Noncommutative solenoids and their projective modules
aa r X i v : . [ m a t h . OA ] N ov NONCOMMUTATIVE SOLENOIDS AND THEIR PROJECTIVEMODULES
FR´ED´ERIC LATR´EMOLI`ERE AND JUDITH A. PACKER
Abstract.
Let p be prime. A noncommutative p -solenoid is the C ∗ -algebra of Z h p i × Z h p i twisted by a multiplier of that group, where Z h p i is the additivesubgroup of the field Q of rational numbers whose denominators are powersof p . In this paper, we survey our classification of these C ∗ -algebras up to*-isomorphism in terms of the multipliers on Z h p i × Z h p i , using techniquesfrom noncommutative topology. Our work relies in part on writing these C ∗ -algebras as direct limits of rotation algebras, i.e. twisted group C*-algebrasof the group Z , thereby providing a mean for computing the K -theory of thenoncommutative solenoids, as well as the range of the trace on the K groups.We also establish a necessary and sufficient condition for the simplicity ofthe noncommutative solenoids. Then, using the computation of the trace on K , we discuss two different ways of constructing projective modules over thenoncommutative solenoids. Introduction
Twisted group algebras and transformation group C ∗ -algebras have been studiedsince the early 1960’s [8] and provide a rich source of examples and problems in C*-algebra theory. Much progress has been made in studying such C ∗ -algebras whenthe groups involved are finitely generated (or compactly generated, in the case of Liegroups). Even when G = Z n , these C ∗ -algebras give a rich class of examples whichhave driven much development in C*-algebra theory, including the foundation ofnoncommutative geometry by Connes [2], the extensive study of the geometry ofquantum tori by Rieffel [14, 16, 17, 18], the expansion of the classification problemfrom AF to AT algebras by G. Elliott and D. Evans [4], and many more (L. Baggettand A. Kleppner [1], and S. Echterhoff and J. Rosenberg [3]).In this paper, we present our work on twisted group C ∗ -algebras of the Cartesiansquare of the discrete group Z h p i of p -adic rationals, i.e. the additive subgroupof Q whose elements have denominators given by powers of a fixed p ∈ N , p ≥ p -solenoid, thereby motivating ourterminology in calling these C ∗ -algebras noncommutative solenoids . We review ourcomputation of the K -groups of these C ∗ -algebras, derived in their full technicalityin [10], and which in and of itself involves an intriguing problem in the theory ofAbelian group extensions. We were also able to compute the range of the trace Date : March 31, 2013.1991
Mathematics Subject Classification.
Primary 46L40, 46L80; Secondary 46L08, 19K14.
Key words and phrases.
C*-algebras; solenoids; projective modules; p -adic analysis. on the K -groups, and use this knowledge to classify these C ∗ -algebras up to ∗ -isomorphism, in [10], and these facts are summarized in a brief survey of [10] in thefirst two sections of this paper.This paper is concerns with the open problem of classifying noncommutativesolenoids up to Morita equivalence. We demonstrate a method of constructing anequivalence bimodule between two noncommutative solenoids using methods dueto M. Rieffel [16], and will note how this method has relationships to the theory ofwavelet frames. These new matters occupy the last three sections of this paper. Acknowledgments.
The authors gratefully acknowledge helpful conversations withJerry Kaminker and Jack Spielberg.2.
Noncommutative Solenoids
This section and the next provide a survey of the main results proven in [10]concerning the computation of the K -theory of noncommutative solenoids and itsapplication to their classification up to *-isomorphism. An interesting connectionbetween the K -theory of noncommutative solenoids and the p -adic integers is un-earthed, and in particular, we prove that the range of the K functor on the classof all noncommutative solenoids is fully described by all Abelian extensions of thegroup of p -adic rationals by Z . These interesting matters are the subject of thenext section, whereas we start in this section with the basic objects of our study.We shall fix, for this section and the next, an arbitrary p ∈ N with p >
1. Ourstory starts with the following groups:
Definition 2.1.
Let p ∈ N , p >
1. The group Z h p i of p -adic rationals is theinductive limit of the sequence of groups: Z z pz −−−−→ Z z pz −−−−→ Z z pz −−−−→ Z z pz −−−−→ · · · which is explicitly given as the group:(2.1) Z (cid:20) p (cid:21) = (cid:26) zp k ∈ Q : z ∈ Z , k ∈ N (cid:27) endowed with the discrete topology.From the description of Z h p i as an injective limit, we obtain the followingresult by functoriality of the Pontryagin duality. We denote by T the unit circle { z ∈ C : | z | = 1 } in the field C of complex numbers. Proposition 2.2.
Let p ∈ N , p > . The Pontryagin dual of the group Z h p i is the p -solenoid group, given by: S p = (cid:8) ( z n ) n ∈ N ∈ T N : ∀ n ∈ N z pn +1 = z n (cid:9) ,endowed with the induced topology from the injection S p ֒ → T N . The dual pairingbetween Q N and S N is given by: (cid:28) qp k , ( z n ) n ∈ N (cid:29) = z qk ,where qp k ∈ Z h p i and ( z n ) n ∈ N ∈ S p . We study in [10] the following C*-algebras.
