Metaplectic transformations and finite group actions on noncommutative tori

VV:N(YYYY), 101–103

METAPLECTIC TRANSFORMATIONS AND FINITE GROUP ACTIONS ON NONCOMMUTATIVE TORI

SAYAN CHAKRABORTY, FRANZ LUEF

Communicated by Editor

ABSTRACT. In this article we describe extensions of some K-theory classes of Heisenberg modules over higher-dimensional noncommutative tori to pro- jective modules over crossed products of noncommutative tori by finite cyclic groups, aka noncommutative orbifolds. The two dimensional case was treated by Echterhoff, Lück, Phillips and Walters. Our approach is based on the theory of metaplectic transformations of the representation theory of the Heisenberg group. We also describe the generators of the K-groups of the crossed prod- ucts of flip actions byZ2on 3-dimensional noncommutative tori.

KEYWORDS: Metaplectic transformations, noncommutative torus, C^∗-crossed prod- uct, group actions.

MSC (2010): 46L35, 22D2.

1. Introduction

Then-dimensional noncommutative torus, A_θ, is defined as the universal C*-algebra generated by unitariesU1, . . . ,Unsubject to the relations

U_kUj=e^2πiθjkUjU_k forj,k=1,· · ·,n,

whereθ = (θ_ij)is a skew-symmetric realn×n matrix. For the 2-dimensional noncommutative torus, sinceθ is determined by only one real number,θ₁₂, we will denoteθ₁₂byθagain and the corresponding 2-dimensional noncommutative torus byA_θ.

Noncommutative tori are central objects in noncommutative geometry. Ri- effel ([17]) constructed projective modules (which are known as Heisenberg mod- ules) over all noncommutative tori. These constitute the framework for studying the geometry of noncommutative tori such as connections, curvature and Dirac operators on noncommutative tori. While the 2-dimensional noncommutative torus is quite well understood, there remain quite a large number of open ques- tions for higher dimensional noncommutative tori.

Crossed product C*-algebras associated to finite group actions on noncom- mutative tori go back to the work of Bratteli, Elliott, Evans and Kishimoto. They considered ([1]) the flip action ofZ2 on two dimensional noncommutative tori and the associated crossed products. Recall that the flip action of Z2 on an 2- dimensional noncommutative torus is given by mapping the generators Ui to U_i⁻¹fori = 1, 2. Kumjian ([10]) computed the K-theory of the crossed product A_θo Z2 for two dimensional noncommutative tori A_θ. (Also see [19] for more detailed analysis onA_θo Z2.) For irrationalθs, Walters in [19] stated generators of the K-theory ofA_θo Z2by showing that the generators of the K0group ofA_θ can be made “flip invariant". Later in [20] and [22], Walters consideredZ4and Z6 actions on two dimensional noncommutative tori. Recall that the following definesZ4andZ6actions on two dimensional noncommutative tori:

U1→U2, U2→U₁⁻¹ (forZ4), U₁→U₂, U₂→e^−πiθU₁⁻¹U₂ (_forZ6)_.

For these actions Walters showed that the generators (as projective modules) of K0ofA_θareZ2andZ4equivariant for irrationalθto construct projective modules overA_θo^Z4andA_θo^Z6, which constitute generators of the correspondingK0

groups.

Later, in [5], Echterhoff, Lück, Phillips and Walters studied 2-dimensional A_θ acted on by a finite cyclic subgroup F of SL2(Z). Note that SL2(Z) has a canonical action onZ², which can be lifted toA_θ. The previous actions ofZ2,Z4

andZ6are implemented by matrices inSL₂(Z). It is demonstrated that the stan- dard canonical projective module overA_θ, aka Bott class (which is a completion ofS(R), Schwartz functions onR), can be made equivariant by the action ofF yielding a projective module over the crossed product algebraA_θoF. (A result of Green and Julg (see Proposition 5.3) shows that equivariant K-theory elements provide elements in the crossed product.) It is also known that the Bott class along with the identity element generate the K₀ group of the noncommutative torus (see the proof of Lemma 4.8 in [5]).

Recently, actions of finite groups on higher-dimensional noncommutative tori have been considered in the article [9]. LetW ∈ GLn(Z)be the generator of the finite cyclic groupF acting onZⁿ withW^TθW = θ. Then the authors in [9]

showed that there exists an action ofFonn-dimensionalA_θ (Section 4). Let us assume thatnis an even number, n = 2m. To analyse projective modules over the corresponding crossed product algebras, we restrict our analysis to the class of Heisenberg modules which are an appropriate completion ofS(R^m), denoted byE. We show that this class of projective modules (which may be thought of as higher dimensional versions of the Bott class) over higher dimensional non- commutative tori can be madeF-equivariant. The metaplectic action (which we introduce in Section 5) is the key tool in our arguments. This generalises the pre- vious results of Walters and [5]. Our main theorem states:

Theorem 1.1. LetW be the generator of the finite cyclic group F acting on Zⁿ withW^TθW =θand hence onA_θ. Then the metaplectic action ofW onS(R^m) extends to an action onE such thatEbecomes anF-equivariantly finitely gener- ated projectiveA_θ module and thus a finitely generated projective module over A_θoF.

Coming back to the flip case, note that this action can be defined for general n-dimensional tori A_θ. In this case any Heisenberg module over A_θ can be ex- tended to a module over the crossed productA_θo Z2(see Section 7). Though, in [7], the authors have computed the K-theory ofA_θo Z2for higher dimensional A_θ, but some computations in [7] are not clear to us (see Remark 7.5). They used an exact sequence by Natsume ([13]) to compute the K-theory of A_θo Z2 (see Section 7) asA_θo Z2can be written asA_θ⁰o(Z2∗Z2)_{, where}A_θ⁰is an(n−1)_- dimensional noncommutative torus. For crossed products like Ao(Z2∗Z2), Natsume’s exact sequence looks like

K0(_A) −−−−→ K0(_Ao Z2)⊕K0(_Ao Z2) −−−−→ K0(_Ao Z2∗Z2) x







 y^e1 K₁(Ao Z2∗Z2) ←−−−− K₁(Ao Z2)⊕K₁(Ao Z2) ←−−−− K₁(A).

