Tracing projective modules over noncommutative orbifolds
aa r X i v : . [ m a t h . OA ] F e b TRACING PROJECTIVE MODULESOVER NONCOMMUTATIVE ORBIFOLDS
SAYAN CHAKRABORTY
Abstract.
For an action of a finite cyclic group F on an n -dimensional noncommutativetorus A θ , we give sufficient conditions when the fundamental projective modules over A θ ,which determine the range of the canonical trace on A θ , extend to projective modules overthe crossed product C*-algebra A θ ⋊ F. Our results allow us to understand the range of thecanonical trace on A θ ⋊ F , and determine it completely for several examples including thecrossed products of 2-dimensional noncommutative tori with finite cyclic groups and theflip action of Z on any n -dimensional noncommutative torus. As an application, for theflip action of Z on a simple n -dimensional torus A θ , we determine the Morita equivalenceclass of A θ ⋊ Z , in terms of the Morita equivalence class of A θ . Introduction
For n ≥
2, let T n denote the space of all n × n real skew-symmetric matrices. The n -dimensional noncommutative torus A θ is the universal C*-algebra generated by unitaries U , U , U , · · · , U n subject to the relations(0.1) U k U j = e πiθ jk U j U k for j, k = 1 , , , · · · , n , where θ := ( θ jk ) ∈ T n . For the 2-dimensional noncommutative tori,since θ is determined by only one real number, θ , we will denote θ by θ again and thecorresponding 2-dimensional noncommutative torus by A θ .There is a canonical action of SL(2 , Z ) on two dimensional noncommutative tori, which isgiven by sending U to e πiacθ U a U c and U to e πibdθ U b U d , for a matrix (cid:18) a bc d (cid:19) ∈ SL(2 , Z ) . This action was further generalised to the higher dimensional noncommutative tori. It waspointed out in [JL15] that the right replacement of the group SL(2 , Z ) isSp( n, Z , θ ) := { W ∈ GL ( n, Z ) : W T θW = θ } . Then there is a natural action of Sp( n, Z , θ ) on the n -dimensional noncommutative torus A θ . It is easy to see that Sp(2 , Z , θ ) is exactly SL(2 , Z ) . The study of crossed product C*-algebras associated to finite group actions on noncommuta-tive tori goes back to the work of Bratteli, Elliott, Evans and Kishimoto ([BEEK91]). Howeverthey only looked at the action of Z on the C*-algebra A θ , for 2-dimensional tori. Recall thatthe action of Z on any n -dimensional A θ , often called the flip action , is defined by sending U i to U − i . Note that the above action is basically given by the matrix − id n ∈ Sp( n, Z , θ ) , whereid n is the n × n unit matrix. Later various other authors studied actions of other finite cyclicsubgroups of SL(2 , Z ) on 2-dimensional noncommutative tori, see [BW07], [ELPW10], [Wal95], Mathematics Subject Classification.
Key words and phrases.
Metaplectic transformations, Morita equivalence, noncommutative torus, C*-crossed product, group actions, classification of C*-algebras. [Wal00]. Motivated by the 2-dimensional results, it is also natural to consider a finite cyclicgroup F inside Sp( n, Z , θ ) and consider the crossed product A θ ⋊ F, for an n -dimensional torus A θ . We may call such a crossed product a noncommutative orbifold.
The authors in [ELPW10] and [BCHL18] considered actions of cyclic subgroups of SL(2 , Z )on 2-dimensional noncommutative tori. Along with K-theory computations of the correspond-ing crossed products, the authors computed the images of the canonical tracial states of suchalgebras. The recent development of the classification program of C*-algebras allowed themto deduce results about isomorphism and Morita equivalence classes of such algebras, whenthe algebras are simple. One of the major facts they used is that the algebras are simple AHalgebras when θ is irrational (for the finite group actions, the algebras are even AF). Then thealgebras are classifiable in the sense of Elliot’s classification program.In [JL15], Jeong and Lee, and in [He19], He studied actions of finite subgroups of Sp( n, Z , θ )on an n -dimensional A θ , and found many of such crossed products to be classifiable, when θ is non-degenerate (see Definition 4.6) so that A θ is simple. However they did not discussisomorphism and Morita equivalence classes of the crossed products. Our paper is a firstattempt towards these kind of results for the higher dimensional cases.To understand isomorphism and Morita equivalence classes of such noncommutative orb-ifolds it is necessary to compute the K-theory of the orbifolds and understand the ranges ofthe canonical tracial states of the algebras. While the dimensions of the K-groups are known(from [LL12]), the tracial ranges are not understood. Our main results help to understandwhich numbers belong to the tracial ranges of the orbifolds, and even determine the tracialranges completely for several examples.To understand the tracial range of an orbifold, one should first understand the same forthe noncommutative torus itself. This was done by Elliott in [Ell84]. To give an overview ofour results, we recall the tracial range result from [Ell84]. For an integer p with 1 ≤ p ≤ n ,if we denote the sub-matrix M θI of θ consisting of rows and columns indexed by the numbers i , i , ..., i p for some i < i < ... < i p , I := ( i , i , . . . , i p ), then Elliott’s result may be statedas Tr (K ( A θ )) = Z + X < | I |≤ n pf( M θI ) Z , where | I | := 2 m for I = ( i , i , . . . , i m ) and pf denotes the pfaffian. Here Tr denotes thecanonical tracial state on A θ . It was observed in [Cha20] that for each such I , there is a projective module E θI over A θ , trace of which is exactly pf( M θI ) , assuming pf( M θI ) = 0 . This module is governed by an element g I, Σ ∈ SO( n, n | Z ) (see Section 2). The modules of such kind are called fundamental projectivemodules .Now coming back to the crossed products of n-dimensional torus A θ with a finite cyclicgroup F ⊂ Sp( n, Z , θ ) , if Tr F denotes the canonical trace on A θ ⋊ F, the regular representation A θ ⋊ F ֒ → M N ( A θ ) givesTr F (K ( A θ ⋊ F )) ⊆ N Tr(K ( A θ )) = 1 N ( Z + X < | I |≤ n pf( M θI ) Z ) . Our main theorem (Theorem 0.1) determines when the term N pf( M θI ) lies in the left handside of the above equation. The proof of the theorem involves extending the modules E θI tomodules over the crossed products using so-called metaplectic operators , which were alreadyused by the author (in a joint work with Luef) in [CL19] to extend a specific type modules( Bott classes ) to modules over the crossed products. racing projective modules over noncommutative orbifolds 3
