NUPHB13327S0550-3213(15)00090-510.1016/j.nuclphysb.2015.03.008The AuthorHigh Energy Physics – Theory

Fig. 1

Lattice points in the complex plane, generated by the vectors 1 and ω, representing permitted κ-topological values of invariant orbifold projections and modules. (It also represents the ρ-topological values with the exception of those of φ3j.)

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Research partly supported by a grant from NSERC.

Toroidal orbifolds of Z3 and Z6 symmetries of noncommutative toriDedicated to George Elliott on his seventieth birthday

Sam

Walters

http://hilbert.unbc.ca

walters@unbc.ca

Department of Mathematics & Statistics, University of Northern B.C., Prince George, B.C. V2N 4Z9, CanadaDepartment of Mathematics & StatisticsUniversity of Northern B.C.

B.C. V2N 4Z9

Prince GeorgeCanada

Editor: Stephan Stieberger

Abstract

The Hexic transform ρ of the noncommutative 2-torus Aθ is the canonical order 6 automorphism defined by ρ(U)=V, ρ(V)=e−πiθU−1V, where U, V are the canonical unitary generators obeying the unitary Heisenberg commutation relation VU=e2πiθUV. The Cubic transform is κ=ρ2. These are canonical analogues of the noncommutative Fourier transform, and their associated fixed point C⁎-algebras Aθρ, Aθκ are noncommutative Z6, Z3 toroidal orbifolds, respectively. For a large class of irrationals θ and rational approximations p/q of θ, a projection e of trace q2θ−pq is constructed in Aθ that is invariant under the Hexic transform. Further, this projection is shown to be a matrix projection in the sense that it is approximately central, the cut down algebra eAθe contains a Hexic invariant q×q matrix algebra M whose unit is e and such that the cut downs eUe, eVe are approximately inside M. It is also shown that these invariant matrix projections are covariant in that they arise from a continuous section E(t) of C∞-projections of the continuous field {At}0<t<1 of noncommutative tori C⁎-algebras such that ρ(E(t))=E(t). It turns out that the projection E(t) is the support projection of a canonical C∞-positive element that has the appearance of a noncommutative 2-dimensional Theta function. The topological invariants (or ‘quantum’ numbers) of E(t), e, and related projections are computed by a new and quicker method than in previous works. (They would also give topological invariants for finitely generated projective modules over noncommutative orbifolds associated to Z6 and Z3 symmetries of noncommutative tori.) We remark that these results have some bearing on research work related to noncommutative orbifolds used in string theory.

Introduction

For each irrational number θ, the irrational rotation C⁎-algebra or noncommutative 2-torus Aθ is the unique C⁎-algebra generated by unitaries U, V enjoying the (unitary) Heisenberg commutation relation(1.1)VU=λUV, where λ=e2πiθ. By convention we always write e(x):=e2πix. As is well-known, if q, p are the position and momentum operators of quantum mechanics satisfying the Heisenberg commutation relation qp−pq=iħ, the unitaries U=eiq, V=eip satisfy (1.1) with 2πθ=ħ.The (noncommutative) Hexic transform is the order 6 automorphism ρ of Aθ defined by(1.2)ρ(U)=V,ρ(V)=λ−1/2U−1V and the Cubic transform κ=ρ2 is its square, the order 3 canonical automorphism κ of Aθ given by11

There are other variations of these canonical order 3 automorphisms – e.g., U→V→V−1U−1. This latter automorphism can, however, be checked to be conjugate to κ by the automorphism U→λ1/6V−1, V→λ1/6U. So our results extend to these other forms as well with a few appropriate changes.

(1.3)κ(U)=λ−1/2U−1V,κ(V)=U−1. The Flip automorphism ϕ is defined by ϕ(U)=U−1, ϕ(V)=V−1, and one quickly notes thatρ2=κ,κρ=ϕ=ρ3. These periodic automorphisms arise naturally from the canonical Brenken–Watatani action of SL(2,Z) on the rotation C⁎-algebra. They are order 3 and 6 analogues of the (noncommutative) Fourier transform (V→U→V−1) studied in [7,20–22,24–26]. (The orders 2, 3, 4, 6 are the only orders possible for such canonical automorphisms since these are the only finite orders of matrices in SL(2,Z). In fact, any finite order automorphism of Aθ induces one on K1(Aθ)=Z2 and hence a matrix in GL(2,Z).)The (noncommutative) toroidal orbifolds associated to the symmetry groups Z3 and Z6 are the fixed point C⁎-subalgebrasAθκ:={x∈Aθ|κ(x)=x} and Aθρ of Aθ. For the case when θ is rational these orbifold algebras take a rather concrete topological form of a 2-sphere S2 with 3 or 4 singularities22

See [1] for the flip case, and [9–11] for the order 3, 4, 6 cases.

