Dynamical systems and commutants in crossed products

Svensson, P.C.; Silvestrov, S.; Jeu, M.F.E. de

Citation

Svensson, P. C., Silvestrov, S., & Jeu, M. F. E. de. (2007). Dynamical systems and commutants

in crossed products. International Journal Of Mathematics, 18, 455-471.

doi:10.1142/S0129167X07004217

Version: Not Applicable (or Unknown)

License: Leiden University Non-exclusive license

Downloaded from: https://hdl.handle.net/1887/62780

Note: To cite this publication please use the final published version (if applicable).

arXiv:math/0604581v5 [math.DS] 22 Oct 2007

CHRISTIANSVENSSON,SERGEISILVESTROV,ANDMARCELDEJEU

Abstra t. Inthispaperwedes ribethe ommutantofanarbitrarysubalge-

bra

A

ofthealgebraoffun tionsonaset

X

ina rossedprodu tof

A

withthe integers,wherethelattera ton

A

bya ompositionautomorphismdenedvia abije tionof

X

. Theresulting onditionswhi harene essaryandsu ient for

A

tobemaximalabelianinthe rossedprodu taresubsequentlyapplied tosituationswherethese onditions anbeshowntobeequivalenttoa ondi-

tionintopologi aldynami s. Asafurtherstep,usingtheGelfandtransform

weobtainfora ommutative ompletelyregularsemi-simpleBana halgebraa

topologi aldynami al onditiononits hara ter spa ewhi hisequivalentto

thealgebrabeingmaximalabelianina rossedprodu twiththeintegers.

1. Introdu tion

The des riptionof ommutativesubalgebras in non- ommutative algebrasand

their properties is an important dire tion of investigation for any lass of non-

ommutativealgebrasandrings,be auseitallowsonetorelaterepresentationthe-

ory,non- ommutativeproperties,gradedstru tures,idealsandsubalgebras,homo-

logi alandotherpropertiesofnon- ommutativealgebrastospe traltheory,duality,

algebrai geometryandtopologynaturallyasso iatedwiththe ommutativesubal-

gebras. Inrepresentationtheory,forexample,oneofthekeysfor the onstru tion

and lassi ationofrepresentationsisthemethodofindu edrepresentations. The

underlying stru tures behind thismethod arethe semi-dire tprodu tsor rossed

produ tsofringsandalgebrasbyvarious a tions. Whenanon- ommutativealge-

braisgiven,onelooksforasubalgebrasu hthatitsrepresentations anbestudied

and lassiedmoreeasily, andsu hthat thewhole algebra anbede omposed as

a rossed produ t of this subalgebra by a suitable a tion. Then the representa-

tionsforthesubalgebraareextendedtorepresentationsofthewholealgebrausing

the a tion and its properties. A des ription of representations is most tra table

for ommutativesubalgebrasasbeing,viathespe traltheoryandduality,dire tly

onne tedtoalgebrai geometry,topologyormeasuretheory.

Ifonehasfoundawaytopresentanon- ommutativealgebraasa rossedprod-

u tofa ommutativesubalgebrabysomea tiononitoftheelementsfromoutside

the subalgebra,then it is important to know whether this subalgebrais maximal

abelianor,ifnot,tondamaximalabeliansubalgebra ontainingthegivensubal-

gebra,sin eifthesele tedsubalgebraisnotmaximalabelian,thenthea tionwill

notbeentirelyresponsible forthe non- ommutativepartasone wouldhope, but

will also havethe ommutative trivialpart taking are of the elements ommut-

ingwith everythingin the hosen ommutativesubalgebra. This maximalityof a

2000Mathemati sSubje tClassi ation. Primary47L65Se ondary16S35,37B05,54H20.

Keywordsandphrases. Crossedprodu t;dynami alsystem; ompletelyregularBana halge-

bra;maximalabeliansubalgebra.

ommutativesubalgebraandrelatedpropertiesofthea tionareintimatelyrelated

to the des ription and lassi ations of representations of the non- ommutative

algebra.

Littleisknowningeneralabout onne tionsbetweenpropertiesofthe ommu-

tativesubalgebrasof rossedprodu talgebrasandpropertiesofdynami alsystems

thatarein manysituations naturallyasso iatedwiththe onstru tion. Aremark-

ableresult in this dire tion is known, however,in the ontextof rossed produ t

C ^∗

^{-algebras. When the algebra is des ribed as the rossed produ t}

C ^∗

^-algebra

C(X) ⋊ α Z

of the algebraof ontinuous fun tions on a ompa t Hausdor spa e

X

^{by an a tionof}

Z

viathe omposition automorphism asso iated with ahome- omorphism

σ

^of

X

^{, it is known that}

C(X)

^{sits inside the}

C ^∗

^{- rossed produ t as}

amaximalabeliansubalgebraifand onlyifforeverypositiveinteger

n

^{theset of}

points in

X

^{having period}

n

^{under iterations of}

σ

^{has no interior points [4℄, [7℄,}

[8℄,[9℄,[10℄. Bythe ategorytheorem,this onditionisequivalenttothea tionof

Z

on

X

^beingtopologi allyfreein thesense thatthenon-periodi pointsof

σ

^are

densein

X

^{. This resultontheinterplaybetweenthe}topologi aldynami s ofthe a tionontheonehand,andthealgebrai propertyofthe ommutativesubalgebra

inthe rossedprodu tbeingmaximalabelianontheotherhand,providedthemain

motivationandstartingpointforourwork.

Inthis arti le,webringsu hinterplayresultsintoamoregeneralalgebrai and

set theoreti al ontext ofa rossedprodu t

A ⋊ α Z

of an arbitrarysubalgebra

A

of thealgebra

C ^X

of fun tions onaset

X

^{(under theusual pointwise}operations) by

Z

, where the latter a ts on

A

^{by a} omposition automorphism dened via a bije tion of

X

^{. In this general algebrai set} theoreti al framework the essen e of the matter is revealed. Topologi al notions are not available here and thus

the ondition of freeness of the dynami s as des ribed above is not appli able,

sothat it hasto be generalizedin aproper way in order to be equivalent to the

maximal ommutativityof

A

^{. Weprovidesu ha}generalization. Infa t,wedes ribe expli itly the (unique) maximal abelian subalgebra ontaining

A

^{(Theorem 3.3),}

and then the general result, stating equivalen e of maximal ommutativity of

A

inthe rossedprodu tandthedesiredgeneralizationoftopologi alfreenessofthe

a tion, followsimmediately (Theorem3.5). It involvesseparationpropertiesof

A

withrespe ttothespa e

X

^{andthea tion.}

Thegeneralsettheoreti alframeworkallowsone toinvestigatetherelationbe-

tween the maximality of the ommutative subalgebra in the rossed produ t on

the one hand, and the properties of the a tion on the spa e on the other hand,

forarbitrary hoi esof theset

X

^{,thesubalgebra}

A

^{andthea tion,dierentfrom}

the previously ited lassi al hoi e of ontinuous fun tions

C(X)

^{on a ompa t}

Hausdor topologi al spa e

X

^{. As arather general} appli ation of our resultswe obtainthat,foraBairetopologi alspa e