ONCOMMUTATIVE SOLENOIDS AND THEIR PROJECTIVE MODULES 3
Definition 2.3. A noncommutative solenoid is a C*-algebra of the form C ∗ (cid:18) Z (cid:20) p (cid:21) × Z (cid:20) p (cid:21) , σ (cid:19) ,where p is a natural number greater or equal to 2 and σ is a multiplier of the group Z h p i × Z h p i .The first matter to attend in the study of these C*-algebras is to describe allthe multipliers of the group Z h p i × Z h p i up to equivalence, where our multipliersare T -valued unless otherwise specified, with T the unit circle in C . Note that thegroup Z h p i has no nontrivial multiplier, so our noncommutative solenoids are thenatural object to consider.Using [8], we compute in [10] the group H (cid:16) Z h p i × Z h p i , T (cid:17) of T -valued mul-tipliers of Z h p i × Z h p i up to equivalence, as follows: Theorem 2.4. [10, Theorem 2.3]
Let p ∈ N , p > . Let: Ξ p = { ( α n ) : α ∈ [0 , ∧ ( ∀ n ∈ N ∃ k ∈ { , . . . , N − } pα n +1 = α n + k ) } which is a group for the pointwise addition modulo one. There exists a group iso-morphism ρ : H (cid:16) Z h p i × Z h p i , T (cid:17) → Ξ p such that if σ ∈ H (cid:16) Z h p i × Z h p i , T (cid:17) and α = ρ ( σ ) , and if f is a multiplier of class σ , then f is cohomologous to: Ψ α : (cid:18)(cid:18) q p k , q p k (cid:19) , (cid:18) q p k , q p k (cid:19)(cid:19) exp (cid:0) iπα ( k + k ) q q (cid:1) . For any p ∈ N , p >
1, the groups Ξ p and S p are obviously isomorphic as topo-logical groups; yet it is easier to perform our computations in the additive group Ξ p in what follows. Thus, as a topological group, H (cid:16) Z h p i × Z h p i , T (cid:17) is isomorphicto S p . Moreover, we observe that a corollary of Theorem (2.4) is that Ψ α and Ψ β are cohomologous if and only if α = β ∈ Ξ p . The proof of Theorem (2.4) involvesthe standard calculations for cohomology classes of multipliers on discrete Abeliangroups, due to A. Kleppner, generalizing results of Backhouse and Bradley.With this understanding of the multipliers of Z h p i × Z h p i , we thus proposeto classify the noncommutative solenoids C ∗ (cid:16) Z h p i × Z h p i , σ (cid:17) . Let us start byrecalling [20] that for any multiplier σ of a discrete group Γ, the C*-algebra C ∗ (Γ , σ )is the C ∗ -completion of the involutive Banach algebra (cid:0) ℓ (Γ) , ∗ σ , · ∗ (cid:1) , where thetwisted convolution ∗ σ is given for any f , f ∈ ℓ (Γ) by f ∗ σ f : γ ∈ Γ X γ ∈ Γ f ( γ ) f ( γ − γ ) σ ( γ , γ − γ ),while the adjoint operation is given by: f ∗ : γ ∈ Γ σ ( γ, − γ ) f ( − γ ).The C*-algebra C ∗ (Γ , σ ) is then shown to be the universal C*-algebra generated bya family ( W γ ) γ ∈ Γ of unitaries such that W γ W δ = σ ( γ, δ ) W γδ for any γ, δ ∈ Γ [20].We shall henceforth refer to these generating unitaries as the canonical unitaries of C ∗ (Γ , σ ). FR´ED´ERIC LATR´EMOLI`ERE AND JUDITH A. PACKER
One checks easily that if σ and η are two cohomologous multipliers of the discretegroup Γ, then C ∗ (Γ , σ ) and C ∗ (Γ , η ) are *-isomorphic [20]. Thus, by Theorem(2.4), we shall henceforth restrict our attention to multipliers of Z h p i × Z h p i ofthe form Ψ α with α ∈ Ξ p . With this in mind, we introduce the following notation: Notation 2.5.
For any p ∈ N , p > α ∈ Ξ p , the C*-algebra C ∗ (cid:16) Z h p i × Z h p i , Ψ α (cid:17) , with Ψ α defined in Theorem (2.4), is denoted by A S α .Noncommutative solenoids, defined in Definition (2.3) as twisted group algebrasof Z h p i × Z h p i , also have a presentation as transformation group C ∗ -algebras, ina manner similar to the situation with rotation C*-algebras: Proposition 2.6. [10, Proposition 3.3]
Let p ∈ N , p > and α ∈ Ξ p . Let θ α be theaction of Z h p i on S p defined for all qp k ∈ Z h p i and for all ( z n ) n ∈ N ∈ S p by: θ α qpk (( z n ) n ∈ N ) = (cid:16) e ( iπα ( k + n ) q ) z n (cid:17) n ∈ N .The C*-crossed-product C ( S p ) ⋊ θ α Z h p i is *-isomorphic to A S α . Whichever way one decides to study them, there are longstanding methods inplace to determine whether or not these C ∗ algebras are simple (see for instance []).For now, we concentrate on methods from the theory of twisted group C ∗ -algebras. Theorem-Definition 2.6.1. [13, Theorem 1.5] The symmetrizer group of a mul-tiplier σ : Γ × Γ → T of a discrete group Γ is given by S σ = (cid:8) γ ∈ Γ : ∀ g ∈ Γ σ ( γ, g ) σ ( g, γ ) − = 1 (cid:9) .The C*-algebra C ∗ (Γ , σ ) is simple if, and only if the symmetrizer group S σ isreduced to the identity of Γ.In [10], we thus characterize when the symmetrizer group of the multipliers of Z h p i × Z h p i given by Theorem (2.4) is non-trivial: Theorem 2.7. [10, Theorem 2.12]
Let p ∈ N , p > . Let α ∈ Ξ p . Denote by Ψ α the multiplier defined in Theorem (2.4). The following are equivalent: (1) the symmetrizer group S Ψ α is non-trivial, (2) the sequence α has finite range, i.e. the set { α j : j ∈ N } is finite, (3) there exists k ∈ N such that ( p k − α ∈ Z , (4) the sequence α is periodic, (5) there exists a positive integer b ∈ N such that: S Ψ α = b Z (cid:20) p (cid:21) × Z (cid:20) p (cid:21) = (cid:26) ( br , br ) , ( r , r ) ∈ Z (cid:20) p (cid:21) × Z (cid:20) p (cid:21)(cid:27) . Theorem (2.8), when applied to noncommutative solenoids via Theorem (2.7),allows us to conclude:
Theorem 2.8. [10, Theorem 3.5]
Let p ∈ N , p > and α ∈ Ξ p . Then the followingare equivalent: (1) the noncommutative solenoid A S α is simple, (2) the set { α j : j ∈ N } is infinite, ONCOMMUTATIVE SOLENOIDS AND THEIR PROJECTIVE MODULES 5 (3) for every k ∈ N , we have ( p k − α Z . In particular, if α ∈ Ξ p is chosen with at least one irrational entry, then bydefinition of Ξ p , all entries of α are irrational, and by Theorem (2.8), the noncom-mutative solenoid A S α is simple. The reader may observe that, even if α ∈ Ξ p onlyhas rational entries, the noncommutative solenoid may yet be simple — as long as α has infinite range. We called this situation the aperiodic rational case in [10]. Example 2.9 (Aperiodic rational case) . Let p = 7, and consider α ∈ Ξ given by α = (cid:18) , , , , · · · (cid:19) = (cid:18) n (cid:19) n ∈ N .Note that α j ∈ Q for all j ∈ N , yet Theorem (2.8) tells us that the noncommutativesolenoid A S α is simple!The following is an example where the symmetrizer subgroup is non-trivial, sothat the corresponding C ∗ -algebra is not simple. Example 2.10 (Periodic rational case) . Let p = 5, and consider α ∈ Ξ given by α = (cid:18) , , , , · · · (cid:19) .Theorem (2.7) shows that the symmetrizer group of the multiplier Ψ α of (cid:0) Z (cid:2) (cid:3)(cid:1) given by Theorem (2.4) is: S α = (cid:26)(cid:18) j k , j k (cid:19) ∈ Q : j , j ∈ Z , k ∈ N (cid:27) .Hence the noncommutative solenoid A S α is not simple by Theorem (2.8).We conclude this section with the following result about the existence of traceson noncommutative solenoids, which follows from [7], since the Pontryagin dual S p × S p of Z h p i × Z h p i acts ergodically on A S α for any α ∈ Ξ p via the dualaction: Theorem 2.11. [10, Theorem 3.8]