each of which takes the form of multiple non-Hausdorff points ‘bundled’ together.33

A simpler example is the spectrum of the infinite dihedral group Z⋊Z2 (see for example [5, II.2.β]) represented as the closed interval [0,1] with its ends split in two:

:—————————:

Such realizations make essential use of the realization of the rational noncommutative torus Ap/q given in [1] as matrix-valued continuous functions on the unit square that satisfy certain boundary conditions at opposite edges of the square. The finite canonical symmetries would in effect ‘fold and paste’ the unit square into spheres with singularities (so such orbifolds are sometimes called noncommutative spheres).These non-Hausdorff points can in some way be gleaned from certain projections in a matrix algebra Mq and by the existence of “unbounded” (noncanonical) traces. In the end, these orbifolds in the rational case look like the algebra of all continuous functions f:S2→Mq which, at the singularities s, commute with certain projections, so that f(s) has a specific block diagonal form at such points.44

Perhaps it's worth noting that the number of these singularities is largely independent of the rational parameter θ, though the sizes of the blocks depend on q where θ=p/q.

Each of the blocks would correspond to each of the multiple non-Hausdorff points at a given singularity, and by taking the trace of any of the blocks of f(s) one obtains the noncanonical traces. We have detailed out computations of this for the noncommutative Fourier transform in [21] (see Sections 3 and 4).Thus, in the rational case we have a reasonably good way of visualizing these orbifolds and their singularities – namely, in terms of projections and noncanonical traces. These orbifolds are, however, more difficult to visualize when θ is irrational, but we can nevertheless treat their projections (and modules) in the orbifold as well as the noncanonical trace maps (which still do exist) as indicative of such singularities in a way similar to the rational case.55

To appreciate the algebraic difference between the rational and irrational cases we point out that the orbifold in the rational case is a type I C⁎-algebra (being a C⁎-subalgebra of matrix-valued continuous functions on a compact Hausdorff space) while the orbifold in the irrational case is a non-type I approximately finite-dimensional algebra as was first shown by Bratteli and Kishimoto [2] for the Flip symmetry.

The projections constructed in this paper and the noncanonical traces (see, for example, Eq. (1.7) below) serve, in part, such purposes for the Cubic and Hexic symmetries.In [7] it was proved that the orbifold algebras Aθκ, Aθρ and their respective (strongly Morita equivalent) C⁎-crossed products Aθ⋊κZ3 and Aθ⋊ρZ6 are approximately finite-dimensional for each irrational θ (extending the Bratteli–Kishimoto Theorem [2] for the Flip case – see also [19]).In this paper we construct projections that are Hexic (hence Cubic) invariant, show that they have a covariance property (see Theorem 1.2), compute their topological invariants, and we prove that from these projections one obtains many matrix projections (the meaning of which is given in Theorem 1.2 below).Our first result is the following theorem; it is based on certain topological maps ψk, φ1, φjk given in Eqs. (1.7) and (1.9)–(1.11) below. We implicitly assume the results of Elliott [8] concerning the continuous field {At}0<t<1 of noncommutative 2-tori.Theorem 1.1There is a continuous section E:(0,1)→{At} of C∞-projections of the continuous field {At}0<t<1 of noncommutative 2-tori C⁎-algebras such that

(1)ρ(E(t))=E(t), κ(E(t))=E(t);

(2)E(t) has canonical trace t and κ-topological numbersψk(E(t))=ω:=12+i23 for k=0,1,2;

(3)E(t) has ρ-topological numbersφ1=3ω−1,φ20=ω,φ21=3ω,φ30=12,φ31=2;

(4)E(t) is the support projection of the noncommutative 2D “Theta function”(1.4)X(t)=t∑m,ne−πt3(m2+n2)e−πt(13−i)mnUtnVtm for 0<t<1; further, X(t) is positive and limt→0+⁡‖E(t)−X(t)‖=0.

In obtaining the next result on matrix (or point) projections, we have restricted ourselves to a concrete class G of irrationals θ∈(0,1) (which contains dense Gδ sets in (0,1)) although the result very likely extends to all irrationals (and corresponding convergents) without the restrictions that we impose here.For example, in the case of the Fourier transform automorphism σ we have shown in [24] that the orbifold Aθσ is approximately finite dimensional for a dense Gδ class of irrationals θ, which was later shown to hold for all irrationals in [7]. In addition, we showed that the point-matrix result (in the Fourier case) holds for all irrationals. We made this restriction here in an effort to make the results of Theorem 1.2 accessible (or else the computations will be considerably longer). (Theorem 1.1 is independent of the class G.)The class G consists of irrationals θ in the open interval (0,1) that can be approximated by infinitely many rational numbers pq (in reduced form) where p=p12 is an even perfect square such that66

For the approximations in Theorem 1.2, the quantity q2θ−pq needs to stay away from 1 – just as in Lemma 7.2 of [24]. The inequality in (1.5) already meets this condition.

