HOUSTON JOURNAL OF MATHEMATICS Volume 17, No. 4, 1991 INVARIANT MEAN CHARACTERIZATIONS OF AMENABLE C*-ALGEBRAS ALAN T. PATERSON Abstract. It is shown that unital amenable and strongly amenable C*-algebras can be characterized by the existence of a right in- variant mean on a certain subspace of œo(cid:127)(H), where H is the unitary group. A fixed-point theorem for amenable C*-algebras is obtained. the main motivation for this paper. Let R be avon Neumann algebra with isometrys emigroupS . Let Bil(cid:127)(R) be the spaceo f boundedb ilinearf orms on R whicha re separatelya, -weaklyc ontinuouosn R. Then R is injectire if and onlyi f theree (cid:127)cistsa meanm on S sucht hatf or all V E Bil(cid:127)(R) and all a (cid:127) R, we have sV (avv)*d, m--f (sVv )( vv*a, )dm(v). Haagerupu ses( 1) in hisp rooft hat nuclearC *-algebraasr ea menable(. An- other proofw hicha voids( 1) and the useo f approximatefi nite dimension- ality hasb een bgyiv Eefnfr os( [7, 8].) Sincei njectivity and amenabilitya re equivalentf or R, it is natural to aski f (1) canb e interpreteda sa ssertintgh e existencoef a suitablyin variant meano n a subspacoef go½(S) with Bil(cid:127)(R). A corresponding question,o f course,c an be askedf or amenableu nital C*-algebrasw ith the unitary group H in place of S. In both cases,t he answeri s positive, and this openst he way to interpretainlgge obpraea rmateonr abihtyi n terms of a classicarli ght invariantm ean( RIM), replacingth e morec omplexn otion of virtual diagonabl y the morea ccessiblaen d better understoodn otion of invariant mean. 551 1. Introduction. The followingre sulto f Haagerup([ 11, Theorem2 .1]) is 552 ALAN L. T. PATERSON In this paper, we examine the C*-case; the author plans to discusst he von Neumann case in another paper. Let A be a unital C*-algebra,a nd Bil(A) be the Banachs paceo f boundedb ilinearf ormso n A. Let Bil22(A) be the subspacoef completely boundedb ilinearf ormsi n Bil(A). Recallt hat a C*-algebraA is called amenablief theree xistsa virtual diagonal-fotrh e definitions, ee( 8) below- for A. This notionw asi ntroducedb y Johnson([ 14]); in his memoir[ 15], Johnsoni ntroduced the notion of a stronglya menableC *-algebra, and Haagerup( [11]) haso bservetdh at this notioni s characterizebdy the exis- tenceo f a speciakl indo f virtuald iagonal(.S eeP ropositio4n. ) Our results can be interpreted as assertingt hat suchv irtual diagonalsc an be taken as arising from a RIM on spaceso f functionso n H. We startb y showingth at amenabilityfo r A is associatewdi th Bil22( A). We show in Proposition 2 that A is amenable if and only if there exists a virtual diagonalo n Bil22(A). This is the analogueo f the result of Effros ([7, 8]) that avon Neumanna lgebraR is amenableif and only if there exists a virtual diagonal on the subspaceo f completely bounded elements of We thent urn to thes ubspaceosf œ (cid:127) (H) whichs upporat RIM whenA is amenableo r strongly amenable. These spacesa re quite simple to define. We definea map A: Bil(A) --, œ(cid:127)(H) by (2) 5(V)(u) = V(u*, u) (u ß Let B(A) be A(Bil(A)) C œoo(H)T. hes ubspacBe2 2(A)o fB (A) is defined: B22(A) = A(Bil22(A)). Both B22(A),B(A) are invarianta nd contain1 . The mainr esulto f thisp aperi s the following(T heorem1 , Theorem2 ): (a) A is stronglya menableif and onlyi f theree xista RIM on B(A) (b) A is amenablief ando nlyi f theree xistsa RIM on B22(A) In the final part of the paper, we prove a fixed-pointt heoremf or amenable C*-algebras. One would expect such a theorem to exist in view of the well knownf act in the theory of amenableg roupst hat sucht heorems area ssociatewdi th invarianmt eanso ns ubspaceosf t oo(G).B unce( [2, 31) proved such a theorem for stronglya menableC *-algebras, and this easily followsb y amenableg roupt echniqueuss ingt he invariantm eanr esult (a) above. We provea fixed-pointth eorema ssociatewd ith (b) above,u sing INVARIANT MEAN CHARACTERIZATIONS 553 the notion of weaklyc ompletelyb oundedA -modules. Here, a locally convex spaceE whichi s a unital A-modulei s calledw eaklyc ompletelyb oundedi f, for every F (cid:127) E* and every x (cid:127) E, the bilinear map Fx, where Fx(a,b) = F(axb) is a completelyb oundedb ilinearf orm on A. This resulte mphasizeas theme of the papert hat amenabilityf or C*-algebrasis a completelyb oundedp he- nomenon.