Let p ∈ N , p > and α ∈ Ξ p . There exists atleast one tracial state on the noncommutative solenoid A S α . Moreover, this tracialstate is unique if, and only if α is not periodic. Moreover, since noncommutative solenoids carry an ergodic action of the com-pact groups S p , if one chooses any continuous length function on S p , then one mayemploy the results found in [18] to equip noncommutative solenoids with quantumcompact metric spaces structures and, for instance, use [19] and [9] to obtain vari-ous results on continuity for the quantum Gromov-Hausdorff distance of the familyof noncommutative solenoids as the multiplier and the length functions are left tovary. In this paper, we shall focus our attention on the noncommutative topologyof our noncommutative solenoids, rather than their metric properties.In [10, Theorem 3.17], we provide a full description of noncommutative solenoidsas bundles of matrix algebras over the space S p , while in contrast, in [10, Propo-sition 3.16], we note that for α with at least (and thus all) irrational entry, thenoncommutative solenoid A S α is an inductive limit of circle algebras (i.e. AT), withreal rank zero. Both these results follow from writing noncommutative solenoids asinductive limits of quantum tori, which is the starting point for our next section. FR´ED´ERIC LATR´EMOLI`ERE AND JUDITH A. PACKER Classification of the noncommutative Solenoids
Noncommutative solenoids are classified by their K -theory; more precisely bytheir K groups and the range of the traces on K . The main content of ourpaper [10] is the computation of the K -theory of noncommutative solenoids and itsapplication to their classification up to *-isomorphism.The starting point of this computation is the identification of noncommutativesolenoids as inductive limits of sequences of noncommutative tori. A noncommuta-tive torus is a twisted group C*-algebra for Z d , with d ∈ N , d > d = 2, we have the following description of noncommutative tori. Any multiplierof Z is cohomologous to one of the form: σ θ : (cid:18)(cid:18) z z (cid:19) , (cid:18) y y (cid:19)(cid:19) exp(2 iπθz y )for some θ ∈ [0 , θ ∈ [0 , C ∗ (cid:0) Z , σ θ (cid:1) is the universal C*-algebra generated by two unitaries U, V such that:
U V = e iπθ V U .We will employ the following notation throughout this paper:
Notation 3.1.
The noncommutative torus C ∗ (cid:0) Z , σ θ (cid:1) , for θ ∈ [0 , A θ . Moreover, the two canonical generators of A θ (i.e. the unitaries correspondingto (1 , , (0 , ∈ Z ), are denoted by U θ and V θ , so that U θ V θ = e iπθ V θ U θ .For any θ ∈ [0 , A θ is *-isomorphic to the crossed-product C*-algebra for the action of Z on the circle T generated by the rotation ofangle 2 iπθ , and thus A θ is also known as the rotation algebra for the rotation ofangle θ — a name by which it was originally known.The following question naturally arises: since A θ is a twisted Z algebra, and Z h p i × Z h p i can be realized as a direct limit group built from embeddings of Z into itself, is it possible to build our noncommutative solenoids A S α as a directlimits of rotation algebras? The answer is positive, and this observation providesmuch structural information regarding noncommutative solenoids. Theorem 3.2. [10, Theorem 3.7]
Let p ∈ N , p > and α ∈ Ξ p . For all n ∈ N , let ϕ n be the unique *-morphism from A α n into A α n +2 given by: ( U α n U pα n +2 V α n V pα n +2 Then: A α ϕ −→ A α ϕ −→ A α ϕ −→ · · · converges to the noncommutative solenoid A S α . Moreover, if ( W r ,r ) ( r ,r ) ∈ Z [ p ] × Z [ p ] is the family of canonical unitary generators of A S α , then, for all n ∈ N , the rota-tion algebra A α n embeds in A S α via the unique extension of the map: U α n W ( pn , ) V α n W ( , pn ) .to a *-morphism, given by the universal property of rotation algebras; one checksthat this embeddings, indeed, commute with the maps ϕ n . ONCOMMUTATIVE SOLENOIDS AND THEIR PROJECTIVE MODULES 7
Our choice of terminology for noncommutative solenoids is inspired, in part, byTheorem (3.2), and the well established terminology of noncommutative torus forrotation algebras. Moreover, as we shall now see, our study of noncommutativesolenoids is firmly set within the framework of noncommutative topology.The main result from our paper [10] under survey in this section and the pre-vious one is the computation of the K -theory of noncommutative solenoid and itsapplication to their classification. An interesting connection between our work onnoncommutative solenoid and classifications of Abelian extensions of Z h p i by Z ,which in turn are classified by means of the group of p -adic integers, emerges as aconsequence of our computation. We shall present this result now, starting withsome reminders about the p -adic integers and Abelian extensions of Z h p i , and referto [10] for the involved proof leading to it. Theorem-Definition 3.2.1.
Let p ∈ N , p >
1. The set: Z p = (cid:8) ( J k ) k ∈ N : J = 0 and ∀ k ∈ N J k +1 ≡ J k mod p k (cid:9) is a group for the operation defined as:( J k ) k ∈ N + ( K k ) K ∈ N = (( J k + K k ) mod p k ) k ∈ N for any ( J k ) k ∈ N , ( K k ) k ∈ N ∈ Z p . This group is the group of p -adic integers. One may define the group of p -adic integer simply as the set of sequences valuedin { , . . . , p − } with the appropriate operation, but our choice of definition willmake our exposition clearer. We note that we have a natural embedding of Z asa subgroup of Z p by sending z ∈ Z to the sequence ( z mod p k ) k ∈ N . We shallhenceforth identify Z with its image in Z p when no confusion may arise.We can associate, to any p -adic integer, a Schur multiplier of Z h p i , i.e. a map ξ j : Z h p i × Z h p i → Z which satisfies the (additive) 2-cocycle identity, in thefollowing manner: Theorem 3.3. [10]
