A LIFTING CHARACTERIZATION OF RFD C*-ALGEBRAS
DON HADWIN∗
Abstract
We prove a conjecture of Terry Loring that characterizes separable RFD C*-algebras in terms of a lifting property. In addition we introduce and study generalizations of RFD algebras. Ifkis an infinite cardinal, we say a C*-algebra is residually less thankdimensional, if the family of representations on Hilbert spaces of dimension less thankseparates the points of the algebra. We give characterizations of this property and prove that this class is closed under free products in the nonunital category. For free products in the unital category, the results depend on the cardinalk.
1. Introduction
A C*-algebra A isresidually finite dimensional (RFD) if the collection of all finite-dimensional representations ofA separates the points ofA; equi- valently, if there is a direct sum of finite-dimensional representations ofA with zero kernel. It is clear that every commutative C*-algebra is RFD. Man- Duen Choi [4] showed that free group C*-algebras are RFD. Ruy Exel and Terry Loring [6] proved that the free product of two RFD algebras is RFD.
Terry Loring [12] proved that projective C*-algebras are RFD. The class of RFD C*-algebras plays an important role in the theory of C*-algebras, e.g., [1], [2], [3], [4], [5], [6], [7], [12], [10].
Suppose{e1, e2, . . .}is an orthonormal basis for a Hilbert spaceH, and, for each integer n ≥ 1, let Pn be the projection onto sp({e1, . . . , en}), let Mn =PnB(2)Pnforn≥1, and (see Lemma 1) let
B =
{Tn} ∈∞
n=1
Mn :∃T ∈B(2)withTn→T (∗-SOT)
,
and let
J = {{Tn} ∈B :Tn→0(∗-SOT)}.
∗The author wishes to thank Tatiana Shulman for bringing Terry Loring’s lifting question to his attention, and he wishes to thank the referee for many valuable comments.
Received 30 May 2012.
Then, (see Lemma 1),B is a unital C*-algebra,J is a closed ideal inB and π({Tn})=(∗-SOT)- lim
n→∞Tn
defines a unital surjective∗-homomorphism fromB toB(H )whose kernel is J. IfA is a separable C*-algebra and every (or even one faithful) represent- ation fromA toB(H )lifts to a representation fromA toB, then it is clear thatA must be RFD.
The idea of using this technique to prove an algebra is RFD first appeared in [7] (see also [12]). It was conjectured by Terry Loring (private communication) that every separable RFD C*-algebra has this lifting property. In this paper we prove Loring’s conjecture (Theorem 11).
We also introduce a related notion. Supposek is an infinite cardinal. We say that a C*-algebraA isresidually less thank-dimensional, conveniently denoted byR<kD, if the class of representations ofA on Hilbert spaces of dimension less thankseparates the points ofA; equivalently, if there is a direct sum of such representations that has zero kernel. Note that whenk = ℵ0, we haveR<kDis the same asRFD. We give characterizations ofR<kDalgebras that show that the free product of an arbitrary collection ofR<kDC*-algebras is R<kD. We also give conditions that ensure that the free product (amalgamated overC) of unital C*-algebras in the category of unital C*-algebras isR<kD;
this always happens when each of the algebras has a one-dimensional unital representation.
The proofs of all of our results rely on a simple result (Lemma 1) and results of the author [8], [9] on approximate unitary equivalence and approximate summands of nonseparable representations of nonseparable C*-algebras.
Supposekandmare infinite cardinals. We say that a C*-algebraA ism- generatedif it is generated by a set with cardinality at mostm. For each cardinal s, we letHs be a Hilbert space whose dimension iss. Ifπ : A → B(H )is a∗-homomorphism, we say that thedimensionofπ is dimπ = dimH. We define Rep≤k(A)to be the set of all representationsπ :A →B(Hs)for some s < k.
IfAis a C*-algebra, thenA+denotes the C*-algebra obtained by adding a unit toA (which is different from the unit inA ifA is unital).
We end this section with our key lemma. SupposeHis a Hilbert space andP is a projection inB(H ). We defineMP =P B(H )P. ThenMP is a unital C*- algebra, but the unit isP, not 1. However,MP is a C*-subalgebra ofB(H ). A unitary element ofMP is an operatorU ∈B(H )such thatUU∗=U∗U =P, and is the direct sum of a unitary operator onP (H ) with 0 on P (H )⊥. If P =1, a unitary operator inMP is never unitary inB(H ).
We use the symbol∗-SOT to denote the∗-strong operator topology.
