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ON THE STABLE RANK AND REAL RANK OF GROUP C

-ALGEBRAS OF NILPOTENT

LOCALLY COMPACT GROUPS

ROBERT J. ARCHBOLD and EBERHARD KANIUTH

Abstract

It is shown that ifGis an almost connected nilpotent group then the stable rank ofC(G)is equal to the rank of the abelian groupG/[G, G]. For a general nilpotent locally compact groupG, it is shown that finiteness of the rank ofG/[G, G] is necessary and sufficient for the finiteness of the stable rank ofC(G)and also for the finiteness of the real rank ofC(G).

Introduction

For aC-algebraA, the real rank RR(A)[3] and the stable rank sr(A)[18] have been defined as numerical invariants giving non-commutative analogues of the real and complex dimension of topological spaces. More precisely, for the continuous functions on a compact Hausdorff spaceXone has RR(C(X)) = dimXand sr(C(X))=1

2dimX

+1, where dimXis the covering dimension ofX[17]. For unitalA, the stable rank sr(A) is either∞or the smallest possible integernsuch that eachn-tuple inAncan be approximated in norm byn-tuples (b1, . . . , bn)such thatn

i=1bibi is invertible. Similarly, the real rank RR(A) is either∞or the smallest non-negative integernsuch that each(n+1)-tuple of self-adjoint elements inAn+1can be approximated in norm by(n+1)-tuples (b0, b1, . . . , bn)of self-adjoint elements such thatn

i=0b2i is invertible. For non-unitalA, these ranks are defined to be those of the unitization ofA.

Several authors have computed or estimated the stable and the real rank of groupC-algebrasC(G)for various classes of locally compact groupsG[1], [6], [7], [13], [15], [19], [20], [21], [22], [23], [23], [25], [26]. For example, for simply connected nilpotent Lie groups, Sudo and Takai [25] (following earlier work of Sheu [20]) have shown that sr(C(G))is the complex dimension of the space of characters of G. On the other hand, for the free groupF2 on 2 generators it has been shown that sr(C(F2)) = RR(C(F2)) = ∞[18],

Supported by a travel grant from the German Research Foundation.

Received May 20, 2004.

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[15], but sr(Cr(F2)) = RR(Cr(F2)) = 1 [6] (wherer indicates the reduced C-algebra of a non-amenable group).

In Section 1 of this paper, the result of Sudo and Takai mentioned above is extended to almost connected nilpotent groupsG. To be specific,

sr(C(G))=1+ 1

2dimG/[G, G]

=1+ 1

2rank(G/[G, G])

<∞ (Theorem 1.5), where the rank of an abelian group is as defined in Sec- tion 1. The method of proof involves structure theory for G together with Nistor’s estimate for the stable rank of C-algebras containing certain con- tinuous trace ideals [16]. As a corollary, it follows that either RR(C(G)) = rank(G/[G, G])(a sufficient condition for this equality is that rank(G/[G, G]) is odd) or RR(C(G))=1+rank(G/[G, G]).

In Section 2, general nilpotent locally compact groupsGare considered.

The main result (Theorem 2.8) is that the finiteness of the rank ofG/[G, G] is necessary and sufficient for the finiteness of sr(C(G)) and also for the finiteness of RR(C(G)). In addition to further structural properties ofG, the proof uses an estimate from Section 1 and also Rieffel’s estimate for the stable rank of crossed products by the integers [18].

The results of Sections 1 and 2 suggest that the stable rank ofC(G), for a nilpotent group G, may depend only on the abelian quotient G/[G, G].

Further evidence in this direction is provided by [24, Theorem 2], which deals with the case of finitely generated, torsion-free, two-step nilpotent (discrete) groups. On the other hand, we give examples to show that the conclusions of Theorems 1.5 and 2.8 may fail to hold if the hypothesis of nilpotency is replaced by solvability.

1. Almost connected nilpotent groups

For any locally compact groupG, letG0denote the connected component of the identity. Recall thatGis said to bealmost connectedif the quotient group G/G0is compact.