Let p ∈ N , p > and let J = ( J k ) k ∈ N ∈ Z p . Define the map ξ J : Z h p i × Z h p i → Z by setting, for any q p k , q p k ∈ Z h p i : ξ J (cid:18) q p k , q p k (cid:19) = − q p k ( J k − J k ) if k > k , − q p k ( J k − J k ) if k > k , qp r ( J k − J r ) if k = k , with qp r = q p k + q p k ,where all fractions are written in their reduced form , i.e. such that the exponentof p at the denominator is minimal (this form is unique). Then: • ξ J is a Schur multiplier of Z h p i [10, Lemma 3.11] . • For any
J, K ∈ Z p , the Schur multipliers ξ J and ξ K are cohomologous if,and only if J − K ∈ Z [10, Theorem 3.14] . • Any Schur multiplier of Z h p i is cohomologous to ξ J for some J ∈ Z p [10,Theorem 3.16] .In particular, Ext (cid:16) Z h p i , Z (cid:17) is isomorphic to Z p / Z . FR´ED´ERIC LATR´EMOLI`ERE AND JUDITH A. PACKER
Schur multipliers provide us with a mean to describe and classify Abelian ex-tensions of Z h p i by Z p . Our interest in Theorem (3.3) lies in the remarkableobservation that the K groups of noncommutative solenoids are exactly given bythese extensions: Theorem 3.4. [10, Theorem 3.12]
Let p ∈ N , p > and let α = ( α k ) k ∈ N ∈ Ξ p . Forany k ∈ N , define J k = p k α k − α , and note that by construction, J ∈ Z p . Let ξ J be the Schur multiplier of Z h p i defined in Theorem (3.3), and let Q J be the groupwith underlying set Z × Z h p i and operation: ( z , r ) ⊞ ( z , r ) = ( z + z + ξ J ( r , r ) , r + r ) for all ( z , r ) , ( z , r ) ∈ Z × Z h p i . By construction, Q J is an Abelian extensionof Z h p i by Z given by the Schur multiplier ξ J .Then: K (cid:0) A S α (cid:1) = Q J and, moreover, all tracial states of A S α lift to a single trace τ on K (cid:0) A S α (cid:1) , char-acterized by: τ : (1 , and (cid:18) , p k (cid:19) α k .Furthermore, we have: K (cid:0) A S α (cid:1) = Z (cid:20) p (cid:21) × Z (cid:20) p (cid:21) . We observe, in particular, that given any Abelian extension of Z h p i by Z , onecan find, by Theorem (3.3), a Schur multiplier of Z h p i of the form ξ J for some J ∈ Z p , and, up to an arbitrary choice of α ∈ [0 , α = (cid:16) α + J k p k (cid:17) k ∈ N , and check that α ∈ Ξ p ; thus all possible Abelian extensions, and only Abelian extensions of Z h p i by Z are given as K groups of noncommutativesolenoids. With this observation, the K groups of noncommutative solenoids areuniquely described by a p -adic integer modulo an integer, and the informationcontained in the pair ( K ( A S α ) , τ ) of the K group of a noncommutative solenoidand its trace, is contained in the pair ( J, α ) with J ∈ Z p / Z as defined in Theorem(3.4). Remark . For any p ∈ N , p > α ∈ Ξ p , the range of the unique trace τ on K ( A S α ), as described by Theorem (3.4), is the subgroup Z ⊕ ⊕ k ∈ N α n Z .Let γ = z + z α n + . . . z k α n k be an arbitrary element of this set, where, to fixnotations, we assume n < . . . < n k . Then, since α n +1 ≡ pα n mod 1 for any n ∈ N , we conclude that we can rewrite γ simply as z ′ + yα n k , for some z ′ , y ∈ Z .Thus the range of our trace on K ( A S α ) is given by: τ (cid:0) K (cid:0) A S α (cid:1)(cid:1) = { z + yα k : z, y ∈ Z , k ∈ N } .We thus have a complete characterization of the K -theory of noncommutativesolenoids. This noncommutative topological invariant, in turn, contains enoughinformation to fully classify noncommutative solenoids in term of their defining ONCOMMUTATIVE SOLENOIDS AND THEIR PROJECTIVE MODULES 9 multipliers. We refer to [10, Theorem 4.2] for the complete statement of this clas-sification; to keep our notations at a minimum, we shall state the corollary of [10,Theorem 4.2] when working with p prime: Theorem 3.6. [10, Corollary 4.3]
Let p, q be two prime numbers and let α ∈ Ξ p and β ∈ Ξ q . Then the following are equivalent: (1) The noncommutative solenoids A S α and A S β are *-isomorphic, (2) p = q and a truncated subsequence of α is a truncated subsequence of β or (1 − β k ) k ∈ N . Theorem (3.6) is given in greater generality in [10, Theorem 4.2], where p, q arenot assumed prime; the second assertion of the Theorem must however be phrasedin a more convoluted manner: essentially, p and q must have the same set of primefactors, and there is an embedding of Ξ p and Ξ q in a larger group Ξ, whose elementsare still sequences in [0 , α and β for these embeddingsare sub-sequences of a single element of Ξ.We conclude this section with an element of the computation of the K groupsin Theorem (3.4). Given γ = (cid:16) , p k (cid:17) ∈ K (cid:0) A S α (cid:1) , if α is irrational, then thereexists a Rieffel-Powers projection in A α k whose image in A S α for the embeddinggiven by Theorem (3.2) has K class the element γ , whose trace is thus naturallygiven by Theorem (3.4). Much work is needed, however, to identify the range of K as the set of all Abelian extensions of Z h p i by Z , and parametrize these, inturn, by Z p / Z , as we have shown in this section.We now turn to the question of the structure of the category of modules overnoncommutative solenoids. In the next two sections, we show how to apply someconstructions of equivalence bimodules to the case of noncommutative solenoidsas a first step toward solving the still open problem of Morita equivalence fornoncommutative solenoids.4. Forming projective modules over noncommutative solenoids fromthe inside out
Projective modules for rotation algebras and higher dimensional noncommuta-tive tori were studied by M. Rieffel ([16]). F. Luef has extended this work to buildmodules with a dense subspace of functions coming from modulation spaces (e.g.,Feichtinger’s algebra) with nice properties ([11], [12]). One approach to buildingprojective modules over noncommutative solenoids is to build the projective mod-ules from the “inside out”.We first make some straightforward observations in this direction. We recall that,by Notation (2.5), for any p ∈ N , p >
1, and for any α ∈ Ξ p , where Ξ p is defined inTheorem (2.4), the C*-algebra C ∗ (cid:16) Z h p i × Z h p i , Ψ α (cid:17) , where the multiplier Ψ α was defined in Theorem (2.4), is denoted by A S α . In this section, we will work with p a prime number. Last, we also recall that by Notation (3.1), the rotation algebrafor the rotation of angle θ ∈ [0 ,
1) is denoted by A θ , while its canonical unitarygenerators are denoted by U θ and V θ , so that U θ V θ = e iπθ V θ U θ .Theorem (3.4) describes the K groups of noncommutative solenoids, and, amongother conclusions, state that there always exists a unique trace on the K of any noncommutative solenoid, lifted from any tracial state on the C*-algebra itself.With this in mind, we state: Proposition 4.1.