Lemma1.Suppose{Pα}is a net of projections inB(H )such thatPα →1 (∗-SOT)and let
B=
{Tα} ∈
α
MPα :∃T ∈B(H ), Tα →T (∗-SOT)
,
and J = {{Tα} ∈B :Tα →0(∗-SOT)},
and defineπ :B →B(H )by
π({Tα})=(∗-SOT)- lim
α Tα. Then
(1) B is a unital C*-algebra,
(2) J is a closed two-sided ideal inB,
(3) IfT ∈B(H ), then{PαTPα} ∈B andπ({PαTPα})=T, (4) πis a unital surjective∗-homomorphism
(5) IfU ∈B(H )is unitary, then there is a unitary{Uα} ∈B such that π({Uα})=U.
Proof. Statements (1)–(4) are easily proved. To prove (5), note that if U ∈ B(H )is unitary, then there is anA = A∗ ∈ B(H )such thatU = eiA. Using (3), we can easily chooseAα = A∗α for eachα so thatπ({Aα}) = A.
Thus, ifUα =eiAα (inMPα), then{Uα}is unitary inBandπ({Uα})=U. Here is a simple application that gives the flavor of our results.
Corollary2.Every free group is RFD.
Proof. SupposeFis a free group andA = C∗(F) = C∗({Ug : g ∈ F}).
Choose a Hilbert space H and a faithful representation ρ : A → B(H ).
Choose a net{Pα}of finite-rank projections such thatPα →1 (∗-SOT). Ap- plying Lemma 1 we have, for each g ∈ F, we can find a unitary element {Ug,α}in B so thatπ({Ug,α}) = Ug. For eachα, we have a unitary group representationσα :F→MPα defined by
σα(g)=Ug,α.
By the definition ofC∗(F), there is a∗-homomorphismτα : A →Mα such thatτα(Ug)=Ug,α. It follows thatτ :A →Bdefined byτ (Ug)= {Ug,a}is
a∗-homomorphism such thatπ◦τ = ρ. Hence the direct sum of theτα’s is faithful, which shows thatA isRFD.
The following corollary is from [3, Exercise 7.1.4].
Corollary 3.Every C*-algebra is a ∗-homomorphic image of an RFD C*-algebra.
Proof. SupposeA is a C*-algebra. We can assume thatA ⊆ B(H )for some Hilbert spaceH. Choose a net{Pα}of finite-rank projections converging
∗-strongly to 1, and let B,J and π be as in Lemma 1. Then B, and thus π−1(A), isRFDandπ(π−1(A))=A.
2. R<kDAlgebras
We now prove our main results on R<kD C*-algebras. The following two lemmas contain the key tools.
Lemma4.Supposeℵ0≤k≤m, andA isR<kDandm-generated. Then (1) We can writeHm = ⊕
λ∈ Xλ with Card = m, and such that, for every λ ∈ , dimXλ < k and there is a unital representationπλ : A+→B(Xλ)such that the representationπ :A+→B(Hm)defined byπ =⊕πλis faithful. Moreover, this can be done so that, for each λ0∈ , we haveCard({λ∈ :πλ≈πλ0})=m.
(2) It is possible to choose the decomposition in(1)so that, for each cardinal s < k, there is aλ∈ such thatdimXλ =s.
Proof. Since A is R<kD, there is a direct sum of representations in Rep≤k(A)whose direct sum is faithful. SupposeDis a generating set forA and Card(D)≤m. We can replaceDby the∗-algebra overQ+iQgenerated byDmaking the cardinality exceedm. For eacha∈Dwe can choose a direct sum of countably many summands from our faithful direct sum that preserves the norm of a. Hence, by choosing ℵ0Card(D) summands, we get a direct sum that is isometric onDand thus isometric onA. Sinceℵ0Card(D)≤m.
we can replace this last direct sum with a direct sum ofmcopies of itself and get a direct sum on a Hilbert space with dimensionm. We can replace this Hilbert space withHmand get a decomposition as in(1). To get(2), note that, sinceA+has a unital one-dimensional representation, we know that, for every cardinals < k, there is a representation ofA+of dimensions. If we take one such representation for eachs < kand take a direct sum ofmcopies of all of them, we get a representation that has has dimension at mostm, so we add this as a summand to the representation we constructed satisfying(1).