In the final section we study this exact sequence especially, the connecting mape₁, and relate it to the classical Pimsner–Voiculescu exact sequence. Recall that for crossed products likeAo Z, the Pimsner–Voiculescu sequence looks like

K0(A) −−−−→ K0(A) −−−−→ K0(Ao Z) x







 y^e2 K₁(Ao Z) ←−−−− K₁(A) ←−−−− K₁(A). The main result of the final section can be stated as follows:

Theorem 1.2. For unitalA, the connecting maps of the above two sequences com- mute in the following sense:

K0(Ao Z2∗Z2) ^{e1 //}

p

%%

K1(A)

K0(Ao Z),

e₂

OO

wherepis the map induced by the natural map fromAo Z2∗Z2∼= (Ao Z)o Z2

toM2(Ao Z).

Using this result, for totally irrationalθ(for definition see 7.2), we discuss the K-theory of crossed products of 3-dimensional A_θ with respect to the flip action and describe the generators of K-theory (see Corollary 7.2). This explains the computations of [7] for the three dimensional case (for totally irrational θ).

Presumably this can be done as well for then-dimensional case, which we plan to discuss in another paper.

Notation:e(x)will always denote the numbere^2πix; the standard symplec- tic matrix onR^2mis defined by J =

0 Im

−Im 0

, where Imis them×munit matrix, andS(R^m)will denote the space of smooth functions of rapid decay on R^m.

Acknowledgments: This research was partially supported by the DFG through SFB 878. The first named author wants to thank Siegfried Echterhoff and Nikolay Ivankov for valuable discussions.

2. Basics on twisted group algebras and noncommutative tori

LetGbe a discrete group. A mapω:G×G→Tis called a2-cocycleif ω(x,y)ω(xy,z) =ω(x,yz)ω(y,z)

wheneverx,y,z∈G, and if

ω(x, 1) =1=ω(1,x) for anyx ∈G.

The ω-twisted left regular representation of the group G is given by the formula:

(Lω(x)f)(y) =ω(x,x⁻¹y)f(x⁻¹y)

for f ∈ l²(G). The reduced twisted group C*-algebraC^∗(G,ω)is defined as the sub-C*-algebra ofB(l²(G))generated by theω-twisted left regular representation of the groupG. More details can be found in [24, 5].

Example 2.1. LetGbe the groupZⁿ. For eachn×nreal skew-symmetric matrix θ, we can construct a 2-cocycle on this group by definingω_θ(x,y) =e(h−θx,yi). The corresponding twisted group C*-algebraC^∗(G,ω_θ) is isomorphic to then- dimensional noncommutative torusA_θ, which was defined in the introduction.

Example 2.2. Suppose W be an n×n matrix of finite order with integer en- tries. Let F := hWi act on Zⁿ by group automorphism andθ is an n×n real skew-symmetric matrix. We assume in addition that W is a θ-symplectic ma- trix, i.e. W^TθW = θ. Then we can define a 2-cocycle ω⁰_θ onG := Zⁿo^F by ω⁰_θ((x,s),(y,t)) =ω_θ(x,s·y). Sometimes one calls the corresponding group C*- algebra,C^∗(G,ω_θ⁰), anoncommutative orbifold.

3. Projective modules over noncommutative tori

We fixn = 2p+qfor p,q ∈ Z≥0. Let us chooseθ :=

θ₁₁ θ₁₂ θ21 θ22

any n×nskew-symmetric matrix partitioned into four sub-matrices θ11,θ12,θ21,θ22

andθ₁₁is a 2p×2pmatrix. We recall the approach of Rieffel [17] to the construc- tion of finitely generated projectiveC^∗(Zⁿ,ω_θ)-modules and follow the presen- tation in [11]. Denoteω_θbyωand define a new cocycleω₁onZⁿbyω₁(x,y) = e(hθ⁰x,yi/2), where

θ⁰=

θ₁₁⁻¹ −θ⁻¹₁₁θ₁₂ θ₂₁θ⁻¹₁₁ θ₂₂−θ₂₁θ⁻¹₁₁θ₁₂

.

Set A = C^∗(Z^n,ω) and B = C^∗(Z^n,ω1). Let M be the group R^p×Z^q, G := M×M^ˆ andh·,·ibe the natural pairing between Mand its dual group ˆM (our notation does not distinguish between the pairing of a group and its dual group, and the standard inner product on linear spaces). Consider the Schwartz spaceE^∞:=S(M)consisting of smooth and rapidly decreasing complex-valued functions onM.

Denote byA^∞= S(Z^n,ω)andB^∞ = S(Z^n,ω1)the dense sub-algebras of A and B, respectively. Let us consider the following (2p+2q)×(2p+q) real valued matrix:

T=





T₁₁ 0 0 Iq

T31 T32



,

whereT11 is the invertible matrix such thatT₁₁^t J0T11 = θ11, J0 :=

0 Ip

−Ip 0

, T₃₁ = θ₂₁ andT₃₂ is the matrix obtained from θ₂₂ replacing the lower diagonal entries by zero.

We also define the following(2p+2q)×(2p+q)real valued matrix:

S=





J0(T₁₁^t )⁻¹ −J0(T₁₁^t )⁻¹T₃₁^t

0 Iq

0 T₃₂^t



. Let

J=





J0 0 0

0 0 Iq

0 −Iq 0





and J⁰be the matrix obtained fromJby replacing the negative entries of it by 0.