(1.5)0<θ−pq<0.995q2. It is easy to see that this class G contains many dense Gδ subsets of (0,1) and that one can give specific examples of irrational numbers in it. To obtain the projection section referred to in Theorem 1.1, one simply takes p=0, q=1 as we shall see in Sections 3 and 4.Our second main result is the following matrix projection approximation.Theorem 1.2Let θ be any irrational number in G. Let pq be a rational number in reduced form (with p≥0, q≥1) such that 0<θ−pq<0.995q2, where p is an even perfect square. Then the projection(1.6)e=ζq,θ(E(q2θ−pq)) in Aθ (has trace q2θ−pq) is ρ invariant, is approximately central, and there exists a ρ-invariant q×q matrix algebra M⊂eAθe with unit e such that: for any finite subset F⊂Aθ and each ϵ>0, there exists large enough q such that exe has distance less than ϵ from M for each x∈F. (The same conclusions hold for the Cubic transform κ.)The canonical morphism ζq,θ is defined by the relations (3.7).Recall that the projection e (as a function of the integer parameter q) is approximately central in Aθ if for any finite subset F⊂Aθ and ϵ>0 there exists large enough q such that ‖xe−ex‖<ϵ, ∀x∈F.We remark that in the matrix approximation of this theorem, the cut downs of the canonical unitaries eUe, eVe are close to order q unitary matrices of M. (This is shown in the proof of Section 4 below.) As the generic unitaries {UmVn} form a total set in the algebra Aθ, the cut down approximation in Theorem 1.2 follows immediately. (In fact, with a little extra work, one can show that they are close to unitary generators of the matrix algebra – as we showed in the Fourier case in [24].)At the end of Section 5 we compute the topological invariants of the matrix projection e of Theorem 1.2 from those in Theorem 1.1.The computation of the topological invariants of projections (or, equivalently, finitely generated projective modules, thanks to the Serre–Swan Theorem!) is based on certain noncanonical (and usually unbounded linear functionals) ‘twisted’ traces defined on the canonical smooth dense ⁎-subalgebra Aθ∞ (the analogue of the algebra of C∞ functions on a manifold).The role that noncommutative tori played in string theory, particularly in compactification of M(atrix) theory, was initiated by Connes, Douglas, and Schwarz [6] and nicely exemplified by Seiberg and Witten in [17] (see, for example their Introduction and Section 6 on gauge theory on a torus). This has lead naturally to the study of orbifolds of noncommutative tori associated to canonical group symmetries.For instance, in [14] Konechny and Schwarz used K-theoretical topological invariants (obtained by the author in [18]) to study the structure of projective modules over non-commutative Z2 orbifolds that admit constant curvature Yang–Mills field as well as obtaining their moduli spaces. And in [13], they study moduli spaces of (equivariant) connections with constant curvature on modules over non-commutative even-dimensional tori and on toroidal orbifolds arising from symmetries of the groups Z2 and Z4 (with respective actions from the flip and Fourier transform). They give a nice and short summary of this in Section 9 of [12] for the Z2 orbifold compactification case.Using the results of the current paper—the construction of the (exotic) projection (or module) E(t) and its topological invariants77

The reason we call E(t) ‘exotic’ is because it is a fundamental generator of K-theory of the orbifold, much as had been done in [7], and in [20] and [21] in the Fourier transform case.

—one could solve the associated problems for canonical orbifolds corresponding to the symmetry groups Z3 and Z6 of the noncommutative tori (induced by the automorphisms κ,ρ). Namely, the problem of studying (modulo gauge transformations) connections of constant curvature on modules over a non-commutative orbifold, since their classes in the K0-group are associated to D-brane charges, following Witten's theory [28].We now explain how we compute these noncanonical trace invariants and recall what they are.Topological numbers: the continuous field method. In this paper we present a new and quicker method for computing the topological invariants (which are certain quantized complex numbers) of the projections constructed here. What makes this possible is that these projections arise from a continuous field of projections E:(0,1)→{At}, as in Theorem 1.1. The result ‖E(t)−X(t)‖→0 in this theorem allows one to compute the topological invariant ψ(E)=ψt(E(t)), which is constant88

Not unlike the fact that if f(t) is a continuous section of projections of the field {At} then the label of its canonical trace is constant:

(f(t))=a+bt where a, b are constant integers.