X

^{andasubalgebra}

A

^of

C(X)

^satisfying

amildseparation ondition,thepropertyof

A

^{beingmaximalabelianinthe rossed}

produ tisequivalenttothea tionbeingtopologi allyfreeinthesenseabove,i.e.,

to thesetofnon-periodi pointsbeingdensein

X

^{(Theorem 3.7). Thisapplies in}

parti ularwhen

X

^{isa ompa tHausdorspa e,sothataresultanalogoustothat}

forthe rossedprodu t

C ^∗

^-algebra

C(X) ⋊ α Z

isobtained. We alsodemonstrate that, for ageneral topologi al spa e

X

^{and a subalgebra}

A

^of

C(X)

^{satisfying a}

less ommonseparation ondition,thesubalgebra

A

^{ismaximalabelianifandonly}

if

σ

^{isnotofniteorder,whi hisamu hless}restri tive onditionthantopologi al

freeness(Theorem3.11). Examplesofthis situationareprovidedby rossedprod-

u ts of subalgebras of holomorphi fun tions on a onne ted omplex manifolds

bybiholomorphi a tions(Corollary3.12). It isinterestingtonote thatthese two

results,Theorem3.7and Theorem3.11,haveno non-trivialsituations in ommon

(Remark3.13).

Inthemotivating Se tion 4.1,weillustrate byseveralexamplesthat, generally

speaking, topologi al freeness and the property of the subalgebra

A

^{being maxi-}

mal abelianin the rossed produ tare unrelated. InSe tion 4.2, we onsider the

rossedprodu t ofa ommutative ompletelyregular semi-simpleBana h algebra

A

^{by the a tion of an} automorphism. Sin e the automorphism of

A

^{indu es a}

homeomorphismof

∆(A)

^{,thesetofall hara terson}

A

^{endowedwiththeGelfand}

topology,su h a rossedprodu tisviatheGelfand transform anoni allyisomor-

phi (bysemi-simpli ity)tothe rossedprodu tofthesubalgebra

A b

^ofthealgebra

C 0 (∆(A))

^{by the indu ed} homeomorphism, where

C 0 (∆(A))

^{denotes the algebra}

of ontinuous fun tions on

∆(A)

^{that vanish at innity. Sin e}

A

^{is also assumed}

tobe ompletelyregular,thegeneralresultinTheorem3.7 anthenbeappliedto

this isomorphi rossed produ t. We thus provethat, when

A

^{is a} ommutative ompletelyregularsemi-simpleBana halgebra,itismaximalabelianinthe rossed

produ tifand onlyiftheasso iateddynami alsystemontheGelfanddual

∆(A)

istopologi allyfreeinthesensethatitsnon-periodi pointsaredenseinthetopo-

logi alspa e

∆(A)

^{(Theorem4.8). Itis thenpossibletounderstand thealgebrai}

propertiesin theexamplesinSe tion4.1asdynami alpropertiesafterall,bynow

looking atthe orre t dynami alsystemon

∆(A)

^{insteadof theoriginallygiven}

dynami al system. Inthese examples, this orresponds to removing parts of the

spa eorenlargingit.

InSe tion 4.3,weapplyTheorem4.8in the aseofanautomorphismof

L 1 (G)

^,

the ommutative ompletelyregularsemi-simpleBana halgebra onsistingofequiv-

alen e lasses of omplexvaluedBorel measurablefun tions onalo ally ompa t

abelian group

G

^{that are integrablewith respe t to aHaar measure, with multi-}

pli ation givenby onvolution. We use a resultsayingthat everyautomorphism

on

L 1 (G)

^{is indu ed by a pie ewise ane} homeomorphism on the dual group of

G

^{(Theorem 4.15) to prove that, for}

G

^{with onne ted dual,}

L 1 (G)

^{is maximal}

abelian in the rossed produ tif and onlyif thegiven automorphismof

L 1 (G)

^is

notofnite order(Theorem4.16). Wealsoprovideanexampleshowingthat this

equivalen emayfail whenthedualof

G

^{isnot onne ted.}

Finally,inSe tion4.4,weprovideanotherappli ationofthe ommutantdes rip-

tionin Theorem3.3andoftheisomorphi rossedprodu tonthe hara terspa e,

showingthat,forasemi-simple ommutativeBana halgebra

A

^{,the ommutantof}

A

^{inthe rossedprodu tisnitely generatedasanalgebraover}

C

ifandonlyif

A

hasnitedimensionas ave torspa eover

C

.

2. Crossed produ tsasso iated withautomorphisms

2.1. Denition. Let

A

^beanasso iative ommutative

C

-algebraandlet

Ψ : A →

A

^beanautomorphism. Considertheset

A ⋊ Ψ Z = {f : Z → A | f (n) = 0

^{ex eptforanitenumberof}

n}.

Weendowitwiththestru tureofanasso iative

C

-algebrabydenings alarmul-

bytwisted onvolution,

∗

^,asfollows;

(f ∗ g)(n) = X

k∈Z

f (k) · Ψ ^k (g(n − k)),

where

Ψ ^k

^denotesthe

k

^-fold ompositionof

Ψ

^{withitself. Itistriviallyveriedthat}

A ⋊ σ Z

is anasso iative

C

-algebraunder these operations. We allit the rossed produ tof

A

^and

Z

under

Ψ

^.

Ausefulwayofworking with

A ⋊ Ψ Z

is towriteelements

f, g ∈ A ⋊ Ψ Z

inthe form

f = P

n∈Z f n δ ⁿ , g = P

m∈Z g m δ ⁿ

^,where

f n = f (n), g m = g(m)

^,additionand

s alarmultipli ation are anoni allydened, and multipli ationis determined by

(f n δ ⁿ ) ∗ (g m δ ^m ) = f n · Ψ ⁿ (g m )δ ^n+m

^,where

n, m ∈ Z

^and

f n , g m ∈ A

^{arearbitrary.}

Using this notation, we may think of the rossed produ t as a omplex Laurent

polynomial algebra in one variable (having

δ

^{as its} indeterminate) over

A

^with

twistedmultipli ation.

2.2. A maximal abelian subalgebra of

A ⋊ Ψ Z

^{. Clearly onemay} anoni ally view

A

^{as anabeliansubalgebra of}

A ⋊ Ψ Z

, namelyas

{f 0 δ ⁰ | f 0 ∈ A}

^{. Here we}

provethatthe ommutantof

A

ⁱⁿ

A ⋊ Ψ Z

,denotedby

A ^′

^,is ommutative,andthus thereexistsauniquemaximalabeliansubalgebraof

A ⋊ Ψ Z

ontaining

A

^,namely

A ^′

^.

Notethatwe an writetheprodu toftwoelementsin

A ⋊ Ψ Z

asfollows;

f ∗ g = ( X

n∈Z

f n δ ⁿ ) ∗ ( X

m∈Z

g m δ ^m ) = X

r,n∈Z

f n · Ψ ⁿ (g r−n )δ ^r .

Weseethatthepre ise onditionfor ommutationoftwosu helements

f

^and

g

^is

(1)

∀r : X

n∈Z

f n · Ψ ⁿ (g r−n ) = X

m∈Z

g m · Ψ ^m (f r−m ).

