Finitely generated nilpotent group C*-algebras have finite nuclear dimension
aa r X i v : . [ m a t h . OA ] M a y FINITELY GENERATED NILPOTENT GROUP C*-ALGEBRASHAVE FINITE NUCLEAR DIMENSION
CALEB ECKHARDT AND PAUL MCKENNEY
Abstract.
We show that group C*-algebras of finitely generated, nilpotent groupshave finite nuclear dimension. It then follows, from a string of deep results, thatthe C*-algebra A generated by an irreducible representation of such a group hasdecomposition rank at most 3. If, in addition, A satisfies the universal coefficienttheorem, another string of deep results shows it is classifiable by its ordered K-theory and is approximately subhomogeneous. We observe that all C*-algebrasgenerated by faithful irreducible representations of finitely generated, torsion freenilpotent groups satisfy the universal coefficient theorem. Introduction
The noncommutative dimension theories of Kirchberg and Winter (decompositionrank) and of Winter and Zacharias (nuclear dimension) play a prominent role in thetheory of nuclear C*-algebras. This is especially apparent in Elliott’s classificationprogram where finite noncommutative dimension is essential for a satisfying classifi-cation theory. In [41], Winter and Zacharias express a hope that nuclear dimensionwill “shed new light on the role of dimension type conditions in other areas of non-commutative geometry.” We share this hope and this work aims to use the theory ofnuclear dimension to shed new light on the representation theory of discrete nilpotentgroups.A discrete group is Type I (and therefore has a “tractable” representation theory)if and only if it has an abelian subgroup of finite index [38]. Therefore being TypeI is a highly restrictive condition for discrete groups and therefore for most discretegroups, leaves many of the tools of classic representation theory out of reach. Recentbreakthroughs of several mathematicians (H. Lin, Z. Niu, H. Matui, Y. Sato and W.Winter to name a few) gave birth to the possibility of classifying the C*-algebrasgenerated by the irreducible representations of nilpotent groups by their ordered K-theory. A key missing ingredient was knowing whether or not the group C*-algebrasof finitely generated nilpotent groups have finite nuclear dimension. Our main result(Theorem 4.4) supplies this ingredient. In particular we show that the nuclear di-mension of C ∗ ( G ) is bounded by 10 h ( G ) − · h ( G )! where h ( G ) is the Hirsch number of G (see Section 2.1.1).Each finitely generated nilpotent group has an algebraic “basis” of sorts and theHirsch number returns the size of this basis–it is therefore not surprising to see h ( G )appear in the nuclear dimension estimate. C.E. was partially supported by NSF grant DMS-1262106.
Fix a finitely generated nilpotent group G and an irreducible representation π of G. The C*-algebra generated by π ( G ) is simple, nuclear, quasidiagonal with unique traceand, by a combination of Theorem 4.4 with many deep results (see Theorem 2.10 fora complete list), has finite decomposition rank. Therefore if C ∗ ( π ( G )) satisfies theuniversal coefficient theorem (see [36]), it is classified by its ordered K-theory andisomorphic to an approximately subhomogeneous C*-algebra by [23, 26, 27] (see [27,Corollary 6.2]). We observe in Section 4 that the work of Rosenberg and Schochet [36]show that if G is torsion free and π is faithful (as a group homomorphism on G ) then C ∗ ( π ( G )) satisfies the universal coefficient theorem.If G is a two-step nilpotent group, it is well-known that the C*-algebras gener-ated by irreducible representations of G are either finite dimensional or A T -algebras.Indeed Phillips showed in [32] that all simple higher dimensional non commutativetori (a class of C*-algebras that include C ∗ ( π ( G )) when G is two-step and π is anirreducible, infinite dimensional representation) are A T algebras. In some sense thisresult forms the base case for our induction proof (see below for a more detailed de-scription). Peeling back a couple layers, we mention that Phillips’ work relies on thatof Elliott and Evans [12] and Kishimoto [21] (see also [3] and [24] for precursors toPhillips’ Theorem).Since in general a discrete nilpotent group G is not Type I we are left with essentiallyno possibility of reasonably classifying its irreducible representations up to unitaryequivalence. On the other hand if every primitive quotient of C ∗ ( G ) satisfies theUCT, then one could classify the C*-algebras generated by these representationsby their ordered K-theory. This provides a dual viewpoint to the prevailing one ofparametrizing irreducible representations by primitive ideals of C ∗ ( G ) or by the spaceof characters of G (see [19])–we thank Nate Brown for sharing this nice observationwith us.Let us provide a broad outline of our proof. First we prefer to deal with torsionfree groups. Since every finitely generated nilpotent group has a finite index torsionfree subgroup we begin in Section 3 by showing that finite nuclear dimension is stableunder finite extensions. We then focus on the torsion free case.We proceed by induction on the Hirsch number (see Section 2.1.1) of the nilpo-tent group G. When dealing with representation theoretic objects, (like a group C*-algebra) induction on the Hirsch number is sometimes more helpful than inductionon, say, the nilpotency class for the simple reason that non-trivial quotients of G haveHirsch number strictly less than G while the nilpotency class of the quotient may beunchanged.By [31] (see also Theorem 2.7 below) we can view C ∗ ( G ) as a continuous field overthe dual of its center, Z ( G ). Since Z ( G ) is a finitely generated abelian group, our taskis to bound the nuclear dimension of the fibers as the base space is already controlled.Since we are proceeding by induction we can more or less focus on those fibers(we call them C ∗ ( G, e γ ) ) induced by characters γ ∈ [ Z ( G ) that are faithful (as grouphomomorphisms) on Z ( G ) . In the case that G is a two step nilpotent group, then C ∗ ( G, e γ ) is a simple higher dimensional noncommutative torus. Phillips showed in UCLEAR DIMENSION OF FINITELY-GENERATED NILPOTENT GROUP C*-ALGEBRAS 3 [32] that any such C*-algebra is an A T -algebra and therefore has nuclear dimension(decomposition rank in fact) bounded by 1. Phillips’ result is crucial for us as it allowsus to reduce to the case that the nilpotency class of G is at least 3 and provides enough“room” for the next step of the proof.We then find a subgroup N of G , with strictly smaller Hirsch number, such that G ∼ = N ⋊ Z . ( N is simply the subgroup generated by Z n − ( G ) and all but one of thegenerators of G/Z n − ( G ); see section 2 below for definitions.) By induction we know C ∗ ( N ) has finite nuclear dimension and so we analyze the action of Z on fibers of C ∗ ( N ) . It turns out that there are two cases: In the first case, the fiber is simple andthe action of Z on the fiber is strongly outer and hence the crossed product (whichis a fiber of C ∗ ( G )) absorbs the Jiang-Su algebra by [28]. This in turn implies finitedecomposition rank of the fiber by a string of deep results (see Theorem 2.10). If thefiber is not simple, we can no longer employ the results of [28], but the non-simplicityforces the action restricted to the center of the fiber to have finite Rokhlin dimensionand so the crossed product has finite nuclear dimension by [15].2. Preliminaries
We assume the reader is familiar with the basics of group C*-algebras and discretecrossed products and refer them to Brown and Ozawa [5] for more information.2.1.
