Operator space structure and amenability for Figa`-Talamanca–Herz algebras

25 Pages·2004·0.33 MB·English

ARTICLE IN PRESS
JournalofFunctionalAnalysis211(2004)245–269
Operator space structure and amenability
for Figa` -Talamanca–Herz
algebras
Anselm Lambert,a Matthias Neufang,b,1 and
Volker Rundec,(cid:1),2
aFachrichtung6.1Mathematik,Universita¨tdesSaarlandes,Postfach151150,66041Saarbru¨cken,
Germany
bSchoolofMathematicsandStatistics,4364HerzbergLaboratories,CarletonUniversity,Ottawa,
Ontario,CanadaK1S5B6
cDepartmentofMathematicalandStatisticalSciences,UniversityofAlberta,CAB632,
Edmonton,Alberta,CanadaT6G2G1
Received29April2003;accepted28August2003
CommunicatedbyD.Sarason
Abstract
Columnandrowoperatorspaces—whichwedenotebyCOLandROW;respectively—over
arbitraryBanachspaceswereintroducedbythefirst-namedauthor;forHilbertspaces,these
definitions coincide with the usual ones. Given a locally compact group G and p;p0Að1;NÞ
with1þ1 ¼1;weusethe operatorspacestructure onCBðCOLðLp0ðGÞÞÞto equiptheFiga`-
p p0
Talamanca–HerzalgebraA ðGÞwithanoperatorspacestructure,turningitintoaquantized
p
Banach algebra. Moreover, we show that, for ppqp2 or 2pqpp and amenable G; the
canonical inclusion A ðGÞCA ðGÞ is completely bounded (with cb-norm at most K2; where
q p G
K isGrothendieck’sconstant).Asanapplication,weshowthatGisamenableifandonlyif
G
(cid:1)Correspondingauthor.Fax:[phone].
E-mailaddresses:[email protected](A.Lambert),[email protected]
(M.Neufang),[email protected](V.Runde).
URLs: http://www.math.uni-sb.de/~ag-wittstock/alambert.html, http://mathstat.carleton.ca/~mneufang/,
http://www.math.ualberta.ca/~runde/.
1Partoftheresearchforthispaperwasdonewhile theauthorheldapostdoctoralfellowship ofthe
PacificInstitutefortheMathematicalSciences(PIMS)attheUniversityofAlberta;thesupportbyPIMS
isacknowledgedwiththanks.
2ResearchsupportedbyNSERCundergrantno.227043-00.
0022-1236/$-seefrontmatterr2003ElsevierInc.Allrightsreserved.
doi:10.1016/j.jfa.2003.08.009
ARTICLE IN PRESS
246 A.Lambertetal./JournalofFunctionalAnalysis211(2004)245–269
A ðGÞ is operator amenable for all—and equivalently for one—pAð1;NÞ; this extends a
p
theorembyRuan.
r2003Elsevier Inc. Allrights reserved.
MSC: primary47L25;secondary43A15;43A30;46B70;46J99;46L07;47L50
Keywords: Operatorspaces;Operatorsequencespaces;Columnandrowspaces;Locallycompactgroups;
Figa`-Talamanca–Herzalgebra;Fourieralgebra;Amenability;Operatoramenability
0. Introduction
The Fourier algebra AðGÞ of a locally compact group G was introduced by
Eymard in [Eym1]. If G is abelian with dual group G; then the Fourier transform
inducesanisometricisomorphismofAðGÞandL1ðGÞ:AlthoughtheFourieralgebra
is an invariant for G—like L1ðGÞ—, its Banach algebraic amenability does not
correspond well to the amenability of G—very much unlike L1ðGÞ: The group G is
amenableifandonlyifL1ðGÞisamenableasaBanachalgebra[Joh1],butthereare
compact groups, among them SOð3Þ; for which AðGÞ fails to be even weakly
amenable [Joh2]. In fact, the only locally compact groups G for which AðGÞ is an
amenable Banach algebra in the sense of [Joh1] are those with an abelian subgroup
of finite index [Run3].