LetGbe a nilpotent locally compact group. ThenGc, the set of all compact elements ofG, is a closed normal subgroup of G [10, Corollary 3.5.1 and Lemma 3.8]. Moreover, Gc is compact wheneverGis compactly generated [10, Theorem 9.7]. From this it can easily be deduced thatG/Gcis compact- free (see [13, Remark 1]). In particular,(G/Gc)0, the connected component ofG/Gc, is a simply connected nilpotent Lie group. Also,G0Gcis open inG andG/G0Gcis torsion-free [10, Theorem 8.3]. SinceG0Gc/Gcis connected and open inG/Gc,G0Gc/Gc =(G/Gc)0. HenceG/Gcis a Lie group. When Gis discrete,Gcis just the subgroup consisting of all elements of finite order

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which is usually denotedGt and called the torsion subgroup ofG. Finally, recall that if G is a torsion-free nilpotent group, then all the subquotients Zj+1(G)/Zj(G)arising from the upper central series ofGare torsion-free as well [2, Corollary 2.11].

We next introduce the group theoreticalrankof a locally compact abelian group. For a discrete torsion-free abelian groupD, rankDmeans the torsion- free rank ofD(see [11]), that is, rankDis the maximal number of independent elements ofDwhen this number is finite and rankD = ∞otherwise. LetG be an arbitrary locally compact abelian group. ThenG/Gc =Rk×D, where Dis torsion-free discrete, and the rank ofGis defined to bek+rankD. Note that rankG <∞wheneverG/Gcis compactly generated. On the other hand, the additive group of rational numbers has rank 1.

Throughout the paper, we shall frequently use the fact that ifGis a locally compact abelian group andH is a closed subgroup ofG, then rank(G/H)≤ rankG. This is easily seen as follows. Define a closed subgroup K of G by KH and K/H = (G/H )c. ThenG/Gc = Rm×D and G/K = (G/H )/(G/H )c = Rn ×E, wherem, n ∈ N0 and D and E are torsion- free abelian discrete groups. Since GcK, the quotient homomorphism GG/H induces a homomorphism q : G/GcG/K. It follows that q(Rm)=Rn, whencenm, and henceqgives rise to a homomorphism from Donto E. Now, it is immediate from the definition of the torsion-free rank that rankE≤rankD. Thus

rank(G/H )=n+rankEm+rankD=rankG.

Denoting byGthe dual group ofG, we have

RR(C(G))=RR(C0(G)) =dimG=rankG and

sr(C(G))=sr(C0(G)) =1+ 1

2dim G

=1+ 1

2rankG

(see [1, Section 2] and the references therein).

In passing, we note that if J is a closed ideal of a C-algebra A then sr(J ),sr(A/J ) ≤ sr(A)[18, Section 4] and similarly for the real rank [9, Théorème 1.4].

Lemma1.1. LetGbe a projective limit of groupsGα =G/Kα, where each Kα is a compact normal subgroup ofG. Then

(i) sr(C(G))=supαsr(C(Gα))andsr(Cr(G))=supαsr(Cr(Gα)). (ii) RR(C(G))=supαRR(C(Gα))andRR(Cr(G))=supαRR(Cr(Gα)).

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Proof. LetK be any compact normal subgroup ofG, and letq : GG/K denote the quotient homomorphism andµK normalized Haar measure on K. Then µK is a central idempotent measure, and the map φ : ffq establishes an isomorphism betweenL1(G/K) and the closed ideal L1(G)µK of L1(G). ForπGandfL1(G/K),π(fq) = π(fq)π(µK)=0 wheneverπG/K ◦q. Notice that ifπ =σ◦qwithσG/K, thenπGr if and only if σ(G/K) r. This implies that fC(G/K) = f ◦qC(G)andfCr(G/K) = f ◦qCr(G), and henceφextends uniquely to isomorphisms fromC(G/K)onto the closed idealL1(G)µKofC(G) and fromCr(G/K)onto the closed idealL1(G)µKr

ofCr(G).