as a function of t (see [7]), by means of the limitψ(E)=limt→0+⁡ψt(X(t)) which will turn out to exist (as computed in Section 5). Here, ψt is any of the topological unbounded traces associated with the Z3 or Z6 symmetries of the noncommutative tori arising from the Cubic or Hexic transforms. (We will write down these traces shortly.) Once this is done for the section E(t), we use the covariance relationship that the morphisms ζq,θ have with the unbounded traces ψ (see for instance Eqs. (5.5)–(5.7)) to quickly obtain the topological invariants of projections e=ζq,θ(E(q2θ−pq)) in Theorem 1.2. This constitutes what we might call the continuous field method for computing topological invariants, thanks to the noncommutative Theta function X(t) in (1.4) since their unbounded traces are amenable to direct calculation (which turn out to involve the Theta functions of classical analysis).The unbounded topological trace maps. Given a (finite order) automorphism β of an algebra A, a (twisted) β-trace is a complex linear map ψ:A→C such thatψ(xy)=ψ(β(y)x) for all x,y∈A. (It is not unlike a KMS state except that we do not have a continuous one-parameter group action.) The restriction of such map ψ to the fixed point subalgebra (orbifold) Aβ={x∈A|β(x)=x} defines a trace, so gives an invariant morphism at the K-theory level ψ⁎:K0(Aβ)→C.In the case of noncommutative tori Aθ, such maps are not defined on the whole algebra but on the canonical dense *-subalgebra Aθ∞ of differentiable elements – namely, Schwartz series ∑amnUmVn where {amn} is rapidly decreasing.In joint work with Julian Buck [3], we computed such twisted traces for the Cubic transform κ and shown [3, Theorem 3.3] that they form a 3-dimensional complex vector space with basis given by the following basic κ-traces(1.7)ψjθ(UmVn)=e(θ6(m−n)2)δ3m−n−j where j=0,1,2, VU=e(θ)UV, and δdm is the divisor delta function given by δdm=1 if d divides m, and δdm=0 otherwise. These noncanonical traces induce group homomorphisms on the K0-group of the Cubic orbifold Aθκ which, together with the usual canonical trace state , give rise to its Connes–Chern character invariant:T3:K0(Aθκ)→C4,T3(x)=(

(x);ψ0(x),ψ1(x),ψ2(x)). For the identity element, for example, we have T3(1)=(1;1,0,0). Recall that the canonical trace is defined by

(∑amnUmVn)=a00.For κ-invariant projections (or finitely generated projective modules over the orbifold), their ψ0, ψ1, ψ2 invariants are quantized numbers (shown in the lattice in Fig. 1

below) and they may be called their κ-topological invariants or numbers.

In [4] (using work of Polishchuk [15]) we showed that the homomorphism T3 is an injective map on K0(Aθκ)=Z8 in the case that θ is irrational – and in the rational case we would have to include Connes' cyclic 2-cocycle that picks out the “label of the trace”. Based on the values in Table 2 of [3, (page 37)],99

We point out that the topological invariants listed in Table 2 of [3] are those of the above mentioned maps ψkθ which differ from the maps we used in [3] by normalization constants – particularly for the maps φ11, φ12 used in [3] which involved the constants e(−θ6), e(−2θ3), respectively, and which should be removed (as we have in fact done so at the end of Section 9 of [3]).

the unbounded traces ψk of Cubic invariant projections take values in the following lattice in the complex plane(1.8)ψjK0(Aθκ)=Z+Zω (for j=0,1,2) where ω is given in Theorem 1.1.1010

In [3], the unbounded trace values differ by a factor of 3 since we were working with the crossed product C⁎-algebra Aθ⋊κZ3, but since this algebra is strongly Morita equivalent to the fixed point C⁎-subalgebra Aθκ the unbounded trace values in Table 2 of [3] need to be multiplied by 3. For the Hexic case one similarly multiplies the unbounded trace values in Table 1 of [3] by 6.

Observe that the range in (1.8) is independent of the “Planck” parameter θ, whereas the canonical bounded trace has range

K0(Aθκ)=Z+Zθ which depends on θ.Remark 1.3We caution that the “ω” used in Tables 1 and 2 of [3] is not the same as the above ω. If ω′ denotes the one used in these tables, which is ω′=e(1/6)=12(1+i3), then in order to make the conversion from crossed product to fixed point subalgebra, one makes use of the relations ω=13ω′1/2, ω′=3ω−1.For the Hexic transform, by Theorem 3.1 [3], there is a unique ρ-trace φ1θ (up to scalar multiples) defined on Aθ∞, a pair of ρ-invariant ρ2-traces φ2jθ, and a pair of ρ-invariant ρ3-traces φ3jθ given by(1.9)φ1θ(UmVn)=e(θ2(m2+n2))(1.10)φ20θ(UmVn)=e(θ6(m−n)2)δ3m−n,φ21θ(UmVn)=e(θ6(m−n)2),(1.11)φ30θ(UmVn)=e(−θ2mn)δ2mδ2n,φ31θ(UmVn)=e(−θ2mn). (Here, of course, ρ2=κ is the Cubic and ρ3=ϕ is the Flip.) When no confusion arises we shall simply write φjkθ=φjk. The Connes–Chern character invariant for the Hexic orbifold Aθρ consists of these together with the canonical trace:T6:K0(Aθρ)→C6,T6(x)=(