Facts about Nilpotent Groups.
Group Theoretic Facts.
We refer the reader to Segal’s book [37] for more infor-mation on polycyclic and nilpotent groups. Here we collect some facts about nilpotentgroups that we will use frequently.Let G be a group and define Z ( G ) = Z ( G ) to be the center of G. Recursivelydefine Z n ( G ) ≤ G to satisfy Z n ( G ) /Z n − ( G ) := Z ( G/Z n − ( G )) . A group G is called nilpotent if Z n ( G ) = G for some n and is called nilpotent of class n if n is theleast integer satisfying Z n ( G ) = G. A group G is polycyclic if it has a normal series { e } E G E · · · E G n − E G n such that each quotient G i +1 /G i is cyclic. The number of times that G i +1 /G i isinfinite is called the Hirsch number of the group G and is denoted by h ( G ) . TheHirsch number is an invariant of the group. If G is polycyclic and N is a normalsubgroup then both N and G/N are polycyclic with h ( G ) = h ( N ) + h ( G/N )Every finitely generated nilpotent group is polycyclic.Let now G be finitely generated and nilpotent. Let G f denote the subgroup con-sisting of those elements with finite conjugacy class. By [1], the center Z ( G ) hasfinite index in G f . If in addition G is torsion free, then by [25] the quotient groups CALEB ECKHARDT AND PAUL MCKENNEY
G/Z i ( G ) are also all torsion free. These facts combine to show that if G is torsionfree, then every non-central conjugacy class is infinite.2.1.2. Representation Theoretic Facts.
Definition 2.1.
Let G be a group and φ : G → C a positive definite function with φ ( e ) = 1 . If φ is constant on conjugacy classes, then we say φ is a trace on G. It isclear that any such φ defines a tracial state on C ∗ ( G ) . Following the representationtheory literature we call an extreme trace a character . Every character gives rise toa factor representation and is therefore multiplicative on Z ( G ) . Let G be a finitely generated nilpotent group. Let J be a primitive ideal of C ∗ ( G )(i.e. the kernel of an irreducible representation). Moore and Rosenberg showed in [29]that J is actually a maximal ideal. Shortly after this result, Howe showed in [16] (seeespecially the introduction of [6]) that every primitive ideal is induced from a unique character, i.e. there is a unique character φ on G such that J = { x ∈ C ∗ ( G ) : φ ( x ∗ x ) = 0 } . Let G be a finitely generated nilpotent group and φ a character on G. Set K ( φ ) = { g ∈ G : φ ( g ) = 1 } . By [16] and [6], G is centrally inductive , i.e. φ vanishes on thecomplement of { g ∈ G : gK ( φ ) ∈ ( G/K ( φ )) f } . In this light we make the followingdefinition.
Definition 2.2.
Let N ≤ G and φ be a positive definite function on N . We denoteby e φ the trivial extension of φ to G , i.e. that extension of φ satisfying e φ ( x ) = 0 for all x N . We use the following well-known fact repeatedly,
Lemma 2.3.
Let G be a finitely generated torsion free nilpotent group and γ a faithfulcharacter on Z ( G ) . Then e γ is a character of G .Proof. Since γ is a character and the extension e γ is a trace on G , by a standard extremepoint argument there is a character ω on G that extends γ. Since G is nilpotent, everynon-trivial normal subgroup intersects the center non-trivially. Since K ( e γ ) ∩ Z ( G )is trivial, we have K ( e γ ) is also trivial. By the preceding section, G f = Z ( G ), i.e. ω vanishes off of Z ( G ) . But this means precisely that ω = e γ . (cid:3) It is clear from the definitions that every nilpotent group is amenable and thereforeby [22], group C*-algebras of nilpotent groups are nuclear. In summary, for everyprimitive ideal J of C ∗ ( G ), the quotient C ∗ ( G ) /J is simple and nuclear with a uniquetrace.2.2. C*-facts.Definition 2.4 ( [28]) . Let G be a group acting on a C*-algebra A with unique trace τ. Since τ is unique the action of G leaves τ invariant and therefore extends to anaction on π τ ( A ) ′′ , the von Neumann algebra generated by the GNS representation of τ. If for each g ∈ G \ { e } the automorphism of π τ ( A ) ′′ corresponding to g is an outerautomorphism, then we say that the action is strongly outer. UCLEAR DIMENSION OF FINITELY-GENERATED NILPOTENT GROUP C*-ALGEBRAS 5
The above definition can be modified to make sense even if A does not have aunique trace (see [28]), but we are only concerned with the unique trace case. Thefollowing theorem provides a key step in our main result. Theorem 2.5 ( [28, Corollary 4.11]) . Let G be a discrete elementary amenable groupacting on a unital, separable, simple, nuclear C*-algebra A with property (SI) andfinitely many extremal traces. If the action of G is strongly outer, then the crossedproduct A ⋊ G is Z -absorbing. The above theorem is actually given in greater generality in [28]. We will onlymake use of it in the case where G is Abelian. Another key idea for us is the factthat discrete amenable group C*-algebras decompose as continuous fields over theircenters. First a definition, Definition 2.6.
Let G be a group and φ a positive definite function on G . Then, wewrite C ∗ ( G, φ ) for the C*-algebra generated by the GNS representation of φ . We recall the following special case of [31, Theorem 1.2]:
Theorem 2.7.
Let G be a discrete amenable group. Then C ∗ ( G ) is a continuous fieldof C*-algebras over [ Z ( G ) , the dual group of Z ( G ) . Moreover, for each multiplicativecharacter γ ∈ [ Z ( G ) the fiber at γ is isomorphic to C ∗ ( G, e γ ) . Definition 2.8.