Let p be a prime number, and fix α ∈ Ξ p , with α Q . Let γ = z + qα N for some z, q ∈ Z and N ∈ N , with γ > . Then there is a leftprojective module over A S α whose K class has trace γ , or equivalently, whose K class is given by (cid:16) z, qp k (cid:17) ∈ Z × Z h p i .Proof. By Remark (3.5), γ is the image of some class in K ( A S α ) for the trace onthis group. Now, since α N +1 = pα N + j for some j ∈ Z by definition of Ξ α , we mayas well assume N is even. As K ( A S α ) is the inductive limit of K ( A α k ) k ∈ N byTheorem (3.2), γ is the trace of an element of K ( A α N )), where A α N is identifiedas a subalgebra of A S α (again using Theorem (3.2). By [14], there is a projection P γ in A α N whose K class has trace γ , and it is then easy to check that the leftprojective module P A S α over A S α fulfills our proposition. (cid:3) So, for example, with the notations of the proof of Proposition (4.1), if P γ is aprojection in A α n ⊂ A S α with trace γ ∈ (0 , A S α − A S α P γ − P γ A S α P γ .From this realization, not much about the structure of P γ A S α P γ can be seen,although it is possible to write this C ∗ -algebra as a direct limit of rotation algebras.Let us now discuss this matter.Suppose we have two directed sequences of C ∗ -algebras: A ϕ −−−−→ A ϕ −−−−→ A ϕ −−−−→ · · · and B ψ −−−−→ B ψ −−−−→ B ψ −−−−→ · · · Suppose further that for each n ∈ N there is an equivalence bimodule X n between A n and B n A n − X n − B n , and that the ( X n ) n ∈ N form a directed system, in the following sense: there existsa direct system of module monomorphisms X i −−−−→ X i −−−−→ X i −−−−→ · · · satisfying, for all f, g ∈ X n and b ∈ B n : h i n ( f ) , i n ( g ) i B n +1 = ψ n ( h f, g i B n )and i n ( f · b ) = i n ( f ) · ψ n ( b ),with analogous but symmetric equalities holding for the X n viewed as left- A n mod-ules.Now let A be the direct limit of ( A n ) n ∈ N , B be the direct limit of ( B n ) n ∈ N and X be the direct limit of ( X n ) n ∈ N (completed in the natural C ∗ -module norm). Then X is an A − B bimodule. If one further assumes that the algebra of adjointableoperators on X viewed as a A − B bimodule, L ( X ) , can be obtained via an appro-priate limiting process from the sequence of adjointable operators {L ( X n ) } ∞ n =1 ( ONCOMMUTATIVE SOLENOIDS AND THEIR PROJECTIVE MODULES 11 where each X n is a A n − B n bimodule), then in addition one has that X is a strongMorita equivalence bimodule between A and B . So suppose that γ ∈ (0 ,
1) is as in the statement of Proposition (4.1), for some α ∈ Ξ p not equal to zero, and suppose that we know that there is a positive integer N and a projection P γ in A α N whose K class has trace γ. Again, without loss ofgenerality, we assume that N is even. Then setting A n = A α N +2 n , X n = A α N +2 n P γ , and B n = P γ A α N +2 n P γ , all of the conditions in the above paragraphs hold a priori , since A S α is a direct limitof the A α N +2 n , so that certainly B = P γ A S α P γ is a direct limit of the P γ A α N +2 n P γ , and X = A S α P γ can be expressed as a direct limit of the X n = A α N +2 n P γ , againby construction, with the desired conditions on the adjointable operators satisfiedby construction.It would be interesting to see how far this set-up could be extended to moregeneral directed systems of Morita equivalence bimodules over directed systems of C ∗ -algebras, but we leave this project to a future endeavor.We discuss very simple examples, to show how the directed system of bimodulesis constructed. Example 4.2.
Fix an irrational α ∈ [0 , p = 2, and consider α ∈ Ξ givenby α = ( α , α = α , α = α , · · · , α n = α n n , · · · , ) , Consider P α ∈ A α ⊂ A α a projection of trace α = 2 α . The bimodule A α − A α · P α − P α A α P α is equivalent to Rieffel’s bimodule A α − C c ( R ) − A α = B . Let β = α . Rieffel’s theory, specifically Theorem 1.1 of [15], again shows there isa bimodule A α − A α · P α − P α A α P α is the same as A α − A α · P α − P α A α P α which is equivalent to Rieffel’s bimodule A α − C c ( R × F ) − C ( T × F ) ⋊ τ Z = B , where F = Z / Z , and the action of Z on T × F is given by multiples of ( β , [1] F ) , for β = α , i.e. multiples of ( α , [1] F ]) , i.e. multiples of ( β , [1] F ) . At the n th stage, using Theorem 1.1 of [15] again, we see that A α n − A α n · P α − P α A α n P α is the same as A α n − A α n · P n α n − P n α n A α n P n α n which is equivalent to A α n − C c ( R × F n ) − C ( T × F n ) ⋊ τ n Z = B n , where the action of Z on T × F n is given by multiples of ( β n n , [1] F n ) , for β n = α n = n α , i.e. multiples of ( α , [1] F n ) , i.e. multiples of ( β , [1] F n ) , for F n = Z / n Z . From calculating the embeddings, we see that for α = ( α , α , · · · , α n , · · · ) ∈ Ξ , we have that A S α is strongly Morita equivalent to a direct limit B of the B n . The structure of B isnot clear in this description, although each B n is seen to be a variant of a rotationalgebra. As expected, one calculates tr ( K ( A S α )) = α · tr ( K ( B )) . Forming projective modules over noncommutative solenoids using p -adic fields Under certain conditions, one can construct equivalence bimodules for A S α ( α ∈ Ξ p , p prime) by using a construction of M. Rieffel [16]. The idea is to first embedΓ = Z h p i × Z h p i as a co-compact ‘lattice’ in a larger group M , and the quotientgroup M /Γ will be exactly the solenoid S p . We thank Jerry Kaminker and JackSpielberg for telling us about this trick.We start with a brief description of the field of p -adic numbers, with p prime.Algebraically, the field Q p is the field of fraction of the ring of p -adic integers Z p —we introduce Z p as a group, though there is a natural multiplication on Z p turningit into a ring. A more analytic approach is to consider Q p as the completion of thefield Q for the p -adic metric d p , defined by d p ( r, r ′ ) = | r − r ′ | p for any r, r ′ ∈ Q ,where | · | p is the p -adic norm defined by: | r | = ( p − n if r = 0 and where r = p n ab with a, b are both relatively prime with p, r = 0.If we endow Q with the metric d p , then series of the form: ∞ X j = k a j p j will converge, for any k ∈ Z and a j ∈ { , . . . , p − } for all j = k, . . . . This isthe p -adic expansion of a p -adic number. One may easily check that addition andmultiplication on Q are uniformly continuous for d p and thus extend uniquely to Q p to give it the structure of a field. Moreover, one may check that the group Z p of p -adic integer defined in Section 3 embeds in Q p as the group of p -adic numbers of theform P ∞ j =0 a j p j with a j ∈ { , . . . , p − } for all j ∈ N . Now, with this embedding,one could also check that Z p is indeed a subring of Q p whose field of fractions is Q p (i.e. Q p is the smallest field containing Z p as a subring) and thus, both constructionsdescribed in this section agree. Last, the quotient of the (additive) group Q p by itssubgroup Z p is the Pr¨ufer p -group Z ( p ∞ ) = { z ∈ T : ∃ n ∈ N z ( p n ) = 1 } .5.1. Embedding Z ( p ) as a lattice in a self-dual group. Since Q p is a metriccompletion of Q and Z h p i is a subgroup of Q , we shall identify, in this section, ONCOMMUTATIVE SOLENOIDS AND THEIR PROJECTIVE MODULES 13 Z h p i as a subgroup of Q p with no further mention. We now define a few grouphomomorphisms to construct a short exact sequence at the core of our construction.Let ω : R → S p be the standard “winding line” defined for any t ∈ R by: ω ( t ) = (cid:16) e πit , e πi tp , e πi tp , · · · , e πi tpn , · · · (cid:17) .Let γ ∈ Q p and write γ = P ∞ j = k a j p j for a (unique) family ( a j ) j = k,... of elementsin { , . . . , p − } . We define the sequence ζ ( γ ) by setting for all j ∈ N : ζ j ( γ ) = e πi (cid:16)P jm = k ampj − m + k (cid:17) with the convention that P kj · · · is zero if k < j .We thus may define the mapΠ : ( Q p × R −→ S p γ Π( γ, t ) = ζ j ( γ ) · ω ( t ).If we set ι : ( Z h p i −→ Q p × R r ι ( r ) = ( r, − r ),then one checks that the following is an exact sequence:1 −−−−→ Z h p i ι −−−−→ Q p × R Π −−−−→ S p −−−−→ −−−−→ Z h p i × Z h p i −−−−→ [ Q p × R ] × [ Q p × R ] −−−−→ S p × S p −−−−→ Z h p i × Z h p i by elements of Q p \ { } and R \ { } to obtain afamily of different embeddings of Z h p i × Z h p i into [ Q p × R ] .We now observe that M = Q p × R is self-dual. We shall use the followingstandard notation: Notation 5.1.