Lemma5.SupposeA is a C*-algebra,k ≤ mare infinite cardinals and D is a generating set for A. Suppose we can write Hm = ⊕
λ∈ Xλ and
π=⊕πλas in part(1)of Lemma 4 so thatCard({λ∈ :πλ≈πλ0})=m for eachλ0 ∈ . Ifρ : A+ → B(Hm)is a unital representation, then, for everyε >0, every finite subsetJ ⊆Dand every finite subsetE⊆Hm, there is a finite subsetF ⊆ , such that, for every finite setGwithF ⊆ G⊆ , if QG is the orthogonal projection onto ⊕
λ∈GXλ, then there is a unitary U ∈QGB(Hm)QGsuch that, for everya∈J ande∈E, we have
[ρ(a)−UG∗π(a)UG]e =
ρ(a)−UG∗
λ∈G
πλ
(a)UG
e < ε.
Proof. Let Rep(A, Hm)denote the set of all unital representations from AintoB(Hm)topologized by the topology of pointwise∗-SOT convergence, and letS denote the closure of the elements in Rep(A, Hm)that are unitarily equivalent to π. We want to prove that ρ ∈ S. It follows that ifa ∈ A and a = 0, then rankπ(a) = m = rank(π ⊕ρ)(a). Hence, by [8], π is approximately unitarily equivalent toπ⊕ρ. Hence there is a unitaryXsuch that X∗(π ⊕ρ)X ∈ S. However, by [8] (and ideas in [9]), ρ is a point-
∗-SOT limit of representations unitarily equivalent to π⊕ρ. Indeed, if for each finite subset α of Hm, Uα : Hm → Hm ⊕Hm is a unitary operator such that Uαx = 0⊕x for each x ∈ α, we have {Uα} forms a net such that{Uα∗(π⊕ρ)(a)Uα}converges∗-strongly toρ(a)for everya∈A. Thus Wα =X∗Uα is a unitary inB(Hm)for eachα, and, for everya∈A,
(∗-SOT)lim
α Wα∗[X∗(π⊕ρ)(a)X]Wα =ρ(a).
Hence,ρ∈S.
Thus there is a net{Vλ}of unitaries inB(Hm)such that{Vλ∗π(·)Vλ}con- verges pointwise∗-strongly toρ. However, the net{QF :F ⊆ , F is finite} is a net of projections converging∗-strongly to 1. Hence, by Lemma 1, each Vλis a∗-SOT limit of unitaries in the union ofQFB(Hm)QF (F ⊆ ,F is finite). The result now easily follows.
Theorem6.Supposeℵ0≤k≤m, andAism-generated with a generating setGwithCardG ≤m. The following are equivalent.
(1) A isR<kD.
(2) There is a faithful unital ∗-homomorphism ρ : A+ → B(Hm) such that, for everyε >0, every finite subsetE⊆Hmand every finite subset W ⊆G, there is a projectionP ∈B(Hm)and a unital∗-homomorphism τ :A →MP =P B(Hm)Psuch that, for everye∈Eand everya∈W we have
[τ (a)−ρ(a)]e< ε.
(3) There is a faithful unital representationρ : A+ → B(Hm)and a net {Pα} of projections in B(Hm), each with rank less than k, such that Pα → 1(∗-SOT)and such that, for eachα, there is a representation πα :A →MPαsuch that, for everya∈A, we have
πα(a)→ρ(a)(∗-SOT).
(4) For every unital representationρ : A+ → B(Hm)there is a net{Pα} of projections inB(Hm), each with rank less thank, such thatPα →1 (∗-SOT)and such that, for eachα, there is a representationπα :A → MPαsuch that, for everya∈A, we have
πα(a)→ρ(a)(∗-SOT).
Proof. (2)⇒(1) LetAbe the set of triples(ε, E, W )ordered by(≥,⊆,⊆).
Ifα =(ε, E, W )letτα :A →PαB(Hm)Pαguaranteed by (2). SinceG=G∗ we have
(∗-SOT)lim
α τα(a)=ρ(a)
for everya∈G. Sinceρand eachτα is a∗-homomorphism, the set ofa∈A for which (∗-SOT)limατα(a)= ρ(a)is a unital C*-algebra and is thusA+. Hence, for everya∈A+, we have
a = ρ(a) ≤sup{τα(a):α∈A}. Therefore the direct sum of theτα’s is faithful and (1) is proved.
(3)⇒(2). This is obvious.
(4)⇒(3). It is clear that we need only show that there is a faithful unital representationρ:A+→B(Hm). Supposeτ :A+→B(M)is an irreducible representation, and supposeDis a generating set with Card(D)≤m. LetA0
be the unital∗-subalgebra ofA+ over the fieldQ+iQof complex rational numbers. ThenA0is norm dense inAand CardA0=CardD≤m. Suppose f ∈M is a unit vector. Sinceτ is irreducible,τ (A0)f must be dense inM. SupposeBis an orthonormal basis forM, and, for eache∈ BletUe be the open ball centered atewith radius√
2/2. EachUemust intersect the dense set τ (A0)f, and since the collection{Ue:e∈B}is disjoint, we conclude that
dimM =CardB≤Cardτ (A0)f ≤Card(A0)≤m.