Since AðGÞ is the predual of the group von Neumann algebra VNðGÞ; it is an
operator space in a natural manner. In [Rua1], Ruan introduced a variant of
amenability for ‘‘quantized’’ Banach algebras—called operator amenability—that
reflects the operator space structures of those algebras. He showed that a locally
compact group G is amenable if and only if AðGÞ is operator amenable [Rua1,
Theorem3.6].Further resultsby Aristov [Ari],Wood[Woo],and Spronk[Spr] lend
additional support to the belief that homological properties of AðGÞ; such as
amenability,biprojectivityorweakamenability,correspondtopropertiesofGmuch
more naturally if the operator space structure is taken into account. Even if one is
only interested in AðGÞ as a Banach algebra, considering the canonical operator
space structure can be valuable: Although the main result of [Run3] is purely
classical in its statement, its proof is operator space theoretic. (Further examples of
classical results obtained with the help of operator space methods can be found in
[F–K–L–S].)
In [Her1], Herz introduced, for pAð1;NÞ; an Lp-analog of the Fourier algebra,
denoted by A ðGÞ: These algebras are called Figa`-Talamanca–Herz algebras. It was
p
asked by the third-named author if there was an analog of Ruan’s theorem for
arbitrary Figa`-Talamanca–Herz algebras [Run1, Problem 34]. The first obstacle
towards a solution of this problem is that there is—at first glance—no natural
operator space structure for A ðGÞ if pAð1;NÞ\f2g:
p
In[Run2],thesameauthorusedtheoperatorspacestructureintroducedbyPisier
on the Lp-spaces via complex interpolation [Pis1] to define operator space analogs
OA ðGÞ of the classical Figa`-Talamanca–Herz algebras for all pAð1;NÞ: These
p
ARTICLE IN PRESS
A.Lambertetal./JournalofFunctionalAnalysis211(2004)245–269 247
operator Figa`-Talamanca–Herz algebras display properties similar to those of their
classical counterparts (see, for example, [Run2, Theorem 4.10]). Nevertheless, the
construction from [Run2] is unsatisfactory for two reasons:
1. Although we have a contractive inclusion A ðGÞCOA ðGÞ for all pAð1;NÞ; the
p p
twoalgebrasA ðGÞandOA ðGÞcanfailtobeidenticalasBanachspacesforpa2:
p p
2. Even though AðGÞ¼OA ðGÞ as Banach spaces, they need not have the same
2
operator space structure.
The reason why AðGÞ and OA ðGÞ need not coincide as operator spaces is that the
2
operator space structure of AðGÞ stems from VNðGÞ and thus from the column
Hilbert space over L2ðGÞ; whereas OA ðGÞ is defined by means of the operator
2
Hilbert space over L2ðGÞ: In order to get asatisfactory operator space structure for
generalFiga`-Talamanca–Herzalgebras,oneshouldthereforestriveforanextension
of the notion of column space from Hilbert spaces to arbitrary Lp-spaces.
In his doctoral dissertation [Lam], the first-named author introduced such a
notion;infact,hedefinedcolumnandrowspacesoverarbitraryBanachspaces.We
first outline his construction and then use it to equip general Figa`-Talamanca–Herz
algebras with a canonical operator space structure, turning them into ‘‘quantized’’
Banach algebras. As an application, we extend [Rua1, Theorem 3.6] to arbitrary
Figa`-Talamanca–Herz algebras.
1. Preliminaries
1.1. Figa`-Talamanca–Herz algebras
LetGbealocallycompactgroup.Foranyfunctionf :G-C;wedefinefˇ:G-Cby
letting fˇðxÞ:¼fðx(cid:9)1Þ for xAG: Let pAð1;NÞ; and let p0Að1;NÞ be dual to p; i.e.