Theorem 0.1. (Theorem 3.2, Theorem 4.2) With all the notations introduced above, assume pf( M θI ) = 0 . Let W ∈ GL( n, Z ) be of finite order such that W t θW = θ and F := h W i . Suppose g I, Σ F ( g I, Σ ) − ⊂ R inside SO( n, n | Z ) . Then E θI becomes a finitely generated, projective moduleover A θ ⋊ F and N pf( M θI ) ∈ Tr F (K ( A θ ⋊ F )) , where N is the order of W . In the above theorem R denotes the subgroup of SO( n, n | Z ) generated by the elements ofGL( n, Z ) . The condition in the above theorem is easy to check for many examples. In fact, we providesome examples with explicit tracial range computations. These examples include the twodimensional cases and the flip action of Z . It is worthwhile to explicitly state the consequencesfor the flip action here in the introduction, since the results were unknown to the author. Forthe tracial range we get, Tr Z (K ( A θ ⋊ Z )) = 12 Tr(K ( A θ )) , for any θ in T n . And as a corollary we have,
Corollary 0.2. (Corollary 4.11) Let θ , θ ∈ T n be non-degenerate. Let Z act on A θ and A θ by the flip actions. Then A θ ⋊ Z is strongly Morita equivalent to A θ ⋊ Z if and only if A θ is strongly Morita equivalent to A θ . It is worth mentioning that the only action of a finite cyclic subgroup of Sp(3 , Z , θ ) on a3-dimensional torus A θ , when θ is non-degenerate, is the flip action ([JL15, Theorem 1.4]).Apart from the applications in classification of C*-algebras, the computations of the rangesof tracial states turn out to be useful in physics (Bellisard’s gap labelling theorem, in partic-ular). Even our results are very much similar to the results which are involved in a twistedversion of the gap labelling theorem, recently conjectured in [BM18]. We hope that our tech-niques will be helpful understanding the conjecture better.This article is organised as follows: in Section 1 we recall the definition of twisted groupC*-algebras and give relevant examples. In Section 2, we discuss the fundamental projectivemodules over noncommutative tori. Section 3 deals with extending the fundamental modules tomodules over orbifolds, and proving Theorem 0.1 (Theorem 3.2). In the last section, Section 4,the proof of Theorem 0.1 (Theorem 4.2) about the ranges of the canonical traces on orbifoldsis discussed along with various examples. We also discuss the results about Morita equivalenceclasses of orbifolds, along with Corollary 0.2 in Section 4. Notation : e ( x ) will always denote the number e πix , and id m will be the m × m unit matrix.1. Twisted group C*-algebras and noncommutative orbifolds
Let G be a discrete group. A map ω : G × G → T is called a if ω ( x, y ) ω ( xy, z ) = ω ( x, yz ) ω ( y, z )whenever x, y, z ∈ G , and if ω ( x,
1) = 1 = ω (1 , x )for any x ∈ G .The ω -twisted left regular representation of the group G is given by the formula:( L ω ( x ) f )( y ) = ω ( x, x − y ) f ( x − y ) , for f ∈ l ( G ). The reduced twisted group C*-algebra C ∗ ( G, ω ) is defined as the sub-C*-algebraof B ( l ( G )) generated by the ω -twisted left regular representation of the group G . Since we do S. Chakraborty not talk about full group C*-algebras in this paper, we simply call C ∗ ( G, ω ) the twisted groupC*-algebra of G with respect to ω. When ω = 1 , C ∗ ( G, ω ) =: C ∗ ( G ) is the usual reduced groupC*-algebra of G. We refer to [ELPW10, Section 1] for more on twisted group C*-algebras andthe details of the above construction.
Example 1.1.
Let G be the group Z n . For each θ ∈ T n , construct a 2-cocycle on G by defining ω θ ( x, y ) = e ( h− θx, y i ). The corresponding twisted group C*-algebra C ∗ ( G, ω θ ) is isomorphicto the n -dimensional noncommutative torus A θ , which was defined in the introduction. Example 1.2.
Suppose W be an invertible n × n matrix of finite order with integer entries.Let F := h W i act on Z n by usual matrix multiplication with vectors. Let us also take θ ∈ T n .We assume in addition that W is a θ -symplectic matrix, i.e. W t θW = θ . Then we can definea 2-cocycle ω ′ θ on G := Z n ⋊ F by ω ′ θ (( x, s ) , ( y, t )) = ω θ ( x, s · y ). Sometimes one calls thecorresponding twisted group C*-algebra, C ∗ ( G, ω ′ θ ), a noncommutative orbifold . We will comeback to this example in Section 3.2. K-theory generators of noncommutative tori
Projective modules over noncommutative tori.
In [RS99], Rieffel and Schwarz de-fined (densely) an action of the group SO( n, n | Z ) on T n . Recall that SO( n, n | Z ) is the subgroupof GL(2 n, R ) , which contains matrices, with integer entries and of determinant 1, of the fol-lowing 2 × g = (cid:18) A BC D (cid:19) , where A, B, C and D are arbitrary n × n matrices satisfying A t C + C t A = 0 , B t D + D t B = 0 and A t D + C t B = id n . The action of SO( n, n | Z ) is then defined as gθ := ( Aθ + B )( Cθ + D ) − whenever Cθ + D is invertible. The subset of T n on which the action of every g ∈ SO( n, n | Z ) isdefined, is dense in T n (see [RS99, page 291]). We have the following theorem due to HanfengLi. Theorem 2.1. ( [Li04, Theorem 1.1] ) For any θ ∈ T n and g ∈ SO( n, n | Z ) , if gθ is defined then A θ and A gθ are strongly Morita equivalent. For any R ∈ GL( n, Z ) , let us denote by ρ ( R ) the matrix (cid:18) R (cid:0) R − (cid:1) t (cid:19) ∈ SO( n, n | Z ) , andfor any N ∈ T n ∩ M n ( Z ) , we denote by µ ( N ) the matrix (cid:18) id n N n (cid:19) ∈ SO( n, n | Z ) . Noticethat the noncommutative tori corresponding to the matrices ρ ( R ) θ = RθR t and µ ( N ) θ = θ + N are both isomorphic to A θ . Also defineSO( n, n | Z ) ∋ σ p := p