Let A be a C*-algebra. Denote by dim nuc ( A ) the nuclear dimen-sion of A (see [41] for the definition of nuclear dimension). It will be crucial for us to view C ∗ ( G ) as a continuous field for our inductive stepto work in the proof of our main theorem. The following allows us to control thenuclear dimension of continuous fields. Theorem 2.9 ( [7, Lemma 3.1], [39, Lemma 5.1]) . Let A be a continuous field ofC*-algebras over the finite dimensional compact space X. For each x ∈ X , let A x denote the fiber of A at x. Then dim nuc ( A ) ≤ (dim( X ) + 1)(sup x ∈ X dim nuc ( A x ) + 1) − . Throughout this paper we never explicitly work with decomposition rank, nucleardimension, property (SI) or the Jiang-Su algebra Z (for example, we never actuallybuild any approximating maps in proving finite nuclear dimension). For this reason wedo not recall the lengthy definitions of these properties but simply refer the reader to[20, Definition 3.1] for decomposition rank, [41, Definition 2.1] for nuclear dimension,[13, 18] for the Jiang-Su algebra and its role in the classification program, and [26,Definition 4.1] for property (SI).Finally, for easy reference we gather together several results into Theorem 2.10 (Rørdam, Winter, Matui and Sato) . Let A be a unital, separable,simple, nuclear, quasidiagonal C*-algebra with a unique tracial state. If A has any ofthe following properties then it has all of them. (i) Finite nuclear dimension.
CALEB ECKHARDT AND PAUL MCKENNEY (ii) Z -stability. (iii) Strict comparison. (iv)
Property (SI) of Matui and Sato. (v)
Decomposition rank at most 3.In particular, if A is a primitive quotient of a finitely generated nilpotent group C*-algebra then it satisfies the hypotheses of this Theorem.Proof. Winter showed (i) implies (ii) in [40]. Results of Matui and Sato [26] andRørdam [34] show that (ii), (iii) and (iv) are all equivalent. Strict comparison andMatui and Sato’s [27] shows (v). Finally (v) to (i) follows trivially from the definitions.If G is a nilpotent group, by [10], any (primitive) quotient of C ∗ ( G ) is quasidiagonaland therefore satisfies the hypotheses of the theorem by the discussion in Section2.1.2. (cid:3) Stability under finite extensions
In this section we show that if a nilpotent group G has a finite index normalsubgroup H such that dim nuc ( C ∗ ( H )) < ∞ , then dim nuc ( C ∗ ( G )) < ∞ . Perhaps sur-prisingly, this portion of the proof is the most involved and relies on several deepresults of C*-algebra theory. Moreover we lean heavily on the assumption that G isnilpotent.This section exists because every finitely generated nilpotent group has a finiteindex subgroup that is torsion free. Reducing to this case gives the reader a clearidea of what is happening without getting bogged down in torsion. We begin withthe following special case that isolates most of the technical details. Theorem 3.1.
Let G be a finitely generated nilpotent group. Suppose H is a normalsubgroup of finite index such that every primitive quotient of C ∗ ( H ) has finite nucleardimension. Then every primitive quotient of C ∗ ( G ) has decomposition rank at most3.Proof. We proceed by induction on | G/H | . Since G is nilpotent, so is G/H.
In particu-lar
G/H has a cyclic group of prime order as a quotient. By our induction hypothesiswe may therefore suppose that
G/H is cyclic of prime order p. Let e, x, x , ..., x p − ∈ G be coset representatives of G/H.
Let α denote the actionof G on ℓ ∞ ( G/H ) by left translation. It is well-known, and easy to prove, that ℓ ∞ ( G/H ) ⋊ α G ∼ = M p ⊗ C ∗ ( H ) and that under this inclusion C ∗ ( H ) ⊆ C ∗ ( G ) ⊆ M p ⊗ C ∗ ( H ) we may realize this copy of C ∗ ( H ) as the C*-algebra generated by thediagonal matrices(3.1) ( λ h , λ xhx − , ..., λ x p − hx − ( p − ) , with h ∈ H. Let ( π, H π ) be an irreducible representation of G. We show that C ∗ ( G ) / ker( π ) hasdecomposition rank less than or equal to 3. Let τ be the unique character on G that induces ker( π ) (see Section 2.1.2) and set K ( τ ) = { x ∈ G : τ ( x ) = 1 } . Byassumption every primitive quotient of H/ ( H ∩ K ( τ )) has finite nuclear dimension.Since (cid:16) G/K ( τ ) (cid:17) / (cid:16) H/ ( H ∩ K ( τ )) (cid:17) is a quotient of G/H it is either trivial, in which
UCLEAR DIMENSION OF FINITELY-GENERATED NILPOTENT GROUP C*-ALGEBRAS 7 case we are done by assumption or it is isomorphic to
G/H.
We may therefore assume,without loss of generality, that(3.2) the character τ that induces ker( π ) is faithful on Z ( G ) . By a well-known application of Stinespring’s Theorem (see [30, Theorem 5.5.1])there is an irreducible representation id p ⊗ σ of M p ⊗ C ∗ ( H ) , such that H π ⊆ H id p ⊗ σ and if P : H id p ⊗ σ → H π is the orthogonal projection, then(3.3) P ((id p ⊗ σ )( x )) P = π ( x ) for all x ∈ C ∗ ( G ) . Let J = ker( σ ) ⊆ C ∗ ( H ) ⊆ C ∗ ( G ) and J G ⊆ C ∗ ( G ) be the ideal of C ∗ ( G ) generatedby J. By (3.3) we have J G ⊆ ker( π ) . We now consider two cases:
Case 1: J G is a maximal ideal of C ∗ ( G ), i.e. J G = ker( π ) . In this case we have C ∗ ( H ) /J ⊆ C ∗ ( G ) /J G ⊆ M p ⊗ ( C ∗ ( H ) /J ) . We would like to reiterate that in general the copy of C ∗ ( H ) /J , is not conjugate tothe diagonal copy 1 p ⊗ ( C ∗ ( H ) /J ), but rather the twisted copy of (3.1). If C ∗ ( H ) /J is conjugate to the diagonal copy, then the proof is quite short (see the last paragraphof Case 1b), so most of the present proof consists of overcoming this difficulty.Let Z p ∼ = G/H denote the cyclic group of order p. Define an action β of Z p on ℓ ∞ ( G/H ) ⋊ α G by β t ( f )( s ) = f ( s − t ) for all f ∈ ℓ ∞ ( G/H ) (i.e. β acts by lefttranslation on ℓ ∞ ( G/H )) and β ( λ g ) = λ g for all g ∈ G. Since the G -action α and Z p -action β commute with each other it is easy to see that β defines an action of Z p on ℓ ∞ ( G/H ) ⋊ α G. Moreover β t ( x ) = x for all t ∈ Z /p Z if and only if x ∈ C ∗ ( G ) . In particular β fixes J G ⊆ C ∗ ( G ) pointwise. Notice that M p ⊗ J is generated bythe β -invariant set { e a + · · · + e p a p | a i ∈ J } , where e i denotes the i th minimalprojection of M p . Therefore, β leaves M p ⊗ J invariant (although not pointwise), andinduces an automorphism of M p ⊗ ( C ∗ ( H ) /J ) . Moreover the fixed-point subalgebraof this induced automorphism is exactly C ∗ ( G ) /J G . We now split further into twosubcases based on the behavior of β. Case 1a:
The action β y M p ⊗ ( C ∗ ( H ) /J ) is strongly outer (Definition 2.4).By assumption, M p ⊗ ( C ∗ ( H ) /J ) has finite nuclear dimension. By Theorem 2.10, M p ⊗ ( C ∗ ( H ) /J ) then has property (SI). Since β is strongly outer, by [28, Corollary4.11], the crossed product M p ⊗ ( C ∗ ( H ) /J ) ⋊ β Z p is Z -stable. Since M p ⊗ ( C ∗ ( H ) /J )has a unique trace and ℓ ∞ ( p ) ⋊ Z p ∼ = M p , it follows that [ M p ⊗ ( C ∗ ( H ) /J )] ⋊ β Z p has unique trace. It follows from [10] that [ M p ⊗ ( C ∗ ( H ) /J )] ⋊ β Z p is quasidiagonal. CALEB ECKHARDT AND PAUL MCKENNEY
Therefore by Theorem 2.10, M p ⊗ ( C ∗ ( H ) /J ) ⋊ β Z p has decomposition rank at most3. By [35], the fixed point algebra of β , i.e. C ∗ ( G ) /J G is isomorphic to a cor-ner of M p ⊗ ( C ∗ ( H ) /J ) ⋊ β Z p , which by Brown’s isomorphism theorem [4] im-plies that C ∗ ( G ) /J G is stably isomorphic to M p ⊗ ( C ∗ ( H ) /J ) ⋊ β Z p and therefore C ∗ ( G ) /J G = C ∗ ( G ) / ker( π ) has decomposition rank at most 3 by [20, Corollary 3.9]. Case 1b:
The action β y M p ⊗ ( C ∗ ( H ) /J ) is not stongly outer (Definition 2.4).Choose a generator t of Z p and set β = β t (note that every β t is strongly outeror none of them are). We will first show that β is actually an inner automorphism of M p ⊗ ( C ∗ ( H ) /J ) . The unique trace on M p ⊗ ( C ∗ ( H ) /J ) restricts to the unique trace τ on C ∗ ( G ) /J G . We will use the common letter τ for both of these traces.Let G f ≤ G be the subgroup consisting of those elements with finite conjugacyclasses. By [1, Lemma 3], G f /Z ( G ) is finite. Let ( π τ , L ) be the GNS representationof ℓ ∞ ( G/H ) ⋊ α G associated with τ. Since τ is multiplicative on Z ( G ) it easily follows that h λ x , λ y i τ = τ ( y − x ) ∈ T , for x, y ∈ Z ( G ) . By the Cauchy-Schwarz inequality it follows that in L we have(3.4) λ x = L τ ( y − x ) λ y for all x, y ∈ Z ( G ) . Let x , ..., x n ∈ G f be coset representatives of G f /Z ( G ) and let C ⊆ G be a set ofcoset representatives for G/G f . The preceding discussion shows that the following set spans a dense subset of L :(3.5) { f λ tx i : f ∈ ℓ ∞ ( G/H ) , t ∈ C, ≤ i ≤ n } . Since τ is β -invariant, for each minimal projection e ∈ ℓ ∞ ( p ) and t ∈ G we have τ ( eλ t ) = τ ( β ( e ) λ t ) . So for each f ∈ ℓ ∞ ( G/H ) and t ∈ G we have τ ( f λ t ) = τ ( f ) τ ( t ).Combining this with the fact that τ vanishes on infinite conjugacy classes, we have(3.6) h f λ t , gλ s i τ = 0 for all f, g ∈ ℓ ∞ ( G/H ) , and t, s ∈ C, t = s. Since β is not strongly outer there is a unitary W ∈ π τ ( ℓ ∞ ( G/H ) ⋊ α G ) ′′ such that W π τ ( x ) W ∗ = π τ ( β ( x )) for all x ∈ ℓ ∞ ( G/H ) ⋊ α G. Since β leaves π τ ( G ) pointwise invariant, W must commute with π τ ( λ t ) for all t ∈ G. For t ∈ G , let Conj( t ) = { sts − : s ∈ G } be the conjugacy class of t. Let s ∈ G \ G f . Suppose first that Conj( s ) intersects infinitely many G/G f cosets.Let ( s n ) ∞ n =1 be a sequence from G such that the cosets s n ss − n G f are all distinct. Let f ∈ ℓ ∞ ( G/H ) . Since W commutes with the λ t ’s, for each n ∈ N we have h W, f λ s i τ = h λ s − n W λ s n , f λ s i τ = h W, α s n ( f ) λ s n ss − n i τ . UCLEAR DIMENSION OF FINITELY-GENERATED NILPOTENT GROUP C*-ALGEBRAS 9