Proposition2.1. The ommutant

A ^′

^{isabelian,andthusitisthe uniquemaximal}

abeliansubalgebra ontaining

A

^.

Proof. Let

f = P

n∈Z f n δ ⁿ , g = P

m∈Z g m δ ^m ∈ A ⋊ Ψ Z

. We shall verify that if

f, g ∈ A ^′

^{, then}

f

^and

g

^{ommute. Using (1) wesee that membership of}

f

^and

g

respe tivelyin

A ^′

^{isequivalenttotheequalities}

∀h ∈ A, ∀n ∈ Z : f n · Ψ ⁿ (h) = f n · h,

(2)

∀h ∈ A, ∀m ∈ Z : g m · Ψ ^m (h) = g m · h.

(3)

Insertionof(2)and(3)into(1)werealizethatthepre ise onditionfor ommuta-

tionofsu h

f

^and

g

^{anberewrittenas}

∀r : X

n∈Z

f n · g r−n = X

m∈Z

g m · f r−m .

This learly holds, and thus

f

^and

g

^{ommute. Fromthis it follows} immediately that

A ^′

^{istheuniquemaximalabeliansubalgebra ontaining}

A

^.



3. Automorphismsindu ed bybije tions

Fix a non-empty set

X

^{, a bije tion}

σ : X → X

^{, and an algebra of fun tions}

A ⊆ C ^X

^{that isinvariantunder}

σ

^and

σ ⁻¹

^{,i.e.,su h thatif}

h ∈ A

^,then

h ◦ σ ∈ A

and

h ◦ σ ⁻¹ ∈ A

^{. Then}

(X, σ)

^{is adis retedynami alsystem(thea tionof}

n ∈ Z

on

x ∈ X

^{is givenby}

n : x 7→ σ ⁿ (x)

^{) and}

σ

^{indu es an}automorphism

eσ : A → A

denedby

eσ(f) = f ◦ σ ⁻¹

^{bywhi h}

Z

a tson

A

^viaiterations.

Inthis se tionwewill onsiderthe rossedprodu t

A ⋊ σ _e Z

fortheabovesetup, andexpli itlydes ribethe ommutant,

A ^′

^{, of}

A

^{andthen enter,}

Z(A ⋊ _e σ Z )

^{. Fur-}

thermore,wewillinvestigateequivalen esbetweenpropertiesofnon-periodi points

ofthesystem

(X, σ)

^{, andpropertiesof}

A ^′

^{. Firstwemakeafew}denitions.

Denition3.1. Foranynonzero

n ∈ Z

^weset

Sep ⁿ _A (X) = {x ∈ X|∃h ∈ A : h(x) 6= eσ ⁿ (h)(x)},

Per ⁿ _A (X) = {x ∈ X|∀h ∈ A : h(x) = eσ ⁿ (h)(x)},

Sep ⁿ (X) = {x ∈ X|x 6= σ ⁿ (x))},

Per ⁿ (X) = {x ∈ X|x = σ ⁿ (x)}.

Furthermore,let

Per ^∞ _A (X) = \

n∈Z\{0}

Sep ⁿ _A (X),

Per ^∞ (X) = \

n∈Z\{0}

Sep ⁿ (X).

Finally,for

f ∈ A

^,put

supp(f ) = {x ∈ X | f (x) 6= 0}.

Itiseasyto he kthatallthesesets,ex eptfor

supp(f )

^,are

Z

-invariantandthat if

A

^{separatesthepointsof}

X

^,then

Sep _A ⁿ (X) = Sep ⁿ (X)

^and

Per ⁿ _A (X) = Per ⁿ (X)

^.

Notealsothat

X \Per ⁿ _A (X) = Sep ⁿ _A (X)

^,and

X \Per ⁿ (X) = Sep ⁿ (X)

^. Furthermore

Sep ⁿ _A (X) = Sep ⁻ⁿ _A (X)

^{with similar equalities for}

n

^and

−n

⁽

n ∈ Z

^{) holding for}

Per ⁿ _A (X)

^,

Sep ⁿ (X)

^and

Per ⁿ (X)

^aswell.

Denition3.2. Wesaythatanon-emptysubsetof

X

^{isadomain of uniqueness}

for

A

^{ifeveryfun tion in}

A

^{that vanishesonit,vanishesonthewhole of}

X

^.

For example,using results from elementary topologyoneeasily showsthat for

a ompletely regulartopologi alspa e

X

^,asubsetof

X

^{isadomainofuniqueness}

for

C(X)

^{ifandonlyifitisdensein}

X

^.

Theorem3.3. The uniquemaximalabeliansubalgebraof

A ⋊ _e σ Z

that ontains

A

ispre isely the setof elements

A ^′ = { X

n∈Z

f n δ ⁿ |

^forall

n ∈ Z : f _n↾Sep ⁿ

A (X) ≡ 0}.

Proof. QuotingapartoftheproofofProposition2.1,wehavethat

P

n∈Z f n δ ⁿ ∈ A ^′

ifandonlyif

∀h ∈ A, ∀n ∈ Z : f _n · e σ ⁿ (h) = f n · h.

Clearlythisisequivalentto

∀h ∈ A, ∀n ∈ Z : f _n↾Sep ⁿ

A (X) ≡ 0.

TheresultnowfollowsfromProposition2.1.



Note that for any non-zerointeger

n

^{, theset}

{f _n ∈ A : f _n↾Sep ⁿ

A (X) ≡ 0}

^{is a}

Z

-invariantidealin

A

^{. Notealsothatif}

σ

^{hasniteorder,thenbyTheorem3.3,}

A

isnotmaximalabelian. Thefollowing orollaryfollowsdire tlyfromTheorem3.3.

Corollary 3.4. If

A

^{separates the points of}

X

^{, then}

A ^′

^{is pre isely the set of}

elements

A ^′ = { X

n∈Z

f n δ ⁿ |

^forall

n ∈ Z : supp(f n ) ⊆ Per ⁿ (X)}.

Proof. ImmediatefromTheorem3.3 andtheremarksfollowingDenition3.1.



ApplyingDenition3.2yieldsthefollowingdire t onsequen eofTheorem3.3.

Theorem3.5. The subalgebra

A

^{is maximalabelian in}

A ⋊ σ _e Z

if andonlyif, for every

n ∈ Z \ {0}

^,

Sep ⁿ _A (X)

^{isadomainof uniquenessfor}

A

^.

Inwhat followsweshall mainly fo us on aseswhere

X

^isatopologi al spa e.

Beforeweturntosu h ontexts,however,weuseTheorem3.3togiveades ription

ofthe enterofthe rossedprodu t,

Z(A ⋊ σ _e Z )

^{,forthegeneralsetupdes ribedin}

thebeginningofthisse tion.

Theorem 3.6. An element

g = P

m∈Z g m δ ^m

^{is in}

Z(A ⋊ _e σ Z )

^{if and only if both}

of thefollowing onditionsare satised:

(i) for all

m ∈ Z

^,

g m

^is

Z

-invariant,and (ii) for all

m ∈ Z

^,

g _m↾Sep ^m

A (X) ≡ 0

^.