00 id n − p p n − p , p n/ . We recall the approach of Rieffel [Rie88] to find the A σ p θ − A θ bimodule and follow thepresentation in [Li04]. racing projective modules over noncommutative orbifolds 5 We fix 1 p n/ , and let q ∈ N such that n = 2 p + q . Let us write θ ∈ T n as (cid:18) θ θ θ θ (cid:19) , partitioned into four sub-matrices θ , θ , θ , θ , and assume θ to be aninvertible 2 p × p matrix. Define a new cocycle ω θ ′ on Z n by ω θ ′ ( x, y ) = e ( h− θ ′ x, y i / θ ′ = (cid:18) θ − − θ − θ θ θ − θ − θ θ − θ (cid:19) = σ p θ. Set A = C ∗ ( Z n , ω θ ) and B = C ∗ ( Z n , ω θ ′ ). Let M be the group R p × Z q , G := M × c M and h· , ·i be the natural pairing between M and its dual group c M (our notation does not distinguishbetween the pairing of a group and its dual group, and the standard inner product on a linearspace). Consider the Schwartz space E ∞ := S ( M ) consisting of smooth and rapidly decreasingcomplex-valued functions on M .Denote by A ∞ = S ( Z n , ω θ ) and B ∞ = S ( Z n , ω θ ′ ) the dense sub-algebras of A and B ,respectively, consisting of formal series (of the variables { U i } ) with rapidly decaying coefficients.Let us consider the following (2 p + 2 q ) × (2 p + q ) real valued matrix:(2.1) T = T
00 id q T T , where T is an invertible matrix such that T t J T = θ , J := (cid:18) p − id p (cid:19) , T = θ and T is any q × q matrix such that θ = T − T t . For our purposes, we take T = θ / . We also define the following (2 p + 2 q ) × (2 p + q ) real valued matrix: S = J ( T t ) − − J ( T t ) − T t q T t . Let J = J q − id q and J ′ be the matrix obtained from J by replacing the negative entries of it by zeroes. Note that T and S can be thought as maps from ( R n ) ∗ to R p × ( R p ) ∗ × R q × ( R q ) ∗ (see the definition of anembedding map in [Li04, Definition 2.1]), and S ( Z n ) , T ( Z n ) ⊆ R p × ( R p ) ∗ × Z q × ( R q ) ∗ . Thenwe can think of S ( Z n ) , T ( Z n ) as in G via composing S | Z n , T | Z n with the natural coveringmap R p × ( R p ) ∗ × Z q × ( R q ) ∗ → G . Let P ′ and P ′′ be the canonical projections of G to M and c M , respectively, and let T ′ := P ′ ◦ T, T ′′ := P ′′ ◦ T, S ′ := P ′ ◦ S, S ′′ := P ′′ ◦ S. Then the following formulas define a B ∞ - A ∞ bimodule structure on E ∞ :(2.2) ( f U θl )( x ) = e ( h− T ( l ) , J ′ T ( l ) / i ) h x, T ′′ ( l ) i f ( x − T ′ ( l )) , (2.3) h f, g i A ∞ ( l ) = e ( h− T ( l ) , J ′ T ( l ) / i ) Z G h x, − T ′′ ( l ) i g ( x + T ′ ( l )) ¯ f ( x ) dx, (2.4) ( U σ p θl f )( x ) = e ( h− S ( l ) , J ′ S ( l ) / i ) h x, − S ′′ ( l ) i f ( x + S ′ ( l )) , S. Chakraborty (2.5) B ∞ h f, g i ( l ) = e ( h S ( l ) , J ′ S ( l ) / i ) Z G h x, S ′′ ( l ) i ¯ g ( x + S ′ ( l )) f ( x ) dx, where U θl , U σ p θl denote the canonical unitaries with respect to the group element l ∈ Z n in A ∞ and B ∞ , respectively. See Proposition 2.2 in [Li04] for the following well-known result. Theorem 2.2 (Rieffel) . The smooth module E ∞ , with the above structures, is an B ∞ - A ∞ Morita equivalence bimodule which can be extended to a strong Morita equivalence between B and A . Let E denote the completion of E ∞ with respect to the C ∗ -valued inner products given above.Now E becomes a right projective A -module which is also finitely generated (see the discussionpreceding Proposition 4.6 of [ELPW10]). Note that E is a Morita equivalence bimodule between B = A σ p θ and A = A θ . Fundamental projective modules.
For a definition of pfaffian of a skew-symmetricmatrix A , pf( A ) , we refer to [Cha20, Definition 3.1]. We start with the following remark. Remark 2.3.
The trace of the module E , which was computed by Rieffel [Rie88], is exactlythe absolute value of the pfaffian of the upper left 2 p × p corner of the matrix θ, which is θ .Indeed, as [Rie88, Proposition 4.3, page 289] says that trace of E is | det e T | , where e T = (cid:18) T
00 id q (cid:19) , the relation T t J T = θ and the fact det( J ) = 1 give the claim.Let p be an integer such that 1 ≤ p ≤ n . Definition 2.4.
A 2 p -pfaffian minor (or just pfaffian minor) of a skew-symmetric matrix A isthe pfaffian of a sub-matrix M AI of A consisting of rows and columns indexed by i , i , ..., i p for some numbers i < i < ... < i p and I := ( i , i , . . . , i p ).Note that the number of 2 p -pfaffian minors is (cid:0) n p (cid:1) and the number of all pfaffian minors is2 n − − θ ∈ T n . We will now see that for each non-zero pfaffian minor of θ, we can constructa projective module over A θ such that the trace of which is exactly the pfaffian minor. Fix1 ≤ p ≤ n . Choose I := ( i , i , . . . , i p ) for i < i < ... < i p , and assume the pfaffianminor pf( M θI ) is non-zero (so that M θI is invertible). Choose a permutation Σ ∈ S n such thatΣ(1) = i , Σ(2) = i , · · · , Σ(2 p ) = i p . If U , U , · · · , U n are generators of A θ , there exists an n × n skew-symmetric matrix, denoted by Σ( θ ), such that U Σ(1) , U
Σ(2) , · · · , U Σ( n ) are generatorsof A Σ( θ ) and A Σ( θ ) ∼ = A θ . Note that the upper left 2 p × p block Σ( θ ) is exactly M θI , which isinvertible. Now consider the projective module constructed as completion of S ( R p × Z n − p )over A Σ( θ ) as in the previous subsection and denote it by E θI . The trace of this module is thepfaffian of M θI by the remark above, which is P ξ ∈ Π ( − | ξ | Q ps =1 θ i ξ (2 s − i ξ (2 s ) . Varying p , andassuming that all the pfaffian minors are non-zero, we get 2 n − − n − − fundamental projective modules .We recall the following fact due to Elliott which will play a key role. Theorem 2.5 (Elliott) . Let θ be a skew-symmetric real n × n matrix. Then Tr(K ( A θ )) isthe range of the exterior exponential exp( θ ) : Λ even Z n → R . racing projective modules over noncommutative orbifolds 7 We refer to ([Ell84, Theorem 3.1]) for the definition of exterior exponential and the proof ofthe above theorem. The range of the exterior exponential is well known and is given below asa corollary of the above theorem:
Corollary 2.6.
