Diagonal automorphisms of the $2$-adic ring $C^*$-algebra

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by  Valeriano Aiello

Diagonal automorphisms of the 2-adic ring C˚-algebra
Valeriano Aiello:, Roberto Conti7, Stefano Rossi6
: Dipartimentodi MatematicaeFisica
7
1 Universita` RomaTre
0 Largo S. Leonardo Murialdo1,00146Roma, Italy.
2
[email protected]

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J 7 Dipartimentodi Scienze diBasee Applicateperl’Ingegneria

5 Sapienza Universita` diRoma
1
ViaA.Scarpa 16, I-00161Roma, Italy.

] [email protected]
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6 DipartimentodiMatematica,
.
h Universita` di RomaTorVergata
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a ViadellaRicerca Scientifica 1,I–00133Roma, Italy.
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[email protected]
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1
v Abstract
3
3 The 2-adic ring C˚-algebra Q2 naturally contains a copy of the Cuntz algebra O2 and,
0 a fortiori, also of its diagonalsubalgebraD2 with Cantor spectrum. This paper is aimed at
04 ostuutdtyhiantgatnhye sgurochupauAtoumtDo2rppQhi2sqmofletahveesauOto2mgolorpbhailslymsinovfarQia2ntfi.xiFnugrtDhe2rmpoorine,twthisee.suIbtgtruorunps
.
1 AutD2pQ2q is shown to be maximal abelian in AutpQ2q. Saying exactly what the group is
0 amountsto understandingwhen an automorphismof O2 that fixes D2 pointwise extendsto
17 Q2. Acompleteanswerisgivenforalllocalizedautomorphisms:thesewillextendifandonly
iftheyarethecompositionofalocalizedinnerautomorphismwithagaugeautomorphism.
:
v
i
X
1 Introduction
r
a
Assoon astheCuntzalgebras wereintroduced in[11],itwasquickly realized thatstudying their
endomorphisms and automorphisms would initiate a fruitful research season. The forecast could
not possibly be more accurate, for after nearly forty years they continue to be a major topic and
a source of inspiration, as demonstrated by the recent literature that has been accumulating at an
impressive rate, see e.g. [9, 6, 8, 7, 2, 4, 3, 5]. Motivated by these works, we found it natural to
ask ourselves whether such a case study would also provide the right tools to analyze the endo-
morphismsandautomorphisms ofotherclassesofC˚-algebras, notably thoserecently associated
with rings, fields and other algebraic objects. In fact, this was one of the main reasons why in
[1]westarted aninvestigation ofthegroupAutpQ2qofunital˚-preserving automorphisms ofthe
so-called dyadic ring C˚-algebra of the integers Q2, a known C˚-algebra (see e.g. [15] and the
references therein)associated tothesemidirect productsemigroup
Z¸t1,2,22,23,...u
that contains a copy of the Cuntz algebra O2 in a canonical way. Inter alia, it was proved in [1]
the useful fact that the canonical diagonal D2 maintains the property of being amaximal abelian
1
subalgebra (MASA) in Q2 also. Actually, more is known, for D2 is even a Cartan subalgebra
of both O2 and Q2 [13, Prop. 4.3]. The study there initiated is further developed in the present
paper. In particular, our main focus is here on the structure of the set of those automorphisms of
Q2 leavingthediagonalMASAD2 pointwisefixed,whichwillalwaysbedenotedbyAutD2pQ2q.
Ratherinterestingly,thisgroupturnsouttobeamaximalabeliansubgroupofAutpQ2q. Moreover,
we show that any of its elements restricts to an automorphism of O2 and it is indeed the unique
extension of its restriction. It immediately follows from the analysis carried out in [10]that such
restrictions are automorphisms of O2 induced by unitaries in D2, henceforth referred to as diag-
onal automorphisms for short. The results here obtained lend further support to the idea, already
expressed in [1], that the group of automorphisms of Q2 is, in a sense, considerably smaller than
that of O2, thus making it reasonable to ask the challenging question whether this group may be
computed explicitly up to inner automorphisms. Indeed, we show that any extensible localized
diagonalautomorphism ofO2 isnecessarily theproduct ofagaugeautomorphism andalocalized
inner diagonal automorphism. The general case is still out of the reach of the techniques used in
this paper instead. Even so, we do spot a necessary and sufficient condition for a diagonal auto-
morphism to extend. Despite all our efforts to exploit the condition, to date this has not aided us
indeciding whethertheonlyextensible diagonal automorphisms ofO2 areproducts ofgaugeand
innerdiagonalautomorphisms. Itisquitepossiblethattheanswertothisproblemwillalsorequire
todelvefurtherintothefineergodicproperties oftheodometermap.