( An eleganta ccounto f the theoryo f completelyb oundedm aps is giveni n [21].) 2. Amenable C*-algebras and invariant means. Let A be a unital C*- algebraT. henB il(A) -- (A(cid:127)A)* is the Banachs paceo f boundedb ilinear formso n A x A. The normo n Bil(A) cana lsob e givenb y: Let Bil22(A) be the subspacoef completelbyo undede lementos f Bil(A). So a bilinearf ormV on A (cid:127) Bil22(A) if it is completelby oundeda s a bilinear map V: A x A -(cid:127) C. (See,f or example[,4 ].) For our purposess,u chf orms can be convenientlsyp ecifieda s follows.L et (a,(cid:127)) (cid:127) a(cid:127) be the universal representatioonf A on its Hilbert space7 -/. Then (cf. [8]) V (cid:127) Bil(A) is completelbyo undeidf and only if theree xist(cid:127), q in 7-/andT (cid:127) B(7-/) such that for all a, b (cid:127) A, (a) We note that sucha representationo f V has been extendedt o the non-scalar caseb y Christenseann d Sinclair( [4])-ane leganat ccounot f this is giveni n [22]. We alson otet hat therea res ubspaceBsil ij(A) for i,j (cid:127) (cid:127)1,2) which arisen aturallya nd are discusseidn [16]. Thesed o not play a role in the presentp aper but are significanti n the yon Neumann case. We will requirea notherc haracterizationo f completelyb oundedb ilin- earf orinsin thep roofo f Propositio5n. Foru (cid:127) A © A, define[[ u[[2_2(cid:127) 0 as follows: (4) u2 2=infE{a jaj- }ll(cid:127)b:bjll«'u=E©ab5 j}' 554 ALAN L. T. PATERSON In [10, 8], the map 11-11i2s2 s hownt o be a norm on A © A and is called the Haagerup norm. It is alsos hownt hat a bilinear. form on A x A is completelyb oundedif and only if it is boundedo n A © A for the Haagerup norm. Recent accountso f the Haagerup norm and other operator space normsa re giveni n [1, 9]. Let R = A** be the envelopingv on Neumanna lgebrao f A realised on 7/. It followsf rom [13, Theorem2 .3] that eachV e Bil(A) extends uniquelyw, ithoutc hangeo f normt o an elementa, lsod enotedV , of Bil(cid:127)(R). (The latter spaceis definedin the Introduction.)S ow e cani dentifyB il(A) with Bil(cid:127)(R) and can identify Bil22(A) with the appropriates ubspace of Bilø(R). This subspacies denotedb y Bil(cid:127)2(R ). The elementsV of Bil(cid:127)2(R ) are alsog ivenb y the formula( 3) with a,b allowedt o lie in R. We recallt hat Bil(A) is a dual BanachA -modulew ith actions ((cid:127)) xV(a, b) - V(a, bx) Vx(a, b) - V(xa, b). Direct checkingin (3) showsth at Bil22(A) is an invariants ubspacoef Bil(A). There is anotheru sefulm odulea ctiono whichw e postponet ill later ((21)). The next results eemsto be well known,b ut for conveniencwe e give the simple proof. Proposition 1. Let V (cid:127) Bil(cid:127)2(R ). Then the mapsx (cid:127) Vx*, x (cid:127) xV are strongo perator-normco ntinuoufsro mR into Bil(cid:127)(R). Proof.' If V is as in (3), then ((cid:127)) IIVx* - Vy*11(cid:127) I1(cid:127)11I ITII IIx(cid:127)- Y(cid:127)11, (7) IIxV- yVII (cid:127) I1(cid:127)11I ITII IIx(cid:127)- yell, The result now follows. [] We now discussa menability for A. This involvest he notion of a virtual diagonaflo r A. Let (cid:127)r: A((cid:127)A -(cid:127) A be the multiplicatiomn ap. An element M of (A(cid:127))A)** is calleda virtuald iagonailf , for all a (cid:127) A: (8) aM- Ma (a e A) 7r**(M)-l. The