Tr(K ( A θ )) is the subgroup of R generated by and the numbers P ξ ( − | ξ | Q ms =1 θ j ξ (2 s − j ξ (2 s ) for ≤ j < j < · · · < j m ≤ n , where the sum is takenover all elements ξ of the permutation group S m such that ξ (2 s − < ξ (2 s ) for all ≤ s ≤ m and ξ (1) < ξ (3) < · · · < ξ (2 m − . Noting that P ξ ( − | ξ | Q ms =1 θ j ξ (2 s − j ξ (2 s ) is exactly the pfaffian of M θI , where I =( i , i , . . . , i m ) , we have(2.6) Tr (K ( A θ )) = Z + X < | I |≤ n pf( M θI ) Z , where | I | := 2 m for I = ( i , i , . . . , i m ) . So for a non-zero pf( M θI ), I = ( i , i , . . . , i p ) , we have constructed a projective mod-ule E θI over A θ , whose trace is pf( M θI ) . A quick thought shows that E θI is an equivalencebimodule between A θ and A g I, Σ θ for some g I, Σ ∈ SO( n, n | Z ) . Indeed, let R Σ I be the permu-tation matrix corresponding to the permutation Σ. Note that Σ( θ ) = ρ (cid:0) R Σ I (cid:1) θ. Then clearly g I, Σ = σ p ρ (cid:0) R Σ I (cid:1) . Explicit generators of K ( A θ ) for a general θ ∈ T n . Consider the matrix Z ∈ T n whose entries above the diagonal are all 1: Z = · · · · · · − − · · · · · · − . Now, for any θ ∈ T n , there exists some positive integer t , such that all the pfaffian minors of µ ( tZ ) θ = θ + tZ are positive (see [Cha20, Proposition 4.6]). Note that A θ + tZ and A θ definethe same noncommutative torus. We then have the following theorem. Theorem 2.7.
The K-theory classes of the fundamental projective modules E θ + tZI , along with [1] generate K ( A θ + tZ ) , and hence K ( A θ ) . Proof.
See [Cha20, Theorem 4.7]. (cid:3) Noncommutative orbifolds and projective modules
Let us recall Example 1.2. Let W := ( a ij ) be an invertible n × n matrix of finite order withinteger entries and F be the finite cyclic group generated by W . In addition, we assume that W t θW = θ . Hence F is a finite subgroup of Sp( n, Z , θ ) := { A ∈ GL ( n, Z ) : A T θA = θ } . By S. Chakraborty
Lemma 2.1 of [ELPW10] we have C ∗ ( Z n ⋊ F, ω ′ θ ) = A θ ⋊ α F, where the action of F on A θ isgiven by (see [JL15, Equation 2.6]):(3.1) α ( U i ) = e ( n X k =2 k − X j =1 a ki a ji θ jk ) U a i · · · U a ni n , where U , ..., U n are the generators of A θ . Sometimes we just write the crossed product as A θ ⋊ F, without the “ α ” decoration.Let us look into the case where n = 2. Note that Sp(2 , Z , θ ) = SL (2 , Z ). Finite cyclicsubgroups of SL (2 , Z ) are up to conjugacy generated by the following 4 matrices: W (2) := (cid:18) − − (cid:19) , W (3) := (cid:18) − −
11 0 (cid:19) ,W (4) := (cid:18) −
11 0 (cid:19) , W (6) := (cid:18) −
11 1 (cid:19) , where the notation W ( r ) indicates that it is a matrix of order r . The actions of the cyclicgroups generated by these matrices are considered already in [ELPW10], where the authorsconstructed projective modules over the corresponding crossed products using the fundamentalprojective modules.For n ≥ W ∈ Sp( n, Z , θ ) is non-trivial. In [JL15], and in[He19], the authors found some of the matrices for n ≥ n there will always be a matrix W of order 2, i.e. − id n . The action by Z = h W i is the flip action, which was already defined in Introduction.One natural question is how does one extend the fundamental projective modules, E θI ,over noncommutative torus A θ to the aforementioned crossed products. In [CL19], this wasanswered when the module E θI is a completion of S ( R p ) , i.e. when the dimension of the torusis even (= 2 p ), and θ is invertible so that E θI is defined. This module is called the Bott class . Inthis section we do this extension for a general E θI . We need the following proposition for suchextensions.
Proposition 3.1.
Suppose F is a finite group acting on a C*-algebra A by the action α .Also suppose that E is a finitely generated projective (right) A -module with a right action T : F → Aut( E ) , written ( ξ, g ) ξT g , such that ξ ( T g ) a = ( ξα g ( a )) T g for all ξ ∈ E , a ∈ A, and g ∈ F . Then E becomes a finitely generated projective A ⋊ F module with action defined by ξ · ( X g ∈ F a g δ g ) = X g ∈ F ( ξa g ) T g . Also, if we restrict the new module to A , we get the original A -module E , with the action of F forgotten.Proof. This is exactly the construction of Green–Julg map. See [ELPW10, Proposition 4.5]. (cid:3)
Let us first recall the approach of [CL19], where the authors define the necessary action of F on the Bott class which allows them to conclude that the Bott class is a projective module overthe crossed product A θ ⋊ F, using Proposition 3.1. Hence assume n (= 2 p ) to be even for themoment. Since F = h W i acts on Z n as before, we have W t θW = θ. In order to define an actionof F on the Bott class, the authors (in [CL19]) used the so-called metaplectic representation ofthe symplectic matrix T W T − , where T t J T = θ as in Equation 2.1. Note that, in this case q = 0 and hence T = T . The main idea is to use the following metaplectic extension: racing projective modules over noncommutative orbifolds 9 (3.2) 0 S Mp c ( n ) Sp( n ) 0where Sp( n ) is the usual symplectic group, and Mp c ( n ) is the complex metaplectic group (see[CL19, Section 5]). For our purposes we do not need much details about the metaplectic group,but we need to know that it has a (metaplectic) representation on S ( R p ) ([CL19, Definition5.1], also see [dG11, Chapter 7]). Now, F ∼ = h T W T − i sits inside Sp( n ) . But also we have thefollowing lift (since H ( F, S ) is trivial, see [CL19, page 158]) possible: F S Mp c ( n ) Sp( n ) 0The above defines an action of F on S ( R p ) which extends to the necessary completion (Bottclass) of S ( R p ) and it satisfies the conditions of Proposition 3.1 (see [CL19, Theorem 5.4]).Hence the Bott class becomes a projective module over A θ ⋊ F. In the following, we shall oftenwrite f W for the above action of W on S ( R p ), for f ∈ S ( R p ). So from [CL19, Equation 5.12]we have(3.3) ( f W ) U l = ( f α W ( U l )) W, f ∈ S ( R p ) , l ∈ Z p , which is the condition in Proposition 3.1.Now we take a general n , not necessarily even. We have R := h ρ ( R ) , R ∈ GL ( n, Z ) i ⊆ SO( n, n | Z ) . Also for W ∈ F we have ρ ( W t ) ∈ SO( n, n | Z ) . In this way F ⊆ SO( n, n | Z ) . Recall g I, Σ = σ p ρ (cid:0) R Σ I (cid:1) . Theorem 3.2.