By (3.6), the vectors { α s n ( f ) λ s n ss − n : n ∈ N } form an orthogonal family of vectors,each with the same L norm. Since W has L norm equal to 1, this implies that h W, f λ s i = 0 for all f ∈ ℓ ∞ ( G/H ) . Suppose now that Conj( s ) intersects only finitely many G/G f cosets. Since Conj( s ) isinfinite, and Z ( G ) has finite index in G f , there is a y ∈ G such that Conj( s ) ∩ yZ ( G )is infinite. Let s n be a sequence from G and t n a sequence of distinct elements from Z ( G ) so s n ss − n = yt n for all n ∈ N . Let f ∈ ℓ ∞ ( G/H ) . Since the set { α g ( f ) : g ∈ G } is finite we may, without lossof generality, suppose that α s n ( f ) = α s m ( f ) for all n, m ∈ N . By (3.4), we have λ yt n = L τ ( t − m t n ) λ yt m for all n, m ∈ N . We then have h W, f λ s i τ = h λ s − n W λ s n , f λ s i τ = h W, α s n ( f ) λ s n ss − n i τ = h W, α s ( f ) λ yt n i τ = τ ( t − t n ) h W, α s ( f ) λ yt i τ In particular, we have τ ( t − t ) h W, α s ( f ) λ yt i τ = τ ( t − t ) h W, α s ( f ) λ yt i τ . Since τ is faithful on Z ( G ) (by (3.2)) we have 1 = | τ ( t − t ) | , and τ ( t − t ) = τ ( t − t ) =1 . Hence h W, f λ s i τ = 0 . We have shown that for all s ∈ G \ G f and f ∈ ℓ ∞ ( G/H ) we have h W, f λ s i τ = 0 . By(3.5) it follows that W ∈ span { π τ ( f λ x i ) : f ∈ ℓ ∞ ( G/H ) , ≤ i ≤ n } ⊆ π τ ( ℓ ∞ ( G/H ) ⋊ G ) ∼ = M p ⊗ ( C ∗ ( H ) /J )By the way that β acts on ℓ ∞ ( G/H ) there are a , ..., a p ∈ C ∗ ( H ) /J so W = e p ⊗ a + p X i =2 e i,i − ⊗ a i . Set U = diag(1 , a ∗ , a ∗ a ∗ , ..., a ∗ a ∗ · · · a ∗ p ) ∈ M p ⊗ ( C ∗ ( H ) /J ) . Since W commutes with C ∗ ( H ) /J , by (3.1) it follows that(3.7) U (cid:16) C ∗ ( H ) /J (cid:17) U ∗ = 1 M p ⊗ C ∗ ( H ) /J ⊆ U ( C ∗ ( G ) /J G ) U ∗ ⊆ M p ⊗ ( C ∗ ( H ) /J ) . Let U be a free ultrafilter on N and for a C*-algebra A , let A U denote the ultrapowerof A. We think of A ⊂ A U via the diagonal embedding and write A ′ ∩ A U for thoseelements of the ultrapower that commute with this diagonal embedding.By Theorem 2.10, C ∗ ( H ) /J is Z -stable. By [33, Theorem 7.2.2], there is an em-bedding φ of Z into ( C ∗ ( H ) /J ) U ∩ ( C ∗ ( H ) /J ) ′ . By both inclusions of (3.7) it is clearthat 1 M p ⊗ φ defines an embedding of Z into ( C ∗ ( G ) /J G ) U ∩ ( C ∗ ( G ) /J G ) ′ . Therefore,again by [33, Theorem 7.2.2] it follows that C ∗ ( G ) /J G is Z -stable. By Theorem 2.10, C ∗ ( G ) /J G has decomposition rank at most 3. Case 2:
The ideal J G is not maximal. Recall the definition of σ from the begin-ning of the proof.For each i = 0 , ..., p − H , σ i ( h ) = σ ( x i hx − i ) . By [10, Sec-tion 3] either all of σ i are unitarily equivalent to each other or none of them are. Wetreat these cases separately. Case 2a:
All of the σ i are unitarily equivalent to each other.By the proof of Lemma 3.4 in [10] it follows that there is a unitary U ∈ M p ⊗ C ∗ ( H ) /J such that U ( C ∗ ( H ) /J ) U ∗ = 1 p ⊗ ( C ∗ ( H ) /J ) . Moreover from the same proof thereis a projection q ∈ M p ⊗ C ∗ ( H ) /J that commutes with U (id M p ⊗ σ ( C ∗ ( G ))) U ∗ so q ( U [id M p ⊗ σ ( C ∗ ( G ))] U ∗ ) ∼ = C ∗ ( G ) / ker( π ) . We therefore have a chain of inclusions q ⊗ C ∗ ( H ) /J ⊆ q ( U [id M p ⊗ σ ( C ∗ ( G ))] U ∗ ) ⊆ qM p q ⊗ C ∗ ( H ) /J. We can now complete the proof as in the end of Case 1b (following nearly verbatimeverything that follows (3.7)).
Case 2b.
None of the σ i are unitarily equivalent to each other.From the proof of Lemma 3.5 in [10] there is a projection q ∈ ℓ ∞ ( p ) ⊗ C ∗ ( H ) /J that commutes with (id M p ⊗ σ )( C ∗ ( G )) so q (id M p ⊗ σ ( C ∗ ( G ))) ∼ = C ∗ ( G ) / ker( π ) . Butsince G acts transitively on G/H (and hence ergodically on ℓ ∞ ( G/H )) we must have q = 1 . But this implies that ker(id M p ⊗ σ | C ∗ ( G ) ) = ker( π ) . Recall the coset representatives e, x, x , ..., x p − of G/H.
Each x ∈ C ∗ ( G ) can bewritten uniquely as P p − i =0 a i λ x i for some a i ∈ C ∗ ( H ) . Fix an 0 ≤ i ≤ p − λ x i ∈ M p ⊗ C ∗ ( H ) . If there is an index ( k, ℓ ) such that the ( k, ℓ )-entry of λ x i is non-zero , then for any j = i the ( k, ℓ )-entry of λ x j must be 0. From thisobservation it follows thatid p ⊗ σ (cid:16) p − X i =0 a i λ x i (cid:17) = 0 , if and only if (id p ⊗ σ )( a i ) = 0 for all 1 ≤ i ≤ p In other words ker( π ) = ker(id M p ⊗ σ | C ∗ ( G ) ) ⊆ J G and we are done by Case 1. (cid:3) Lemma 3.2.
Let G be a finitely generated nilpotent group and N a finite indexsubgroup of Z ( G ) and φ a faithful multiplicative character on N. Then there is afinite set F of characters of G such that e φ (see Definition 2.6) is in the convex hullof F . Proof.