algebraA is called amenableif there existsa virtual diagonalf or A. INVARIANT MEAN CHARACTERIZATIONS 555 The subspac(cid:127)re* (A*) can easilyb e identifiedw ith A* by associating (cid:127)r*(;b)- V½w ith ;b,w here = It is simple to check that the natural A-module structure of A* coincides with the submodulset ructuret hat it inheritsa s a subspacoef Bil(A), and that A* C Bil22(A). Further,r egardingA C A**, the seconde qualityo f (8) becomes: The first equalityo f (8) can be reformulated: (9) v*Mv= M (v ß H). Indeed, (9) is equivalentto Mv = vM for all v ß H, which in turn is equivalent to aM = Ma for all a ß A sinceH spansA . Virtual diagonalfso r submoduleosf Bil(A) containingA * are defined in the obviousw ay. There is a natural H-action on Bil(A) associatedw ith the module actionso f (5) and (21). We define: b)= Clearly,B il(A) is a BanachH -module.U sing( 5), we have (11) v.V =vVv*. SinceB il22(A)i s an A-submoduolef Bil(A), it followsth at it is alsoa n H-submodule. Note alsot hat for ;bß A*, we havev .V(cid:127), = Vvcv-,a nd sincev *v = 1, we alsoh aveV (cid:127).v = V(cid:127). In particularA, * is an H-submoduloe f Bil(A). In the dual H-module actiono n A**, wherew e regardA C A**, we have v.l = l = l.v for all v ß H. The actionso f (10) of coursed uahset o givea n H-modulea ctiono n (Bil(A))*. Thesea ctionsw ill be denotedb y: (v,M) -(cid:127) v.M (M, v) -(cid:127) M.v. Note that, using( 11): (12) M.v = v*Mv. The followingp ropositions howst hat for amenabilityf or A, we requirea virtual diagonaol nlyo n Bil22(A). 556 ALAN L. T. PATERSON Proposition 2. The C*-algebra A is amenablei f and only if there exists a virtual diagonaol n Bil22(A). Proof'. Supposet hat there exists a virtual diagonal M on the space Bil22(A). Let G be the unitaryg roupo f R. Let V ß Bil(cid:127)2(R ). Let v ß G and {u(cid:127)} be a net in H sucht hat us -(cid:127) v stronglyin R ([23, The- orem 2.3.3]). Now sincet he stronga nd weako peratort opologiesc oincide on G ([26, p. 84]), it followsth at the map u -(cid:127) u* is strongo perator continuous,a nd using Propositioni and the triangular inequality, we have u*(cid:127)Vuo(cid:127) - v*V v[[ --(cid:127) O. Hence vMv*(V) - limu,(cid:127)Mu*(cid:127)(V) = M ands oi dentifyingB il(A) with Bil(cid:127)2(R), wes eet hat M is a virtuald iagonal on Bil(cid:127)aR ). By a resulto f [7, 8], R is amenablea nd so injective. So A is amenable(= nuclear)b y the well-knowrne sult( duet o Connesa nd Choi- Effros):A is nucleari f and onlyi f A** is injectire. The rest of the proof is trivial. [] We now discussin variant meanso n groups. Let G be a group. Con- volutiono n t?(xG ) dualisesto givea G-actiono n (fso)(s)-- f(sos) (sof)(s)-- f(sso) for all so,s ß G and all f ß t?(cid:127)(G). A right invariantm ean (RIM) on t?(cid:127)½(G)i s a mean (=state) on t?(cid:127)½(G)w hichi s right invariantu ndert he right dual G-actiono n (t(cid:127)½(G))*. Soa meanm on G is a RIM if and only if m(sf)-m(f) for all f ß t(cid:127)½(G) and all s ß G. The groupG is calledr ight amenablief theree xistsa RIM on t(cid:127)½(G). Left amenabilitayn d two-sidedam enability for G are defined in the obviousw ays. Recent accountso f amenability theorya re giveni n [19, 24, 25]. A subspacXe of t(cid:127)½(G) is ca.liedle ft invarianti f sf ß X for all f ß X and all s ß G. If X is left invariant and contains1 , then a RIM on X is an elementm ß X* satisfyingre (l) = i = Ilmlla nd m(sf) = re(f) for all f ß X and all s ß G. Similarlyw e can definel eft invariantm eans( LIM's) for right invariantu nital subspaceosf t?(cid:127)(G). We will be concernewd ith INVARIANT MEAN CHARACTERIZATIONS 557 invariantm eanso n subspaceosf œoo(H).S inceH is sol argea nd (usually) highly non-commutative,i t is rarely going to be amenable, and we are interested in the existenceo f invariant means on certain smaller, though significants,u bspaceosf /?