With all the notations introduced before, assume pf( M θI ) = 0 . Let W ∈ GL( n, Z ) be of finite order such that W t θW = θ and F := h W i . Suppose g I, Σ F ( g I, Σ ) − ⊂ R inside SO( n, n | Z ) . Then E θI becomes a finitely generated, projective module over A θ ⋊ F .Proof. g I, Σ F ( g I, Σ ) − ⊂ R means σ p ρ (cid:0) R Σ I W t ( R Σ I ) − (cid:1) σ p ∈ R , noting that the inverse of σ p is σ p again. Now (cid:0) R Σ I W t ( R Σ I ) − (cid:1) t is a ρ ( R Σ I ) θ -symplectic matrix and A ρ ( R Σ I ) θ are A θ isomorphic F -equivariantly, where F acts on A ρ ( R Σ I ) θ by viewing F = h (cid:0) R Σ I W t ( R Σ I ) − (cid:1) t i . Soit is enough to replace R Σ I W t ( R Σ I ) − by W t again and view E θI as an A θ -module, not A ρ ( R Σ I ) θ -module.From σ p ρ ( W t ) σ p ∈ R , we have(3.4) σ p ρ (cid:0) W t (cid:1) σ p = (cid:18) S
00 ( S − ) t (cid:19) , for some S ∈ GL ( n, Z ). Writing W = (cid:18) W W W W (cid:19) , where W is the 2 p × p block, a simplecomputation shows that W = W = 0, and S = (cid:18) W − W t (cid:19) . So we have(**) W = (cid:18) W W (cid:19) . Writing θ = (cid:18) θ θ θ θ (cid:19) as before, W t θW = θ gives the following compatibility relations:(3.5) W t θ W = θ W t θ W = θ W t θ W = θ W t θ W = θ Let us first write down Equation 2.2, which is(3.6) ( f U l )( x ) = e ( h− T ( l ) , J ′ T ( l ) / i ) h x, T ′′ ( l ) i f ( x − T ′ ( l )) , more explicitly. Writing l = ( l , l ) ∈ Z n , for l ∈ Z p , l ∈ Z q , we have, T ( l ) = T
00 id q θ θ (cid:18) l l (cid:19) = T l l θ l + θ l . Let J ′ be the matrix obtained by replacing the negative entries of J by zeroes. Also, if Q ′ and Q ′′ be the canonical projections of R p × c R p to R p and c R p , respectively, denote T ′ := Q ′ ◦ T , T ′′ := Q ′′ ◦ T . Then e ( h− T ( l ) , J ′ T ( l ) / i ) = e − T l l θ l + θ l · J ′ q T l l θ l + θ l / = e − T l l θ l + θ l · J ′ T l θ l + θ l / = e ( − T l · J ′ T l / e (cid:18) − l · θ l − l · θ l / (cid:19) = C ( l ) A ( l , l ) , where C ( l ) := e ( − T l · J ′ T l / , A ( l , l ) := e (cid:0) − l · θ l − l · θ l / (cid:1) . h x, T ′′ ( l ) i = (cid:28)(cid:18) x x (cid:19) , (cid:18) T ′′ l θ l + θ l (cid:19)(cid:29) = h x , T ′′ l i (cid:28) x , θ l + θ l (cid:29) = C ( x , l ) B ( x , l , l ) , f ( x − T ′ ( l )) = f (cid:18) x − T ′ l x − l (cid:19) , where C ( x , l ) := h x , T ′′ l i , B ( x , l , l ) := (cid:10) x , θ l + θ l (cid:11) . Now, for f ∈ S ( R p × Z q ) , we define(3.7) ( f W )( x , x ) := p det( W )( f ♯ W )( x ) , where f ♯ ∈ S ( R p ) defined as f ♯ ( x ′ ) = f ( x ′ , W x ) . Note that here we have used the metaplec-tic action of W on f ♯ . We first want to show that f → f W extends to a unitary operator on racing projective modules over noncommutative orbifolds 11 L ( R p × Z q ) using the fact that the metaplectic operators are unitary. To this end we checkthat(3.8) h f W, g i L = h f, gW − i L , which follows from the following computation. h f W, g i L = Z G ( f W )( x , x ) g ( x , x ) dx = p det( W ) Z G ( f ♯ W )( x ) g ( x , x ) dx = p det( W ) Z G f ♯ ( x )( g ′ W − )( x ) dx dx (where g ′ ( x ) = g ( x , x ))= p det( W ) det( W ) − Z G f ( x , x ) g ♯ W − ( x ) dx dx (change x to W − x )= Z G f ( x , x ) gW − ( x , x ) dx dx = h f, gW − i L . We want to show(3.9) ( f W ) U l = ( f α W ( U l )) W = ( f U W l ) W. From Equation 3.3 we already have(3.10) ( f ♯ W ) U l = ( f ♯ U W l ) W . Now( f U
W l ) ♯ ( x ) = ( f U W l )( x , W x ) , = C ( W l ) A ( W l , W l ) C ( x , W l ) B ( W x , W l , W l ) f (cid:18) x − T ′ W l W x − W l (cid:19) . C ( W l ) C ( x , W l ) A ( l , l ) B ( x , l , l ) f (cid:18) x − T ′ W l W ( x − l ) (cid:19) = A ( l , l ) B ( x , l , l )( g ♯ U W l )( x ) , where g ( x ) = f ( x , x − l ) . So the RHS of 3.9 becomes p det( W )(( f U W l ) ♯ ) W ( x ) = p det( W ) A ( l , l ) B ( x , l , l )(( g ♯ U W l ) W )( x )3 . p det( W ) A ( l , l ) B ( x , l , l )( g ♯ W ) U l ( x ) . Now the LHS( f W ) U l ( x ) = C ( l ) C ( x , l ) A ( l , l ) B ( x , l , l ) f W (cid:18) x − T ′ l x − l (cid:19) = p det( W ) C ( l ) C ( x , l ) A ( l , l ) B ( x , l , l )( g ♯ W )( x − T ′ l )= p det( W ) A ( l , l ) B ( x , l , l )( g ♯ W ) U l ( x ) . Thus we have proved Equation 3.9. We finish the proof with the compatibility of the actionwith the h ., . i A ∞ as defined in (2.3): h f W, gW i A ∞ = α W − ( h f, g i A ∞ ) . This will make sure that the action of F on S ( R p × Z q ) defined through Equation 3.7 has aunique extension to E θI , and hence we can use Proposition 3.1. Now replacing f by f W − , itsuffices to check:(3.11) h f, gW i A ∞ = α W − ( h f W − , g i A ∞ ) . Note that(3.12) h f, g i A ∞ ( l ) = h gU − l , f i L for h f, g i L = R G f ( x ) g ( x ) dx , and hence(3.13) α W − ( h f, g i A ∞ )( l ) = h gα W ( U θ − l ) , f i L . Now h f, gW i A ∞ ( l ) 3 . h ( gW ) U − l , f i L . Z R p × Z q ( gα W ( U − l ) W ( x )) f ( x ) dx, . Z R p × Z q ( gα W ( U − l ))( x )( f W − )( x )) dx, . α W − ( h f W − , g i A ∞ )( l ) . which is the desired identity. (cid:3) Remark 3.3.