2 Preliminaries and notations
We recall the basic definitions and properties of the 2-adic ring C˚-algebra so as to make the
reading of the present paper suitable for a broader, not necessarily specialized, audience. This
is the universal C˚-algebra Q2 generated by an isometry S2 and a unitary U such that S2S2˚ `
US2S2˚U˚ “ 1 and S2U “ U2S2. A very informative account of its most relevant properties is
givenin[15]aswellasinourformerwork[1]. Asfarasthepurposesofthisworkareconcerned,
it is important to mention that the Cuntz algebra O2, i.e. the universal C˚-algebra generated by
two isometries X1 and X2 such that X1X1˚ `X2X2˚ “ 1, can be thought of as a subalgebra of
Q2 via the injective unital ˚-homomorphism that sends X1 to US2 and X2 to S2. Asof now this
˚-homomorphism willalways be understood without explicit mention, therefore wesimply write
O2 Ă Q2 to refer to the copy of O2 embedded in Q2 in this way. The 2-adic ring C˚-algebra is
actually a kind of a more symmetric version of O2, in which the Cuntz isometries S1 and S2 are
nowintertwined bytheunitary U,towitS2U “ US1. Thiscircumstance introduces ahigher de-
greeofrigidity, whichisultimately responsible forashortage ofouter automorphisms. Although
not completely computed, the group OutpQ2q is nevertheless closely looked over in [1], where
it is shown to be considerably smaller than OutpO2q, while being still uncountable and noncom-
mutative. To get a better idea of to which extent the former group is smaller than the latter it is
worthwhiletopointupthatendomorphisms orautomorphisms ofO2 willnotingeneral extendto
Q2,aswidelydiscussedin[1]. Amongthosethatdoextendtherearethecanonicalendomorphism,
thegaugeautomorphisms andtheflip-flop. Ofthese onlythefirsttwowillplayanimportant role
in this work. In particular, the gauge automorphisms will actually play an overriding role, which
meanstheyneedabitmoreexhaustive introduction.
On O2 these are the automorphisms αθ acting on each isometry simply multiplying it by eiθ, i.e.
α pS q “ eiθS fori “ 1,2, whereθ isanyrealnumber. Theactionoftheone-dimensional torus
θ i i
TonO2 providedbythegaugeautomorphismsenablestospeakofthegaugeinvariantsubalgebra
F2 Ă O2,whichisbydefinition theC˚-subalgebra whoseelementsarefixedbyalltheαθ’s. Itis
well known that F2 is the UHF algebra of type 28. Now the gauge automorphisms are immedi-
atelyseentoextendtoautomorphismsαθ ofthewholeQ2 withαθpUq “ U. Lessobviously, each
α is an outer automorphism when it is not trivial, which is proved in [1]. Furthermore, the ex-
θ
Ă Ă
Ă
2
tendedgaugeautomorphismsallowustoconsiderthegaugeinvariantsubalgebra ofQ2,whichwe
denotebyQT. Amongotherthings,QT isknowntobeaBunce-Deddensalgebra. Itisnotparticu-
2 2
larlyhardtoprovethatQT2 canalsobedescribedastheC˚-subalgebraofQ2generatedfromeither
F2 or D2 and U, where D2 Ă F2 is a notable commutative subalgebra of the Cuntz algebra O2.
Commonly referred to as the diagonal subalgebra, D2 is in fact the subalgebra generated by the
.
diagonal projections Pα “ SαSα˚, where for any multi-index α “ pα1,α2,...,αkq P nt1,2un
the isometry S is the product S S ...S . The multi-index notation is rather convenient
α α1 α2 αk Ť
when making computations in Cuntz algebras and willbe used extensively throughout the paper.
In particular, we need to recall that |α| is the length of the multi-index α. By its very definition,
the diagonal D2 is also the inductive limit of the increasing sequence of the finite-dimensional
subalgebras Dk2 Ă D2k`1 Ă D2 given by Dk2 “. spantPα : |α| “ ku, k P N. That D2 is quite a
remarkable subalgebra is then seen at the level of its spectrum, for the latter is the Cantor set K,
of which many a concrete topological realization is known. However, in what follows we shall
alwaysthinkofitastheTychonovinfiniteproductt1,2uN.
Not of less importance is the canonical endomorphism ϕ P EndpO2q, which is defined on each
element x P O2 as ϕpxq “ S1xS1˚ `S2xS2˚. By its very definition it clearly extends to Q2, on
which it still acts as a strongly ergodic map, namely ϕnpQ2q “ C1, as shown in [1]. The
n
intertwining rules Six “ ϕpxqSi for any x P Q2 with i “ 1,2 still hold true. In addition, the
Ş
canonical endomorphism preservesthediagonal D2,actingonitsspectrum astheusualshiftmap
ont1,2uN.
