced

in [4], and not on the choice of a magnetic potential. However, this would lead us too

far from our purpose since we would have to consider \mathb {Z}^{d} as a graph endowed with edges

between every pair of vertices.
Let us now adopt a very pragmatic point of view and recall only the strictly nec‐

essary information on twisted crossed product C^{*} ‐algebras. More can be found in thefundamental papers [9, 10] or in the review paper [7]. Since the group we are dealing

with is simply \mathb {Z}^{d} , most of the necessary information can also Ue found in [14].

Considerl^{1}(\mathbb{Z}^{d};l^{\infty}(\mathbb{Z}^{d}) with the,norm\displaystyle \Vert f\Vert_{1,\infty}:=\sum_{x\in \mathrm{Z}^{d} \sup_{q\in \mathrm{Z}^{d} |f(q;x)| \foral f\in l^{1}(\mathb {Z}^{d};l^{\infty}(\mathb {Z}^{d})and with the twisted product and the involution defined for any f,by,g\in l^{1}(\mathbb{Z}^{d};l^{\infty}(\mathbb{Z}^{d})[f\displaystyle \mathrm{o}g](q;x):=\sum_{y\in \mathrm{Z}^{d} f(q;y)g(q+y;x-y) $\omega$(q;y, x-y)

andf^{ $\theta$}(q;x)=\overline{f(q+x;-x)}.
Endowed with this multiplication and with this involution,l^{1}(\mathbb{Z}^{d};l^{\infty}(\mathbb{Z}^{d}) isaBanach‐algebra. The corresponding enveloping C^{*} ‐algebra will be denoted by \mathb {C}( $\omega$) Recall
that this algebra corresponds to the completion ofl^{1}(\mathbb{Z}^{d};l^{\infty}(\mathbb{Z}^{d}) with respect to the*.

C‐norm defined as the supremum over all the faithful representations of
As a consequence,\Vert f\Vert\leq\Vert f\Vert_{1,\infty}.

l^{1}(\mathbb{Z}^{d};l^{\infty}(\mathbb{Z}^{d})l^{1}(\mathbb{Z}^{d};l^{\infty}(\mathbb{Z}^{d}).

is dense in \not\subset( $\omega$) and the new C^{*} ‐norm \Vert\cdot\Vert satisfies

Note that the above construction holds for any normalized 2‐cocycle satisfying
the additional property (iii). In fact, it is proved in [11] that any such 2‐cocycle canbe obtained by a magnetic potential, i.e. there always exists a magnetic potential $\phi$satisfying (2.1).

Let us now look at a faithful representation of the algebra \mathb {C}( $\omega$) in the Hilbert space\mathcal{H}=l^{2}(\mathbb{Z}^{d}) For that purpose, consider any magnetic potential $\phi$ satisfying.

\exp\{i[ $\phi$(q, q+x)+$\phi$'(q+x, q+x+y)+$\phi$'(q+x+y, q)]\}= $\omega$(q;x, y)
Then for any

h\in l^{1}(\mathbb{Z}^{d};l^{\infty}(\mathbb{Z}^{d}) any u\in \mathcal{H} and any x\in \mathbb{Z}^{d} one sets,[\displaystyle \Re \mathrm{e}\mathfrak{p}^{$\phi$'}(h)u](x):=\sum_{y\in \mathb {Z}^{d} h(x;y-x)\mathrm{e}^{i$\phi$'(X_{1y)} .u(y)

.

.

(2.2)

74

Clearly, this expression is similar to the one contained in (1.1) and this fact partiallyjustifies the entire construction. The main properties of this representation are gatheredin the following statement, which corresponds to [7, Prop. 2. 16 & 2.17] adapted to our
setting.