Now, in the situation of the lemma, letIα and Jα denote the closure of L1(G)µKαinC(G)andCr(G), respectively. Then∪αIαis dense inC(G) and∪αJα is dense inCr(G)since∪αL1(G)µKα is dense inL1(G). Since the sets{Iα}and{Jα}are suitably directed, it follows from [18, Theorem 5.1]

and [13, Lemma 4(i)] that sr(C(G))≤sup

α sr(Iα)=sup

α sr(C(Gα))≤sr(C(G)) and

RR(C(G))≤sup

α RR(Iα)=sup

α RR(C(Gα))≤RR(C(G)), and similarly for the reducedC-algebras. This yields (i) and (ii).

Alternatively, to prove Lemma 1.1, we could exploit Proposition 2.2 of [14].

Lemma1.2. LetGbe a Lie group such thatG0is nilpotent andG/G0is finite. Then

sr(C(G))≤max

2,1+ 1

2rank(G0/[G0, G0]) .

Proof. The connected nilpotent Lie group G0 is type I, and every irre- ducible representation ofG0is either 1-dimensional or infinite dimensional.

This follows from the fact that every irreducible representation of a connected nilpotent Lie group is induced from a character of some closed subgroup [4].

SinceG/G0is finite,Gis type I and dimπ ≤[G:G0] for every finite dimen- sional irreducible representationπofG. Indeed, for any suchπ,π≤indGG0χ for some characterχ ofG0. In particular, the setGfin consisting of all finite dimensional representations inGis closed inG. SinceGfin has the property that ifπ , ρGfinthen every irreducible subrepresentation ofπ⊗ ¯ρbelongs toGfin, it follows thatGfin=G/N for some closed normal subgroupNofG.

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Actually,N = [G0, G0]. In fact, ifx ∈ [G0, G0] thenχ(x)= 1 for all char- actersχ ofG0and henceπ(x)is the identity operator for everyπGfin, and conversely, ifxN then indGG0χ(x)is the identity operator, whencexG0

andχ(x)=1 for every characterχ ofG0.

We claim thatC(G)has a composition series of finite length in which the successive quotients have continuous trace. By a result of Dixmier [5], theC-algebra of any simply connected nilpotent Lie group has this property.

Hence the same is true for theC-algebra of any connected nilpotent Lie group because itsC-algebra is a quotient of theC-algebra of its simply connected covering group. SinceG/G0is finite, it follows thatC(G)has a composition series of finite length in which all the successive quotients have continuous trace [8, Corollary 1].

Now, letI be the closed ideal ofC(G)such that I = G\Gfin. ThenI is a separable type IC-algebra all of whose irreducible representations are infinite dimensional. By intersecting the ideals in the composition series for C(G)withI, we obtain a sequence

I =I0I1⊇ · · · ⊇Ir+1= {0}

of closed ideals ofC(G)such thatIj/Ij+1has continuous trace for 0≤jr. Applying Lemma 2 of [16] repeatedly, we get

sr(C(G))≤max{2,sr(C(G)/Ir)} ≤max{2,max{2,sr(C(G)/Ir−1)}}

. . .≤max{2,sr(C(G)/I)}.

Recall thatC(G)/I = C(G/N) = C(G/[G0, G0]). SinceG0/[G0, G0] is an abelian normal subgroup of finite index inG/[G0, G0] andGis second countable, by [19, Corollary 3.3]

sr(C(G/N))≤sr(C(G0/[G0, G0]))=1+ 1

2rank(G0/[G0, G0])

, which finishes the proof.

Proposition1.3. Let Gbe an almost connected locally compact group such thatG0is nilpotent. Then

sr(C(G))≤max

2,1+ 1

2rank(G0/[G0, G0]) .

Proof. By [12]Gis a projective limit of Lie groupsG/Kα, and by Lem- ma 1.1

sr(C(G))=sup

α sr(C(G/Kα)).