To complete the description of our framework we still need to sort out a distinguished represen-
tation of Q2 among the many, namely the so-called canonical representation [15]. This is the
representation in which S2 and U are concrete operators acting on the Hilbert space ℓ2pZq as
S2ek “ e2k andUek “ ek`1,foreveryk P Z,wheretek : k P Zuisthecanonical basisofℓ2pZq.
Finally,inthisrepresentation D2 canbeseenasanorm-closed subalgebra ofℓ8pZq,thediagonal
operators withrespect tothe canonical basis. Forany d P ℓ pZqwedenote by dpkq its kthdiag-
8
.
onal entries, that is dpkq “ pe ,de q, k P Z. It is obvious that AdpUq leaves ℓ pZq invariant. It
k k 8
isslightly less obvious that it also leaves D2 globally invariant. Moreover, the spectrum of D2 is
acted on by AdpUq through the homeomorphism given by the so-called odometer on the Cantor
set,whichisknowntobeauniquelyergodic map,see[12]. Finally,theactionofAdpUqonD2 is
compatible withitsbeinganinductivelimit,i.e. AdpUqpDkq “ Dk foreveryk P N.
2 2
3 General structure results on Aut pQ q
D2 2
Given an inclusion of C˚-algebras A Ď B, we shall denote by AutpB,Aq the group of those
automorphisms of B leaving A globally invariant, and by Aut pBq the group of those fixing A
A
pointwise. Both endomorphisms and automorphism are tacitly assumed to be unital whenever
our C˚-algebras are unital, as D2 and Q2 are. Having a general content, the following result
may possibly be known, cf. [10] for instance. Because we have no explicit reference, a precise
statementisnevertheless included alongwithitsproof. Tostateitasclearly aspossible, however,
we still need to set some notations. In particular, we recall that if H is any subgroup of a group
G, its normalizer is the largest subgroup N pGq in which H is contained as a normal subgroup.
H
More explicitly, N pGq can be identified with the set of those g P G such that gHg´1 “ H.
H
Finally,ifS isanysubsetofAutpBq,wedenotebyBS thesub-C˚-algebra ofB whoseelements
arefixedbyallautomorphisms ofS. Herefollowstheresult.
Proposition3.1. LetA Ď B beaunitalinclusion ofC˚-algebras. Then
1. AutpB,Aq Ď NAutApBqpAutpBqq;
2. IfBAutApBq “ AonehasAutpB,Aq “ NAutApBqpAutpBqq;
3
3. IfAisaMASAinB,thenBAutApBq “AandAutpB,Aq “ NAutApBqpAutpBqq.
Proof. For the first property, let γ P AutpB,Aq and β P Aut pBq. Since γ ˝ β ˝ γ´1paq “
A
a for all a P A, we get that γ belongs to NAutApBqpAutpBqq. For the second, consider γ P
NAutApBqpAutpBqq and β P AutApBq. We have that γ ˝β ˝γ´1paq “ a for all a P A, hence
β ˝ γ´1paq “ γ´1paq. This means that γ´1paq P BAutApBq which is by hypothesis equal to
A, thus γ P AutpB,Aq. Finally, for the third we observe that any u P UpAq gives rise to an
element ofAutApBq,namely Adpuq. Thismeans thatforanyx P BAutApBq wehave uxu˚ “ x.
Since A is a MASA, then x P A and the second property in turn implies that AutpB,Aq “
NAutApBqpAutpBqq. ˝
Byapplying theformer result to theinclusion D2 Ă Q2,the set equality QA2utD2pQ2q “ D2 is
immediatelygotto. NowAutD2pQ2qcontainsbothtαθ |θ P RuandtAdpuq, uP UpD2qu,where
αθ istheuniqueextensiontoQ2 ofthegaugeautomorphism αθ P AutpO2q,see[1]. Furthermore,
the intersection AutD2pQ2q InnpQ2q is easily seenĂto reduce to tAdpuq,u P UpD2qu thanks to
mĂaximalityofD2 again. SinŞceD2 isglobally invariant underAdpUq,foranyα P AutD2pQ2qwe
have
αpUdU˚q “UdU˚ “ αpUqdαpUq˚foralld P D2
thatisU˚αpUqcommuteswitheveryd P UpD2qandtherefore bymaximality U˚αpUq “ dα for
somedα P UpD2q,whichwerewriteas
αpUq “ Ud
α
Itisalsoclearthat
α´1pUq “Ud˚
α
To take but one example, when α is an inner automorphism, say Adpuq for some u in UpD2q,
the corresponding d is nothing but U˚uUu˚. More importantly, the map α ÞÑ d is eas-
Adpuq α
ily recognized to be a group homomorphism between AutD2pQ2q and UpD2q. Obviously, its
kernel coincides with AutQTpQ2q. Our next goal is to show that the Cuntz-Takesaki unitary
. 2
uα “ αpS1qS1˚`αpS2qS2˚ belongstoD2 aswell.