Proposition 2.2. Let $\phi$ be any magnetic potential satisfying (2.2) for a given normal‐
ized 2‐cocycle $\omega$ with the additional condition (iii) of Lemma 2.1. Then,(i) The representation \Re \mathrm{e}\mathfrak{p}^{$\phi$'} of \mathb {C}( $\omega$) is irreducible and faithful(ii) Any other choice for $\phi$' leads to a unitarily equivalent representation,(iii) If h\in l^{1}(\mathbb{Z}^{d};l^{\infty}(\mathbb{Z}^{d})

3

,

then

\Re e\mathrm{p}^{$\phi$'}(h)\dot{u} self‐adjoint if h^{ $\theta$}=h.

A continuous field of C^{*} ‐algebras

In this section we consider a family of discrete magnetic systems which are parame‐terized by the elements $\epsilon$ of a compact Hausdorff space $\Omega$ The necessary continuity.

relations between the various objects is encoded in the structure of a field of twisted
crossed product C^{*} ‐algebras, as introduced in [12] and already used in a similar con‐[1]. Again, let us be very pragmatic and introduce only the strictly necessary
information, and refer to [11, Sec. 3] for more details.
text inFor fixed x, y\in \mathbb{Z}^{d} , let us consider a continuous map$\Omega$\ni $\epsilon$\mapsto$\omega$^{ $\epsilon$}(x, y)\in \mathscr{U}(\mathbb{Z}^{d})such that each element $\omega$^{$\epsilon$}(3.1)) is a 2‐cocycle on \mathbb{Z}^{d} with values % (\mathbb{Z}^{d}) and which satisfies

the.additional property (iii) of Lemma 2.1. Note that equivalently one can consider anormalized 2‐cocycle $\omega$ on \mathb {Z}^{d} and with values in the C^{*} ‐algebra

setting the additional condition reads
relation clearly holds:

C( $\Omega$;l^{\infty}(\mathbb{Z}^{d}) In this.

$\omega$(x, -x)=1 for any x\in \mathbb{Z}^{d} , and the following$\omega$^{ $\epsilon$}(q;x, y)\equiv[$\omega$^{ $\epsilon$}(x, y)](q)=[ $\omega$(x, y)]( $\epsilon$, q)\equiv $\omega$( $\epsilon$, q;x, y).

Let us stress that the continuity assumption mentioned in the previous paragraphis precisely the sufficient one for the continuity of the spectrum. More precisely, thiscontinuity condition corresponds to a gauge‐independent condition, and does not cor‐
respond to any direct requirement on any magnetic potential $\phi$^{$\epsilon$} So, let us now state.

the main result of this note:
Theorem 3.1. Let \{$\omega$^{ $\epsilon$}\}_{ $\epsilon$\in $\Omega$} be a family of normalized 2‐cocycle satisfying the additional

property (iii) of Lemma 2.1 and such that the map defined by (3.1) is continuous. Con‐sider a family\{h^{ $\epsilon$}\}_{ $\epsilon$\in $\Omega$}\subset l^{1}(\mathbb{Z}^{d};l^{\infty}(\mathbb{Z}^{d}) such that the following conditions are satisfied:(i) \displaystyle \sum_{x\in \mathbb{Z}^{d} \sup_{q\in \mathbb{Z}^{d} \sup_{ $\epsilon$\in $\Omega$}|h^{ $\epsilon$}(q;x)|<\infty,

75(ii) For any fixed x\in \mathbb{Z}^{d},\displaystyle \lim_{ $\epsilon$\rightar ow $\epsilon$}\sup_{q\in \mathb {Z}^{d} |h^{$\epsilon$'}(q;x)-h^{ $\epsilon$}(q;x)|=0,(iii) (h^{ $\epsilon$})^{ $\theta$}=h^{ $\epsilon$}.
Then, for any family of magnetic potential $\phi$^{$\epsilon$} satisfying[$\omega$^{ $\epsilon$}(x, y)](q)=\exp\{i[$\phi$^{ $\epsilon$}(q, q+x)+$\phi$^{ $\epsilon$}(q+x, q+x+y)+$\phi$^{ $\epsilon$}(q+x+y, q)]\},
the family of spectra
at every point $\epsilon$\{$\sigma$(\Re\mathrm{t}\mathfrak{p}^{$\phi$^{$\epsilon$}(h^{$\epsilon$})\}_{$\epsilon$\in$\Omega$} forms

an outer and an inner continuous

familyof $\Omega$.