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Thus it suffices to show that ifKis any compact normal subgroup ofGsuch thatG/Kis a Lie group, then

sr(C(G/K))≤max

2,1+ 1

2rank(G0/[G0, G0]) .

Observe next that (G/K)0 = G0K/K. Indeed, G0K/K is connected and (G/K)/(G0K/K)=G/G0Kis a totally disconnected Lie group and hence discrete. Also

(G/K)0/[(G/K)0, (G/K)0]=(G0K/K)/[G0K/K, G0K/K], which is a quotient ofG0/[G0, G0]. Thus

rank((G/K)0/[(G/K)0, (G/K)0])≤rank(G0/[G0, G0]).

Since(G/K)0is nilpotent and has finite index inG/K, Lemma 1.2 yields that sr(C(G/K))≤max

2,1+

1

2rank((G/K)0/[(G/K)0, (G/K)0])

≤max

2,1+ 1

2rank(G0/[G0, G0]) , as required.

WhenGitself rather than justG0is nilpotent, we can prove a considerably stronger result (Theorem 1.5) which generalizes the result of Sudo and Takai mentioned in the introduction. For that, we need the following fairly simple lemma.

Lemma1.4. LetGbe an almost connected nilpotent locally compact group.

Then

rank(G/[G, G])=rank(G0/[G0, G0]).

Proof. Choose any compact normal subgroupKofGsuch thatG/Kis a Lie group. Since

(G/K)/[G/K, G/K]=(G/K)/([G, G]K)/K =G/[G, G]K and [G, G]K/[G, G] is compact, we obtain

rank((G/K)/[G/K, G/K])=rank(G/[G, G]).

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ReplacingGwithG0andKwithKG0, we get

rank(G0/[G0, G0])=rank(G0/(G0K)/[G0/(G0K), G0/(G0K)])

=rank((G0K/K)/[G0K/K, G0K/K])

=rank((G/K)0/[(G/K)0, (G/K)0])

(compare the proof of Proposition 1.3). Thus it suffices to prove the lemma whenGis a Lie group, so thatG/G0is finite.

To that end, we observe first that ifHis a torsion-free nilpotent group having an abelian normal subgroup of finite index, thenH is abelian. This is shown by induction on the length of the upper central series ofH. SinceH/Z(H )is torsion-free and has an abelian subgroup of finite index, it is abelian. ThusH is 2-step nilpotent, torsion-free and has an abelian normal subgroupAof finite indexd, say. Forx, yH, it follows that

[x, y]d2 =[x, yd]d =[xd, yd]=e.

SinceH is torsion-free, we obtain that [H, H]= {e}.

Now define a closed normal subgroup N of G by N ⊇ [G0, G0] and N/[G0, G0]=(G/[G0, G0])c. Then, since(N∩G0)/[G0, G0] has only com- pact elements,

rank(G0/[G0, G0])=rank(G0/(NG0))=rank(G0N/N).

Moreover, G0N/N is an abelian subgroup of finite index in the compact- free nilpotent groupG/N. By the above observation, G/N is abelian. Thus N ⊇ [G, G] ⊇ [G0, G0] and N/[G, G] has only compact elements. This implies

rank(G/[G, G])=rank(G/N)=rank(G0N/N)=rank(G0/[G0, G0]), which concludes the proof.

Before proceeding we mention that ifGis an almost connected nilpotent locally compact group andGis non-abelian, then rank(G/[G, G])≥2. This can be seen as follows.

By Lemma 1.4, we can assume that G is connected. Moreover, we can assume thatG is a Lie group since for any compact normal subgroupC of G,(G/C)/[G/C, G/C] is a quotient ofG/[G, G]. LetH denote the simply connected covering group ofGandq:HGthe covering homomorphism.