Proposition3.2. LetαbeinAutD2pQ2q. Thenthecorresponding unitaryuα liesinD2.
Proof. Firstweobserve thatαpSiq “ uαSi fori “ 1,2. BymaximalityofD2 Ă Q2,itisenough
toprovethatuαcommuteswiththegeneratingprojectionsPi1i2...ik ofD2. Thiscanbeeasilyseen
by induction on k, as done by Cuntz for O2. The case of length one reduces to the computation
P “ αpP q “ αpS S˚q “ u S S˚u˚ “ u P u˚. Thecaseoflenght twoentails thecomputation
i i i i α i i α α i α
P “ αpP q “ αpS S S˚S˚q “ u S pu S S˚u˚qS˚u˚ “ u S S S˚S˚u˚ “ u P u˚. It is
ij ij i j j i α i α j j α i α α i j j i α α ij α
nowclearhowtogoon. ˝
The following result can be derived at once from the foregoing proposition. Therefore, we
can safely state it without proof, limiting ourselves to point out that uniqueness relies on results
obtained in[1].
Corollary 3.3. Anyα P AutD2pQ2qrestricts toan automorphism of theCuntz algebra O2 and it
istheuniqueextension ofsuchrestriction. Inparticular, thegroupAutD2pQ2qisabelian.
Every element in AutD2pQ2q can thus be written as the unique extension of an element λd P
AutD2pO2q, for some d P UpD2q. We denote such extension as λd. We need hardly say that
the converse is also true, that is for any d P UpD2q the associated automorphism λd fixes D2
pointwise. Thus far we have seen that any α P AutD2pQ2q acts onĂU as αpUq “ Udα for some
dα P D2. Moreover, it is possible to rewrite this relation in the form αpUq “ dˇαU, where dˇα is
4
simply given by UdαU˚ and is still a unitary of D2. As α “ λd, we can simply write dˇinstead
ˇ ˇ
ofd . Forthesamereason asabove,themapd ÞÑ disagrouphomomorphism. Infact, thismap
α
will turn out to be vital in the next sections. Contrary to whatĂone might expect, though, it has
proved to be a difficult task to establish a priori whether it is norm continuous, possibly because
determining its domain is just another way to recast our main problem. Nevertheless, its kernel
canbedescribed explicitly.
ˇ
Proposition 3.4. The kernel of the map d ÞÑ d is the subgroup of the gauge automorphisms.
Actually,onehas
AutQT2pQ2q “ AutF2pQ2q“ tαθ : θ P Ru.
Proof. Clearly the condition dˇ“ 1 is the same as λdpUqĂ“ U, which implies λd P AutQTpQ2q.
2
The conclusion now readily follows from the fact that the automorphisms of O2 fixing F2 point-
wiseareprecisely thegaugeautomorphisms, asprovedbyCuntzin[10]. ˝
In particular, the restriction map AutD2pQ2q Q λ Ñ λ æO2P AutD2pO2q induces a group
embedding which allows us to think of the fomer group as a subgroup of the latter. Therefore,
as of now we will simply write AutD2pQ2q Ă AutD2pO2q to mean that. Of course the inclusion
is proper. In other words, not all the automorphisms of O2 that leave the diagonal D2 glob-
ally invariant will extend. As a matter of fact, very few automorphisms can be extended. Al-
though we do not have a general explicit description of all extensible automorphisms yet, we do
have a complete description for a particular class of automorphisms. This is just the subgroup
AutD2pO2qloc of those localized automorphisms we mentioned above in passing. Actually, the
terminology comes from Quantum Field Theory. Roughly speaking, an automorphism is local-
ized when it preserves the union of the matrix subalgebras. More precisely, an automorphism
λ P AutpO q is said to be localized when the corresponding unitary u P UpO q belongs to the
u n n
algebraicdensesubalgebra Fn Ă O ,whereFnisgeneratedbytheelementsoftheformS S˚
k k n k α β
with α,β P t1,...,nuk. FŤurthermore, the inclusion AutD2pQ2q Ă AutD2pO2q allows us to de-
fineasubgroup AutD2pQ2qloc astheintersection AutD2pQ2q AutpO2qloc. Asmaintained inthe
abstract, wewillprovethatAutD2pQ2qloc issosmallthattheŞsolelocalizedautomorphisms fixing
D2thatextendarethecompositionofalocalizedinnerautomorphismwithagaugeautomorphism.