The Pontryagin dual of a locally compact group G is denoted by b G . The dual pairing between a group and its dual is denoted by h· , ·i : G × b G → T .Let us show that M ∼ = c M . To every x ∈ Q p , we can associate the character χ x : q ∈ Q p e iπi { x · q } where { x · q } p is the fractional part of the product x · q in Q p , i.e. it is the sum ofthe terms involving the negative powers of p in the p -adic expansion of x · q . Themap x ∈ Q p χ x ∈ c Q p is an isomorphism of topological group. Similarly, everycharacter of R is of the form χ r : t ∈ R e iπrt for some r ∈ R . Therefore everycharacter of M is given by χ ( x,r ) : ( q, t ) ∈ Q p × R χ x ( q ) χ r ( t )for some ( x, r ) ∈ Q p × R (see [6]) for further details on characters of specific locallycompact abelian groups). It is possible to check that the map ( x, r ) χ ( x,r ) is agroup isomorphism between M and c M , so that M = Q p × R is indeed self-dual. The Heisenberg representation and the Heisenberg equivalence bi-module of Rieffel.
In this section, we write Γ = Z h p i × Z h p i where p is someprime number, and now let M = [ Q p × R ]. We have shown in the previous sectionthat M is self-dual, since both Q p and R are self-dual. Now suppose there is anembedding ι : Γ → M × c M .
Let the image ι (Γ) be denoted by D . In the case we areconsidering, D is a discrete co-compact subgroup of M × c M . Following the methodof M. Rieffel [16], the Heisenberg multiplier η : ( M × c M ) × ( M × c M ) → T isdefined by: η (( m, s ) , ( n, t )) = h m, t i , ( m, s ) , ( n, t ) ∈ M × c M . (We note we use the Greek letter ‘ η ’ rather than Rieffel’s ‘ β ’, because we have used‘ β ’ elsewhere. Following Rieffel, the symmetrized version of η is denoted by theletter ρ, and is the multiplier defined by: ρ (( m, s ) , ( n, t )) = η (( m, s ) , ( n, t )) η (( n, t ) , ( m, s )) , ( m, s ) , ( n, t ) ∈ M × c M .M. Rieffel [16] has shown that C C ( M ) can be given the structure of a left C ∗ ( D, η ) module, as follows. One first constructs an η -representation of M × c M on L ( M ) , defined as π, where π ( m,s ) ( f )( n ) = h n, s i f ( n + m ) , ( m, s ) ∈ M × c M , n ∈ M .When the representation π is restricted to D, we still have a projective η -representationof D, on L ( M ) , and its integrated form gives C C ( M ) the structure of a left C ∗ ( D, η )module, i.e. for Φ ∈ C C ( D, η ) , f ∈ C C ( M ) ,π (Φ) · f ( n ) = X ( d,χ ) ∈ D Φ(( d, χ )) π ( d,χ ) ( f )( n )= X ( d,χ ) ∈ D Φ(( d, χ )) h n, χ i f ( n + d ).There is also a C C ( D, η ) valued inner product defined on C C ( M ) given by: h f, g i C C ( D,η ) = Z M f ( n ) π ( d,χ ) ( g )( n ) dn = Z M f ( n ) h n, χ i g ( n + d ) dn. Moreover, Rieffel has shown that setting D ⊥ = { ( n, t ) ∈ M × c M : ∀ ( m, s ) ∈ D ρ (( m, s ) , ( n, t )) = 1 } , C C ( M ) has the structure of a right C ∗ ( D ⊥ , η ) module. Here the right modulestructure is given for all f ∈ C c ( M ), Ω ∈ C c ( D ⊥ ) and n ∈ M by: f · Ω( n ) = X ( c,ξ ) ∈ D ⊥ π ∗ ( c,ξ ) ( f )( n )Ω( c, ξ ) , and the C C ( D ⊥ , η )-valued inner product is given by h f, g i C C ( D ⊥ ,η ) ( c, ξ ) = Z M f ( n ) π ( c,ξ ) ( g )( n ) dn = Z M f ( n ) h n, ξ i g ( n + c ) dn ,where f, g ∈ C C ( M ) , Ω ∈ C C ( D ⊥ , η ) , and ( c, ξ ) ∈ D ⊥ . Moreover, Rieffel shows in [16, Theorem 2.12] that C ∗ ( D, η ) and C ∗ ( D ⊥ , η ) arestrongly Morita equivalent, with the equivalence bimodule being the completion of C C ( M ) in the norm defined by the above inner products. ONCOMMUTATIVE SOLENOIDS AND THEIR PROJECTIVE MODULES 15
In order to construct explicit bimodules, we first define the multiplier η moreprecisely, and then discuss different embeddings of Z h p i × Z h p i into M × c M .
In the case examined here, the Heisenberg multiplier η : [ Q p × R ] × [ Q p × R ] → T is given by: Definition 5.2.
The Heisenberg multiplier η : [ Q p × R ] × [ Q p × R ] → T is definedby η [(( q , r ) , ( q , r )) , (( q , r ) , ( q , r ))] = e πir r e πi { q q } p ,where { q q } p is the fractional part of the product q · q , i.e. the sum of the termsinvolving the negative powers of p in the p -adic expansion of q q . The following embeddings of Z h p i × Z h p i in [ Q p × R ] will prove interesting: Definition 5.3.