LetZn+1(G)=GandZn(G)=G. Then [G, G]⊆Zn(G)andq1(Zj(G))= Zj(H ), 1≤jn+1. Now, it is well-known that sinceHis simply connected, H/Zn(H )=Rdfor somed≥2. So the above claim follows.

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Theorem 1.5. Let Gbe an almost connected nilpotent locally compact group. Then

sr(C(G))=1+ 1

2rank(G/[G, G])

=1+ 1

2rank(G0/[G0, G0])

=sr(C(G0)) <∞.

Proof. We can assume that Gis non-abelian. By Proposition 1.3, Lem- ma 1.4 and the fact thatC(G/[G, G])is a quotient ofC(G),

sr(C(G))≤1+ 1

2rank(G0/[G0, G0])

=1+ 1

2rank(G/[G, G])

=sr(C(G/[G, G]))≤ sr(C(G)).

This proves the first equality. ReplacingGwithG0and applying Lemma 1.4 again we obtain the remaining equalities. Finally, note thatGis compactly generated and that the rank of any compactly generated abelian group is finite.

Corollary1.6. LetGbe an almost connected nilpotent locally compact group.

(i) Ifrank(G/[G, G])is odd, then

RR(C(G))=rank(G/[G, G]).

(ii) Ifrank(G/[G, G])is even, then

rank(G/[G, G])≤RR(C(G))≤1+rank(G/[G, G]).

Proof. Both (i) and (ii) are immediate consequences of Theorem 1.5 and the estimate RR(A) ≤ 2 sr(A)−1 which holds for arbitraryC-algebrasA [3, Proposition 1.2].

In [26, Example 4.2] the so-called split oscillator groupG (a semidirect product ofRwith the Heisenberg group) was considered. It was shown that sr(C(G)) = 2 although rank(G/[G, G]) = 1. So the last equality of The- orem 1.5 does not hold in general for simply connected solvable Lie groups of type I.

The following example shows that all the other equalities of Theorem 1.5 may also fail for solvable almost connected groups.

Example1.7. LetGbe the semidirect productG=Z2Rn, whereZ2= {1,−1}acts onRnby coordinatewise multiplication. ThenG0=[G, G]=Rn

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and hence

sr(C(G0))=1+n 2

,rank(G0/[G0, G0])=n and rank(G/[G, G])=0.

On the other hand, since every non-trivial character ofRninduces a 2-dimen- sional irreducible representation ofG,C(G)has a closed idealJ isomorphic to C0((Rn \ {0})/Z2, M2(C)) with quotientC(G)/J = C2. Since n is the maximal dimension of compact subsets of the quotient space(Rn\ {0})/Z2,

sr(C(G))=max

sr(C0((Rn\ {0})/Z2), M2(C)),sr(C2)

=1+ 1

2 n

2

(see [1, Lemmas 1.3 and 1.2]).

2. Finiteness of the ranks for nilpotent groups

In this section we are going to characterize, for a general nilpotent locally compact groupG, the finiteness of both sr(C(G))and RR(C(G))in terms of a simple and purely group-theoretic condition, the finiteness of the rank of the abelian locally compact groupG/[G, G] (Theorem 2.8). To establish this result, we need to show that ifGis a (discrete) nilpotent group andG/[G, G] has finite rank then all the subquotients arising from the lower central series of Galso have finite rank. The analogous conclusion is known to be true when the finite rank condition is replaced by finite generation. The arguments in that case have influenced our proofs (in particular, that of Lemma 2.3). Note that in Lemmas 2.3 and 2.5 we have temporarily suspended the convention thatG0is the connected component of the identity of a locally compact group. Instead, we use the notationG=G0G1⊇ · · · ⊇Gn+1= {e}for the lower central series of an(n+1)-step nilpotent discrete group.

Lemma2.1. LetAbe an abelian group. ThenrankAr if and only if every finitely generated subgroupBof Ais of the formB = Zk ×F, where krandF is a finite group.

Proof. Suppose first that rankAr and let B be a finitely generated subgroup ofA. ThenB = Zk ×F wherek ≥ 0 andF is finite. ThusA/At contains a copy ofZk and sor ≥rank(A/At)k.