(cid:127)(H). The subspaceBs2 2(A) and B(A) that will concernu s are associated with the followingm ap A: Bil(cid:127)(A) -(cid:127)/?(cid:127)(S): (13) zx(v)½) = v½*,,O. We definet he followings ubspaceosf œ(cid:127)(G): (14) B(A) = A(Bil(A)) B22(A)= A(Bil22(A)). We giveH the relativea (A,A*) (i.e. the weak)t opologyT. hen ([20]) H is a topologicagl roup.T he invariantu, nital C*-algebraL UC(H) (resp RUC(H)) is the seto f functionsf &/?(cid:127)(H) sucht hat the map s -(cid:127) sf (resp s -(cid:127) rs) is normc ontinuousS. ince1 & H, eachf (cid:127) LUC(H) is continuous. We now collects omes implef acts relating to the spacesB (A) and B22(A). Proposition 3. (a) The map A is an H-equivariant,n orm decreasing, //near map onto B(A). Further, the spacesB (A),B22(A) are invariant subspaceosf (cid:127)(H), and A(A*) = C1. (b) A*(m) is a uirtual diagonaflo r eueryR IM m on B(A). (c) Both subspaceBs (A) and B22(A) are dosed under the complexc onjugationm ap f -(cid:127) f. (d) 1 e B22(A)C LUC(H). Proof: ((cid:127)) (cid:127)or V e mr(.4), u,,(cid:127) e n, w(cid:127) h(cid:127)v(cid:127) a(v.(cid:127))((cid:127)) = v.(cid:127)((cid:127)*, (cid:127))= v((cid:127)*(cid:127)*, (cid:127)) = a(v)((cid:127))= a(v)(cid:127)((cid:127)) a((cid:127).v)((cid:127)) = (cid:127).v((cid:127)*, (cid:127)) = v((cid:127)*(cid:127)*, (cid:127)) = a(v)((cid:127)) = (cid:127)a(v)((cid:127)) so that A is H-equivariant. Obviously,A is norm-decreasinagn d linear. SinceA is equivarianatn dt he spaceBs il(A),Bi122(A)a re H-modulesi,t followsth at B(A) and B22(A)a re invariant.F inally,i f (cid:127)b e A*, then (15) A(V(cid:127))(v) = V(cid:127)(v*, v) = (cid:127))(v*v) = (cid:127)b(1) 558 ALAN L. T. PATERSON sot hat A(V(cid:127)) = (cid:127)b(1)1. (b) If m is a RIM on B(A), then,f or v e H, (cid:127)b e A*, using( a), (12) and (15)' v*a*((cid:127))v = a*((cid:127)).v = a*(mv) = A*(m)(V+) = m(A(V+)) = (cid:127)5(1) = I(V,). So using( 9), A* (m) is a virtual diagonal. (c) For V e BiI(A), defineV * (cid:127) BiI(A) by: v*½,(cid:127)) = v((cid:127),,(cid:127),). Then A(V)= A(V*), and B(A) is closedu nderc omplex-conjugatioTnh.e samep ropertyh oldsf or B22(A): we observeth at the conjugatef of f (cid:127) B22( A) is obtainedb y replacingth e T in (3) by its adjointa ndi nterchanging (cid:127) and 9. (d) SinceA * C Bi122(A),i t followsfr om (a) that 1 (cid:127) B22(A). If V (cid:127) Bi122(A),t hen for u, v (cid:127) H, (16) uA(V) - (cid:127)A(V) (cid:127) IIVu* - v(cid:127)*11 + uV - vV . Now u, (cid:127) u weakly in A if and only if u, -(cid:127) u in the strong operator topologyo f R - A**. It followsfr om (16) and PropositionI that A(V) (cid:127) (cid:127)vc((cid:127)/). [] The next result givesa n invariant mean characterizationo f amenable C*-algebras. Theorem 1. The followings tatementsa re equivalent: (a) A is amenable (b) ,he(cid:127)e e(cid:127)t(cid:127) (cid:127) (cid:127) o(cid:127) (cid:127)((cid:127)) (c) theree xista RIM on LUC(H) Proof: The equivalencoef (a) and (c) followsb y [20]. SinceB 22(A) C LUC(H) by (d) of Propositio3n, weh avet hat (c) implies(b ). Nows uppose that (b) holdsa ndl et R- A**. Let m be a RIM on B2(cid:127)(A). By (3), each f e B(cid:127)(A) is of the formf T(cid:127)n where: (17) fT(cid:127)n(u) = u*Tu(cid:127).rl. INVARIANT MEAN CHARACTERIZATIONS 559 For g 6 B22(A), defineg * 6 too(H) by settingg *(u) = f(u-1), and let Y ='{g*: g 6 B22(A)}. Then m*, wherem *(g*) = re(g), is a left invariant mean( LIM) on Y. Now H is stronglyd ensein the unitaryg roupG of R, and G is a topologicaglr oupi n the strongo peratotro pology([ 12]). From (17), eachg = fT(, extendsu niquelyb y continuityto a continuoufsu nction g' on G-just allowu in (17) to belongt o G. Let Y'= {g': g 