Proof. If

g

^isin

Z(A ⋊ _e σ Z )

^{then ertainly}

g ∈ A ^′

^{, andhen e ondition(ii)follows}

from Theorem 3.3. For ondition(i), note that

g

^{is in the enter if andonly if}

g

ommuteswitheveryelementontheform

f n δ ⁿ

^. Multiplyingout,orlookingat(1), weseethat thismeansthat

g = X

m∈Z

g m δ ^m

^isin

Z(A ⋊ _e σ Z ) ⇐⇒

∀n ∈ Z, ∀m ∈ Z, ∀f ∈ A : f · eσ ⁿ (g m ) = g m · eσ ^m (f ).

Wex

m ∈ Z

^andan

x ∈ Per ^m _A (X)

^{. Thenforall}

f ∈ A : f (x) = eσ ^m (f )(x)

^{. Ifthere}

is a fun tion

f ∈ A

^{that does notvanish in}

x

^{, then for}

g

^{to bein the enter we}

learlymusthavethatforall

n ∈ Z : g m (x) = eσ ⁿ (g m )(x)

^{. Ifall}

f ∈ A

^vanishin

x

^,

theninparti ularboth

g m

^and

eσ ⁿ (g m )

^{do. Thusforallpoints}

x ∈ Per ^m _A (X)

^wehave

that

g m

^{is onstantalongtheorbitof}

x

^(i.e.,forall

n ∈ Z : g m (x) = eσ ⁿ (g m )(x)

^)for

all

m ∈ Z

^{, sin e}

m

^{wasarbitrary in ourabove}dis ussion. It remains to onsider

x ∈ Sep ^m _A (X)

^{. For su h}

x

^{, we have on luded that}

g m (x) = 0

^{. If there exists}

f ∈ A

^{that doesnotvanishin}

x

^{, weseethat inorder fortheequalityaboveto be}

satisedwemust have

eσ ⁿ (g m )(x) = 0

^forall

n

^{, andifall}

f ∈ A

^vanishin

x

^{, then}

inparti ular

eσ ⁿ (g m )

^doesforall

n

^{andtheresultfollows.}



Wenow fo ussolely ontopologi al ontexts. Thefollowingtheorem makesuse

ofCorollary3.4.

Theorem3.7. Let

X

^{be aBaire spa e, andlet}

σ : X → X

^beahomeomorphism indu ing,asusual,anautomorphism

eσ

^of

C(X)

^{. Suppose}

A

^{isasubalgebraof}

C(X)

thatisinvariant under

eσ

^{anditsinverse, separatesthepointsof}

X

^{andissu hthat}

for every non-empty open set

U ⊆ X

^{, thereis anon-zero}

f ∈ A

^{that vanishes on}

the omplement of

U

^{. Then}

A

^{is a maximal abelian subalgebraof}

A ⋊ _e σ Z

if and onlyif

Per ^∞ (X)

^{isdense in}

X

^.

Proof. Assume rst that

Per ^∞ (X)

^{is dense in}

X

^{. This means in parti ular that}

any ontinuous fun tion that vanishes on

Per ^∞ (X)

^{vanishes on the whole of}

X

^.

ThusCorollary3.4tellsusthat

A

^{isamaximalabeliansubalgebraof}

A ⋊ _e σ Z

. Now assumethat

Per ^∞ (X)

^{isnot densein}

X

^{. This meansthat}

T

n∈Z >0 (X \ Per ⁿ (X))

isnotdense. Notethat thesets

X \ Per ⁿ (X)

^,

n ∈ Z >0

^{areallopen. Sin e}

X

^{is a}

Bairespa e thereexists an

n 0 ∈ Z >0

^{su h that}

Per ^{n 0} (X)

^{hasnon-emptyinterior,}

say

U ⊆ Per ^{n 0} (X)

^{. Bythe assumptionon}

A

^{, there isanonzerofun tion}

f n 0 ∈ A

thatvanishesoutside

U

^{. Hen eCorollary3.4showsthat}

A

^{isnot maximalabelian}

inthe rossedprodu t.



Example 3.8. Let

X

^{bea lo ally ompa tHausdor spa e, and}

σ : X → X

^a

homeomorphism. Then

X

^{is aBairespa e, and}

C c (X), C 0 (X), C b (X)

^and

C(X)

allsatisfytherequired onditionsfor

A

^{inTheorem3.7. Fordetails,seeforexample}

[5℄. Hen ethese fun tion algebrasaremaximalabelian intheirrespe tive rossed

produ tswith

Z

under

σ

^ifandonlyif

Per ^∞ (X)

^isdensein

X

^.

Example 3.9. Let

X = T

^{bethe unit ir le in the omplexplane, and let}

σ

^be

ounter lo kwiserotationbyanangle whi h isanirrationalmultiple of

2π

^{. Then}

everypointisnon-periodi andthus, byTheorem 3.7,

C(T)

^{ismaximalabelianin}

theasso iated rossedprodu t.

Example3.10. Let

X = T

^and

σ

ounter lo kwiserotationbyananglewhi hisa rationalmultipleof

2π

^,say

2πp/q

^(where

p, q

^{arerelativelyprimepositiveintegers).}

Theneverypointonthe ir le hasperiodpre isely

q

^,andthenon-periodi points are ertainlynotdense. UsingCorollary3.4weseethat

C(T) ^′ = { X

n∈I

f nq δ ^nq | f nq ∈ C(T)}.

Wewillusethefollowingtheoremtodisplayanexampledierentinnaturefrom

theonesalready onsidered.

Theorem3.11. Let

X

^beatopologi al spa e,

σ : X → X

^ahomeomorphism, and

A

^{anon-zerosubalgebraof}

C(X)

^{,invariantboth undertheusualindu edautomor-}

phism

eσ : C(X) → C(X)

^{andunderitsinverse. Assumethat}

A

^{separatesthepoints}

of

X

^{andissu hthat everynon-emptyopenset}

U ⊆ X

^{isadomainof uniqueness}

for

A

^{. Then}

A

^{ismaximal abelian in}

A ⋊ σ _e Z

if andonlyif

σ

^{isnot ofnite order}

(that is,

σ ⁿ 6= id _X

^{for anynon-zerointeger}

n

^).

Proof. ByCorollary 3.4,

A

^{being maximal abelian implies that}

σ

^{is not of nite}

order. Indeed, if

σ ^p = id X

^{, where}

p

^{is the smallest su h positive integer, then}

X = Per ^p (X)

^{and hen e}

f δ ^p ∈ A ^′

^{for any}

f ∈ A

^{. Forthe onverse, assume that}

σ

^{doesnothavenite order. Thesets}

Sep ⁿ (X)

^{arenon-emptyopensubsetsof}

X

forall

n 6= 0

^{andthusdomainsofuniqueness for}

A

^{byassumptionofthetheorem.}

Theimpli ationnowfollowsdire tlyfrom Corollary3.4.