Conversely, let{x1, . . . , xl}be a subset ofAwhose image inA/At is an independent set. Assuming the stated condition on finitely generated subgroups of A, the subgroup generated by x1, . . . , xl is of the form Zk ×F where krandF is finite. Passing toA/At, we obtain a copy ofZk containing an independent set oflelements. Hencelkr, and so rankAr.

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Lemma2.2. LetH be a nilpotent group and let{e} =H0H1⊆ · · · ⊆ Hn = H be a sequence of normal subgroups ofH such that Hj/Hj−1 = Zdj ×Fj for some finite groupFj anddj ∈N0(1jn). ThenH contains a normal subgroupNsuch thatH/Nis finite andN has a composition series N0= {e} ⊆N1⊆ · · · ⊆Nn=NwhereNj−1is normal inNj,Nj/Nj−1=Zej andn

j=1ejn

j=1dj. In particular,Nis generated byn

j=1djelements.

Proof. Ifn = 1, we may simply putN = Zd1. Now suppose thatn > 1 and that the result has been established up to stagen−1. SinceH is finitely generated, it contains a torsion-free normal subgroup K of finite index [2, Theorem 2.1]. LetKj =K∩Hjforj =1, . . . , n. Then the quotientsKj/Kj−1

have similar structure to the quotientsHj/Hj−1, with no increase in thedj. Applying the induction hypothesis toK/K1, we obtain a normal subgroupL of finite index inKand a series

K1=L1L2⊆ · · · ⊆Ln =L

such thatLj−1is normal inLj,Lj/Lj−1is isomorphic toZlj (2≤jn) and n

j=2ljn

j=2dj.

SinceLhas finite index inH, there exists a normal subgroupNofH such thatNLandH/Nis finite. LetNj =LjN,1≤jn. ThenNj/Nj−1

is isomorphic to a subgroup ofKj/Kj−1and hence is isomorphic toZej where ejlj (2 ≤ jn). Finally, sinceN1 = L1N is isomorphic toZd1, the inductive step is complete.

Lemma2.3. LetGbe a nilpotent group and letGn, n=0,1, . . .denote the lower central series ofG. Suppose thatGn+1= {e}and that all the quotient groupsGk−1/Gk,1≤kn, are of finite rank. ThenGnhas finite rank.

Proof. Let a1, . . . , amGn = [G, Gn−1] be given. Then eachai,1 ≤ im, can be written as

ai =

mi

j=1

[bij, cij],

wherebijG, cijGn−1,1 ≤ jmi, mi ∈ N. Since all the quotients Gk−1/Gk, 1 ≤ kn, have finite rank, by Lemma 2.2 there exist N ∈ N and elementsu1, . . . , usG, where s depends only on G/Gn, such that, denoting by U the subgroup generated by u1, . . . , us, bNijUGn for all 1≤im,1≤jmi. Similarly, sinceGn−1/Gnhas finite rank, there exist M ∈Nandv1, . . . , vtGn−1, wheret depends only onGn−1/Gn, such that cMijV Gn(V the subgroup ofGn−1generated byv1, . . . , vt). Thus

bNij =uijxij and cMij =vijyij,

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whereuijU, vijV, xij, yijGn. SinceGn is contained in the centre of G, we have [x, y1y2] = [x, y1][x, y2] for xG, y1, y2Gn−1 and [x1x2, y]=[x1, y][x2, y] forx1, x2G, yGn−1. It follows that

aiNM = mi

j=1

[bij, cij] NM

=

mi

j=1

[bNij, cMij]=

mi

j=1

[uijxij, vijyij]=

mi

j=1

[uij, vij]. So, ifais any element of the subgroupAgenerated bya1, . . . , am, thenaNM is contained in the subgroupH ofGngenerated by the set of all commutators [uk, vl],1≤ks,1≤lt.