(x);φ1(x),φ20(x),φ21(x),φ30(x),φ31(x)) For the identity one has T6(1)=(1;1,1,1,1,1). In this case, the ranges of these noncanonical traces on the K0-group of the ρ-orbifold subalgebra take the following valuesφ1K0(Aθρ)=Z+Zω,φ2jK0(Aθρ)=Z+Zω,φ3jK0(Aθρ)=12Z (j=0,1) and are independent of θ. For the canonical trace,

K0(Aθρ)=Z+Zθ.Observe the following relations between the maps φ2⁎ and ψj(1.12)φ20=ψ0,φ21=ψ0+ψ1+ψ2 which will be useful in giving the φ2⁎ topological invariants once we have determined the ψj values.The following order 3 (toral) automorphism commutes with the Cubic transform:γ(U)=e(13)U,γ(V)=e(−13)V (as well as γ−1=γ2). (Of all canonical toral T2-actions on Aθ, only γ, γ2 and the identity commute with κ.) The relationship between γ and the κ-traces ψk is as follows(1.13)ψ0γ=ψ0,ψ1γ=e(13)ψ1,ψ2γ=e(23)ψ2, or ψjγ=e(j3)ψj. These enable one to immediately obtain the κ invariants for the corresponding fields γE(t) and γ2E(t) in view of (2) of Theorem 1.1:ψj(γE(t))=e(j3)ψj(E(t))=e(j3)ω∈{ω,ω−1,1−2ω} (the latter values are for j=0,1,2, respectively).Remark 1.4Note, however, that unlike E, the projection field γE is not Flip invariant (as the Flip automorphism and γ do not commute) so is not Hexic invariant.2

Rieffel's bimodule construction and cubic integral transform

In this section we recall the main aspects of Rieffel's equivalence bimodule construction [16] and apply his Theorem 2.15 to our situation with canonical symmetries κ, ρ.We begin with a locally compact Abelian group M, with Mˆ denoting its Pontryagin dual consisting of all characters M→T, and we form the self-dual direct product group G=M×Mˆ. Denote by 〈,〉 the canonical pairing map M×Mˆ→T given by 〈m,s〉=s(m) where m∈M, s∈Mˆ. The Heisenberg bicharacter on G is the canonical map h:G×G→T defined byh((m,s),(n,t))=〈m,t〉 for m,n∈M, s,t∈Mˆ.The Heisenberg projective unitary representation of G is defined byπ:G→L(L2(M)),[π(m,s)f](n)=〈n,s〉f(n+m) for m,n∈M, s∈Mˆ and f∈L2(M). (Here, L2(M) is the Hilbert space of square-integrable complex functions on M with respect to Haar measure of M.) These unitary operators πx satisfy the Heisenberg commutation relation(2.1)πxπy=h(x,y)πx+y=h(x,y)h(y,x)¯πyπx and their adjoints satisfy(2.2)πx⁎=h(x,x)π−x for x,y∈G.If D is any lattice subgroup of G (i.e., a discrete cocompact subgroup of G), its covolume |G/D| is the Haar measure of a fundamental domain for D in G. The associated twisted group C⁎-algebra C⁎(D,h) is defined as the C⁎-subalgebra of the bounded operators on L2(M) generated by the unitaries πx for x∈D:C⁎(D,h)=C⁎{πx|x∈D}. It