This result follows from Thoma’s work on characters [38] (see also the discus-sion on page 355 of [19]). For the convenience of the reader we outline a proof andkeep the notation of [19]. Let G f ≤ G denote the group consisting of elements withfinite conjugacy class. By [1] Z ( G ) (and hence N ) has finite index in G f . UCLEAR DIMENSION OF FINITELY-GENERATED NILPOTENT GROUP C*-ALGEBRAS 11
Let π be the GNS representation of G f associated with e φ | G f . Since N has finiteindex in G f and π ( N ) ⊆ C it follows that π ( G f ) generates a finite dimensional C*-algebra. Therefore there are finitely many characters ω , ..., ω n of G f that extend φ (trivially as π ( N ) ⊆ C ) and a sequence of positive scalars λ i such that(3.8) e φ | G f = n X i =1 λ i ω i The positive definite functions e ω i on G need not be traces, but this is easily remediedby the following averaging process (which we took from [19] and [38]).Let x ∈ G f . Then the centralizer of x in G , denoted C G ( x ), has finite index. Let A x be a complete set of coset representatives of G/C G ( x ) . For each 1 ≤ i ≤ n and x ∈ G f define(3.9) e ω Gi ( x ) = 1[ G : C G ( x )] X a ∈ A x e ω i ( axa − ) . Then each e ω Gi is extreme in the space of G -invariant traces on G f (see [19, Page 355]).Since each e ω Gi is G -invariant, the trivial extension to G (which we still denote by e ω Gi )is a trace on G. We show that the e ω Gi are actually characters on G. By a standard convexity argument, there is a character ω on G which extends e ω iG .Let K ( ω ) = { x ∈ G : ω ( x ) = 1 } . Then K ( ω ) ⊆ G f . Indeed if there is a g ∈ K ( ω ) \ G f ,then g necessarily has infinite order (otherwise the torsion subgroup of G would beinfinite). Since every finitely generated nilpotent group has a finite index torsion freesubgroup and every nontrivial subgroup of a nilpotent group intersects the centernon-trivially, this would force K ( ω ) ∩ Z ( G ) to have non-zero Hirsch number, but byassumption φ = ω | N is faithful on a finite index subgroup of Z ( G ) . Since G is finitely generated and nilpotent, it is centrally inductive (see [6]). Thismeans that ω vanishes outside of G f ( ω ) = { x ∈ G : xK ( ω ) ∈ ( G/K ( ω )) f } . But since K ( ω ) ⊆ G f and ω | N = φ , it follows that K ( ω ) is finite. From this it follows that G f ( ω ) = G f , i.e. ω must vanish outside of G f . But this means precisely that ω = e ω Gi . By (3.8) and (3.9) for x ∈ G f we have n X i =1 λ i e ω Gi ( x ) = n X i =1 λ i (cid:16) G : C G ( x )] X a ∈ A x e ω i ( axa − ) (cid:17) = 1[ G : C G ( x )] X a ∈ A x (cid:16) n X i =1 λ i e ω i ( axa − ) (cid:17) = 1[ G : C G ( x )] X a ∈ A x e φ ( x )= e φ ( x ) . For x G f we clearly have P ni =1 λ i e ω Gi ( x ) = 0 = e φ ( x ) . (cid:3) Lemma 3.3.
Set f ( n ) = 10 n − n ! . Let G be a finitely generated nilpotent group. Let H E G be normal of finite index. If dim nuc ( C ∗ ( H/N )) ≤ f ( h ( H/N )) for every normalsubgroup of H , then dim nuc ( C ∗ ( G/K )) ≤ f ( h ( G/K )) for every normal subgroup of G. Proof.
We proceed by induction on h ( G ) . If h ( G ) = 0, then G is finite and thereis nothing to prove. So assume that for every finitely generated nilpotent group A with h ( A ) < h ( G ) that satisfies dim nuc ( C ∗ ( A/N )) ≤ f ( h ( A/N )) for every normalsubgroup N of A , we have dim nuc ( C ∗ ( A ′ /N ′ )) ≤ f ( h ( A ′ /N ′ )) where A ′ is a finitenormal extension of A and N ′ is a normal subgroup of A ′ . Let now H be a finite index normal subgroup of G that satisfies the hypotheses. If G/Z ( G ) is finite, then dim nuc ( C ∗ ( G )) = h ( G ) ≤ f ( h ( G )) by Theorem 2.9. Supposethat G/Z ( G ) is infinite, i.e. h ( Z ( G )) < h ( G ) . Since for any quotient
G/K of G , thegroup H/ ( H ∩ K ) has finite index in G/K it suffices to show that dim nuc ( C ∗ ( G )) ≤ f ( G ) . We use Theorem 2.7 to view C ∗ ( G ) as a continuous field over [ Z ( G ) . We estimate thenuclear dimension of the fibers. Let γ ∈ [ Z ( G ) . Suppose first that h (ker( γ )) > . Thefiber C ∗ ( G, e γ ) is a quotient of the group C*-algebra C ∗ ( G/ ker( γ )) . By our inductionhypothesis, dim nuc ( C ∗ ( G/ ker( γ ))) ≤ f ( h ( G ) − . Now suppose on the other hand that F = ker( γ ) is finite. If x ∈ G with [ x, y ] ∈ F for all y ∈ G , then yx | F | y − = x | F | for all y ∈ G, i.e. x | F | ∈ Z ( G ) . Hence Z ( G/F ) / ( Z ( G ) /F ) is a finitely generated, nilpotent torsion group, hence is finite.We therefore replace G with G/F and suppose that Z ( G ) contains a finite indexsubgroup N such that γ is a faithful homomorphism on N (Notice we can not saythat γ is faithful on Z ( G ) since it may not be the case that Z ( G/F ) = ( Z ( G ) /F )).By Lemma 3.2, there are finitely many distinct characters ω , ..., ω n on G such that e γ is a convex combination of the ω i . The GNS representation associated with e γ isthen the direct sum of the GNS representations associated with the ω i . Let J i be themaximal ideal of C ∗ ( G ) induced by ω i (Section 2.1.2). Since all of the J i are maximaland distinct it follows (from purely algebraic considerations) that C ∗ ( G, e γ ) ∼ = n M i =1 C ∗ ( G, ω i ) . Since each ω i is a character on G , it follows by Theorem 3.1 that the decompositionrank of C ∗ ( G, ω i ) is bounded by 3 which also bounds the decomposition rank of C ∗ ( G, e γ ) by 3.Since the nuclear dimension of all the fibers of C ∗ ( G ) are bounded by f ( h ( G ) − nuc ( C ∗ ( G )) ≤ h ( G ) f ( h ( G ) − ≤ f ( h ( G )). (cid:3) Main Result
The work of Matui and Sato on strongly outer actions (see Theorem 2.5) and ofHirshberg, Winter and Zacharias [15] on Rokhlin dimension are both crucial to the
UCLEAR DIMENSION OF FINITELY-GENERATED NILPOTENT GROUP C*-ALGEBRAS 13 proof of our main result. In our case, their results turned extremely difficult problemsinto ones with more or less straightforward solutions.
Lemma 4.1.
Let α be an outer automorphism of a torsion free nilpotent group G .Then for every a ∈ G , the following set is infinite. { s − aα ( s ) : s ∈ G } . Proof.
Suppose that for some a , the above set is finite. Then for any s ∈ G there are0 ≤ m < n so that s − m aα ( s m ) = s − n aα ( s n ) or a − s n − m a = α ( s n − m ) . Therefore s n − m = aα ( s n − m ) a − = ( aα ( s ) a − ) n − m . Since G is nilpotent and torsion free it has unique roots (see [25] or [2, Lemma 2.1]),i.e. s = aα ( s ) a − , or α is an inner automorphism. (cid:3) Lemma 4.2.