Before going on with our discussion, we would like to point out a remark for the sake of
completeness.
Remark 3.5. Let d P UpD2q and consider the associated automorphism λd of O2. If λd extends
to an endomorphism λd of Q2, then λd is actually an automorphism. In fact, λdpQ2q contains
λdpO2q “ O2 andλdpdˇ˚Uq“ dˇ˚dˇU “ U withdˇP UpD2q.
Ă Ă Ă
GoingbacktoAĂutD2pQ2q,wehaveshownitisabelian,butwewanttoimproveourknowledge
by proving it is also maximal abelian in AutpQ2q, in a way that closely resembles what happens
fortheCuntzalgebra O2 [10]. Herefollowstheproof.
Theorem3.6. Thesubgroup AutD2pQ2qismaximalabelianinAutpQ2q.
Proof. LetαbeanautomorphismofQ2thatcommuteswithAutD2pQ2q. Inparticularrα,Adpuqs “
0 for every u P UpD2q, to wit Adpαpuqq “ Adpuq. As the center of Q2 is trivial, we see that
αpuq “ χpuqu for every u P UpD2q, where χ is a character of the group UpD2q. Our result
will be proved once we show χpuq “ 1 for every u P UpD2q. Tothis aim, note that the equality
αpuq “ χpuqusaysthatD2 isatleastgloballyinvariantundertheactionofα. Withaslightabuse
of notation, we still denote by α the restriction of α to D2 – CpKq, where K is the Cantor set.
LetΦ P HomeopKqsuch thatαpfq “ f ˝Φforeveryf P CpKq. Theidentity obtained above is
thenrecastintermsofΦasf ˝Φ “ χpfqf foreveryf P CpK,Tq. WeclaimthatχpUpD2qq Ă T
5
isatmostcountable. Ifso,thetheorem cannowbeeasilyinferred. Indeed, ifΦisnottheidentity
map, then there exists x P K such that Φpxq ‰ x. Then pick a function f P CpK,Tq such that
fpxq “ 1 and fpΦpxqq “ eiθ. The equality f ˝ Φ “ χpfqf evaluated at x gives χpfq “ eiθ,
thatisχisontoT. Toreallyachieve theresultwearethusleftwiththetaskofproving theclaim.
Thisshould bequite astandard fact from ergodic theory. However, wedogive acomplete proof.
Ifµ isanyBorel Φ-invariant measure onK ,wecan consider theHilbert space L2pK,µq, which
is separable because K is metrizable, and the Koopman unitary operator U associated with Φ,
φ
whose action is simply given by UΦpfq “ f ˝Φ a.e. for every f P L2pKq. As eigenfunctions
of UΦ associated with different eigenvalues are orthogonal and χpfq is an eigenvalue for every
f P CpK,Tq,weseethattχpfq: f P CpK,Tquisacountable setbyvirtueofseparability. ˝
Remark 3.7. We can also provide an alternative argument for the above result, proving more
directly that αpP q “ P for all the multi-indeces β. First of all we observe that the relation
β β
αpdq “ χpdqd implies that the spectrum of the unitary d is invariant under the rotation of χpdq.
We begin with the case of P1. Consider the unitary d1 “ P1 ` ei2πθP2 with θ ‰ ˘1. On the
one hand, we know that αpd1q “ χpd1qd1. On the other hand, since the spectrum of d1 is not
invariant under non-trivial rotations, we find that χpd1q must be 1. The same reasoning applies
to the unitary d1 “ P1 ´ eiθP2 too, and so we get the equality χpd1q “ 1, hence αpP1q “
α d1`2d1 “ P1. WenowdealwiththegeneralcaseofaPβ withβ beingamulti-indexoflength
r r
k ´in murch¯the same way. Consider the two unitary operators d “ eiθP `P and
β |γ|“k,γ‰β γ β
d˜ “ ´ eiθP `P . By the same argument as above we still find both αpd q “ d
β |γ|“k,γ‰β γ β ř β β
˜
andαpdβřq “ dβ,andthusαpPβq “ α dβ`2dβ “ Pβ andwearedone.
´ ¯
r r
4 Necessary and sufficient conditions for extendability
Thankstotheresultsachievedinthelastsection,givingacompletenon-tautologicaldescriptionof
AutD2pQ2qentailsstudyingthoseunitariesd P UpD2qforwhichthecorrespondingλd P AutpO2q
may be extended to Q2. This section is mainly concerned with problems of this sort. When an
automorphism λ extends, wewillsayeverysooftenthatthecorresponding disextensible itself.
d
Thisisundoubtedly aslightabuseofterminology, butitaidsbrevity. Herefollowsourfirstresult.