Corollary3.12. Let

M

^{bea onne ted omplexmanifoldandsupposethefun tion}

σ : M → M

^is biholomorphi . If

A ⊆ H(M )

^{is a subalgebra of the algebra of}

holomorphi fun tionsthat separatesthe pointsof

M

^{andwhi h isinvariant under}

theindu edautomorphism

eσ

^of

C(M )

^{anditsinverse,then}

A ⊆ A ⋊ _e σ Z

ismaximal abelianif andonlyif

σ

^{isnot ofnite order.}

Proof. On onne ted omplexmanifolds, opensets are domainsof uniqueness for

H(M )

^{. Seeforexample[2℄.}



Remark3.13. Itisimportanttopointoutthattherequired onditionsin Theo-

rem3.7and Theorem3.11 anonlybesimultaneouslysatisedin ase

X

^onsists

of asinglepointand

A = C

^{. Tosee thisweassumethat both onditionsare sat-}

ised. Noterstofallthat thisimpliesthateverynon-emptyopensubsetof

X

^is

dense. Assumetothe ontrarythatthereisanon-emptyopensubset

U ⊆ X

^that

isnotdensein

X

^{. Wemaythen hooseanon-zero}

f ∈ A

^{that vanisheson}

X \ U

^.

Certainly,

f

^{must then vanish on}

V = X \ U

^{. As}

U

^{is not dense, however, this}

implies that

f

^is identi ally zerosin e the non-emptyopen set

V

^{is a domain of}

uniquenessbyassumption. Hen ewehavea ontradi tionand on ludethatevery

non-emptyopensubsetof

X

^{isdense. Se ondly,wenotethatsin e}

C

isHausdor

and

A ⊆ C(X)

^{separatesthepointsof}

X

^,

X

^{mustbeHausdor. Assumenowthat}

therearetwodistin tpoints

p, q ∈ X

^{. As}

X

^{isHausdorwe anseparatethemby}

twodisjointopen sets. Sin eeverynon-emptyopensubsetis dense,however,this

isnotpossibleandhen e

X

^{onsistsofonepoint. If}

X = {p}

^,theonlypossibilities for

A

^are

{0}

^and

C

. Asthe onditionsinTheorem3.7implyexisten eofnon-zero fun tions in

A

^{we on ludethat}

A = C

^. Conversely, if

X = {p}

^and

A = C

^{, the}

onditionsinbothTheorem3.7andTheorem 3.11aresatised.

4. Automorphismsof ommutative ompletelyregular semi-simple

Bana halgebras

4.1. Motivation. In the setup in Theorem 3.7 we on luded that we have an

appealingequivalen ebetweendensityofthenon-periodi pointsin

X

^and

A

^being

maximal abelian in the asso iated rossed produ t, under a ertain ondition on

A

^{. Example 3.9 and Example 3.10,} respe tively, show an instan e where both theseequivalentstatementsaretrue,andwheretheyarefalse. Generallyspeaking,

however,thedensity ofthenon-periodi pointsand

A

^{beingmaximal abelianare}

unrelatedproperties;allfourlogi alpossibilities ano ur. Weshowthisbygiving

twoadditionalexamples.

Example 4.1. As in Example 3.9, let

X = T

^{be the unit ir le and}

σ

^ounter-

lo kwiserotationbyananglethatisanirrationalmultipleof

2π

^{. Ifweuse}

A = C

insteadof

C(T)

^{,thewhole rossedprodu tis} ommutativeandthus

A

^{is learlynot}

maximalabelianin it. Thenon-periodi points,however,are of oursestilldense.

Here wesimply hose asubalgebra of

C(T)

^{sosmall that the} homeomorphism

σ

wasnolongervisiblein the rossedprodu t.

Example4.2. Let

X = T ∪ {0}

^{withtheusualsubspa etopologyfrom}

C

,andlet

σ ^′

^{besu hthatitxestheoriginandrotatespointsonthe ir le} ounter lo kwise withanangle thatis anirrationalmultipleof

2π

^{. Asfun tion algebra}

A

^,wetake

C(T)

^{andextendeveryfun tionin itto}

X

^{sothat itvanishesin theorigin. Thisis}

obviouslyanalgebraoffun tions being ontinuouson

X

^{,whi h isinvariantunder}

σ ^′

^{anditsinverse. Sin e}

A

^{separates pointsin}

X

^{, Corollary3.4assures usthat}

A

ismaximalabelianinthe rossedprodu t,eventhoughthenon-periodi pointsare

notdensein

X

^.

Inthefollowingexample,theequivalen einTheorem3.7failsinthesamefashion

asinExample 4.2. Itis in luded,however,sin eitwill beilluminatingto referto

itinwhat follows.

Example 4.3. As in Example 4.2, let

X = T ∪ {0}

^and

σ ^′

^{the map dened as}

ounter lo kwise rotation by an angle that is an irrational multiple of

2π

^on

T

and

σ ^′ (0) = 0

^{. Let}

A

^{be the} restri tion to

X

^{of all ontinuous fun tions on the}

losed unit dis that are holomorphi on the open unit dis . By the maximum

modulus theorem,noneof thesefun tions arenon-zerosolely in theorigin. Thus,

by Corollary3.4, we again obtain a ase where the non-periodi points are not

dense,but

A

^{is amaximalabeliansubalgebrainthe rossedprodu t.}

In summary, wehave nowdisplayed three examples where we do not havean

equivalen e betweenalgebraand topologi al dynami sas in Theorem 3.7. Inthe

followingsubse tionweproveageneralresult-inthe ontextofautomorphismsof

Bana halgebras-thatinparti ularshowsthatfora ertain lassofpairsofdis rete

dynami alsystemsand

Z

-invariantfun tionalgebrasonit,

((X, σ), A)

^,yieldingthe

asso iated rossedprodu t

A ⋊ _e σ Z

asusual,one analwaysndanothersu hpair

((Y, φ), B)

^{withasso iated rossedprodu t}

B⋊ _{φ e} Z

anoni allyisomorphi to

A⋊ σ _e Z

, wheretheequivalen edoes hold: thenon-periodi pointsof

Y

^{aredenseifandonly}

if

B

^{is maximal abelian in}

B ⋊ _{φ e} Z

(whi h it is, by the anoni al isomorphism,if and only if

A

^{is maximal abelian in}

A ⋊ σ _e Z

). In this way, theequivalen e of an algebrai propertywithatopologi aldynami alpropertyisrestored. Examples4.1

through4.3allfallintothismentioned lassofpairs,aswewillseeinExample4.9

-4.11.

4.2. A system on the hara ter spa e. We will now fo us solely on Bana h

algebras, and start by re alling a number of basi results on erning them. We

refer to [3℄ for details. All Bana h algebras under onsideration will be omplex

and ommutative.

Denition4.4. Let

A

^{bea omplex} ommutativeBana halgebra. Thesetofall non-zeromultipli ativelinearfun tionals on

A

^{is denotedby}

∆(A)

^{and alledthe}

hara terspa e of

A

^.

Denition 4.5. Givenany

a ∈ A

^{, wedene afun tion}

ba : ∆(A) → C

^by

ba(µ) =

µ(a) (µ ∈ ∆(A))

^{. The fun tion}

ba

^{is alled theGelfand transform of}

a

^{. Let}

A = b

{ba | a ∈ A}

^{. The hara terspa e}

∆(A)

^{isendowedwiththetopologygeneratedby}

A b

^{, whi h is alled theGelfand topology on}

∆(A)

^{. TheGelfand topologyis lo ally}

ompa tand Hausdor. A ommutativeBana h algebra

A

^{for whi h theGelfand}

transform,i.e.,themapsending

a

^to

ba

^{,isinje tive,is alled}semi-simple.