Let G be a torsion free nilpotent group of class n ≥ . Suppose that G = h N, x i where x ∈ G \ Z n − ( G ) , N ∩ h x i = { e } and Z ( G ) = Z ( N ) . Let φ be atrace on G that is multiplicative on Z ( G ) and that vanishes off of Z ( G ) . Let α be theautomorphism of C ∗ ( N, φ | N ) induced by conjugation by x. Then α is strongly outer.Proof. We first show that α induces an outer automorphism of N/Z ( G ) . If not, thenthere is a z ∈ N such that xax − Z ( G ) = z − azZ ( G ) for all a ∈ N. In other words zx ∈ Z ( G ) . Since x Z n − ( G ) we also have z Z n − ( G ) . Then z − Z n − ( G ) = xZ n − ( G ), from which it follows that z N. Since G is torsion free so is N. Since φ is multiplicative on Z ( N ) and vanishes out-side of Z ( N ) it follows that φ is a character (see Lemma 2.3) and therefore C ∗ ( N, φ )has a unique trace (see Section 2.1). Let ( π φ , L ( N, φ )) be the GNS representationassociated with φ. By way of contradiction suppose there is a W ∈ π φ ( N ) ′′ such that W π φ ( g ) W ∗ = π φ ( α ( g )) for all g ∈ N. For each s ∈ N we let δ s ∈ L ( N, φ ) be its canonical image. Notice that if aZ ( N ) = bZ ( N ), then δ a and δ b are orthogonal. In particular for any complete choice of cosetrepresentatives C ⊆ N for N/Z ( G ), the set { δ c : c ∈ C } is an orthonormal basisfor L ( N, φ ) . Since W is in the weak closure of the GNS representation π φ it is in L ( N, φ ) with norm 1. Therefore there is some a ∈ N such that h W, δ a i 6 = 0 . We now have, for all s ∈ N , h W, δ a i = h W π φ ( s ) , δ as i = h π φ ( α ( s )) W, δ as i = h W, δ α ( s ) − as i . Since α is outer on N/Z ( N ), by Lemma 4.1 the set { α ( s ) − asZ ( N ) : s ∈ N } isinfinite. Since distinct cosets provide orthogonal vectors of norm 1, this contradictsthe fact that W ∈ L ( N, φ ) . Essentially the same argument shows that any power of α is also not inner on π τ ( N ) ′′ , i.e. the action is strongly outer. (cid:3) We refer the reader to the paper [15] for information on Rokhlin dimension of ac-tions on C*-algebras. For our purposes we do not even need to know what it is,simply that our actions have finite Rokhlin dimension. Therefore we omit the some-what lengthy definition [15, Definition 2.3]. We do mention the following corollaryto the definition of Rokhlin dimension: If α is an automorphism of a C*-algebra A and there is an α -invariant, unital subalgebra B ⊆ Z ( A ) such that the action of α on B has Rokhlin dimension bounded by d , then the action of α on A also has Rokhlindimension bounded by d. Lemma 4.3.
Let G be a finitely generated, torsion free nilpotent group. Supposethat G = h N, x i where N ⊳ G , N ∩ h x i = { e } , Z ( G ) ⊆ Z ( N ) , and Z ( G ) = Z ( N ) . Let φ be a trace on G that is multiplicative on Z ( G ) and vanishes off Z ( G ) . Let α be the automorphism of C ∗ ( N, φ | N ) defined by conjugation by x. Then the Rokhlindimension of α is 1.Proof. Consider the action of α restricted to Z ( N ) . Since Z ( N ) = Z ( G ), α is not theidentity on Z ( N ) . Since Z ( N ) is a free abelian group we have α ∈ GL ( Z , d ) where d is the rank of Z ( N ) . Since G is nilpotent, so is the group Z ( N ) ⋊ α Z . Therefore(1 − α ) d = 0 . In particular there is a y ∈ Z ( N ) such that (1 − α )( y ) = 0 but(1 − α ) ( y ) = 0 . From this we deduce that α ( y ) = y + z for some z ∈ Z ( G ) \ { } . Therefore the action of α on C ∗ ( π φ ( y )) ∼ = C ( T ) is a rotation by φ ( z ) . Since φ isfaithful we have φ ( z ) = e πiθ for some irrational θ. By [15, Theorem 6.1] irrationalrotations of the circle have Rokhlin dimension 1. Since C ∗ ( π φ ( y )) ⊆ Z ( C ∗ ( N, φ )),the remark preceding this lemma shows that the Rokhlin dimension of α acting on C ∗ ( N, φ ) is also equal to 1. (cid:3)
Theorem 4.4.
Define f : N → N by f ( n ) = 10 n − n ! . Let G be a finitely generatednilpotent group. Then dim nuc ( C ∗ ( G )) ≤ f ( h ( G )) .Proof. We proceed by induction on the Hirsch number of G. If h ( G ) = 0, there isnothing to prove. It is well-known that G contains a finite index torsion-free subgroup N (see section 2.1.1). Therefore by Lemma 3.3 we may assume that G is torsion free.We decompose C ∗ ( G ) as a continuous field over [ Z ( G ) as in Theorem 2.7. Let γ ∈ [ Z ( G ) . If γ is not faithful on Z ( G ), then since G is torsion free we have h (ker( γ )) > . By our induction hypothesis we then have dim nuc ( C ∗ ( G, e γ )) ≤ f ( h ( G ) − . Suppose now that γ is faithful on Z ( G ) . If G is a 2 step nilpotent group, then C ∗ ( G, e γ ) is a simple higher dimensional noncommutative torus and therefore an A T algebra by [32]. A T algebras have nuclear dimension (decomposition rank in fact)bounded by 1 [20].Suppose then that G is nilpotent of class n ≥ Z i ( G ) denote its uppercentral series. By [25] (see also [17, Theorem 1.2] ), the group G/Z n − ( G ) is a freeabelian group. Let xZ n − ( G ) , x Z n − ( G ) , ..., x d Z n − ( G ) be a free basis for G/Z n − ( G ) . Let N be the group generated by Z n − ( G ) and { x , ..., x d } . Then N is a normalsubgroup of G with h ( N ) = h ( G ) − G = N ⋊ α Z where α is conjugation by x. UCLEAR DIMENSION OF FINITELY-GENERATED NILPOTENT GROUP C*-ALGEBRAS 15
Suppose first that Z ( N ) = Z ( G ) . Since h ( N ) = h ( G ) −
1, the group C*-algebra C ∗ ( N ) has finite nuclear dimension by our induction hypothesis. Assuming Z ( N ) = Z ( G ) means that e γ is a character on N , i.e. C ∗ ( N, e γ ) is primitive quotient of C ∗ ( N ) . It then enjoys all of the properties of Theorem 2.10.By Lemma 4.2, the action of α on C ∗ ( N, e γ ) is strongly outer. By [28, Corollary4.11], the crossed product C ∗ ( N, e γ ) ⋊ α Z ∼ = C ∗ ( G, e γ ) is Z -stable, hence C ∗ ( G, e γ ) hasdecomposition rank bounded by 3 by Theorem 2.10.Suppose now that Z ( G ) is strictly contained in Z ( N ) . By [15, Theorem 4.1] andLemma 4.3 we havedim nuc ( C ∗ ( G, e γ )) = dim nuc ( C ∗ ( N, e γ ) ⋊ α Z ) ≤ nuc ( C ∗ ( N, e γ )) + 1) ≤ f ( h ( N )) . Therefore the nuclear dimension of every fiber of C ∗ ( G ) is bounded by 9 f ( h ( G ) − h ( G ) −
1. By Theorem 2.9, we havedim nuc ( C ∗ ( G )) ≤ h ( G ) f ( h ( G ) −
1) = f ( h ( G )) . (cid:3) Application to the Classification Program.