Note thatTandScan be thought as mapsR^p×R^∗p×Z^q →G(see the definition 2.1 of the embedding map in [11]). LetP⁰andP⁰⁰be the canonical projections of GtoMand ˆM, respectively, and let

T⁰ :=P⁰◦T, T⁰⁰:=P⁰⁰◦T, S⁰ :=P⁰◦S, S⁰⁰:=P⁰⁰◦S.

Then the following formulas define aA^∞-B^∞bimodule structure onE^∞:

(3.1) (f U_l^θ)(x) =e(h−T(l),J⁰T(l)/2i)hx,T⁰⁰(l)if(x−T⁰(l)),

(3.2) hf,giA^∞(l) =e(h−T(l)_,J⁰T(l)_/2i) Z

Ghx,−T⁰⁰(l)ig(x+T⁰(l))f^¯(x)dx, (3.3) (V_l^θf)(x) =e(h−S(l)_,J⁰S(l)_/2i)hx,−S⁰⁰(l)if(x+S⁰(l))_,

(3.4) B^∞hf,gi(l) =e(hS(l),J⁰S(l)/2i) Z

Ghx,S⁰⁰(l)ig¯(x+S⁰(l))f(x)dx, whereU_l^θ,V_l^θ denote the canonical unitaries with respect to the group element l∈ZⁿinA^∞andB^∞, respectively.

See Proposition 2.2 in [11] for the following well-known result.

Theorem 3.1(Rieffel). The smooth moduleE^∞, with the above structures, is an A^∞-B^∞Morita equivalence bimodule which can be extended to a strong Morita equivalence betweenAandB.

LetEbe the completion ofE^∞with respect to theC^∗-valued inner products given above. NowE becomes a right projective A-module which is also finitely generated (see the discussion preceding Proposition 4.6 of [5]). The projective module corresponding toq = 0 is called theBott class. Note that this Bott class appears only for even dimensional tori.

Remark 3.2. The trace of the moduleE, which was computed by Rieffel [17], is exactly the absolute value of the pfaffian of the upper left 2p×2pcorner of the matrixθ, which isθ₁₁. Indeed, as [17, Proposition 4.3, page 289] says that trace of E is|detTe|, where

Te=

T11 0 0 Iq

,

the relationT₁₁^t J₀T₁₁ =θ₁₁and the fact det(J₀) =1 give the claim.

4. A quick look into noncommutative orbifolds

LetW := (a_ij)be ann×nmatrix of finite order with integer entries acting onZⁿandFbe the cyclic group generated byW. In addition, we assume thatWis aθ-symplectic matrix as in section 2. HenceFis a finite subgroup ofSP(n,Z,θ):= {A ∈ GL(n,Z) : A^TθA = θ}. By Lemma 2.1 of [5] we have C^∗(ZⁿoF,ω⁰_θ) = A_θoαFwith respect to the action ([9, Equation 2.6]) :

(4.1) α(U_i) =e(

∑

k−1

∑

j=1

a_kia_jiθ_jk)U₁^a1i· · ·U_n^ani,

whereU₁, ...,Unare the generators ofA_θ.

Let us look into the case wheren =2. Note thatSP(2,Z^,θ) =SL(2,Z). Fi- nite cyclic subgroups ofSL(2,Z)are up to conjugacy generated by the following 4 matrices:

W2:=

−1 0 0 −1

,W3:=

−1 −1

1 0

,

W4:=

0 −1

1 0

,W6:=

0 −1

1 1

, where the notationWrindicates that it is a matrix of orderr.

The actions of these matrices are considered already in [5], where the au- thors constructed projective modules over the corresponding crossed products and used these projective modules to prove some classification results for these crossed products.

For n ≥ 3 finding finite order matrixW ∈ SP(n,Z,θ) is non-trivial. For n = 3, there is only one such matrix (−I3) acting on all A_θ’s. In [9] the authors found someW’s and associated actions forn≥4 such that the crossed products are well defined.

5. Projective modules over noncommutative orbifolds

One natural question is how does one extend the projective modules over noncommutative tori to the aforementioned crossed products? Our main theo- rem addresses this question for the Bott classes.

In the following sections (except Section 7) we consider n to be an even number,n=2m. SupposeF=hWiis a finite cyclic group acting onZ^{n. We want} to build some projective modules overC^∗(ZⁿoF,ω_θ⁰). Note thatWneeds to be aθ-symplectic matrix, i.eW^TθW=θ, as noted earlier.

In order to construct projective modules overC^∗(ZⁿoF,ω⁰_θ), we will use the so-called metaplectic representation of the symplectic matrixW. Whenθis the standard skew-symmetric matrixJthenWis also a standard symplectic matrix.

We denote the group of all J-symplectic matrices (also known as standard symplectic matrices) byS P(n), which is called the symplectic group. We refer to Chapter 2 of [8] for preliminaries on symplectic groups and their metaplectic representations. We recall the metaplectic action associated to the symplectic ma- trixW. Any symplectic matrix can be written as product of two free symplectic matrices (see page 38, [8]) which is by definition a symplectic matrix

A B C D

,

such that det(B) 6=0. LetWto be a free symplectic matrix. We now associate to Wthegenerating function:

(5.1) W(x,x⁰) = ¹

2hDB⁻¹x,xi − hB⁻¹x,x⁰i+¹

2hB⁻¹Ax⁰,x⁰i, whenx,x⁰∈R^m.

In what follows, for y ∈ Q, we denote the complex number i^y for some choice of element in the range of i^y for the multivalued functioni^z,z ∈ C^{. In} other words, we fix a branch fori^z.

Definition 5.1. Themetaplectic operator(metaplectic transformation) associated to WonS(R^m)is given by

F_Wf(x) =i^s−m2 q

|det(B⁻¹)|

Z

R^m

e(W(x,x⁰))f(x⁰)dx⁰;

the integers(sometimes called Maslov index) corresponds to a choice of the ar- gument arg of detB⁻¹:

sπ≡arg(detB⁻¹) mod 2π.