6 Y}. Then Y' C RUe(G): this is easilyc heckeads in Proposition1. (Seea lso[ 12].) As in [20, Proposition1 ], theree xistsa n LIM on Y', and a resulto f de la Sarpe (cf [19, p. 78]) givest hat R is injective.H enceA is nucleara nd so amenable.S o (b) implies( a). [] We will show in Theorem 2 below that strong amenability for A is equivalentt o the existenceo f a RIM on B(A). For conveniencew, e write (cid:127)-dS for the weak* closureo f the convexh ull of a subset$ of a Banach space dual X*, and for any Banachs paceX , will regard X C X**. Recallt hat ([15]) the algebraA is calleds tronglya menablief, when- ever X is a unital Banach A-module and D: A -(cid:127) X* is a derivation, then theree xistsc (cid:127)0 in (cid:127)-d{u*D(u): u 6 H} sucht hat D(a) = for all a 6 A. Haagerup([ 11, Lemma3 .4 seq])r emarkst hat the following characterizationo f stronga menabilityh olds. Proposition 4. The C*-a/gebraA is stronglya menableif and onlyf f there existsa virtual diagonaMl in (cid:127)-d{u* © u: u (cid:127) H}. Theorem 2. The C*-algebra A is stronglya menablei f and only if there existsa RIM on B(A). Proof: Supposeth at m is a RIM on B(A). From (b) of Proposition3 , A*(m) is a virtual diagonaflo r A. For u (cid:127) G, let (cid:127) (cid:127) eo(G)* be given by: (cid:127)(qb) = qb(u).I t is easilyc hecketdh at A*((cid:127)) = u* © u. Sincem is in 2-6{fi:u (cid:127) G}, it followtsh at A*(m) is in 2-6{u©* u:u (cid:127) G} in (ACA)*. By Proposition 4, A is strongly amenable. Converselys,u pposeth at A is stronglya menablea, nd let M be as in Propositio4n. Thent heree xistsa net {f,} in P(G) suchth at in the weak* topology 560 ALAN L. T. PATERSON In particular,i f V (cid:127) Bil(A), then (18) © (19) Definem (A(V)) = M(V). Then m is wen-defineadn di s a meano n B(A). Let v 6 G. By (9) and (11), M(v.V) = M(V). Further,b y (a) of Proposi- tion 3, A(v.V) = vA(V). It follows that m is a RIM. [] We concludeb y discussingh ow some characterizationso f amenable and strongly menable C*-algebras can be interpreted as fixed-point or extensiont heoremso f classicaal menabihtyt ype. In particular, usingm od- ulesw ith a certain completelyb oundedp roperty,w e win provea fixed-point theorem for amenableC *-algebrasw hich fills a gap in the hterature. We begin with stronglym enable C*-algebrasf or which the literature is more complete. In [2, 3], Bunceg ivess ix characterizationosf strongly amenableC *-algebras.A n accounot f the resultso f Buncei s giveni n [25, Chapter2 ]. Three of thesec an be interpreteda s fixed-pointt heoremsfo r the unitary group H analogoust o the classicf ixed-point theorem of Day. A fourth can be interpreteda s a strongerv ersiono f a resulti n [17] which is validf or amenablBe anacha lgebras(.S ee[ 6] for an eleganpt roof.)T he remaining two give invariant extensionc haracterizations. An of these char- acterizationsc an be readily provedu singT heorem 2 and the approacho f the fixed-pointth eoremsfo r amenableg roups([ 19, (2.16) if.]). We are particularly interested in the following fixed-point theorem of Bunce. Theorem 3. The C*-algebraA is stronglya menableif and only iœw hen- ever X is a unital Banach A-module and S is weak*-closed convex subset oœX * such that v*Sv = S for ali v (cid:127) H, then there existsg (cid:127) S such that v* gv = g œora 11v (cid:127) H. Bunce givest wo characterizationso f amenableC *-algebras. As in the strongly amenablec ase,o ne of thesei s of the Khelemskii-typea nd the other is an invariant extensionr esult. Both have Banach algebrav ersions,t he extensiovne rsiona ppearingin [18]. We nowd iscusas fixed-pointth eorem for amenableC *-algebrasc orrespondintgo Theorem3 . Let E be a locally convex space which is a unital A-module. The module E is called weaklyc ompletelyb oundedif , for every F 6 E* and