1þ 1 ¼1: The Figa`-Talamanca–Herz algebra A ðGÞ consists of those functions
p p0 p
f :G-Csuchthattherearesequencesðx ÞN inLpðGÞandðZ ÞN inLp0ðGÞsuchthat
n n¼1 n n¼1
N
X
jjxnjjLpðGÞjjZnjjLp0ðGÞoN ð1Þ
n¼1
and
N
X
f ¼ x (cid:1)Z$ : ð2Þ
n n
n¼1
ThenormonA ðGÞisdefinedastheinfimumoverallsums(1)suchthat(2)holds.It
p
is clear that A ðGÞ is a Banach space that embeds contractively into C ðGÞ: It was
p 0
shown by Herz [Her1] that A ðGÞ is closed under pointwise multiplication and, in
p
fact, a Banach algebra. The case where p¼q¼2 had previously been studied by
Eymard [Eym1]; in this case AðGÞ:¼A ðGÞ is called the Fourier algebra of G:
2
ARTICLE IN PRESS
248 A.Lambertetal./JournalofFunctionalAnalysis211(2004)245–269
Let lp0 :G-BðLp0ðGÞÞ be the regular left representation of G on Lp0ðGÞ: Via
integration, l extends to a representation of L1ðGÞ on Lp0ðGÞ: The algebra of p0-
p0
pseudomeasures PMp0ðGÞ is defined as the w(cid:1)-closure of lp0ðL1ðGÞÞ in BðLp0ðGÞÞ:
There is a canonical duality PM ðGÞDA ðGÞ(cid:1) via
p0 p
/x(cid:1)Z$;TS:¼/TZ;xS ðxALp0ðGÞ;ZALpðGÞ;TAPM ðGÞÞ:
p0
Ifp¼2;thenVNðGÞ:¼PM ðGÞisavonNeumannalgebra,thegroupvonNeumann
2
algebra of G:
For more information, see [Eym1], [Eym2], [Her1], [Her2], and [Pie].
1.2. Operator spaces and quantized Banach algebras
There are now two booklength monographs available on the subject of operator
spaces([E–R]and[Pis2]);furthermore,averyaccessiblesurveyarticleisavailableon
the internet [Wit]. We therefore refrain from repeating here the basic definitions of
operator space theory. In our choice of notation, we mostly follow [E–R]. In
particular, the projective and injective tensor product of Banach spaces are denoted
by#gand#l;respectively,whereas## and#& standfortheprojectiveandinjective
tensor product of operator spaces.
We deviate from [E–R] with our notation in two points:
1. TheminimalandmaximaloperatorspaceoveragivenBanachspaceE isdenoted
by MINðEÞ and MAXðEÞ; respectively.
2. The column and row space over a Hilbert space H is denoted by COLðHÞ and
ROWðHÞ; respectively.
Definition 1.1. A quantized Banach algebra is an algebra which is also an operator
space such that multiplication is completely bounded.
Remark. We do not require multiplication to be completely contractive (such
algebrasarecalledcompletelycontractiveBanachalgebras;see[Rua1]).Inourchoice
of terminology, we follow [Ari].
Examples. 1. For any Banach algebra A—not necessarily with contractive
multiplication—the maximal operator space MAXðAÞ is a quantized Banach
algebra.
2. If H is a Hilbert space, then any closed subalgebra of BðHÞ is completely
contractive.
3. We denote the W(cid:1)-tensor product by #% : A Hopf–von Neumann algebra is a
pair ðM;rÞ; where M is a von Neumann algebra, and r is a co-multiplication: a
unital, w(cid:1)-continuous, and injective (cid:1)-homomorphism r :M-M#% M which is
ARTICLE IN PRESS
A.Lambertetal./JournalofFunctionalAnalysis211(2004)245–269 249
co-associative, i.e. the diagram
commutes. Let M denote the unique predual of M: By [E–R, Theorem 7.2.4],
(cid:1)
we have M#% MDðM ## M Þ(cid:1): Thus r induces a complete contraction
(cid:1) (cid:1)
r :M ## M -M turning M into a completely contractive Banach algebra.
(cid:1) (cid:1) (cid:1) (cid:1) (cid:1)
4. Let G be a locally compact group. Then the representation
G/BðL2ðG(cid:12)GÞÞ; x/l ðxÞ#l ðxÞ
2 2
induces a co-multiplication r : VNðGÞ-VNðG(cid:12)GÞDVNðGÞ#% VNðGÞ: Hence,
AðGÞwithitscanonicaloperator spacestructureisacompletelycontractiveBanach
algebra.
2. Operator sequence spaces
In [Math], Mathes characterized the column operator space COLðHÞ over a
Hilbert space H through the isometries
M ðCOLðHÞÞ¼M ðMAXðHÞÞ and M ðCOLðHÞÞ¼M ðMINðHÞÞ
n;1 n;1 1;n 1;n
for all nAN; i.e. COLðHÞ is maximal on the columns and minimal on the rows.