is in fact the universal C⁎-algebra generated by unitaries satisfying the commutation relation (2.1).The complementary subgroup D⊥, which turns out to also be a lattice subgroup of G (Lemma 3.1 of [16]), is defined byD⊥={y∈G:h(x,y)h(y,x)¯=1,∀x∈D}. By taking the adjoints of the relations in (2.1), we see that the ‘dual’ unitaries πy⁎ for y∈D⊥ satisfy the same relations (2.1) but with the conjugate cocycle h¯ in place of h with the opposite multiplication:(2.3)πx⁎•πy⁎=h¯(x,y)πx+y⁎=h¯(x,y)h¯(y,x)¯πy⁎•πx⁎,(πx⁎)⁎=h¯(x,x)π−x⁎ for x,y∈D⊥, where we used • for the opposite multiplication (a•b:=ba). Therefore, the unitaries πy⁎, for y∈D⊥, generate the twisted group C⁎-algebra C⁎(D⊥,h¯) but with the understanding that it is now equipped with opposite multiplication:C⁎(D⊥,h¯)=C⁎{πy⁎|y∈D⊥}.Rieffel's Theorem 2.15 in [16] states that if D is a lattice subgroup of G=M×Mˆ, then the completion of the Schwartz space S(M) of M (under the norm (2.9) defined below) is an equivalence C⁎(D,h)-C⁎(D⊥,h¯)-bimodule in a natural way with appropriate C⁎-inner products. We now recall the module actions and the C⁎-inner products that accompany this equivalence.The module actions of the C⁎-algebras C⁎(D,h) and C⁎(D⊥,h¯) are given by(2.4)af=∫Da(x)πx(f)dx=|G/D|∑x∈Da(x)πx(f)(2.5)fb=∫D⊥b(y)πy⁎(f)dy=∑y∈D⊥b(y)πy⁎(f) where f∈S(M), a∈C⁎(D,h), b∈C⁎(D⊥,h¯), and where the measure (dx) of each point of D is |G/D| and each point of D⊥ has measure 1. (Here, of course, a is generically represented as a=∑x∈Da(x)πx, where a(x) are its complex coefficients.)The C⁎-valued inner products on S(M) with values in the algebras C⁎(D,h) and C⁎(D⊥,h¯) are given by〈f,g〉D=|G/D|∑w∈D〈f,g〉D(w)πw,〈f,g〉D⊥=∑z∈D⊥〈f,g〉D⊥(z)πz⁎ where the complex coefficients in these sums are(2.6)〈f,g〉D(w1,w2)=〈f,π(w1,w2)g〉L2(M)=∫Mf(x)g(x+w1)¯〈x,w2〉¯dx(2.7)〈f,g〉D⊥(z1,z2)=∫Mf(x)¯g(x+z1)〈x,z2〉dx where (w1,w2)∈D and (z1,z2)∈D⊥. These C⁎-inner products satisfy the associativity condition(2.8)〈f,g〉Dh=f〈g,h〉D⊥ for all f,g,h∈S(M). (See [16, pages 266 and 269].) Further, the module actions and C⁎-inner products satisfy the properties〈af,g〉D=a〈f,g〉D,〈f,gb〉D⊥=〈f,g〉D⊥•b=b〈f,g〉D⊥ for a∈C⁎(D,h) and b∈C⁎(D⊥,h¯), and also for the adjoints of inner products〈f,g〉D⁎=〈g,f〉D,〈f,g〉D⊥⁎=〈g,f〉D⊥.The Schwartz space S(M) gives rise to an equivalence C⁎(D,h)–C⁎(D⊥,h¯) bimodule when it is completed under the norm(2.9)‖f‖:=‖〈f,f〉D‖1/2=‖〈f,f〉D⊥‖1/2. (The last equality is a theorem of Rieffel – cited in [16].)Finally, the twisted groups C⁎-algebras C⁎(D,h) and C⁎(D⊥,h¯) have canonical normalized traces defined on them by