Lemma4.1. Letdbe inUpD2q. Thenλd P AutpO2qextends toQ2 ifand only ifthere exists adˇ
inUpD2qsuchthat
ˇ ˇ
dUdS1 “ dS2dU (1)
ˇ
dUdS2 “ dS1 (2)
Moreover,suchanextension isautomatically anautomorphism wheneveritexists.
Proof. Ifλ extends, thenthetwoequalities inthestatementareeasily obtained ifoneapplies its
d
extension λd to US1 “ S2U and US2 “ S1 respectively also taking into account that λdpUq “
ˇ ˇ
dU. The converse is dealt with analogously by noting that the pair (dU,dS2q still satisfies the
definingreĂlationsofQ2. Thelastobservation isnothingbutRemark3.5. Ă ˝
ˇ
Atthispoint,thereadermaybewonderingwhetheriteverhappensthatd “ d. Infact,itturns
outthatthisisneverthecaseunlessd “ 1,namelywehavethefollowingresult.
Proposition4.2. Theunitary d“ 1istheuniquefixedpointofthemapd ÞÑ dˇ.
6
Proof. Ifweworkinthecanonicalrepresentation, wesimplyneedtoshowthatdpkq “ 1forevery
k P Z. Wefirst handle the even entries of d. Formula (2)becomes dUdS2 “ dS1, which in turn
gives
dS2 “ U˚S1 “ S2 .
Now by computing the above equality on the vectors of the canonical basis of ℓ2pZq we get
dp2kq “ 1 for all k P Z. As for the odd entries, Formula (1) leads to dUdS1 “ dS2dU, which
yields dp2k `1q “ dpk `1q for all k. At this point, the claim easily follows by induction and
usingthefactthattheevenentriesofdaretrivial. ˝
AlthoughmorefocusedontheCuntzalgebraO2,thenextusefulresultisincludedallthesame.
Infact,wedobelievethatitmayshedsomelightonapplications yettocome.
Proposition 4.3. Let λd P AutpO2q be an extensible automorphism. Then either λd is a gauge
automorphism orλdUd˚U˚ isouter.
Proof. Firstofallweprovethat
ϕpdqS1 “ S1 . (3)
To this aim, rewrite Formula (2) as dˇUd~S2 “ dˇUdU˚S1 “ dS1. Then, by using the identity
dϕpdq˚ “ dUd˚U˚ andthemultiplicativity ofthemapˇ onefinds
­ dˇS1 “ dUd˚U˚S1 “ dϕpdq˚S1 “ dˇϕpdq˚S1 (4)
whence the claim. Likewise, Formula (1)yields­dˇUdS1 “ dˇU­dU˚S2U “ dS2dˇU “ dϕpdˇqS2U
andthusdˇUdU˚S2 “ dϕpdˇqS2,thatis
dˇd˚UdU˚ϕpdˇ˚qS2 “ S2 . (5)
The former equality actually shows that the automorphism Adpdˇ˚q ˝λdUd˚U˚ fixes S2, i.e.
Adpdˇ˚q˝λdUd˚U˚rS2s“ S2. By[16,CorollaryB],theneitherAdpdˇ˚q˝λdUd˚U˚ istheidentity
or Adpdˇ˚q ˝λdUd˚U˚ is an outer automorphism of O2. In the first case dˇϕpdˇq˚ “ dUd˚U˚ “
dϕpdq˚ “ dˇϕpdˇq˚ “ dˇϕpdq˚,hence
ˇ
­ ~ ϕpdq“ ϕpdq.
Byapplying thelastequality to(3)weget ~
ˇ ˇ
S1 “ ϕpdqS1 “ϕpdqS1 “ S1d
whichprovesthatdˇ“ 1,thusλ isaga~ugeautomorphism byProposition3.4. Inthesecondcase,
d
clearlyλdUd˚U˚ isanouterautomorphism ofO2. ˝
Atanyrate,afirstapplication canbegivenatonce.
Corollary 4.4. Ifλd,dP UpD2qisanon-trivial innerautomorphism ofO2,thenλUdU˚ isouter.
Wecannowresumetoourgeneraldiscussion. Withthisinmind,westartbyspottingauseful
necessary condition ondforthecorresponding λ toextend.
d
Proposition4.5. LetdbeaunitaryinD2. Ifλd extends toQ2,thendp0q “ dp´1q.