For θ ∈ R , θ = 0 , we define ι θ : Z h p i × Z h p i → [ Q p × R ] by ι θ ( r , r ) = [( r , θ · r ) , ( r , r )].We examine the structure of the multiplier η more precisely and then discussdifferent embeddings of Z h p i × Z h p i into [ Q p × R ] and their influence on thedifferent equivalence bimodules they allow us to construct.We start by observing that for r , r , r , r ∈ Z h p i : η ( ι θ ( r , r )) , ι θ ( r , r )) = e πi { r r } p e πiθr r = e πir r e πiθr r = e πi ( θ +1) r r .(Here we used the fact that for r i , r j ∈ Z ( p ) , { r i r j } p ≡ r i r j modulo Z . )One checks that setting D θ = ι θ (cid:16) Z h p i × Z h p i(cid:17) , the C ∗ -algebra C ∗ ( D θ , η ) isexactly *-isomorphic to the noncommutative solenoid A S α for α = (cid:18) θ + 1 , θ + 1 p , · · · , θ + 1 p n , · · · (cid:19) = (cid:18) θ + 1 p n (cid:19) n ∈ N .For this particular embedding of Z h p i × Z h p i as the discrete subgroup D inside M × c M , we calculate that D ⊥ θ = (cid:26)(cid:16) r , − r θ (cid:17) , ( r , − r ) : r , r ∈ Z (cid:20) p (cid:21)(cid:27) .Moreover, η (cid:16)h(cid:16) r , − r θ (cid:17) , ( r , − r ) i , h(cid:16) r , − r θ (cid:17) , ( r , − r ) i(cid:17) = e − πi ( θ +1) r r .It is evident that C ∗ ( D ⊥ θ , η ) is also a non-commutative solenoid A S β where β = (cid:16) − θ +1 p n θ (cid:17) n ∈ N .Note that for α = (cid:18) θ + 1 , θ + 1 p , · · · , θ + 1 p n , · · · (cid:19) , and β = (cid:18) − θ + 1 p n θ (cid:19) n ∈ N , we have θ · τ (cid:0) K (cid:0) A S α (cid:1)(cid:1) = τ (cid:0) K (cid:0) A S β (cid:1)(cid:1) with the notations of Theorem (3.4). Thus in this case we do see the desiredrelationship mentioned in Section 4: the range of the trace on the K groups of thetwo C ∗ -algebras are related via multiplication by a positive constant.We can now generalize our construction above as follows. Definition 5.4.
For any x ∈ Q p \ { } , and any θ ∈ R \ { } , there is an embedding ι x,θ : Z (cid:20) p (cid:21) × Z (cid:20) p (cid:21) → [ Q p × R ] defined for all r , r ∈ Z h p i by ι x,θ ( r , r ) = [( x · r , θ · r ) , ( r , r )].Then, we shall prove that for all α ∈ Ξ p there exists x ∈ Q p \ { } and θ ∈ R \ { } such that, by setting D x,θ = ι x,θ (cid:18) Z (cid:20) p (cid:21) × Z (cid:20) p (cid:21)(cid:19) the twisted group C*-algebra C ∗ ( D, η ) is *-isomorphic to A S α .As a first step, we prove: Lemma 5.5.
Let p be prime, and let M = Q p × R . Let ( x, θ ) ∈ [ Q p \ { } ] × [ R \ { } ] , and define ι x,θ : Z h p i × Z h p i → [ Q p × R ] ∼ = M × c M by: ι x,θ ( r , r ) = [( x · r , θ · r ) , ( r , r )] for all r , r ∈ Z (cid:20) p (cid:21) .Let η denote the Heisenberg cocycle defined on [ M × c M ] and let D = ι x,θ (cid:18) Z (cid:20) p (cid:21) × Z (cid:20) p (cid:21)(cid:19) .Then D ⊥ x,θ = (cid:26)(cid:20) ( t , − t ) , (cid:18) x − t , − t θ (cid:19)(cid:21) : t , t ∈ Z (cid:20) p (cid:21)(cid:27) . Proof.
By definition, D ⊥ x,θ = (cid:26) [( q , s ) , ( q , s )] : ∀ r , r ∈ Z (cid:20) p (cid:21) ρ ([ ι x,θ ( r , r )] , [( q , s ) , ( q , s )]) = 1 (cid:27) = (cid:26) [( q , s ) , ( q , s )] : ∀ r , r ∈ Z (cid:20) p (cid:21) ρ ([( x · r , θ · r ) , ( r , r )] , [( q , s ) , ( q , s )]) = 1 (cid:27) = (cid:26) [( q , s ) , ( q , s )] : ∀ r , r ∈ Z (cid:20) p (cid:21) e πiθr s e πi { x · r q } p e πis r e πi { q r } p = 1 (cid:27) .Now if r = 0 , and r = p n , for any n ∈ Z , this implies ∀ n ∈ Z e πiθp n s e πi { x · p n q } p = 1 , so that if we choose s = − t θ for some t ∈ Z h p i ⊆ R , we need q = x − t . Likewise, if we take r = 0 , and r = p n , for any n ∈ Z , we need ( q , s ) such that ∀ n ∈ Z e πis p n e πi { q p n } p = 1 . ONCOMMUTATIVE SOLENOIDS AND THEIR PROJECTIVE MODULES 17
Again fixing q = t ∈ Z h p i , this forces s = − t . Thus D ⊥ x,θ = (cid:26)(cid:20) ( t , − t ) , (cid:18) x − t , − t θ (cid:19)(cid:21) : t , t ∈ Z (cid:20) p (cid:21)(cid:27) ,as we desired to show. (cid:3) One thus sees that the two C ∗ -algebras C ∗ ( D x,θ , η ) and C ∗ ( D ⊥ x,θ , η ) are stronglyMorita equivalent (but not isomorphic, in general), and also the proof of this lemmashows that C ∗ ( D ⊥ x,θ , η ) is a noncommutative solenoid.We can use Lemma (5.5) to prove the following Theorem: Theorem 5.6.