We know that for every x ∈ A0 there is an irreducible representation τx : A+→B(Mx)such thatτx(x) = x. Since dim⊕
x∈A0Mx≤m·m=m, there is a representationρ : A+ → B(Hm)that is unitarily equivalent to a
direct sum ofmcopies of⊕
x∈A0τx. Henceρis isometric on the dense subset A0, which impliesρis faithful.
(1)⇒(3). Since A is R<kD, we can choose a decomposition Hm = ⊕
λ∈ Xλ and representationπ = ⊕
λ∈ πλ as in part (1) of Lemma 4. Now (3) follows from Lemma 5.
We see that the class ofR<kDalgebras is closed under arbitrary free products in the nonunital category of C*-algebras.
Theorem7.Supposek is an infinite cardinal and{Aι :i ∈I}is a family ofR<kDC*-algebras. Then the free product∗Ai isR<kD.
Proof. Choose an infinite cardinalm≥k+
i∈ICard(Ai). Since∗i∈IAi
is generated byG=
i∈IAi
\{0} ⊆ ∗i∈IAi, clearly∗i∈IAiism-generated.
Choose a set with Card( )= mand letSbe the set of cardinals less than k. Write
Hm=
⊕ s∈S λ∈
Xs,λ
where dimXs,λ=sfor everys∈Sandλ∈ . It follows that, for eachi∈I, we can find a representationπi :Ai →B(Hm)such that
πi =
⊕ s∈S λ∈
πs,λi
satisfying (1) and (2) of Lemma 4. Suppose ε > 0, E ⊆ Hm is finite and W ⊆ G is finite. We can write W as a disjoint union ofWi1, . . . , Win with Wi =W∩Ai. Letρi be the restriction ofρtoAi. Applying Lemma 5 toAij
andρij andπij for 1 ≤ j ≤ n, we can find one finite subsetG⊆ S× so that ifP is the projection on ⊕
(s,λ)∈GXs,λ, then there are unitary operators Ui1, . . . , Uin ∈MP =P B(Hm)P so that, for 1≤j ≤n,a ∈Wj,e∈E, we have [ρij(a)−Uij∗πij(a)Uij]e< ε.
Defineτij :Ai+j →MP by
τij(a)=Uij∗πij(a)Uij, and fori∈I\ {i1, . . . , in}defineτi :Ai →MP by
τi(a)=P πi(a)P .
Then, by the definition of free product, there is a representationτ :∗i∈IAi+→ MP such thatτ|Ai =τi for everyi ∈I. It follows that, for everye∈Eand
everya∈W,
[ρ(a)−τ (a)]e< ε.
It follows from part (2) of Theorem 6 that∗i∈IAi isR<kD.
Corollary 8. Suppose k is an infinite cardinal and {Aι : i ∈ I}is a family ofR<kDC*-algebras such that eachAi has a one-dimensional unital representation. Then the unital free product∗Ci∈IAi isR<kD.
Proof. This follows from the fact that if τi : Ai → C is a unital ∗- homomorphism for eachi∈I, then∗Ci∈IAiis∗-isomorphic to(∗i∈Ikerτi)+. Without the condition on unital one-dimensional representations, the pre- ceding corollary may fail. For example,∗Cn∈NMn(C)is notRFD(=R<ℵ0D), even though eachMn(C)isRFD. The reason is that each unital representation of the free product must be injective on eachMn(C)and must have infinite- dimensional range.
However, there is something we can say about the general situation. Ifkis a limit cardinal (i.e.,k is the supremum of all the cardinals less thank), the cofinalityofkis the smallest cardinalsfor which there is a setEof cardinals less thankwhose supremum isk. Clearly, the cofinality ofkis at mostk. Ifkis not a limit cardinal, then there is a cardinalssuch thatkis the smallest cardinal larger thans, and ifEis a set of cardinals less thank, then sup(E)≤s < k.
Theorem9.Supposek is an infinite cardinal and{Aι :i ∈I}is a family of unitalR<kDC*-algebras. Then
(1) Ifk is a limit cardinal andCard(I )is less than the cofinality ofk, then the free product∗Ci∈IAi isR<kD.
(2) Ifkis not a limit cardinal, then the free product∗Ci∈IAi isR<kD.