These operators can be extended to L²(R^m) giving unitary operators on L²(R^m) (see page 81, [8]). We denote byMP(n) the group of metaplectic op- erators which is a subgroup of the group of unitary operators ofL²(R^m).

Theorem 5.2. There exists an exact sequence:

0 Z2 MP(n) S P(n) 0,

where the mapMP(n)→ S P(n)is uniquely determined by the mapF_W →W.

Proof. See page 84, [8].

One also defines the circle extension ofS P(n),MP^c(n). This is defined to be the groupMP(n)×_Z2S¹ : (MP(n)×S¹)_/∆(Z2)_,∆(Z2)being the diagonal Z2×Z2sitting insideMP(n)×S¹. This gives rise to the exact sequence

0 S¹ MP^c(n) S P(n) 0,

whereS¹denotes the circle group.

In the following, we shall often write f WforF_W(f).

Following [8, Section 3.2.2] the following matrices generate (as a group) all J-symplectic matrices:

J:=

0 I

−I 0

, ML:=

L 0 0 (L^T)⁻¹

,VP:=

I 0 P I

, for a symmetricm×mmatrixPand an invertiblem×mmatrixL.

Following [8, Section 7.1.2], we write down the metaplectic operators (up to some constant which will not matter in the proof) corresponding toJ,MLandVP:

(5.2) (f J)(x) = Z

R^m

e(h−x,x⁰i)f(x⁰)dx⁰

(5.3) (f ML)(x) =

q

det(L)f(L(x))

(5.4) (f V_P)(x) =e(¹

2hPx,xi)f(x)_.

Hence it suffices to check statements of multiplicative type about metaplec- tic transformations for the above three operators and also note that the Schwartz space is invariant under metaplectic transformations, see [8, Corollary 63]. For our result we always assumeθto be a non-degenerate matrix.

We recall the following proposition from [5, Proposition 4.5].

Proposition 5.3. SupposeFis a finite group acting on a C*-algebraAby the action α. Also suppose thatE is a finitely generated projective (right) A-module with a right actionT:F→Aut(E), written(ξ,g)ξTg, such thatξ(Tg)a= (ξαg(a))Tg

for allξ ∈ E,a ∈ A, andg ∈ F. ThenE becomes a finitely generated projective AoFmodule with action defined by

ξ·(

∑

g∈F

agδ_g) =

∑

g∈F

(ξag)Tg

.

Also, if we restrict the new module toA, we get the original A-moduleE, with the action ofFforgotten.

Proof. This is exactly the construction of Green–Julg map. For a proof, see [5, Proposition 4.5].

Now we are in the position to formulate our main theorem. LetS(R^m)_be as in Section 3.

Theorem 5.4. LetW be the generator of the finite cyclic group F acting on Zⁿ withW^TθW =θand hence onC^∗(Zⁿ,ω_θ). Then the metaplectic action ofW on S(R^m)extends to an action onEsuch thatEbecomes anF-equivariantly finitely generated projectiveC^∗(Z^n,ωθ)module and thus a finitely generated projective module overC^∗(Z^n,ωθ)o^F.

Proof. We divide the proof in two parts.

First part: (the case θ = −J): Recall that from (3.1) for the choice of T := −I 0

0 I

the action ofS(Z^n,ω−J) =C^∞(Tⁿ)onS(R^m)is given by the fol- lowing:

(5.5) f U_i^p(y₁,y2, . . . ,ym) = f(y₁,y2, . . . ,y_i+p, . . . ,ym), ifi≤m,

(5.6) f U_i^p(y₁,y₂, . . . ,ym) =e(pyi−m)f(y₁,y₂, . . . ,ym), ifi>m,

whereU_i’s are the generators ofn-dimensional smooth torusC^∞(Tⁿ). [Note that θ = −J is chosen instead ofθ = J to keep the formulas somewhat similar to [5]]. Letα_Wdenotes the action of the matrixXonS(Zⁿ):α_W(φ)(x) =φ(W⁻¹x). According to Proposition 5.3, we still have to check the following equation to complete the proof:

(5.7) f(W)φ= (fαW(φ))W,

for allf ∈_S(R^m)andφ∈_S(Z^n,ω−J), which will then imply thatEbecomesF- equivariant. Also, sinceS(Zⁿ,ω_−J)is generated byU1,U2, . . . ,Un, it is enough to check (5.7) forφ=U1,U2, . . . ,Un.

So we are left with checking the following equations:

(5.8) f JUi = (fα_J(Ui))J, (5.9) f M_LU_i = (fα_ML(U_i))M_L, (5.10) f VPUi = (fα_VP(Ui))VP, for all 1≤i≤n.

First we check the equations (5.8), (5.9) and (5.10) for 1 ≤ i ≤ m. The left hand side (LHS) of (5.8) is

(f JU_i)(x₁,x₂, . . . ,x_m) = (f J)(x₁,x₂, . . . ,x_i+_{1 . . . ,}x_m)_,

= Z

R^m

e(−h(x₁,x₂, . . . ,x_i+1 . . . ,xm),(x⁰₁,x⁰₂, . . . ,x⁰_m)i)f(x⁰)dx⁰,

= Z

R^m

e(−h(x₁,x₂, . . . ,xm),(x⁰₁,x⁰₂, . . . ,x⁰_m)i).e(−x⁰_i)f(x⁰)dx⁰; and the right hand side (RHS):

(_fαJ(_Ui))_J(_{x1,x2, . . . ,xm}) = Z

R^m

e(−h(x1,x2, . . . ,xm)_,(_x1⁰_,x2⁰_{, . . . ,x}⁰_m)i)fαJ(_Ui)(_x⁰)_dx⁰_,

= Z

R^m

e(−h(x₁,x2, . . . ,xm),(x₁⁰,x₂⁰, . . . ,x⁰_m)i)(f U_i+m⁻¹ )(x⁰)dx⁰,

= Z

R^m

e(−h(x₁,x₂, . . . ,xm),(x₁⁰,x₂⁰, . . . ,x⁰_m)i).e(−x_i⁰)f(x⁰)dx⁰.