The proof of this statement as well as the proof of Theorem 1.2 are given in [11].

Let us simply mention that it is based on the construction of the enveloping C^{*} ‐algebra\mathbb{C}( $\omega$) of the Banach *‐algebra l^{1}(\mathbb{Z}^{d};C( $\Omega$;l^{\infty}(\mathbb{Z}^{d}) ) endowed with a product and an
Properties of the evaluation maps e_{$\epsilon$} : \mathrm{C}( $\omega$)\rightar ow \mathrm{C}($\omega$^{ $\epsilon$})involution defined in terms of $\omega$at any $\epsilon$\in $\Omega$ lead rather.

straightforwardly to our main theorem, once the abstract

results obtained in [12] are taken into account.

References[1] N. Athmouni, M. Măntoiu, R. Purice, On the continuity of spectra for families ofmagnetic pseudodifferential operators, J. Math. Phys. 51 no. 8 (2010), 083517, 15
pp.

[2] S. Beckus, J. Bellissard, Continuity of the spectrum of a field of self‐ajdoint opera‐
tors, Preprint ArXiv 1507.04641.

[3] J. Bellissard, Lipshitz continuity of gap boundaries for Hofstadter‐like spectra,
Comm. Math. Phys. 160 no. 3 (1994), 599‐613.
[4] Y. Colin de Verdière, N. Torki‐Hamza, F. Truc, Essential self‐adjointness for com‐binatorial Schrölinger operators III‐Magnetic fields, Ann. Fac. Sci. Toulouse Math.
(6) 20 no. 3 (2011), 599‐611.

[5] H. Cornean, On the Lipschitz continuity of spectral bands of Harper‐like and mag‐netic Schrölinger operators, Ann. Henri Poincare 11 no. 5 (2010), 973‐990.
[6] M. Kotani, Lipschitz continuity of the spectra of the magnetic transition operators
on a crystal lattice, J. Geom. Phys. 47 no. 2‐3(2003), 323‐342.
[7] M Măntoiu, R. Purice, S. Richard, Twisted crossed products and magnetic pseu‐
dodifferential operators, Advances in operator algebras and mathematical physics,
137‐172, Theta Ser. Adv. Math. 5, Theta, Bucharest, 2005.

76[8] G. Nenciu, On the smoothness of gap boundaries for generalized Harper operators,
Advances in operator algebras and mathematical physics, 173‐182, Theta Ser. Adv.
Math. 5, Theta, Bucharest, 2005.

[9] J. Packer, I. Raeburn, Twisted crossed products of C^{*} ‐algebras, Math. Proc. Cam‐bridge Philos. Soc. 106,no.2 (1989), 293‐311.
[10] J. Packer, I. Raeburn, Twisted crossed products of C^{*} ‐algebras II, Math. Ann. 287
no. 4 (1990), 595‐612.
[11] D. Parra, S. Richard, Continuity of the spectra for families of magnetic operatorson\mathb {Z}^{d},

to appear in Anal. Math. Phys., 17 pages.

[12] M. Rieffel, Continuous fields of C^{*} ‐algebras coming from group cocycles and ac‐tions, Math. Ann. 283 no. 4 (1989), 631‐643.
[13] T. Sunada, A discrete analogue of periodic magnetic Schrödinger operators, inGeometry of the spectrum (Seattle, WA, 1993), 283‐299, Contemp. Math. 173,
Amer. Math. Soc., Providence, RI, 1994.

[14] G. Zeller‐Meier, Produits croises dune C^{*} ‐algèbre par un groupe dautomorphismes,
J. Math. Pures Appl. 47 (1968), 101‐239.