D(∑w∈Dawπw)=a0,

D⊥(∑z∈D⊥bzπz⁎)=b0, (aw,bz∈C) respectively, and they satisfy the trace equation(2.10)

D(〈f,g〉D)=|G/D|

D⊥(〈g,f〉D⊥) for all f,g∈S(M). (See the equation just before Theorem 3.5 in [16].)From Rieffel's bimodule theorem it follows that if ξ is a Schwartz function such that 〈ξ,ξ〉D⊥=1, then e=〈ξ,ξ〉D is a projection in C⁎(D,h) of trace |G/D|. In this case, one gets an isomorphism(2.11)η:eC⁎(D,h)e→C⁎(D⊥,h¯),η(eae)=〈ξ,aξ〉D⊥,η−1(b)=〈ξb,ξ〉D, where a∈C⁎(D,h) and b∈C⁎(D⊥,h¯). (Note that eξ=ξ in view of the associative property (2.8) above.)Self-dual locally compact Abelian groups. We shall now add the symmetry structure to Rieffel's bimodule construction and prove some canonical results from it (see Propositions 2.3 and 2.4 below) that we shall need.We begin by taking M to be a self-dual locally compact Abelian group, so that we have an isomorphism Mˆ≅M arising from a pairing 〈,〉:M×M→T. We shall require that this pairing be symmetric: 〈m,n〉=〈n,m〉 for all m,n∈M. This in fact holds for the cases that we are interested in, namely the groups R, Zq, or direct products therefrom. Indeed, as we have recently shown in [27], this symmetric condition can always be arranged for all compactly generated self-dual (locally compact) Abelian groups.Under this circumstance, we have the Fourier transform fˆ of a Schwartz function f∈S(M) given byfˆ(s)=∫Mf(x)〈x,s〉¯dx for s∈M. The square of the Fourier transform gives the flip: fˆˆ(t)=f(−t).The self-duality of M permits us to define a canonical order 3 automorphism of G=M×M byC:G→G,C(u,v)=(v,−u−v) (for u,v∈M) which we call the Cubic map since it will give rise to the Cubic transform automorphism κ given in (1.3). If D is a lattice subgroup of G that is invariant under C (thus, C(D)=D), then so is D⊥. In this case, there are corresponding order 3 automorphisms κ and κ′ of the group C⁎-algebras C⁎(D,h) and C⁎(D⊥,h¯), respectively, given byκ(πx)=χ(x)πCx,κ′(πy)=χ(y)πCy (x∈D, y∈D⊥) for a suitable “projective character” χ:G→T, which we shall soon endeavor to obtain and justify.To obtain χ, we resort to our recent work [27, Theorem 1.2] where it was shown that for compactly generated locally compact Abelian groups M there exists a projective character ν:M→T — that is, a continuous map (even smooth when M is a Lie group) satisfying the conditions(2.12)ν(m+n)=ν(m)ν(n)〈m,n〉,ν(−n)=ν(n),ν(0)=1, for m,n∈M. Taking n=−m it is clear that ν(m)2=〈m,m〉 so that ν is a ‘square root’ of the quadratic form 〈m,m〉.One now defines the map(2.13)χ:G→T,χ(r,s)=ν(s)¯⋅〈r,s〉¯ for (r,s)∈G=M×M. It can be checked that this map has the following projective property with respect to the Heisenberg character and the Cubic map C:(2.14)χ(x+y)=χ(x)χ(y)⋅h(x,y)¯h(Cx,Cy) for x,y∈G. Thereby one obtains the order 3 automorphism κ of the group C⁎-algebra C⁎(D,h) given by(2.15)κ(πx)=χ(x)πCx for x∈D.The existence of the automorphism κ is established by checking that the universal property of the twisted group C⁎-algebra C⁎(D,h) — namely that its unitary generators {πx}x∈D satisfy the projective commutation relation πxπy=h(x,y)πx+y for each x,y∈D — remains satisfied when each unitary generator πx is replaced by the associated unitary operator χ(x)πCx. This is easy to check in view of (2.14) and (2.1).To check that κ has order 3, it is enough to check thatχ(x)χ(Cx)χ(C2x)=1 for all x∈G, and this, again, is straightforward to check. (Here, one makes use of the equation ν(m)2=〈m,m〉 for m∈M.)In a similar vain one checks that the Cubic transform is also defined on the Morita equivalent group C⁎-algebra C⁎(D⊥,h¯) by(2.16)κ′(πy⁎)=χ(y)¯πCy⁎ for y∈D⊥, and that it has order 3. (Indeed, in the universal property for the unitaries πy⁎ (for y∈D⊥) defining the C⁎-algebra C⁎(D⊥,h¯), the replacement πy⁎→χ(y)¯πCy⁎ preserves the defining commutation relation (2.3), hence the automorphism κ′ exists and has order 3.)As shown in [27], the map ν allows one to define the Cubic transform of Schwartz functions on the group M according to the following prescription(2.17)fc(m):=Kν(m)fˆ(−m)=Kν(m)∫Mf(t)〈t,m〉dt for m∈M, where K is a suitable normalizing constant with |K|=1. Theorem 1.3 of [27] says that this transform has order 3. As was shown in the proof of this