A similar characterization holds for ROWðHÞ:
InordertodefinecolumnandrowoperatorspacesoverarbitraryBanachspacesin
the next section, we first introduce and discuss an axiomatic characterization of the
columns of operator spaces: the operator sequence spaces. Those spaces were
introduced by the first-named author in his doctoral dissertation [Lam, Definition
1.1.1]. They form a category somewhere in between Banach and operator spaces.
Afullaccountofthetheoryofoperatorsequencespaceswillbepublishedelsewhere.
In this section, we content ourselves with presenting the concepts and results we
needfortheremainderofthepaper:todefinecolumnandrowoperatorspacesover
arbitraryBanachspacesandtousethoseoperatorspacestructurestoturntheFiga`-
Talamanca–Herz algebras into quantized Banach algebras. We are somewhat
sketchy with our proofs—especially if they consist mainly of routine calculations or
straightforward adaptations of proofs of the corresponding Banach or operator
space results.
N
Definition 2.1. A sequential norm over a Banach space E is a sequence ðjj(cid:13)jj_nÞn¼1
such that jj(cid:13)jj_1 is the given norm on E and jj(cid:13)jj_n is, for each nAN; a norm on En
ARTICLE IN PRESS
250 A.Lambertetal./JournalofFunctionalAnalysis211(2004)245–269
such that
(cid:4)(cid:4)(cid:2)x(cid:3)(cid:4)(cid:4)
(cid:4)(cid:4)(cid:4)(cid:4)(cid:4)(cid:4) 0 (cid:4)(cid:4)(cid:4)(cid:4)(cid:4)(cid:4) _ ¼jjxjjm_ ðm;nAN;xAEmÞ;
mþn
(cid:4)(cid:4)(cid:2)x(cid:3)(cid:4)(cid:4)2
(cid:4)(cid:4)(cid:4)(cid:4)(cid:4)(cid:4) y (cid:4)(cid:4)(cid:4)(cid:4)(cid:4)(cid:4) _ pjjxjj2m_þjjyjj2_n ðm;nAN;xAEm;yAFnÞ
mþn
and
jjaxjjm_pjjajjjjxjj_n ðm;nAN;xAEn;aAMm;nÞ:
FornAN;wewriteE_n todenoteEn equippedwithjj(cid:13)jj_n:ThespaceE togetherwith
N
the sequential norm ðjj(cid:13)jj_nÞn¼1 is called an operator sequence space (over E).
Examples. 1. Letting C_n :¼c2 for nAN; we define the (unique) operator sequence
n
space over C:
2.LetE beanyBanachspace.TheminimaloperatorsequencespaceminðEÞoverE
is defined by letting minðEÞ_n :¼Bðc2;EÞ for nAN; the adjective minimal will be
n
justified below.
3.LetE beanyBanachspace.ThemaximaloperatorsequencespacemaxðEÞover
E is defined as follows: For nAN and xAEn; define
jjxjj_n :¼inffjjajjjjyjjc2mðEÞ:mAN;aAMn;m;yAEm;x¼ayg:
As in the case of minðEÞ; the use of the adjective maximal will soon become clear.
4.LetE beanoperatorsequencespace,andletmAN:WedefineðEm_Þ_n :¼Em_n for
nAN: This turns Em_ into an operator sequence space.
5. Let E be an operator space. Define an operator sequence space CðEÞ over
M ðEÞ by letting CðEÞ_n :¼M ðEÞ for nAN: It will become apparent in the next
1 n;1
section that every operator sequence space occurs in that fashion.
Having introduced the objects of the category of operator sequence spaces, we
now turn to defining its morphisms.
Given two operator sequence spaces E and F; a linear map T:E-F; and nAN;
let
T_n :E_n-F_n; ½x(cid:15) /½Tx(cid:15)
j j¼1;y;n j j¼1;y;n
denote the nth amplification of T:
ARTICLE IN PRESS
A.Lambertetal./JournalofFunctionalAnalysis211(2004)245–269 251
_ _
Definition 2.2. Let E and F be operator sequence spaces. Then TABðE1;F1Þ is
called sequentially bounded if
jjTjjsb :¼snAuNp jjT_njjBðE_n;F_nÞoN:
_n
If jjTjj p1; we call T a sequential contraction, and if T is an isometry for each
sb
nAN; we call T a sequential isometry. The collection of all sequentially bounded
maps from E to F is denoted by SBðE;FÞ:
Remarks. 1. It is straightforward that jj(cid:13)jj turns SBðE;FÞ into a Banach space.
sb
2. We write SBðEÞ instead of SBðE;EÞ:
Examples. 1. Let E be an operator sequence space, and let F be a Banach space.