SinceA=Zk×F, wherek ≥0 andF is finite, and{aNM :aA} ⊆H, it follows thatH contains a subgroup isomorphic toZk. Hencekst and so rankGnstby Lemma 2.1.

Corollary2.4. LetGbe a nilpotent group. IfG/[G, G]is of finite rank, then so are all the subquotients arising from the lower central series ofG.

Proof. This follows by induction from Lemma 2.3.

Lemma2.5. LetGbe a torsion-free nilpotent group of lengthn+1such thatGj/Gj+1 is of finite rankrj,0 ≤ jn. Then there exists a sequence {e} =N0N1⊆ · · · ⊆Nn+1=Gof normal subgroups such thatNj+1/Nj, 0≤ jn, is contained in the centre ofG/Nj and is torsion-free of ranksj

wheren

j=0sjn

j=0rj.

Proof. We prove the lemma by induction on the lengthl(G)of the lower central series. Ifl(G)=1, nothing has to be shown. Suppose the statement is true whenl(G)n, and letGbe as in the lemma withl(G)=n+1. LetZ denote the centre ofGand let

N = {x∈Z:xGn(Z/Gn)t}.

ThenZ/N is torsion-free and sinceG/Zis also torsion-free,G/N is torsion- free. Moreover,l(G/N)nsinceGnZ. Let(G/N)j, j =0,1, . . ., denote the lower central series ofG/N. Then(G/N)n= {N}and(G/N)j/(G/N)j+1

is a quotient ofGj/Gj+1. Thus

rank((G/N)j/(G/N)j+1)rj (0≤jn−1).

SetN1 = N. By the inductive hypothesis, there exists a sequence of normal subgroupsNkofG,k =2, . . . , n+1, such thatN1N2⊆ · · · ⊆Nn+1=G, Nk+1/Nk is torsion-free and contained in the centre ofG/Nk, andNk+1/Nk

has rank sk, where n

k=1skn−1

j=0rj. Since N1Z, it only remains to notice that rankN =rankGn.

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To that end, let{x1, . . . , xd}be an independent subset ofN. Then, since N/Gnis a torsion group,x1s, . . . , xdsGnfor somes∈N, and these elements are independent inGn. Thusd ≤rankGn. This finishes the proof.

Proposition2.6. LetG be a nilpotent locally compact group such that G/G0=(G/G0)c. Then

sr(C(G))≤max

2,1+ 1

2rank(G0/[G0, G0]) .

Proof. LetH denote the collection of all compactly generated open sub- groups ofG. ThenG= ∪H∈HHand, for eachHH,H0=G0andH/G0is compact since every compact subset ofG/G0generates a compact subgroup of G/G0. SinceC(G)is the inductive limit ofC-subalgebrasC(H), HH, it follows from Proposition 1.3 that

sr(C(G))≤ sup

H∈Hsr(C(H ))≤ sup

H∈Hmax

2,1+ 1

2rank(H0/[H0, H0])

=max

2,1+ 1

2rank(G0/[G0, G0]) . Lemma2.7. LetNbe an open normal subgroup of a locally compact group G, and suppose thatG/N is abelian and torsion-free of finite rankr. Then

sr(C(G))≤sr(C(N))+r.

Proof. Let H be a subgroup of GcontainingN such that H/N is iso- morphic toZm for somem ∈ N. ThenC(H) can be written as a repeated crossed product

C(H )=(. . . (C(N)×α1Z) . . .)×αmZ,

and hence Theorem 7.1 of [17] yields that sr(C(H))≤sr(C(N))+m. Let H be the collection of all subgroupsH ofGsuch thatNH and H/Nis finitely generated. Then, for each suchH,H/Nis isomorphic toZm, wheremr, whence sr(C(H )) ≤ sr(C(N))+r by the first paragraph.

Finally,

sr(C(G))≤ sup

H∈Hsr(C(H))≤sr(C(N))+r.

Theorem2.8. LetGbe a nilpotent locally compact group. Then the fol- lowing conditions are equivalent.