Proof. (1). Choose m ≥ k +
i∈ICard(Ai), and choose a set with Card( )=m. Using Lemma 4 we can, for eachi ∈I, find a faithful repres- entationπi =
λ∈ πλ,iso that dimπi =mand, for everyi ∈I andλ∈ , we have dimπλ,i< k. Since Card(I )is less than the cofinality ofk, we have, for eachλ∈ , a cardinalsλ< ksuch that supi∈Idimπλ,i ≤sλ. If we replace eachπλ,iwith a direct sum ofsλcopies of itself, we get a new decomposition which we will denote by the same names such that, for eachi and eachλwe have dimπλ,i=sλ. Hence we may write direct sum decompositions of theπi’s with respect to a common decompositionHm=
λ∈ Xλwhere dimXλ=sλ for everyλ∈ . The rest now follows as in the proof of Lemma 7.
(2). Ifkis not a limit cardinal, there is a largest cardinals < k. Repeat the proof of part (1) withsλ=sfor everyλ∈ .
Remark 10. We cannot remove the condition on Card(I ) in part (1) of Theorem 9. Supposekis a limit cardinal andIis a set of cardinals less thank
whose cardinality equals the cofinality ofkand such that sup(I )=k. For each infinite cardinalm, choose a set mwith cardinalitym, and letSmdenote the universal unital C*-algebra generated by{vλ:λ∈ m}with the conditions
(1) vλ∗vλ=1 for everyλ∈ m, (2) vλ1vλ∗
1vλ2v∗λ
2 =0 forλ1=λ2in m.
SinceSm ism-generated, it follows that every irreducible representation of Smis at mostm-dimensional (see the proof of (4)⇒(3) in Theorem 6). Hence Smis separated bym-dimensional representations. On the other hand, ifπis a unital representation ofSm, then{π(vλ;vλ∗):λ∈ m}is an orthogonal family of nonzero projections, which implies that the dimension ofπis at leastm.It follows that eachSs isR<kDfors∈I. However, any unital representationπ of the free product∗Cs∈ISs must induce a unital representation of eachSs, so its dimension is at least supI =k. Hence∗Cs∈ISs is notR<kD.
3. Separable RFD Algebras
In this section we show that for a separable C*-algebra being RFD is equivalent to a lifting property.
Suppose{e1, e2, . . .}is an orthonormal basis for a Hilbert space2, and, for each integer n ≥ 1, let Pn be the projection onto sp({e1, . . . , en}). Let Mn =PnB(2)Pnforn≥1, and, following Lemma 1, let
B =
{Tn} ∈ ∞ n=1
Mn:∃T ∈B(2)withTn→T (∗-SOT)
,
and let
J ={{Tn} ∈B :Tn→0(∗-SOT)}.
Then, by Lemma 1, we have thatBis a unital C*-algebra,J is a closed ideal inB and
π({Tn})=(∗-SOT)- lim
n→∞Tn
defines a unital surjective∗-homomorphism fromB toB(H )whose kernel is J. We can now give our characterization of RFD for separable C*-algebras.
Theorem 11. Suppose A is a separable C*-algebra. The following are equivalent.
(1) A is RFD
(2) For every unital∗-homomorphismρ : A+ →B(2)there is a unital
∗-homomorphismτ :A+→B such thatπ◦τ =ρ.
Proof. The implication (2)⇒(1) is clear.
(1)⇒(2). SupposeA =C∗({a1, a2, . . .})isRFDandρ:A+→B(2)is a unital∗-homomorphism. It follows from Theorem 6 that there is an increasing sequence{nk}of positive integers and unital∗-homomorphismsτk : A → Mnk such that
[τk(aj)−ρ(aj)]ei<1/k
for 1≤i, j ≤k. It follows thatτnk(a)→ρ(a)(∗-SOT) for everya ∈A+. If nk< n < nk+1we defineτn:A+→Mnby
τn(a)=
⎛
⎜⎜
⎝ τnk(a)
β(a) ...
β(a)
⎞
⎟⎟
⎠,
whereβ:A+→Cis the unique∗-homomorphism with kerβ=A, relative to the decomposition
Pn(2)=Pnk(2)⊕Ce1+nk⊕ · · · ⊕Ce−1+nk+1.
It is easily seen thatτn(a)→ρ(a)(∗-SOT) for everya ∈ A+. If we define τ :A →B by
τ (a)= {τn(a)}, we see thatπ◦τ =ρ.
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DEPARTMENT OF MATHEMATICS AND STATISTICS UNIVERSITY OF NEW HAMPSHIRE
DURHAM, NH, 03824 USA
E-mail:don@math.unh.edu