Let

A

^bea ommutativesemi-simple omplexBana halgebraand

eσ : A → A

^an

algebraautomorphism. Then

eσ

^{indu es abije tion}

σ : ∆(A) → ∆(A)

^{dened by}

σ(µ) = µ ◦ eσ ⁻¹ , (µ ∈ ∆(A))

^{,whi his}automati allyahomeomorphismwhen

∆(A)

hastheGelfandtopology. Notethatbysemi-simpli ityof

A

^,themap

φ : Aut(A) → {σ ∈ Homeo(∆(A)) | ba ◦ σ, ba ◦ σ ⁻¹ ∈ b A

^forall

a ∈ A}

denedby

φ( eσ)(µ) = µ ◦ eσ ⁻¹

is an isomorphismof groups. In turn,

σ

^{indu es an} automorphism

bσ

^on

A b

^{as in}

Se tion3,namely

bσ(ba) = ba ◦ σ ⁻¹ = d eσ(a)

^.

Thefollowingresultshowsthat inthe ontextof asemi-simpleBana h algebra

onemaypassto an isomorphi rossedprodu t, but nowwith analgebraof on-

tinuousfun tions onatopologi alspa e. It isherethat topologi aldynami s an

Theorem 4.6. Let

A

^{be a} ommutative semi-simple Bana h algebra and

eσ

^an

automorphism, indu ing an automorphism

σ : b b A → b A

^{as above. Then the map}

Φ : A ⋊ σ _e Z → b A ⋊ σ _b Z

dened by

P

n∈Z a n δ ⁿ 7→ P

n∈Z c a n δ ⁿ

^{is an}isomorphism of algebrasmapping

A

^onto

A b

^.

Denition4.7. A ommutativeBana halgebra

A

^{issaidtobe ompletelyregular}

ifforeverysubset

F ⊆ ∆(A)

^{thatis losedin the}Gelfand-topologyandfor every

φ 0 ∈ ∆(A)\ F

^{thereexistsan}

a ∈ A

^{su hthat}

ba(φ) = 0

^forall

φ ∈ F

^and

ba(φ ⁰ ) 6= 0

^.

InBana halgebratheoryitisprovedthat

A

^{is ompletelyregularifandonlyifthe}

hull-kerneltopologyon

∆(A)

^{oin ideswiththeGelfandtopology,seeforexample}

[1℄.

Theorem 4.8. Let

A

^{be a} ommutative ompletely regular semi-simple Bana h algebra,

eσ : A → A

^{an algebra} automorphism and

σ

^the homeomorphism on

∆(A)

in the Gelfand topology indu ed by

eσ

^{as des ribed above. Then the} non-periodi points of

(∆(A), σ)

^{are dense if and only if}

A b

^{is a maximal abelian subalgebra of}

A⋊ b _b σ Z

. Inparti ular,

A

^{ismaximalabelianin}

A⋊ σ _e Z

ifandonlyifthenon-periodi pointsof

(∆(A), σ)

^aredense.

Proof. As mentioned in Denition 4.5,

∆(A)

^{is lo ally ompa tHausdor in the}

Gelfand topology, and learly

A b

^{is by denition aseparating fun tion algebraon}

it. Sin e we assumed

A

^{to be ompletely regular, we see that all the onditions}

assumedinTheorem3.7aresatised,andthusthistheoremyieldstheequivalen e.

Furthermore,byTheorem 4.6,

A

^{ismaximal abelianin}

A ⋊ _{e σ} Z

ifandonlyif

A b

^is

maximalabelianin

A ⋊ b σ _b Z

.



WeshallnowrevisitExamples4.1through4.3anduseTheorem4.8to on lude

algebrai propertiesfrom topologi aldynami safterall.

Example 4.9. Consider again Example 4.1. Obviously

∆(C) = {id C }

^{. Thus}

C

is a ommutativesemi-simple ompletely regular Bana h algebra. Trivially,a set

with only oneelementhas no non-periodi point, so that

A = C

^{is not maximal}

abelianbyTheorem4.8.

Example4.10. ConsideragainExample4.2. Clearlythefun tionalgebra

A

^on

X

isisometri allyisomorphi to

C(T)

^{(whenbothalgebrasareendowedwiththesup-}

norm) and thus a ommutative ompletely regular semi-simple Bana h algebra.

It is furthermore a well known result from the theory of Bana h algebras that

∆(C(X)) = {µ x | x ∈ X}

^{forany ompa t Hausdor spa e}

X

^{, where}

µ x

^denotes

thepointevaluationin

x

^,andthat

∆(C(X))

^{equippedwiththeGelfandtopologyis}

homeomorphi to

X

^{(seeforexample[3℄). Thus learlyhere}

∆(A) = ∆(C(T)) = T

^.

Of ourse the indu ed map

σ

^{dened on}

∆(A)

^by

σ(µ x ) = µ σ(x)

^{ansimply be}

identiedwithrotation

σ ^′

^{ofthe ir lebyananglethatisanirrationalmultipleof}

2π

^{. Soon}

∆(A)

^thenon-periodi pointsaredense,andhen e

A

^{ismaximalabelian}

byTheorem 4.8. Herepassingto thesystemonthe hara ter spa e orresponded

to deleting the origin from

X

^{, thus re overing}

T

, and restri ting

A

^to

T

, hen e re overing

C(T)

^{sothatin theendwere overedthesetupin Example3.9.}

Example4.11. ConsideragainExample4.3. Usingthemaximummodulustheo-

remoneseesthat

A

^isisometri allyisomorphi to

A(D)

^{,thealgebraofallfun tions}

that are ontinuousontheunit ir leand holomorphi ontheopenunit dis (de-

noted by

D

), as

A

^{is the} restri tions of su h fun tions to

T ∪ {0}

^{. Hen e}

A

^is

a ommutative ompletely regularsemi-simpleBana halgebra. Furthermoreit is

a standard result from Bana h algebratheory that

∆(A(D))

^{(endowed with the}

Gelfandtopology)is anoni allyhomeomorphi to

D

;theelementsin

∆(A(D))

^are

pre iselythepointevaluationsin

D

(see[3℄). Sowe on ludethat

∆(A)

^isalsoequal

to

D

. Theindu edhomeomorphism

σ

^on

∆(A) = D

^{isrotationbythesameangle}

asfor

σ ^′

^{. Here the}non-periodi pointsare obviouslydense,so that

A

^{is maximal}

abelianbyTheorem4.8.

Note thedieren e in nature betweenExamples 4.10and 4.11. Inthe former,

passingtothesystemonthe hara terspa e orrespondstodeleting apointfrom

theoriginalsystemandrestri tingthefun tionalgebraandhomeomorphism,while

inthelatterit orrespondstoadding(alot of)pointsandextending.

We on ludethis subse tionbygivingyetanotherexampleofanappli ationof

Theorem 4.8, re overingone of the results we obtained in Example 3.8 by using

Theorem3.7.