Hence we have proved (5.8). The LHS of (5.9) equals

(f MLU_i)(x₁,x2, . . . ,xm) = (f ML)(x₁,x2, . . . ,x_i+1, . . . ,xm),

= q

det(L)f(L(x1,x2, . . . ,x_i+1, . . . ,xn));

and the RHS is

(fα_ML(U_i))M_L(x₁,x₂, . . . ,xm) =^qdet(L)(fα_ML(U_i))L(x₁,x₂, . . . ,xm)_,

= q

det(L)f(L(x₁,x2, . . . ,xm) +L(x_i)),

= q

det(L)f(L(x1,x2, . . . ,xi+1, . . . ,xm)). Hence we have demonstrated (5.9). We have for the LHS (5.10):

(f V_PU_i)(x₁,x₂, . . . ,xm) = (f V_P)(x₁,x₂, . . . ,x_i+1, . . . ,xm),

=e(¹

2hP(x₁,x₂, . . . ,x_i+_{1, . . . ,}xm)_,· · · (x1,x2, . . . ,xi+1, . . . ,xm)i)· · ·

f(x1,x2, . . . ,xi+1, . . . ,xm), and the RHS is

(fα_VP(U_i))V_P(x₁,x₂, . . . ,xm) =e(¹

2(Px·x))(fα_VP(U_i))(x),

=e(¹

2hP(x₁,x₂, . . . ,x_i+_{1, . . . ,}xm)_,· · · (x1,x2, . . . ,xi+1, . . . ,xm)i)· · ·

f(x1,x2, . . . ,xi+1, . . . ,xm). Hence we have shown (5.10).

Now, letm<i ≤n. We check the equations (5.8), (5.9) and (5.10) for these values ofi.

For (5.8) the LHS is

(f JU_i)(x₁,x₂, . . . ,xm) =e(xi−m)(f J)(x₁,x₂, . . . ,xm),

=e(xi−m) Z

R^m

e(−hx,x⁰i)f(x⁰)dx⁰; and the RHS

(fαJ(U_i))J(x₁,x₂, . . . ,x_m) = Z

R^m

e(−hx,x⁰i)(fαJ(U_i))(x⁰)dx⁰,

= Z

R^m

e(−hx,x⁰i)(f(Ui−m))(x⁰)dx⁰,

= Z

R^m

e(−hx,x⁰i)f(x⁰₁,x⁰₂, . . . ,x⁰_i−m+1, . . . ,x⁰_m)dx⁰,

=e(x_i−m) Z

R^m

e(−hx,x⁰i)f(x⁰₁,x⁰₂, . . . ,x⁰_i−m, . . . ,x⁰_m)dx⁰,

=e(x_i−m) Z

R^m

e(−hx,x⁰i)f(x⁰)dx⁰.

Hence we have proved (5.8).

For (5.9), the LHS

(f MLUi)(x1,x2, . . . ,xm) =e(xi−m)(f ML)(x1,x2, . . . ,xm),

=^q_det(_L)_e(_xi−m)_f(_L(_{x1,x2, . . . ,xm}))_; and the RHS

(fαML(U_i))ML(x₁,x2, . . . ,xm) = q

det(L)(fαML(U_i))(L(x₁,x2, . . . ,xm)),

= q

det(L)e(h(L⁻¹)^T(ei−m),L(x₁,x₂, . . . ,xm)i)· · · f(L(x₁,x₂, . . . ,xm)),

= q

det(L)e(hei−m,(L⁻¹L)(x₁,x₂, . . . ,xm)i)· · · f(L(x1,x2, . . . ,xm)),

= q

det(L)e(xi−m)f(L(x₁,x2, . . . ,xm)).

Thus (5.9) is verified.

For (5.10), the LHS

(f V_PU_i)(x₁,x₂, . . . ,xm) =e(xi−m)(f V_P)(x₁,x₂, . . . ,xm),

=e(xi−m)e(¹

2hPx,xi)f(x); and the RHS:

(fαV_P(Ui))VP(x1,x2, . . . ,xm) =e(¹

2hPx,xi)(fα(Ui)(x),

=e(¹

2hPx,xi)(f U_i)(x),

=e(xi−m)e(¹

2hPx,xi)f(x). Hence we have proved (5.10).

Now we have the following diagram:

F

0 Z2 MP(n) S P(n) 0.

In the above diagram it is not assured that the inclusionF ,→ S P(n)lifts to an inclusionF,→ MP(n). SinceFis cyclic the following lift is always possible:

F

0 S¹ MP^c(n) S P(n) 0,

whereMP^c(n)is the circle extension ofS P(n). Indeed, for the generatorW∈F, we can choose a scalar z ∈ T to get that the order of the operator z· F_W ∈ MP^c(n)is same as the order of the elementW ∈ F. Hence the inclusionF ,→ MP^c(n)gives the required action ofWonS(R^m).

Second part (the general case):

Let θ be a general non-degenerate skew-symmetric matrix. In this case W_θ^TθW_θ = θ. We recall the construction of the projective modules in this case.

Sinceθis non-degenerate, there exists an invertible matrixTsuch thatT^TJT=θ.

Recall that the action ofU_l^θ(forl∈Zⁿ) onS(R^m)is defined by

(5.11) (f U_l^θ)(x) =e((−T(l)·J⁰T(l)/2))e(hx,T⁰⁰(l)i)f(x−T⁰(l)). .