Then
_
1
SBðE;minðFÞÞ¼BðE ;FÞ ð3Þ
holds isometrically. (This justifies the name minimal operator space.)
2. Let E be a Banach space, and let F be an operator sequence space. Then
_
1
SBðmaxðEÞ;FÞ¼BðE;F Þ ð4Þ
holds isometrically. (This justifies the adjective maximal.)
3.LetE beaninfinite-dimensionalBanachspace.Thenid : MINðEÞ-MAXðEÞ
E
is not completely bounded [Pau1, Theorem 2.12]. Interestingly, the situation
for operator sequence spaces and sequentially bounded maps is different: For
example, if A is a C(cid:1)-algebra, then A is subhomogeneous if and only if
id ASBðminðAÞ;maxðAÞÞ [Lam, Satz 2.2.25].
A
Next, we sketch the duality theory for operator sequence spaces. We first
introduceacanonicaloperatorsequencespacestructureovertheBanachspaceofall
sequentially bounded maps between operator sequence spaces:
Proposition 2.3. Let E and F be operator sequence spaces. Letting
SBðE;FÞ_n :¼SBðE;F_nÞ ðnANÞ
defines an operator sequence space over SBðE;FÞ:
Weskiptheproofwhichparallelstheoneofthecorrespondingresultforoperator
spaces.
We require an analog of Smith’s lemma [E–R, Proposition 2.2.2] for operator
sequence spaces:
ARTICLE IN PRESS
252 A.Lambertetal./JournalofFunctionalAnalysis211(2004)245–269
Lemma2.4 (Smith’slemmaforoperator sequence spaces). LetE andF beoperator
_ _
sequence spaces such that dimF ¼moN: Then BðE1;F1Þ¼SBðE;FÞ holds with
jjTjj ¼jjTm_jj for each TABðE;FÞ:
sb
Proof. LetTABðE;FÞ;andletnAN:Letx¼½x(cid:15) AE_n;andlety ;y;y bea
j j¼1;y;n 1 m
basis for F: Then there are a AC for j ¼1;y;n and k¼1;y;m such that
j;k
m
X
Tx ¼ a y :
j j;k k
k¼1
Let y:¼½yk(cid:15)k¼1;y;mAFm_ and a:¼½aj;k(cid:15)j¼1;y;nAMn;m: It follows that
k¼1;y;m
_n
T x¼ay:
1
Let vAMn;m be a partial isometry such that a¼vjaj; where jaj:¼ða(cid:1)aÞ2: It follows
that
jjayjj _ ¼jjvjajyjj _pjjjajyjj _ ð5Þ
Fn Fn Fm
and
jjjajyjj _ ¼jjv(cid:1)ayjj _pjjayjj _ ð6Þ
Fm Fm Fn
so that in (5) and (6) equality holds. Consequently, we have
_n
jjT xjj _ ¼jjayjj _
Fn Fn
¼jjv(cid:1)ayjj _
Fm
¼jjv(cid:1)T_nxjj _
Fm
¼jjTm_ðv(cid:1)xÞjj _
Fm
pjjTm_jjjjv(cid:1)xjj _
Fm
m_
pjjT jjjjxjj _;
Fn
so that jjT_njjpjjTm_jj: Since nAN was arbitrary, this yields jjTjj pjjTm_jj: The
sb
converse inequality is trivial. &
ARTICLE IN PRESS
A.Lambertetal./JournalofFunctionalAnalysis211(2004)245–269 253
Remark. In view of [E–R, Proposition 2.2.2], one might have suspected that
Lemma 2.4 holds true for F ¼c2 only. The fact that there is a stronger version of
m
Smith’slemmaforoperatorsequencespacesthanforoperatorspaceshasinteresting
consequences. For example, the principle of local reflexivity, which is a cornerstone
of the local theory of Banach spaces [D–F, 6.6], but fails to have an analog for
operator spaces [E–R, Corollary 14.3.8], still works in the category of operator
sequence spaces [Lam, Satz 1.3.26].