Example 4.12. Let

X

^{be a lo ally ompa t Hausdor spa e, and}

σ : X → X

a homeomorphism. Let

A = C 0 (X)

^{; then}

σ

^{indu es an} automorphism

eσ

^on

A

asin Se tion 3. Here

∆(A)

^is anoni allyhomeomorphi

X

^and

A

^is anoni ally isomorphi (withrespe t tothehomeomorphism between

∆(A)

^and

X

^)to

A b

^(see

forexample[3℄). Hen ebyTheorem4.8

A

^{ismaximalabelianin}

A ⋊ _e σ Z

ifandonly ifthenon-periodi pointsof

X

^{aredense,asalreadymentionedinExample3.8.}

4.3. Integrable fun tions on lo ally ompa t abelian groups. Inthis sub-

se tionwe onsiderthe rossedprodu t

L 1 (G) ⋊ Ψ Z

,where

G

^{isalo ally ompa t}

abeliangroupand

Ψ : L 1 (G) → L 1 (G)

^anautomorphism. Wewill showthat un- der anadditional onditionon

G

^{, a strongerresultthan Theorem 4.8 istrue ( f.}

Theorem4.16).

We startby re alling anumber of standardresultsfrom thetheory of Fourier

analysisongroups,andreferto [3℄and[6℄fordetails. Let

G

^{bealo ally ompa t}

abeliangroup. Re allthat

L 1 (G)

^onsistsofequivalen e lassesof omplexvalued Borelmeasurablefun tionsof

G

^{thatareintegrablewithrespe ttoaHaarmeasure}

on

G

^{, and that}

L 1 (G)

^{equipped with} onvolutionprodu t is a ommutative om- pletely regularsemi-simple Bana h algebra. A group homomorphism

γ : G → T

from alo ally ompa tabelian group

G

^{to theunit ir le is alleda hara ter of}

G

^{. Thesetofall ontinuous hara tersof}

G

^formsagroup

Γ

^{,thedualgroup of}

G

^,

ifthegroupoperationisdenedby

(γ 1 + γ 2 )(x) = γ 1 (x)γ 2 (x) (x ∈ G; γ 1 , γ 2 ∈ Γ).

If

γ ∈ Γ

^andif

f (γ) = b

Z

G

f (x)γ(−x)dx (f ∈ L 1 (G)),

then the map

f 7→ b f (γ)

^{is a non-zero omplex} homomorphism of

L 1 (G)

^{. Con-}

versely,everynon-zero omplexhomomorphismof

L 1 (G)

^{is obtainedin this way,}

and distin t hara ters indu e distin t homomorpisms. Thus we may identify

Γ

with

∆(L 1 (G))

^{. The fun tion}

f : Γ → C b

^{dened as above is alled the Fourier}

transform of

f, f ∈ L 1 (G)

^{, andishen epre iselytheGelfandtransformof}

f

^{. We}

denotetheset ofall su h

f b

^by

A(Γ)

^{. F}urthermore,

Γ

^{is alo ally ompa tabelian}

Denition 4.13. Givenaset

X

^{, aring of subsets of}

X

^{is a olle tionofsubsets}

of

X

^{whi his losedunder theformationofnite unions,nite}interse tions, and omplements(in

X

^{). Notethatany}interse tionofringsisagainaring. The oset- ring of

Γ

^{is denedto bethesmallestringofsubsetsof}

Γ

^{whi h ontainsallopen}

osets, i.e., all subsets of

Γ

^{of the form}

a + U

^{, where}

a ∈ Γ

^and

U

^{is an open}

subgroupof

Γ

^.

Wearenowready todeneaparti ulartypeofmap onthe osetringof

Γ

^{( f.}

[6℄).

Denition 4.14. Let

E

^{be a oset in}

Γ

^{. A ontinuous map}

σ : E → Γ

^{whi h}

satisestheidentity

σ(γ + γ ^′ − γ ^′′ ) = σ(γ) + σ(γ ^′ ) − σ(γ ^′′ ) (γ, γ ^′ , γ ^′′ ∈ E)

is alledane. Supposethat

(i)

S 1 , . . . , S n

^{arepairwisedisjointsetsbelongingtothe oset-ringof}

Γ

^;

(ii) ea h

S i

^{is ontainedinanopen oset}

K i

ⁱⁿ

Γ

^;

(iii) forea h

i

^,

σ i

^{isananemap of}

K i

^into

Γ

^;

(iv)

σ

^isthemapof

Y = S 1 ∪ . . . ∪ S n

^into

Γ

^{whi h oin ideson}

S i

^with

σ i

^.

Then

σ

^{issaidtobeapie ewise ane map from}

Y

^to

Γ

^.

The following theorem is a key result for what follows. It states that every

automorphismof

L 1 (G)

^{isindu edbyapie ewiseane}homeomorphism,andthat apie ewiseanehomeomorphismindu esaninje tivehomomorphismfrom

L 1 (G)

toitself.

Theorem 4.15. Let

eσ : L ¹ (G) → L 1 (G)

^{be an} automorphism. Then for every

f ∈ L 1 (G)

^{wehave that}

eσ(f) = b d f ◦ σ

^,where

σ : Γ → Γ

^{isa xedpie ewise ane}

homeomorphism. Also, if

σ : Γ → Γ

^{is a pie ewise ane} homeomorphism, then

f ◦ σ ∈ A(Γ) b

^{for every}

f ∈ A(Γ) b

^.

Proof. FollowsfromthemoregeneralTheorems4.1.3and4.6.2in [6℄.



Nowlet

eσ : L ¹ (G) → L 1 (G)

^beanautomorphismand onsiderthe rossedprod- u t

L 1 (G) ⋊ _e σ Z

. Letting

eσ

^{indu e a} homeomorphism as des ribed for arbitrary ommutative ompletelyregularsemi-simpleBana halgebrasintheparagraphfol-

lowing Denition 4.5, we obtain

σ ⁻¹ : Γ → Γ

^{, where}

σ

^{is the pie ewise ane}

homeomorphismindu ing

e σ

^{ina ordan ewithTheorem4.15.}

Theorem4.16.Let

G

^{bealo ally ompa tabeliangroupwith onne teddualgroup}

andlet

eσ : L ¹ (G) → L 1 (G)

^bean automorphism. Then

L 1 (G)

^{ismaximal abelian}

in

L 1 (G) ⋊ _e σ Z

if andonlyif

eσ

^{isnot ofnite order.}

Proof. Denoteby

Γ

^{thedualgroupof}

G

^{. ByTheorem4.8,}

σ

^,thehomeomorphism indu edby

eσ

^{in a ordan ewith thedis ussion followingDenition 4.5, is notof}

nite orderif

L 1 (G)

^{ismaximal abelian. Assumenowthat}

L 1 (G)

^{is notmaximal}

abelian. ByTheorem4.8, thisimpliesthat

Per ^∞ (Γ)

^{is notdense in}

Γ

^{. As argued}

in theproofofTheorem3.7, theremustthen exist

n 0 ∈ N

^{su hthat}

Per ^{n 0} (Γ)