First we note thatW := TW_θT⁻¹ is a J-symplectic matrix (a matrix A is J-symplectic ifA^TJ A=J). Thus we can define

f W = f(TW_θT⁻¹):=F_TW

θT⁻¹(f)

for f ∈ _S(R^m)and f W_θ to be the function f W. Consequently, in this case we have to check the following equation:

(5.12) (f W_θ)U_l^θ(x) = (fα_W

θ(U_l^θ))W_θ(x), x ∈R^m.

(f W_θ)U^θ_l(x) =e(h−T(l),J⁰T(l)/2i)e(hx,T⁰⁰(l)i)(f W_θ)(x−T⁰(l)),

=e(h−T(l),J⁰T(l)/2i)e(hx,T⁰⁰(l)i)(f W)(x−T⁰(l)),

=e(h−T(l),J⁰Id(Tl)/2i)e(hx, Id⁰⁰(Tl)i)(f W)(x−Id⁰(Tl)),

= (f W)U_Tl^J (x),

= (fα_W(U_Tl^J ))W(x) (using (5.10))

and the RHS (fα_W

θ(U_l^θ))W_θ(x) = (fα_W

θ(U_l^θ))W(x),

= Z

R^m

e(W(x,x⁰))(fαW_θ(U^θ_l))(x⁰)dx⁰,

= Z

R^m

e(W(x,x⁰))(f(U_W^θ

θ(l)))(x⁰)dx⁰,

= Z

R^m

e(W(x,x⁰))(f(U_W(Tl)^J ))(x⁰)dx⁰, (using (5.13))

= Z

R^m

e(W(x,x⁰))(fαW(U_Tl^J ))(x⁰)dx⁰, where

(5.13) (f(U_W^θ

θ(l)))(x⁰) = (f(U_W(Tl)^J ))(x⁰), x⁰∈R^m because of the fact:

(f(U_W^θ

θ(l)))(x⁰) =e(h−T(W_θ(l)),J⁰T(W_θ(l))/2i)e(hx⁰,T⁰⁰(W_θ(l))i)f(x⁰−T⁰(W_θ(l))), which is equal to

e(h−T(W_θ(T⁻¹Tl)),J⁰T(W_θ(T⁻¹Tl))/2i)e(hx⁰,T⁰⁰(W_θ(T⁻¹Tl))i)f(x⁰−T⁰(W_θ(T⁻¹Tl))),

= e(h−(TW_θT⁻¹)(Tl),J⁰(TW_θT⁻¹)(Tl)/2i)e(hx⁰,(TW_θT⁻¹)⁰⁰(Tl)i)f(x⁰−(TW_θT⁻¹)⁰(Tl)),

= e(h−W(Tl),J⁰W(Tl)/2i)e(hx⁰, Id⁰⁰(W(Tl))i)f(x⁰−Id⁰(W(Tl))),

= (f(U_W(Tl)^J ))(x⁰).

We finish the proof with the compatibility of the action with theh., .iA^∞ as defined in (3.2):

hf W_θ,gW_θi_A∞ =α_W−1 θ

(hf,gi_A∞). Replacing f byf W_θ, it suffices to check:

(5.14) hf,gW_θiA∞=α_W−1 θ

(hf W_θ⁻¹,giA∞).

The argument is based on some observations: (i) the explicit description ofh., .i_A∞

in terms of the right action ofA^∞onS(R^m)_:

hf,gi_A∞(l) =hgU_−l^θ ,fi_L2, forhf,gi_L2 =R

R^m f(x)g(x)dx, and (ii) the relations:

α_W−1 θ

(hf,gi_A∞)(l) =hgα_Wθ(U_−l^θ ),fi_L2.

The realization ofh., .iA^∞in terms of the right action allows us to use equa- tion (5.12):

(f W_θ)U_l^θ(x) = (fα_Wθ(U_l^θ))W_θ(x), x∈R^m

in the proof of (5.14):

hf,gW_θiA^∞(l) = h(gW_θ)U_−l^θ ,fi_L2,

= Z

R^m

(gαW_θ(U_−l^θ )W_θ(x))f(x)dx,

= Z

R^m

(gαW_θ(U_−l^θ ))(x)(f W_θ⁻¹)(x))dx,

= α_W−1 θ

( Z

R^m

(gU^θ_−l)(x)(f W_θ⁻¹)(x))dx),

= α_W−1 θ

(hf W_θ⁻¹,gi_A∞), which is the desired identity.

Note that we have always written the operatorTW up to some constant of modulus one, which is not essential for our discussion.

6. The 2-dimensional case - revisited

The results for the 2-dimensional case [5] are revisited from the perspective of metaplectic transformations. As mentioned before, there are up to conjugation four matrices of finite order in SL₂(Z)generatingZ2,Z3,Z4,Z6. For theZ2action onS(R), given by f → f^˜, where ˜f(x) = f(−x), the corresponding module, called the flip module, over A_θo Z2 is quite well studied by Walters [21]. In the next section we discuss in more detail flip modules in the higher-dimensional setting. TheZ4action is given by the Fourier automorphism f → f^˜where ˜f(x) = R

Re(hx,x⁰i)f(x⁰)dx⁰. Walters has studied these modules extensively and among other things he computed the Chern character for the flip modules and Fourier modules. TheZ3andZ6actions are similar so we only treat theZ6action.

The cyclic groupZ6is generated by the matrixW6 :=

1 −1

1 0

that we denote byW. Note that thisW₆slightly differs fromW₆from section 4. We choose thisW6to keep the final formula similar to the formula forW6:=

0 −1

1 1

in [5]. One should note that the action of finite group on the projective module as in Proposition 5.3 is not unique.