Corollary 2.5. Let E be an operator sequence space. Then E(cid:1) ¼SBðE;CÞ holds
isometrically.
With Corollary 2.5 at hand, we can now equip the (Banach space) dual of an
operator sequence space with a canonical operator sequence space structure.
Taking the adjoint of a sequentially bounded operator yields again a sequentially
bounded operator. But more is true:
Theorem 2.6. Let E and F be operator sequence spaces. Then
jjðT(cid:1)Þ_njj¼jjT_njj ðnAN;TASBðE;FÞÞ
holds. Moreover,
SBðE;FÞ-SBðF(cid:1);E(cid:1)Þ; T/T(cid:1) ð7Þ
is a sequential isometry.
Proof. Thefirst part ofthetheoremis [Lam, Satz1.3.14] andhasaproofsimilarto
its operator space analog [E–R, Proposition 3.2.2]. In particular, (7) is an isometry.
To see that (7) is in fact a sequential isometry, fix nAN and note that, we have a
(sequential) isometric canonical isomorphism
SBððF_nÞ(cid:1);E(cid:1)ÞDSBðF(cid:1);ðE(cid:1)Þ_nÞ ð8Þ
by [Lam, Satz 1.3.10 and Satz 1.3.12]. Hence, we have the following canonical
isometries:
_n _n
SBðE;FÞ ¼SBðE;F Þ
+SBððF_nÞ(cid:1);E(cid:1)Þ; by ½Lam; Satz 1:3:14(cid:15);
DSBðF(cid:1);ðE(cid:1)Þ_nÞ; by ð8Þ;
¼SBðF(cid:1);E(cid:1)Þ_n:
This completes the proof. &
ARTICLE IN PRESS
254 A.Lambertetal./JournalofFunctionalAnalysis211(2004)245–269
We conclude this section with the analog of the MIN2MAX duality [Lam, Satz
2.1.11]:
Theorem 2.7 (min2max duality). For any Banach space E; we have the sequentially
isometric isomorphisms
minðEÞ(cid:1) ¼maxðE(cid:1)Þ and maxðEÞ(cid:1) ¼minðE(cid:1)Þ:
Proof. Sincethecompatibilityofbidualsforoperatorspaces[Ble,Theorem2.5]has
an analog in the category of operator sequence spaces [Lam, Satz 1.3.19] with an
almostidenticalproof,themin2maxdualitycanbeprovenbymoreorlessfollowing
the proof of the MIN2MAX duality in [Ble]. &
3. Column and row spaces over arbitrary Banach spaces
With the preparations made in the previous section, we can now define column
and row spaces overarbitrary Banach spaces. As before, thematerial isfrom [Lam]
and will appear elsewhere in fuller detail.
Definition3.1. LetEbeanoperatorsequencespace.Thentheminimaloperatorspace
MinðEÞ over E is defined by letting M ðMinðEÞÞ:¼Bðc2;E_nÞ:
n n
Remarks. 1. By [Lam, Satz 4.1.2], MinðEÞ is an operator space for any operator
sequence space E:
2. For any operator space E; and for any operator sequence space F;
CBðE;MinðFÞÞ¼SBðCðEÞ;FÞ
holds isometrically [Lam, Satz 4.1.6].
3. Let F be a Banach space. Then the previous remark and the isometric identity
(3) combined yield that
CBðE;MinðminðFÞÞÞ¼BðM ðEÞ;FÞ
1
holds isometrically for each operator space E; so that we have MinðminðFÞÞ¼
MINðFÞ:
The following definition generalizes V. I. Paulsen’s formula for the maximal
operator space norm over a Banach space [Pau2, Theorem 2.1]:
Definition 3.2. Let E be an operator sequence space. Then the maximal operator
space MaxðEÞ over E is defined by letting, for xAM ðEÞ;
n
jjxjj :¼inffjjajjjjbjj:x¼adiagðv ;y;v Þbg;
MnðMaxðEÞÞ 1 k