^has

non-empty interior. Namely, sin e in this ase

T

n∈Z >0 (Γ \ Per ⁿ (Γ))

^{is notdense}

andthatthesets

Γ \ Per ⁿ (Γ)

^,

n ∈ Z >0

^{areallopen,thefa tthat}

Γ

^{isaBairespa e}

(beinglo ally ompa tandHausdor)impliesexisten eofan

n 0 ∈ Z >0

^{su h that}

Per ^{n 0} (Γ)

^{hasnon-emptyinterior. Notethat}

Γ

^{being onne tedimpliesthat}

σ ⁻¹

^,the

pie ewiseanehomeomorphismof

Γ

^{indu ing}

eσ

^{ina ordan ewithTheorem4.15,}

must beane by onne tednessof

Γ

^{(the oset-ringistrivially}

{∅, Γ}

^{)and hen e}

sois

σ

^{. It isreadilyveriedthat themap}

σ ^{n 0} − I

^{isthenalso ane. Now learly}

Per ^{n 0} (Γ) = (σ ^{n 0} −I) ⁻¹ ({0})

^{. Theanenatureof}

σ ^{n 0} −I

^{assuresusthat}

Per ^{n 0} (Γ)

^is

a oset. Inatopologi algroup,however, ontinuityofthegroupoperationsimplies

that osetswithnon-emptyinteriorareopen,hen ealso losed. We on ludethat

Per ^{n 0} (Γ)

^{is a non-empty losed and open set.} Conne tedness of

Γ

^{now implies}

that everypoint in

Γ

^is

n 0

^{-periodi under}

σ

^{. Hen e, by the dis ussion following}

Denition 4.5,

eσ ^{n 0}

^{istheidentitymapon}

L 1 (G)

^.



Thefollowingexampleshowsthatifthedualof

G

^{isnot onne ted,theequiva-}

len ein Theorem4.16neednothold.

Example 4.17. Let

G = T

^{bethe ir le group. Here}

Γ = Z

^{(see [3℄ for details),}

whi h is not onne ted. Dene

σ : Z → Z

^by

σ(n) = n (n ∈ 2Z)

^and

σ(m) =

m+2 (m ∈ 1+2Z)

^{. Obviously}

σ

^and

σ ⁻¹

^{arethenpie ewiseane}homeomorphisms that are notof niteorder. By Theorem4.15,

σ

^{indu esan} automorphism

σ g ⁻¹ :

L 1 (T) → L 1 (T)

^{, whi h in turn indu es the} homeomorphism

σ ⁻¹ : Z → Z

^{. Now}

A(Z)

^{isnotmaximalabelianin}

A(Z) ⋊ _{σ d −} 1 Z

sin ebyCorollary3.4wehave

A(Z) ^′ = { X

n∈Z

c f n δ ⁿ |

^forall

n ∈ Z \ {0} : supp(c f n ) ⊆ 2Z},

andhen ebyTheorem 4.6

L 1 (T) ^′ = { X

n∈Z

f n δ ⁿ |

^forall

n ∈ Z \ {0} : supp(c f n ) ⊆ 2Z}.

Notethat

{ X

n∈Z

f n δ ⁿ |

^forall

n ∈ Z \ {0} : f n ∈ C[z ² , z ⁻² ]} ⊆ L 1 (T) ^′ ,

andthuswe on ludethat

L 1 (T)

^{isnotmaximalabelian.}

4.4. Atheoremon generatorsforthe ommutant. Anaturalquestiontoask

iswhetherornotthe ommutant

A ^′ ⊆ A⋊ _{e σ} Z

isnitelygeneratedasanalgebraover

C

ornot. Herewegiveananswerinthe asewhen

A

^isasemi-simple ommutative Bana halgebra.

Theorem 4.18. Let

A

^{be a} semi-simple ommutative Bana h algebra andlet

e σ :

A → A

^bean automorphism. Then

A ^′

^{isnitely generatedasan algebraover}

C

if andonlyif

A

^{hasnite dimension asave torspa e over}

C

.

Proof. Let theindu ed homeomorphism

σ : ∆(A) → ∆(A)

^{beas usual. Assume}

rstthat

A

^{hasinnitedimension. Bybasi theoryofBana hspa es,}

A

^mustthen

haveun ountable dimension. If

A ^′

^{weregeneratedbynitelymanyelements,then}

A ^′

^{,andinparti ular}

A

^{,wouldhave ountabledimension,whi hisa} ontradi tion.

Hen e

A ^′

^{isnotnitely generatedasanalgebraover}

C

. Forthe onverse,weneed tworesultsfrom Bana halgebratheory. Supposethat

A

^{hasnite dimension. By}

Proposition26.7in [1℄,

∆(A)

^{isthenaniteset,andthuseverypointin}

(∆(A), σ)

hasa ommonniteperiod,

n 0

^{say. F}urthermore,byCorollary21.6in [1℄

A

^must

thenalsobeunital. Wepassnowtothe rossedprodu t

A⋊ b _b σ Z

,whi hisisomorphi to

A⋊ _e σ Z

byTheorem4.6. Clearly

A b

^{isunitalandhasnitelineardimensionsin e}

A

does. ByCorollary3.4,forageneralelement

P

n∈Z c a n δ ⁿ ∈ ( b A) ^′

^{thesetofpossible}

oe ientsof

δ ⁿ

^{isave torsubspa e(andevenanideal)of}

A b

^,

K n

^{say,and hen e}

of nite dimension. Sin e allelementsof

(∆(A), σ)

^haveperiod

n 0

^{, Corollary3.4}

also tellsus that

K r+l·n 0 = K r

^{for all}

r, l ∈ Z

^{. Now note that sin e}

A b

^{is unital,}

δ ^{n 0} , δ ^{−n 0} ∈ ( b A) ^′

^{. Thus, denoting a basis for a}

K l

^by

{e (l,1) , . . . , e (l,l r ) }

^(where

l r ≤ s

^{),theabovereasoningassures usthat}

S n 0

l=1

S l r

j=1 {e (l,j) δ ^l }

^generates

( b A) ^′

^as

analgebraover

C

. ByTheorem 4.6this impliesthat also

A ^′ ⊆ A ⋊ _e σ Z

is nitely

generatedasanalgebraover

C

.



A knowledgments

Thisworkwassupportedbyavisitor'sgrantoftheNetherlandsOrganisationfor

S ienti Resear h(NWO),TheSwedishFoundationforInternationalCooperation

in Resear h andHigher Edu ation (STINT), Crafoord Foundation andThe Royal

Physiographi So iety inLund.

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Mathemati alInstitute,Leiden University,P.O.Box9512,2300RALeiden,The

Netherlands,andCentreforMathemati alS ien es,LundUniversity,Box118,SE-

22100Lund,Sweden

E-mailaddress: hrissmath.leidenuniv. nl

Centre for Mathemati al S ien es, Lund University,Box118, SE-221 00 Lund,

Sweden

E-mailaddress: Sergei.Silvestrovmath. lth .se

Mathemati alInstitute,Leiden University,P.O.Box9512,2300RALeiden,The

Netherlands

E-mailaddress: mdejeumath.leidenuniv. nl