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(i) sr(C(G)) <. (ii) RR(C(G)) <.

(iii) The abelian locally compact groupG/[G, G]has finite rank.

Proof. (i) ⇒(ii) follows from RR(A) ≤ 2 sr(A)− 1 for anyC-alge- braA[3].

Suppose that (ii) holds and letH =G/[G, G]. SinceC(H)is a quotient of C(G), RR(C(H )) <∞. NowH/Hcis the direct product of a vector group Rnand a torsion-free discrete groupD. By definition, rank(H)=n+rank(D). Thus it suffices to observe thatDhas finite rank. SinceC(D)is a quotient of C(H ), we have

rank(D)=RR(C(D))≤RR(C(H)) <∞.

To show (iii)⇒(i), letq:GG/G0denote the quotient homomorphism and letN =q1((G/G0)c). By Proposition 2.6,

sr(C(N))≤max

2,1+ 1

2rank(G0/[G0, G0]) .

N is open in GbecauseG/G0 is totally disconnected and hence has com- pact open subgroups. Moreover, G/N is torsion-free. By hypothesis (iii), (G/N)/[G/N, G/N] has finite rank. Then, by Corollary 2.4 and Lemma 2.5, there exists a sequence N0 = NN1 ⊆ · · · ⊆ Nn+1 = G of nor- mal subgroups of G such that each Nj+1/Nj is abelian, torsion-free and has finite rank rj, say (0 ≤ jn). Climbing up the ascending series Nj, j = 1, . . . , n+ 1, and applying Lemma 2.7 at each step, we obtain sr(C(G))≤sr(C(N))+n

j=0rj.

The implications (iii)⇒(i) and (iii)⇒(ii) of the preceding theorem do not hold for arbitrary locally compact groups. We conclude with a simple example of a discrete group G which has an abelian subgroup of index 2 with the property thatG/[G, G] has rank zero, whereas

sr(C(G))=RR(C(G))= ∞.

Example2.9. LetN denote the direct sum of infinitely many copies ofZ andGthe semidirect productG=Z2N, where the action ofZ2= {1,−1} onNis given by(−1)·(x1, x2, . . .)=(−x1,−x2, . . .). Then [G, G] consists of all elements(x1, x2, . . .)ofN such that allxj are even. ThusG/[G, G] is the direct sum of infinitely many copies ofZ2 and hence is a torsion group.

So rank(G/[G, G])= 0. However, the finite conjugacy class subgroup ofG

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equalsN. Then, by Theorem 3.4 of [1], RR(C(G))

rank(N) 2[G:N]−1

= 1

3rank(N)

= ∞, and hence also sr(C(G))= ∞.

REFERENCES

1. Archbold, R. J., and Kaniuth, E.,Stable rank and real rank for some classes of groupC- algebras, Trans. Amer. Math. Soc. 357 (2005), 2165–2186 .

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Math. Soc., Providence, 1971.

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4. Corwin, L., and Greenleaf, F. P.,Representations of Nilpotent Lie Groups and their Applica- tions. Par I: Basic Theory and Examples, Cambridge University Press, 1990.

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22. Sudo, T.,Stable rank of theC-algebras of amenable Lie groups of type I, Math. Scand. 84 (1999), 231–242.

23. Sudo, T.,Dimension theory of groupC-algebras of connected Lie groups of type I, J. Math.

Soc. Japan 42 (2000), 583–590.

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Math. 6 (1995), 439–446.

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Operator Theory 38 (1997), 67–86.

DEPARTMENT OF MATHEMATICAL SCIENCES UNIVERSITY OF ABERDEEN

ABERDEEN AB24 3UE SCOTLAND, UK

E-mail:r.archbold@maths.abdn.ac.uk

INSTITUT FÜR MATHEMATIK UNIVERSITÄT PADERBORN D-33095 PADERBORN GERMANY

E-mail:kaniuth@math.uni-paderborn.de

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