7
Proof. Owingtotheextendability ofλd theisometries dS1 “ λdpS1qanddS2 “ λdpS2qarestill
intertwined,i.e. theyareunitarilyequivalent. Inparticular, theirpointspectramustcoincide. Now
theequalities σppdS1q “ tdp1qu andσppdS2q “ tdp0qu areboth easily checked, hence thethesis
follows. ˝
It goes without saying that the condition is only necessary. Even so, it does have the merit
of highlighting a property of which we will have to make an extensive use. Therefore, unless
otherwisestated, ourunitaries d P UpD2qwillalwayssatisfythecondition
dp0q “ dp´1q.
In addition, there is no lack of generality ifwe further assume that both dp0q and dp´1q equal 1.
For if this were not the case, we could always multiply λ with a suitable gauge automorphism,
d
whichofcoursewouldnotaffecttheextendability ofλ ,sincegaugeautomorphismsdoextendto
d
Q2. Finally,weadopttheDiracbra-ketnotationforrank-oneoperators: foranygivenu,v P ℓ2pZq
the operator w Ñ pv,wqu is denoted by |uyxv|. This said, we can state the main result of the
presentsection.
Theorem4.6. ForaunitarydinD2 suchthatdp0q “ dp´1q “ 1thefollowingareequivalent:
1. Theautomorphism λd P AutpO2qextends toanautomorphism ofQ2,namelythere exists a
unitarydˇP D2 suchthatdS2 anddˇU satisfy thedefining relations ofQ2.
2. Thestronglimitofthesequence
k´1
xk “. |e0yxe0|`dϕpdqϕ2pdq...ϕk´1pdq S2iS1S1˚pS2˚qi Uϕk´1pd˚q...ϕpd˚qd˚U˚
˜i“0 ¸
ÿ
belongs toD2,inwhichcaseitcoincides withthedˇabove.
Proof. To begin with, we prove that the sequence referred to in the statement is always strongly
convergent to some δ in ℓ pZq. If we set d “. dϕpdqϕ2pdq...ϕk´1pdq, we see that x “
8 k k
d Ud˚U˚ and then this sequence is convergent since d pjq “ d pjq for all k ě |j| thanks to
k k k |j|
ϕpdqpkq “ dprk{2sq, k P Z, and dp0q “ dp´1q “ 1. In particular, these allow us to write
the equality dk “ dϕpdk´1q, k P N, which is necessary to carry out some of the following
computations.
Fortheimplication2 ñ 1,itisenoughtoshowthatS˜2U˜ “ U˜2S˜2andS˜1 “ U˜S˜2,whereU˜ “ δU
˜
andS “ dS . Asforthefirst,werewriteδU as
i i
k´1
δU “ |e0yxe´1|`limdϕpdqϕ2pdq...ϕk´1pdq S2iS1S1˚pS2˚qi Uϕk´1pd˚q...ϕpd˚qd˚
k ˜i“0 ¸
ÿ
where |e0yxe´1|pvq “ vp´1qe0 for all v P ℓ2pZq. If we now use the expression obtained above,
8
˜2 ˜
wecancomputeU S2 as
k´1 k´1
U˜2S˜2 “ |e0yxe´1|`limdk S2iS1S1˚pS2˚qi U S2iS1S1˚pS2˚qi Uϕpd˚k´1qS2
k ˜i“0 ¸ ˜i“0 ¸
ÿ ÿ
k´1 k´1
“ |e0yxe´1|`limdk S2iS1S1˚pS2˚qi U S2iS1S1˚pS2˚qi US2d˚k´1
k ˜i“0 ¸ ˜i“0 ¸
ÿ ÿ
k´1 k´1
“ |e0yxe´1|`limdk S2iS1S1˚pS2˚qi U S2iS1S1˚pS2˚qi S1d˚k´1
k ˜i“0 ¸ ˜i“0 ¸
ÿ ÿ
k´1
“ |e0yxe´1|`limdk S2iS1S1˚pS2˚qi US1d˚k´1
k ˜i“0 ¸
ÿ
k´1
“ |e0yxe´1|`limdk S2iS1S1˚pS2˚qi S2Ud˚k´1
k ˜i“0 ¸
ÿ
k´2
“ |e0yxe´1|`limdkS2 S2iS1S1˚pS2˚qi Ud˚k´1
k ˜i“0 ¸
ÿ
˜ ˜
whilstS2U isgivenby
k´1
S˜2U˜ “ dS2 |e0yxe´1|`limdk S2iS1S1˚pS2˚qi Ud˚k
˜ k ˜i“0 ¸ ¸
ÿ
k´1
“ |e0yxe´1|`limdS2dk S2iS1S1˚pS2˚qi Ud˚k
k ˜i“0 ¸
ÿ
k´1
“ |e0yxe´1|`limdk`1S2 S2iS1S1˚pS2˚qi Ud˚k
k ˜i“0 ¸
ÿ
and the two expressions coincide, as maintained. The second is dealt with by means of similar
computations:
k´1
U˜S˜2 “ |e0yxe0|`limdk S2iS1S1˚pS2˚qi Ud˚kU˚ UdS2
˜ k ˜i“0 ¸ ¸
ÿ
k´1
“ limdk S2iS1S1˚pS2˚qi Ud˚kU˚UdS2
k ˜i“0 ¸
ÿ
k´1
“ limdk S2iS1S1˚pS2˚qi Uϕpd˚k´1qS2
k ˜i“0 ¸
ÿ
k´1
“ limdk S2iS1S1˚pS2˚qi S1d˚k´1
k ˜i“0 ¸
ÿ
“ limdkS1d˚k´1 “ limdϕpdk´1qS1d˚k´1
k k
“ limdS1dk´1d˚k´1 “ dS1 “ S˜1
k
For1 ñ 2, ifwe think of λd as a representation of O2 on ℓ2pZqProposition 4.1 in[15]says that
thisextendstoarepresentationofQ2onthesameHilbertspace. Ingeneralsuchanextensionfrom
O2toQ2neednotbeunique,butunderourhypothesisitis(cf. Remark4.2intheaforesaidpaper).