Combining Theorem 4.4 withresults of Lin and Niu [23] and Matui and Sato [26, 27] (see especially Corollary 6.2of [27]) we display the reach of Elliott’s classification program:
Theorem 4.5.
Let G be a finitely generated nilpotent group and J a primitive ideal of C ∗ ( G ) . If C ∗ ( G ) /J satisfies the universal coefficient theorem, then C ∗ ( G ) /J is classi-fiable by its ordered K-theory and is isomorphic to an approximately subhomogeneousC*-algebra. We do not know if every quotient of a nilpotent group C*-algebra satisfies the UCT,but Rosenberg and Schochet show that satisfying the UCT is closed under Z -actionswhich covers most cases of interest; Theorem 4.6.
Let G be a finitely generated, torsion free nilpotent group and π afaithful irreducible representation of G. Then C ∗ ( π ( G )) is classifiable by its orderedK-theory and is an ASH algebra.Proof. By Theorem 4.5 we only need to show that C ∗ ( π ( G )) satisfies the UCT. Let τ be the unique character inducing ker( π ) (see Section 2.1.2). Since G is torsion freeso is G/Z ( G ) (Section 2.1.1). Therefore we have a normal series Z ( G ) = G E G E · · · E G n = G such that G i /G i − ∼ = Z for all i = 1 , ..., n. Let x , ..., x n ∈ G be such that x i G i − generates G i /G i − for i = 1 , ..., n. For i = 1 , ..., n let α i be the automorphism of C ∗ ( π ( G i − )) defined by conjugation by x i . Since C ∗ ( π ( G )) is an iterated crossedproduct by the Z -actions α i , a repeated application of [36, Proposition 2.7] showsthat C ∗ ( π ( G )) satisfies the UCT. (cid:3) Unitriangular groups and K-theory.
In light of Theorem 4.6 it is natural towonder about K-theory calculations for specifc groups and representations. Let usannounce a little progress in this direction. Let d ≥ U d = n A ∈ GL d ( Z ) : A ii = 1 and A ij = 0 for i > j o . The center Z ( U d ) ∼ = Z is identified with those elements whose only non-zero, non-diagonal entry can occur in the (1 , d ) matrix entry. U d is a finitely generated d − θ and consider the character τ θ on U d induced fromthe multiplicative character n e πnθi . Then C ∗ ( U d , τ θ ) is covered by Theorem 4.6.In the case of d = 4, we show in [11], together with Craig Kleski that the Elliottinvariant of C ∗ ( U , τ θ ) is given by K = K = Z with the order on K given bythose vectors x ∈ Z satisfying h x, (1 , θ, θ , , ..., i > . Questions and Comments
Very broadly this section contains one question: Is there a group theoretic char-acterization of finitely generated groups whose group C*-algebras have finite nucleardimension? We parcel this into more manageable chunks.Since finite decomposition rank implies strong quasidiagonality (see [20]) there aremany easy examples of finitely generated group C*-algebras with infinite decomposi-tion rank [8]. There are also difficult examples of finitely generated amenable groupswith infinite nuclear dimension.
Theorem 5.1 (Giol and Kerr [14]) . The nuclear dimension of C ∗ ( Z ≀ Z ) is infinite.Proof. Giol and Kerr construct several C*-algebras of the form C ( X ) ⋊ Z with infinitenuclear dimension. One notices that some of these algebras are actually quotients of C ∗ ( Z ≀ Z ), forcing the nuclear dimension of C ∗ ( Z ≀ Z ) to be infinite by [41, Proposition2.3]. (cid:3) On the other hand if we restrict to polycyclic groups–given the finite-dimensionalfeel of the Hirsch number and the role it played in the present work–it seems plau-sible that all of these groups have finite nuclear dimension. Note that in [9] thereare numerous examples of polycyclic groups that are not strongly quasidiagonal andtherefore have infinite decomposition rank.
Question 5.2.
Are there any polycyclic groups with infinite nuclear dimension?
Question 5.3.
Are there any polycyclic, non virtually nilpotent groups with finitedecomposition rank or finite nuclear dimension?
The paper [9] also provides examples of non virtually nilpotent, polycyclic groupswhose group C*-algebras are strongly quasidiagonal. It seems that these groups couldbe a good starting point for a general investigation into nuclear dimension of polycyclicgroups. The difficulty here is that these groups have trivial center and we thereforedo not have a useful continuous field characterization of their group C*-algebras asin Theorem 2.7.
UCLEAR DIMENSION OF FINITELY-GENERATED NILPOTENT GROUP C*-ALGEBRAS 17
Unfortunately we were unable to extend our results to the virtually nilpotent case,and thus are left with the following question.
Question 5.4.
Do virtually nilpotent group C*-algebras have finite nuclear dimen-sion?
Finally we have
Question 5.5. If G is finitely generated and nilpotent, does C ∗ ( G ) have finite de-composition rank? The careful reader will notice that the only part of our proof where we cannotdeduce finite decomposition rank is in the second case of the proof of Theorem 4.4.There is definitely a need for both cases as there exist torsion free nilpotent groups G such that whenever G/N ∼ = Z with Z ( G ) ≤ N , then Z ( N ) is strictly bigger than Z ( G ) (we thank the user YCor on mathoverflow.net for kindly pointing this out tous). In general if a C*-algebra A has finite decomposition rank and an automorphismhas finite Rokhlin dimension, one cannot deduce that the crossed product has finitedecomposition rank (for example, consider α ⊗ β where α is a shift on a Cantor spaceand β is an irrational rotation of T ). References [1] Reinhold Baer. Finiteness properties of groups.
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Department of Mathematics, Miami University, Oxford, OH, 45056
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