The generating function associated toW=W6is given by W(x,x⁰) =xx⁰−¹

2x⁰²,

which follows from (5.1). The corresponding metaplectic transformation (for the choices=1) is

F_W(f)(x) =√ i

Z

R

e(xx⁰−¹

2x⁰²)f(x⁰)dx⁰, f ∈_S(R)

The following proposition is due to Walters:

Proposition 6.1.

(F_W)⁶=−I.

We modify the operator F_W to i⁻¹³F_W, which amounts to including the Maslov index of the transformation. Then F_W⁶ = I. The corresponding pro- jective module over AJo Z6 is called the hexic module by Walters, where J = 0 1

−1 0

is the standard symplectic form onR². For a generalA_θ, choosing T:=

−θ 0

0 1

, we get from the main theorem:

F_Wθ(_f)(_x) =_i¹⁶θ⁻

1 2

Z

R

e( ¹

2θ(_2xx⁰−x⁰²))_f(_x⁰)_dx⁰_,f ∈S(R)_; which is the exactly the formula for theZ6action considered in [5].

7. K-theory of the crossed product ofn-dimensional noncommutative tori with the flip action

We consider then×nmatrixW = −In which generates the two element group. Suppose this group acts on an = 2p+q-dimensional noncommutative torus with respect to the parameterθwithθ :=

θ₁₁ θ₁₂ θ21 θ22

,θ₁₁ being the left 2p×2p corner, which amounts to the conditionW^TθW = θ that holds in this case. We call this action the flip action.

We define the following operator onS(R^p×Z^q)with respect toW:

(7.1) T_W(f)(x,t):= f(−x,−t).

S(R^p×Z^q)with respect to this action is aA^∞_θ o Z2module which can be com- pleted to anA_θo Z2module. For a totally irrational 3×3 skew-symmetric matrix θ(the definition of which we will introduce shortly), we will see that any gener- ator of K0(A_θ)can be given by completions of modules of the typeS(R^p×Z^q). Hence it will follow that all the generators of K0(A_θ) can be extended to pro- vide classes in K0 of A_θo Z2. We will also show that K-theory classes of these modules can be extended to a generating set of K0(A_θo Z2)for 3-dimensional noncommutative tori. Our results will show that this should also be the case for the generaln-dimensional case, but at this moment we are unable to compute the generators of K0(A_θo Z2)forn-dimensional noncommutative tori A_θ. It should be noted that K1(A_θo Z2)is trivial ([5]).

Letθbe a real skew-symmetricn×nmatrix andθ⁰ be the upper left(n− 1)×(_n−1)_{block of}θ. In this case,A_θcan be written as a crossed productA_θ⁰o Z, where the actionγofZ^onAθ⁰given (by the generator ofZ^{) byU}i → e(θ_in)U_i, fori=1,· · ·,n−1. NowA_θo Z2= A_θ⁰o Z o Z2= A_θ⁰o Z2∗Z2, sinceZ2∗Z2

is isomorphic toZ o Z2as groups (see [7, Proposition 6] for more details). Note that one copy ofZ2acts onA_θ⁰ by flip actionβand the other byα= γ◦β. Our next step is to understand the K-theory ofA_θ⁰o(Z2∗Z2).

For a general crossed product Ao Z2, we first define a map pwhich goes fromAo Z2toM2(A)such that

p(a+bW) =

a b WbW WaW

,

whereWis the unitary inAo Z2implementing the action of the generator ofZ2. This induces a mapp∗ : K0(Ao Z2)→K0(A), which is known to be the inverse of Green–Julg map. We recall the six term exact sequence by Natsume [13] which was used by Farsi–Watling [7] to compute the K-theory ofA_θ⁰o(Z2∗Z2). For a free productH₁∗H₂acting on a C*-algebra A, Natsume obtained the following exact sequence:

K0(A) −−−−→^{i1∗−i2∗} K0(AoH₁)⊕K0(AoH2) −−−−→^j1∗+j2∗ K0(AoH₁∗H2) x







 y^e1 K₁(AoH₁∗H₂) ←−−−−

j1∗+j2∗

K₁(AoH₁)⊕K₁(AoH₂) ←−−−−

i1∗−i2∗

K₁(A),

wherei₁,i₂,j₁,j₂are the natural inclusion maps. The right vertical mape₁, which we will describe in a while, is constructed in Natsume’s paper. We call it ex- ponential map since it is based on the exponential map in K-theory. We want to compare the above sequence with the six-term exact sequence obtained from the classical Toeplitz exact sequence (with coefficient in A) which is same as the Pimsner–Voiculescu exact sequence for actions ofZon the C*-algebraA.

From the definition of the crossed product, any crossed product algebra, AoαG, for a unital C*-algebra Aand a discrete groupG, has a natural represen- tation (also called regular representation)ιon the Hilbert modulel²(G,A)which is given by

ι(a)(ξ)(g) =α_g−1(a)ξ(g), ι(h)(ξ)(g) =ξ(h⁻¹g),

for a ∈ A and g,h ∈ G. Let Z act on a unital C*-algebra A by an actionγ.

The classical Toeplitz algebraT^Awith coefficients inAis defined as follows: we restrict the natural representation ιof A froml²(Z,A)to l²(Z≥0,A) (note that the restriction is well defined). Call this restricted representationι₁. When there is no confusion, we just call ι(_a)_and ι₁(_a) _by a. Take the right shift operator S on l²(Z≥0,A) which is given by S(ξ)(n) = ξ(n−1),ξ(−1) = 0. ThenT^A is generated by the elements a ∈ l²(Z≥0,A) andS ∈ l²(Z≥0,A). We have the following exact sequence:

(7.2) 0 −−−−→ K(l²(Z^≥0,A)) −−−−→ T^{ϕ A} −−−−→^ψ Ao Z −−−−→ 0

by definingψ(a) =aandψ(S) =U, whereUis the unitary in the crossed product Ao Zcoming from the generator ofZ. It can be easily checked that ker(ψ) = A⊗ K. This is the so-called Pimsner–Voiculescu exact sequence which gives rise to the Pimsner–Voiculescu six term exact sequence. Now we define the mape2to be the exponential map in K-theory for the above exact sequence. So we have

K0(K(l²(Z≥0,A))) −−−−→ K0(T^A) −−−−→ K0(Ao Z) x







 y^e2 K₁(Ao Z) ←−−−− K₁(T^A) ←−−−− K₁(K(l²(Z≥0,A))).