The condition of the above theorem holds if and only if R Σ I W t ( R Σ I ) − is of theform **. However, the condition reveals more information: Equation 3.4 really means that onecan define an action of h S t i on A σ p θ , and the Morita equivalence between A θ and A σ p θ canbe lifted to an equivalence between the corresponding orbifolds. This will appear in a jointwork with Ullisch ([CU21]).As an immediately corollary we have, Corollary 3.4.
Let Z and t be as in Subsection 2.3. With all the notations introduced before,assume pf( M θ + tZI ) = 0 . Let W ∈ GL( n, Z ) be of finite order such that W t θW = θ, W t ZW = Z and let F := h W i . Suppose g I, Σ F ( g I, Σ ) − ⊂ R inside SO( n, n | Z ) . Then E θ + tZI becomes afinitely generated, projective module over A θ ⋊ F .Proof. Follows immediately from the preceding theorem and Theorem 2.7, noting that theisomorphism between A θ + tZ and A θ is F-equivariant. (cid:3) The above corollary shows that under the extra assumption W t ZW = Z, all the fundamentalprojective modules over A θ + tZ ∼ = A θ become finitely generated, projective modules over thecrossed product.4. Some applications: Morita equivalence of noncommutative orbifolds
Trace of the extended module.
Let F be a finite group acting on a C*-algebra A .Also suppose that τ be an F -invariant trace on A . Then we can define a trace τ F on A ⋊ F by τ F ( X g ∈ F a g δ g ) := τ ( a e ) . Let E F denote the finitely generated, projective A ⋊ F -module, which is obtained from a finitelygenerated, projective A -module E , as in Proposition 3.1. racing projective modules over noncommutative orbifolds 13 Lemma 4.1. τ F ([ E F ]) = τ ([ E ]) | F | , where [ E F ] and [ E ] denote the K-theory classes of [ E F ] and [ E ] , respectively.Proof. Let p F denote the projection corresponding to E F , and p corresponding to E . Definethe canonical injection (regular representation) Ψ from A ⋊ F to A ⊗ B (cid:0) l ( F ) (cid:1) by mapping a to P g ∈ F g · a ⊗ p g (where p g is the projection onto the functions supported on { g } ) andby mapping δ g to 1 ⊗ ρ ( g ) , where ρ is the right regular representation. It is well known thatthe above map defines an inverse to the Green–Julg map in F -equivariant K-theory (see e.g.[HG04, page 191]), when A ⋊ F is equipped with the trivial F -action. If p F is in A ⋊ F , letus write p F = P g ∈ F a g δ g . Then τ F ([ p F ]) = τ ( a e ) . On the other hand, [Ψ( p F )] = [ p ] in K G ( A )and hence, τ ([Ψ( p F )]) = τ ([ p ]) . But τ ([Ψ( p F )]) = | F | τ ( a e ) , using the above formula of Ψ andthe fact that τ is F -invariant. Hence τ F ([ p F ]) = τ ([ p ]) | F | . A similar computation holds when p F is in some matrix algebra over A ⋊ F . (cid:3) Images of the canonical traces of noncommutative orbifolds.
Let us come backto the noncommutative orbifolds. As in Example 1.2, take a finite order matrix W ∈ GL( n, Z )such that W t θW = θ . Assume that the order of W is N. We then have C ∗ ( Z n ⋊ F, ω ′ θ ) = A θ ⋊ α F, F := h W i . For A θ ⋊ α F, the regular representation Ψ : A θ ⋊ α F ֒ → M N ( A θ ) is given by the following:(4.1) Ψ( N − X i =0 a i W i ) = a a a · · · a N − α ( a N − ) α ( a ) α ( a ) · · · α ( a N − ) α (cid:0) a N − α ( a N − ) α ( a ) · · · ...... . . . . . . . . . α N − ( a ) α n − ( a ) α N − ( a ) · · · α N − ( a N − ) α N − ( a ) . The canonical trace Tr on A θ is clearly F -invariant. Now the canonical trace on A θ ⋊ F isgiven by Tr F ( N − X i =0 a i W i ) := Tr( a ) . If we identify A θ ⋊ F inside M N ( A θ ) via the map Ψ , the trace Tr F is the normalised trace onM N ( A θ ) . This immediately gives(4.2) Tr F (K ( A θ ⋊ F )) ⊆ N Tr(K ( A θ )) . So from Equation 2.6 we haveTr F (K ( A θ ⋊ F )) ⊆ N ( Z + X < | I |≤ n pf( M θI ) Z ) . Our main theorem, Theorem 3.2, gives sufficient conditions on W so that N pf( M θI ) ∈ Tr F (K ( A θ ⋊ F )) as we have the following theorem. Theorem 4.2.
With all the notations introduced before, let W ∈ GL( n, Z ) be of finite ordersuch that W t θW = θ and F := h W i . Suppose g I, Σ F ( g I, Σ ) − ⊂ R inside SO( n, n | Z ) . Then N pf( M θI ) ∈ Tr F (K ( A θ ⋊ F )) , where N is the order of W . Proof.