ˇ
Thisallows ustoconclude that dand δ must bethe samething. Therefore, the strong limitof x
k
isstillinD2. ˝
9
In the following we keep the same hypotheses as in the theorem proved above. In particular,
we shall continue to consistently denote the product dϕpdqϕ2pdq...ϕk´1pdq by d . We should
k
also mention that dk P D2 actually occurs quite often in the literature to do with Cuntz algebras,
whereitisusuallycalledthekthcocycleofdandisdenotedbyd ,justaswedohere. Inaddition,
k
weremarked intheabove proof thatthe sequence td ustrongly converges toadiagonal operator
k
inℓ pZq,whichishenceforth denotedbyd .
8 8
Remark 4.7. In the course of the proof given above wehave proved that λ exists if and only if
d
the strong limit of the sequence dkUd˚k is still in Q2, which is the case if and only if d8Ud˚8 is
in Q2. The second condition is particularly worth stressing, for it also coĂvers the case of gauge
automorphisms, although theseareruledoutbyourassumptions.
Nowweknowthatthefollowinglimit,too,existsinthestrongoperator topology
limd Ud˚U˚ .
k k
k
beingnothingbutd Ud˚ U˚. Wealsohavetheoperator equality
8 8
k´1
|e0yxe0|`lim S2iS1S1˚pS2˚qi “ 1
k
i“0
ÿ
whichcanbeprovedeitherbydirectcomputations orbymeansoftheidentityˇ1 “ 1.
Furthermore, thefollowingequalities hold
k´1
|e0yxe0|`limdk S2iS1S1˚pS2˚qi Ud˚kU˚
k ˜i“0 ¸
ÿ
k´1
“ |e0yxe0|`lim S2iS1S1˚pS2˚qi limdkUd˚kU˚
˜ k i“0 ¸ˆ k ˙
ÿ
“ limd Ud˚U˚ .
k k
k
Therearenowtwocasestobediscussed according aswhetherd8 belongstoD2 ornot. Ifso,
wehavethatdˇ“ d8Ud˚8U˚ P D2,withd8andUd˚8U˚ P D2,whichmeansthatd “ d8ϕpd8q˚
and thus λ is inner. We observe that if d “ d1ϕpd1q˚ (and thus d1ϕpd1q˚ “ d1Ud1˚U˚), then
d
d1 “ zd for z P T. Infact, given x, y P ℓ pZq, itcan be easily seen that xUx˚U˚ “ yUy˚U˚
8 8
implies that x “ zy for z P T. This shows that whenever λd is inner­, then d8 P D2. Unluckily,
it is not clear how the second could be dealt with. We observe that by the former discussion if
theautomorphisms λ exists, then itisnotinner. However, itisweakly inner (withrespect tothe
d
canonical representation) under the usual assumption that dp0q “ dp´1q “ 1. More precisely, a
simplecomputationĂyieldstheequality λ “ Adpd q. Indeed,
d 8
d S d˚ “ limd S limd˚
8 i 8 k i k
k k
ˆ ˙ ˆ ˙
“ limd S d˚
k i k
k
ˆ ˙
“ limdSidk´1d˚k
k
ˆ ˙
“ limdS ϕk´1pd˚q
i
k
ˆ ˙
“ dS “ λ pS q
i d i
10