Let p be prime, and let α = ( α i ) i ∈ N ∈ Ξ p , with α ∈ (0 , . Thenthere exists ( x, θ ) ∈ [ Q p \{ } ] × [ R \{ } ] with C ∗ ( D x,θ , η ) isomorphic to the noncom-mutative solenoid A S α , where D x,θ = ι x,θ (cid:16) Z h p i × Z h p i(cid:17) . Moreover, the methodof Rieffel produces an equivalence bimodule between A S α and another unital C ∗ -algebra B , and B is itself isomorphic to a noncommutative solenoid.Proof. By definition of Ξ p , for all j ∈ N there exists b j ∈ { , . . . , p − } such that pα j +1 = α j + b j . We construct an element of the p -adic integers, x = P ∞ j =0 b j p j ∈ Z p ⊂ Q p . Let θ = α , and now consider for this specific x and this specific θ the C ∗ -algebra C ∗ ( D x,θ , η ). By Definition (5.4), ι x,θ ( r , r ) = [( x · r , θ · r ) , ( r , r )] , for r , r ∈ Z h p i . Then η ( ι x,θ ( r , r ) , ι x,θ ( r , r )) = η ([( x · r , θ · r ) , ( r , r )] , [( x · r , θ · r ) , ( r , r )])= e πiθr r e πi { xr r } p , r , r , r , r ∈ Z (cid:20) p (cid:21) ,and, setting r i = j i p ki , ≤ i ≤ , and setting θ = α , we obtain η (cid:18) ι x,α (cid:18) j p k , j p k (cid:19) , ι x,α (cid:18) j p k , j p k (cid:19)(cid:19) = e πiα j j pk k e πi { x j j pk k } p for all j p k , j p k , j p k , j p k ∈ Z (cid:20) p (cid:21) .We now note that the relation pα j +1 = α j + b j , b j ∈ { , , · · · , p − } allows us toprove inductively that ∀ n ≥ α n = α + P n − j =0 b j p j p n . By Theorem (2.4), the multiplier Ψ α on Z h p i × Z h p i is defined by:Ψ α (cid:18)(cid:18) j p k , j p k (cid:19) , (cid:18) j p k , j p k (cid:19)(cid:19) = e πi ( α ( k k j j ) = e πi α j j pk k e πi ( P k k − j =0 b j p j j j ) /p k k . A p -adic calculation now shows that for j p k and j p k ∈ Z h p i and x = P ∞ j =0 b j p j ∈ Z p , we have { x j j p k k } p = ( P k + k − j =0 b j p j ) · j j p k k modulo Z , so that e πi { x j j pk k } p = e πi ( P k k − j =0 b j p j j j ) /p k k . We thus obtain η ( ι x,θ ( r , r ) , ι x,θ ( r , r )) = Ψ α (( r , r ) , ( r , r ))for all r , r , r , r ∈ Z h p i ,as desired.To prove the final statement of the Theorem, we use Lemma 5.5. We haveshown A S α is isomorphic to C ∗ ( D x,θ , η ) , and the discussion prior to the statement ofLemma 5.5 shows that C ∗ ( D x,θ , η ) is strongly Morita equivalent to C ∗ ( D ⊥ x,θ , η ) = B . But the proof of Lemma 5.5 gives that D ⊥ x,θ is isomorphic to Z h p i × Z h p i , so that C ∗ ( D ⊥ x,θ , η ) = B is a noncommutative solenoid, as we desired to show. (cid:3) Remark . It remains an open question to give necessary and sufficient conditionsunder which two noncommutative solenoids A S α and A S β would be strongly Moritaequivalent, although it is evident that a necessary that the range of the trace on K of one of the C ∗ -algebras should be a constant multiple of the range of the traceon the K group of the other. By changing the value of θ to be α + j, j ∈ Z , and adjusting the value of x ∈ Q p accordingly, one can use the method of Theorem5.6 to construct a variety of embeddings ι x,θ of Z h p i × Z h p i into [ Q p × R ] thatprovide lattices D x,θ such that C ∗ ( D x,θ , η ) and A S α are ∗ -isomorphic, but suchthat the strongly Morita equivalent solenoids C ∗ ( D ⊥ x,θ , η ) vary in structure. Thismight lend some insight into classifying the noncommutative solenoids up to strongMorita equivalence, as might a study between the relationship between the twodifferent methods of building equivalence modules described in Sections 4 and 5. Remark . In the case where the lattice Z n embeds into R n × c R n , F. Luefhas used the Heisenberg equivalence bimodule construction of Rieffel to constructdifferent families of Gabor frames in modulation spaces of L ( R n ) for modulationand translation by Z n ([11], [12]). It is of interest to see how far this analogy canbe taken when studying modulation and translation operators of Z h p i acting on L ( Q p × R ) , and we are working on this problem at present. References [1] L. Baggett and A. Kleppner,
Multiplier representations of abelian groups , J. Functional Anal-ysis (1973), 299-324.[2] A. Connes, C*–alg`ebres et g´eom´etrie differentielle , C. R. de l’academie des Sciences de Paris(1980), no. series A-B, 290.[3] S. Echterhoff and J. Rosenberg,
Fine structure of the Mackey machine for actions of abeliangroups with constant Mackey obstruction , Pacific J. Math. (1995), 17-52.[4] G. Elliott and D. Evans,
Structure of the irrational rotation C ∗ -algebras , Annals of Mathe-matics (1993), 477–501.[5] L. Fuchs, Infinite Abelian Groups, Volume I , Academic Press, New York and London, 1970.[6] E. Hewitt and K. Ross,
Abstract Harmonic Analysis, Volume II , Springer-Verlag Berlin,1970.[7] R. Hoegh-Krohn, M. B. Landstad, and E. Stormer,
Compact ergodic groups of automor-phisms , Annals of Mathematics (1981), 75–86.[8] A. Kleppner,
Multipliers on Abelian groups , Mathematishen Annalen (1965), 11–34.[9] F. Latr´emoli`ere,
Approximation of the quantum tori by finite quantum tori for the quantumgromov-hausdorff distance , Journal of Funct. Anal. (2005), 365–395, math.OA/0310214.[10] F. Latr´emoli`ere and J. Packer,
Noncommutative solenoids , Submitted (2011), 30 pages,ArXiv: 1110.6227.
ONCOMMUTATIVE SOLENOIDS AND THEIR PROJECTIVE MODULES 19 [11] F. Luef,
Projective modules over noncommutative tori and multi-window Gabor frames formodulation spaces , J. Funct. Anal. (2009), 1921–1946.[12] ,
Projections in noncommutative tori and Gabor frames , Proc. Amer. Math. Soc. (2011), 571–582.[13] J. Packer and I. Raeburn,
On the structure of twisted group C ∗ -algebras , Trans. Amer. Math.Soc. (1992), no. 2, 685–718.[14] M. A. Rieffel, C*-algebras associated with irrational rotations , Pacific Journal of Mathematics (1981), 415–429.[15] , The cancellation theorem for the projective modules over irrational rotation C ∗ -algebras , Proc. London Math. Soc. (1983), 285–302.[16] , Projective modules over higher-dimensional non-commutative tori , Can. J. Math. XL (1988), no. 2, 257–338.[17] , Non-commutative tori — a case study of non-commutative differentiable manifolds ,Contemporary Math (1990), 191–211.[18] ,
Metrics on states from actions of compact groups , Documenta Mathematica (1998),215–229, math.OA/9807084.[19] , Gromov-Hausdorff distance for quantum metric spaces , Mem. Amer. Math. Soc. (March 2004), no. 796, math.OA/0011063.[20] G. Zeller-Meier,
Produits crois´es d’une C*-alg`ebre par un groupe d’ Automorphismes , J.Math. pures et appl. (1968), no. 2, 101–239. Department of Mathematics, University of Denver, 80208
E-mail address : [email protected] Department of Mathematics, Campus Box 395, University of Colorado, Boulder, CO,80309-0395
E-mail address ::