If pf( M θI ) = 0 , using Theorem 3.2, E θI becomes a projective A θ ⋊ F -module. Since thetrace of E θI is pf( M θI ) , use Lemma 4.1. If pf( M θI ) = 0 , the statement is obvious. (cid:3) We now discuss various examples. We observe that the trace of the projection p := N (1 + W + W + · · · + W N − ) ∈ K ( A θ ⋊ F ) is N . Also for an even n with n = 2 p , and I :=(1 , , · · · , p ) , Σ must be trivial. In this case g I, Σ F ( g I, Σ ) − ⊂ R , since g I, Σ ρ ( W t )( g I, Σ ) − = ρ ( W − ) . Hence N pf( M θI ) = N pf( θ ) ∈ Tr F (K ( A θ ⋊ F )) , for | I | = n = 2 p. For an odd n , weof course have N pf( θ ) = 0 ∈ Tr F (K ( A θ ⋊ F )) . Example 4.3. (2-dimensional cases)
Let θ be a real number. For the 2-dimensional torus A θ , we have actions of F := h W i , where W = W (2) , W (3) , W (4) , W (6) ∈ SL(2 , Z ) , on A θ as in Section3. From the above observations, we have N and N pf (cid:18)(cid:18) θ − θ (cid:19)(cid:19) ∈ Tr F (K ( A θ ⋊ F )) , where N = 2 , , , W = W (2) , W (3) , W (4) , W (6) , respectively. But pf (cid:18)(cid:18) θ − θ (cid:19)(cid:19) = θ. Hence(4.3) Tr F (K ( A θ ⋊ F )) = 1 N ( Z + θ Z ) = 1 N Tr(K ( A θ )) . Example 4.4. (diagonal actions on 4-dimensional tori)
First take θ ∈ T n ( R ) and θ ∈T n ( R ). Let W , W be θ -symplectic and θ -symplectic matrices of order N and N , respec-tively. Then clearly W := (cid:18) W W (cid:19) is a θ := (cid:18) θ θ (cid:19) -symplectic matrix of order oforder N := lcm ( N , N ) . Hence F := h W i acts on A θ . Clearly pf( θ ) , pf( θ ) , pf( θ ) belong toTr(K ( A θ )) . Now assume that these three terms are non-zero. So n and n must be even.Then pf( θ ) N is in Tr F (K ( A θ ⋊ F )) , from the previous observation. For, I = (1 , , · · · , n ) and I = ( n + 1 , n + 1 , · · · , n + n ), one can choose R Σ I = id n + n and (cid:18) n id n (cid:19) , respec-tively. Then one easily checks that in both cases, g I, Σ F ( g I, Σ ) − ⊂ R . Hence pf( θ ) N , pf( θ ) N arein Tr F (K ( A θ ⋊ F )) . Let us specialise this example to n = n = 2 . The 4 × θ is then given by θ = θ − θ θ − θ . In this case Tr(K ( A θ )) = Z + P < | I |≤ pf (cid:0) M θI (cid:1) Z = Z + pf (cid:16) M θ (1 , (cid:17) Z + pf (cid:16) M θ (3 , (cid:17) Z +pf (cid:16) M θ (1 , , , (cid:17) Z = Z + θ Z + θ Z + θ θ Z . Let us also take W , W ∈ SL(2 , Z ) of finiteorder (say N and N , respectively) as in the previous example. Then F := h W i acts on A θ , where W := (cid:18) W W (cid:19) . Using the above,Tr F (K ( A θ ⋊ F )) = 1 N Tr(K ( A θ )) , for N := lcm ( N , N ) . One may look at [He19] for more examples of similar kind, where onecan compute the ranges of the traces explicitly just like the above. racing projective modules over noncommutative orbifolds 15
Example 4.5. (flip actions on n-dimensional noncommutative tori)
Let us consider the flipaction ( W = − id n ) of Z on an n -dimensional noncommutative torus A θ . In this case, g I, Σ ρ ( W t )( g I, Σ ) − = ρ ( W t ) . Hence g I, Σ F ( g I, Σ ) − ⊂ R trivially, for every I and Σ. Hence(4.4) Tr Z (K ( A θ ⋊ Z )) = 12 ( Z + X < | I |≤ n pf (cid:0) M θI (cid:1) Z ) = 12 Tr(K ( A θ )) . Note that Example 4.3 and Example 4.5 also satisfy the conditions of Corollary 3.4.4.3.
Morita equivalence of noncommutative tori and orbifolds.
To obtain results aboutclassification, we will restrict ourselves to simple C*-algebras. We start with the followingdefinition.
Definition 4.6.
A skew symmetric real n × n matrix θ is called non-degenerate if whenever x ∈ Z n satisfies e ( h x, θy i ) = 1 for all y ∈ Z n , then x = 0 . Let us denote the canonical trace of A θ by Tr θ . We want to prove the following theorem.
Theorem 4.7.
Let θ and θ be non-degenerate inside T n . Let W ∈ GL( n, Z ) be of finite ordersuch that W t θ W = θ and W t θ W = θ . Also assume that the action of F := h W i on Z n isfree outside the origin ∈ Z n . Then A θ ⋊ F is strongly Morita equivalent to A θ ⋊ F if andonly if there exists a λ > such that Tr Fθ and λ Tr Fθ have the same range. It is clear that the actions in Example 4.3, and Example 4.5 and the 4-dimensional examplein Example 4.4 are free outside the origin 0 ∈ Z n . Also in [JL15] and [He19], various examplesof W are constructed which have the same property.The proof of Theorem 4.7 needs some preparation. Let us first recall the following proposi-tion. Proposition 4.8. ( [Phi06, Proposition 3.7] ) Let A be a simple infinite dimensional separableunital nuclear C*-algebra with tracial rank zero and which satisfies the Universal CoefficientTheorem. Then A is a simple AH algebra with real rank zero and no dimension growth. If K ∗ ( A ) is torsion free, A is an AT algebra. If, in addition, K ( A ) = 0 , then A is an AF algebra. Let θ ∈ T n be non-degenerate. Then the following are known. • A θ is a simple C*-algebra (even the converse is true: simplicity of A θ implies θ mustbe non-degenerate) with a unique tracial state ([Phi06, Theorem 1.9]); • A θ has tracial rank zero ([Phi06, Theorem 3.6]); • If β is an action of a finite group on A θ which has the tracial Rokhlin property (see[ELPW10, Section 5]), A θ ⋊ β F is a simple C*-algebra with tracial rank zero ([Phi11,Corollary 1.6, Theorem 2.6]). Also, A θ ⋊ β F has a unique tracial state ([ELPW10,Proposition 5.7]); • Let W ∈ GL( n, Z ) be of finite order such that W t θW = θ . The the action α of F := h W i on A θ has the tracial Rokhlin property ([ELPW10, Lemma 5.10 and Theorem5.5]); • For the action α, A θ ⋊ α F satisfies the Universal Coefficient Theorem ([JL15, Propo-sition 3.1]); • K ∗ ( A θ ⋊ α F ) = K ∗ ( C ∗ ( Z n ⋊ F, ω ′ θ )) = K ∗ ( C ∗ ( Z n ⋊ F )) ([ELPW10, Theorem 0.3]).For the K-groups of C ∗ ( Z n ⋊ F ) , the following result is known. Theorem 4.9. ( [LL12, Theorem 0.1] ) Let n, m ∈ N . Consider the extension of groups → Z n → Z n ⋊ Z m → Z m → such that the action of Z m on Z n is free outside the origin ∈ Z n . Then K ( C ∗ ( Z n ⋊ Z m )) ∼ = Z s for some s ∈ Z and K ( C ∗ ( Z n ⋊ Z m )) ∼ = Z s for s ∈ Z . If m is even, s = 0 . We are now ready to prove Theorem 4.7.