Pimsner–Voiculescu also proved thatT^Ais KK-equivalent to the algebraA. This gives

K0(A) ^id−γ

−1∗

−−−−→ K0(A) −−−−→^i∗ K0(Ao Z) x







 y K₁(Ao Z) ←−−−−

i∗ K₁(A) ←−−−−

id−γ⁻¹∗

K₁(A), whereiis the inclusion.

The exact sequence 7.2 also gives rise to the following exact sequence (ten- soring withM₂)

0 −−−−→ M2(K(l²(Z≥0,A))) −−−−→^ϕ M2(T ^A) −−−−→^ψ M2(Ao Z) −−−−→ 0.

We now describe the map e1using an exact sequence like the one above.

Let the groupZ2∗Z2be generated bygands, i.e. gandsgenerate the first and second copy ofZ2inZ2∗Z2, respectively. Natsume obtained the exact sequence

0 −−−−→ K(l²(P)) −−−−→ T^η p π

−−−−→ C^∗(Z2∗Z2) −−−−→ 0,

with P = P⁰∪ {e}, where P⁰ is the the set of all non-empty words inZ2∗Z2, which end ingandT_pis generated byµ(g)andv(s), whereµ(g)is the restriction of the left regular representation tol²(P)andv(s) =λ(s)q(P), whereλ(s)is the restriction of the left regular representation to l²(P⁰)and q(P) is the projection onto the subspace generated by the inclusionl²(P⁰) ⊂l²(P). When all these are defined, there is a mapπsendingµ(g)toλ(g)andv(s)toλ(s). Denoting ker(π) to beI, it can be shown thatI is isomorphic toK(l²(P)). More details may be found in the paper by Natsume [13]. Let e1be the map from K0(C^∗(Z2∗Z2)) to K1(K(l²(P)) coming from the six-term exact sequence corresponding to the above exact sequence in K-theory.

The above construction can be easily extended to the case of crossed prod- uct. LetZ2∗Z2act on a unitalAwith the actionαandβonAfromhsiandhgi, respectively. We call the action ofZ2∗Z2byα,βand denote the crossed product

byAoα,βZ2∗Z2.T_p^Ais constructed as follows. We have the natural representa- tionι⁰ofAoα,βZ2∗Z2onl²(Z2∗Z2,A)which we restrict to the Hilbert module l²(P,A)in the following sense: if we denote the restriction byι₂,a∈A,g,sact by the operators

ι2(a)(ξ)(x) = (α,β)_x−1(a)ξ(x), ι2(g)(ξ)(x) =ξ(gx), ι2(s)(ξ)(x) =ξ(sx) (by settingξ(s) =0). ThenT_p^Ais the C*-algebra generated byι2(a)(a∈ A),ι2(g) andι2(s). Now we have the following exact sequence (see Lemma A.3 [13]):

0 −−−−→ K(l²(P,A)) −−−−→ T^η _p^A −−−−→^π Aoα,βZ2∗Z2 −−−−→ 0 by definingπ(ι2(a)) = ι⁰(a),π(ι2(g)) = ι⁰(g)and π(ι2(s)) = ι⁰(s). Note that for the case A = C,ι₂(g) = µ(g)(as a generator of Tp), ι₂(s) = v(s),ι⁰(g) = µ(g)_,ι⁰(s) = µ(s). Denote the exponential map (of K-theory) of the above ex- act sequence still bye1. The above exact sequence gives rise to Natsume’s exact sequence in K-theory. So we have

K0(K(l²(P,A))) −−−−→ K0(T_p^A) −−−−→ K0(Aoα,βZ2∗Z2) x







 y^e1 K₁(Aoα,βZ2∗Z2) ←−−−− K₁(T_p^A)) ←−−−− K₁(K(l²(P,A))).

T_p^A can be shown to be KK-equivalent to(AoαZ2)⊕(AoβZ2)(see [13], also [16]).

LetS = Z o Z2 with generatorsaandbi.e ageneratesZ^,bgeneratesZ2, andZ2acts onZby flip. NowSis isomorphic toZ2∗Z2, where the later group is generated by g ands and the isomorphism identifiesaand bwith sgand g, respectively. Nowl²(P,A)could be identified withl²(P1,A)⊕l²(P2,A), where

P2={g,gsg,gsgsg,gsgsgsg, . . .}, P1={e,sg,sgsg,sgsgsg, . . .}.

Counting the number of sg’s, P1 and P2 have natural identifications withZ≥0. Under this identificationl²(P,A)becomesl²(Z≥0,A)⊕l²(Z≥0,A),ι2(s)becomes 0 S

S^∗ 0

,ι₂(g)becomes

0 1 1 0

andι₂(a)becomes

γ⁻¹(a) 0 0 γ⁻¹β(a)

, whereγis the actionα◦βonA. Now under the above identification we have

K(l²(P,A)) =K(l²(P1,A)⊕l²(P2,A))

=K(l²(Z≥0,A)⊕l²(Z≥0,A))

=M2(K((l²(Z≥0,A))). Also note that, from the generators and relations we have

Aoα,βZ2∗Z2= (AoγZ)o Z2,

where, in the right hand side, the action ofZ2 = hgionAis the given actionβ and onZ^{is flip.}