Proof of Theorem 4.7.
The freeness condition shows that the K-groups are torsion free, usingTheorem 4.9. Then the above list of results along with Proposition 4.8 shows that A θ ⋊ F and A θ ⋊ F are AT algebras.Assume there is a λ > Fθ and λ Tr Fθ have the same range. Now it is enoughto find an isomorphism g : K ( A θ ⋊ F ) → K ( A θ ⋊ F ) such that λ Tr Fθ ◦ g = Tr Fθ . Indeed, g is then an order isomorphism by [BCHL18, Proposition 3.7], and g ([1]) ∈ K ( A θ ⋊ F ) + . So there is a q ∈ N and a projection p ∈ M q ( A θ ⋊ F ) such that [ p ] = g ([1]) . Since A θ ⋊ F is simple, p is full. Then A θ ⋊ F and p M q ( A θ ⋊ F ) p have isomorphic Elliott invariants ofAT algebras, so A θ ⋊ F ∼ = p M q ( A θ ⋊ F ) p by classification ([Ell93], [Lin04, Theorem 5.2]).Clearly the right hand side algebra is Morita equivalent to A θ ⋊ F. Let us now see the existence of the isomorphism g . Denote the ranges of Tr Fθ and Tr Fθ by R and R , respectively. Since R and R are finitely generated subgroups of R , they arefree. Also R = λR implies that they have the same rank. Now we have the following exactsequences : 0 ker (cid:0) Tr Fθ (cid:1) K ( A θ ⋊ F ) R Tr Fθ (cid:0) Tr Fθ (cid:1) K ( A θ ⋊ F ) R Tr Fθ Note that the above sequences split and since the K-groups are torsion free. Now ker (cid:0) Tr Fθ (cid:1) and ker (cid:0) Tr Fθ (cid:1) are finitely generated abelian groups of the same rank. So there exists anisomorphism ψ between them. Now g is defined as ψ ⊕ φ, where φ is the map between R and R given by multiplication with λ . Clearly λ Tr Fθ ◦ g = Tr Fθ , since the following diagramcommutes. K ( A θ ⋊ F ) R K ( A θ ⋊ F ) R Fθ g φ Tr Fθ For the only if part, assume A := A θ ⋊ F is strongly Morita equivalent to B := A θ ⋊ F. Let X be an A − B -imprimitivity bimodule. Define a positive tracial functional τ X on B by τ X ( h x, y i B ) := Tr Fθ ( A h y, x i ) , x, y ∈ X. By [Rie81, Corollary 2.6], Tr Fθ and τ X have the same range. Since B has a unique trace Tr Fθ , τ X must be a scalar multiple of that trace. (cid:3) Now if Tr Fθ and Tr θ have the same range upto a factor | F | , and if the same holds for Tr Fθ and Tr θ , we have: Tr Fθ and λ Tr Fθ have the same range iff Tr θ and λ Tr θ have the samerange, for some λ > . But the last condition holds iff A θ is strongly Morita equivalent to A θ , using Theorem 4.7. This observation gives us the following corollaries. racing projective modules over noncommutative orbifolds 17 Corollary 4.10.
Let θ and θ be irrational numbers and W be one of the matrices W (2) ,W (3) , W (4) , W (6) as in Section 3 (see also Example 4.3). If F := h W i , then A θ ⋊ F is stronglyMorita equivalent to A θ ⋊ F if and only if A θ is strongly Morita equivalent to A θ . Proof.
Since the action of F on Z is free outside the origin 0 ∈ Z , the result follows from thetracial range computation in Example 4.3, and Theorem 4.7. (cid:3) The above corollary is not new, see [BCHL18, Theorem 5.3].
Corollary 4.11.
Let θ , θ ∈ T n be non-degenerate. Let Z act on A θ and A θ by the flipactions. Then A θ ⋊ Z is strongly Morita equivalent to A θ ⋊ Z if and only if A θ is stronglyMorita equivalent to A θ . Proof.
Follows similarly as in Corollary 4.10 from the tracial range computation in Example 4.5,and noting that the action of Z is free outside the origin 0 ∈ Z n . (cid:3) One can definitely build more examples for which similar results can be stated. For example,a similar result is true for the 4-dimensional example in Example 4.4.From Theorem 2.1, A θ and A gθ are Morita equivalent, if gθ := Aθ + BCθ + D is well defined for g ∈ SO( n, n | Z ) . Note that if θ is non-degenerate, then A θ is simple, and hence A gθ has tobe simple so that gθ is non-degenerate. In the case of two dimensional tori, we have even astronger result due to Marc Rieffel. Rieffel (in [Rie81]) showed that A θ and A θ ′ are Moritaequivalent if and only if θ and θ ′ are in the same GL(2 , Z ) orbit, that is, θ ′ = aθ + bcθ + d for somematrix (cid:18) a bc d (cid:19) in GL(2 , Z ) . It is also known that for a non-degenerate θ, the fixed point algebra A Fθ is Morita equivalentto the crossed product algebra A θ ⋊ F (see the proposition in [Ros79]). Hence as a consequenceof Corollary 4.11 we have the following. Corollary 4.12.
Let θ , θ and Z be as in Corollary 4.11. Then A Z θ is strongly Moritaequivalent to A Z θ if and only if A θ is strongly Morita equivalent to A θ . A similar statement is true for the 2-dimensional cases (Example 4.3), which follows fromCorollary 4.10.
Acknowledgements
This research was supported by DST, Government of India under the
DST-INSPIRE FacultyScheme with Faculty Reg. No. IFA19-MA139.
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