theorem, K=νˆ(0)−1/3, and in the two cases that concern us in this paper, for the reals R the constant is K=i−1/6, and in the cyclic group Zq case, K=Kq depends on the mod 4 class of q as summarized in Example 2.2 below. It is convenient at this point to give examples of ν and K for the two cases that will concern us here.Example 2.1For M=R, the map ν(x)=e(12x2) satisfies (2.12) with respect to the canonical pairing 〈x,y〉=e(xy)=e2πixy. As shown in [23], the Cubic transform on the reals then takes the formfc(x)=i−1/6e(12x2)∫−∞∞f(t)e(xt)dt where K=i−1/6=e(−124). (Further, that fccc=f for each Schwartz function f.) In this paper we shall use the complex Gaussian function h(x)=e−πz1x2 where z1=12(3−i)=i−1/3=e(−1/12) because it is Cubic invariant: hc=h.Example 2.2For the cyclic group M=Zq, one has (as in [27]) the projective characterνq(m)=(−1)me(m22q) with respect to the canonical pairing 〈m,n〉=e(mnq). In this case the Cubic transform for cyclic groups takes the formfc(m)=Kqνq(m)fˆ(−m)=Kqq(−1)me(m22q)∑n=0q−1f(n)e(mnq) where the normalizing constant Kq has been worked out (see Section 4 of [27]) with the result:(2.18)Kq={e(q−124)if q is even,e(1−q248)if q≡1 mod 4,i−1/3e(1−q248)if q≡3 mod 4.Since in the next section we will be interested in the group M=R×Zq×Zq, we can take the direct product of the projective characters ν of each factor to obtain the projective character on M according toν(x,m,n)=e(12x2)νq(m)νq(n). This leads to the associated Cubic transform as given by (2.17).We now prove covariance properties that the Cubic transform has with respect to the module actions and the C⁎-inner products.Proposition 2.3One has the following bimodule properties for the Cubic transform:(af)c=κ(a)fc,(fb)c=fcκ′(b) for a∈C⁎(D,h), b∈C⁎(D⊥,h¯) and for all Schwartz functions f∈S(M).ProofWe will check the first one since the second is quite similar. Further, it is enough to check it for a=πx where x=(x1,x2). In view of (2.17), we have for any w∈M(πxf)c(w)=Kν(w)∫M(π(x1,x2)f)(y)〈y,w〉dy=Kν(w)∫Mf(y+x1)〈y,w+x2〉dy=ν(w)〈−x1,w+x2〉⋅K∫Mf(y)〈y,w+x2〉dy=ν(w)〈−x1,w+x2〉ν(w+x2)¯fc(w+x2)=〈−x1,w+x2〉ν(x2)¯〈w,x2〉¯fc(w+x2)=χ(x1,x2)〈−x1−x2,w〉fc(w+x2)=[κ(πx)fc](w) which ends the proof. □Proposition 2.4One hasκ(〈f,g〉D)=〈fc,gc〉D,κ′(〈f,g〉D⊥)=〈fc,gc〉D⊥ for all Schwartz functions f, g on M. ProofWe prove the first equality and then show how the second one quickly follows from it and Proposition 2.3. For fixed (u,v)∈G, we have〈fc,gc〉D(u,v)=∫Mfc(x)gc(x+u)¯〈x,v〉¯dx=∫Mν(x)fˆ(−x)ν(x+u)gˆ(−x−u)¯〈x,v〉¯dx=ν(u)¯∫Mfˆ(−x)gˆ(−x−u)¯〈x,u+v〉¯dx (since the constant K has modulus 1), using inversion x→−x one gets=ν(u)¯∫Mfˆ(x)gˆ(x−u)¯〈x,u+v〉dx=ν(u)¯∫Mfˆ(x)hˆ(x)¯dx=ν(u)¯∫Mf(x)h(x)¯dx where we used the unitarity of the Fourier transform and h is the function such thathˆ(x)=gˆ(x−u)〈x,u+v〉¯. We compute h by taking the inverse Fourier transform of hˆ:h(x)=∫Mhˆ(t)〈t,x〉dt=∫Mgˆ(t−u)〈t,x−u−v〉dt=〈u,x−u−v〉∫Mgˆ(t)〈t,x−u−v〉dt=〈u,x−u−v〉g(x−u−v) which gives us h. Therefore, from above we obtain〈fc,gc〉D(u,v)=ν(u)¯∫Mf(x)g(x−u−v)¯〈u,x−u−v〉¯dx=〈f,g〉D(−u−v,u)⋅ν(u)¯〈u,u+v〉=〈f,g〉D(C−1(u,v))⋅ν(u)¯〈u,u+v〉 This gives〈fc,gc〉D=|G/D|∑(u,v)∈D〈fc,gc〉D(u,v)⋅π(u,v)=|G/D|∑(u,v)∈D〈f,g〉D(C−1(u,v))⋅ν(u)¯〈u,u+v〉⋅π(u,v) which after making the replacement (u,v)→(v,−u−v)=C(u,v) becomes〈fc,gc〉D=|G/D|∑(u,v)∈D〈f,g〉D(u,v)⋅ν(v)¯〈v,−u〉⋅πC(u,v) which is indeed equal toκ(〈f,g〉D)=|G/D|∑(u,v)∈D〈f,g〉D(u,v)⋅κ(π(u,v)) by definition of κ since κ(π(u,v))=χ(u,v)πC(u,v)=ν(v)¯〈u,v〉¯πC(u,v).We can now see that κ′(〈f,g〉D⊥)=〈fc,gc〉D⊥ follows from what we just proved. In view of Proposition 2.3, for each Schwartz function h we havehcκ′(〈f,g〉D⊥)=(h〈f,g〉D⊥)c=(〈h,f〉Dg)c=κ(〈h,f〉D)gc=〈hc,fc〉Dgc=hc〈fc,gc〉D⊥, hence the result. □The Flip symmetry. The Flip automorphisms are defined on these twisted group C⁎-algebras by ϕ(πx)=π−x, ϕ′(πy)=π−y for x∈D, y∈D⊥. One can in fact easily check that they satisfy analogues of Proposition 2.4, namely thatϕ(〈f,g〉D)=〈f˜,g˜〉D,ϕ′(〈f,g〉D⊥)=〈f˜,g˜〉D⊥ where f˜(x)=f(−x). (We also have the analogue of Proposition 2.3 for the Flip.) This, together with Proposition 2.4, gives the result for the Hexic transform as well:ρ(〈f,g〉D)=〈f˜cc,g˜cc〉D,ρ′(〈f,g〉D⊥)=〈f˜cc,g˜cc〉D⊥ where f˜cc is the Hexic transform of f (of order 6). (Note that (f˜)c=fc˜.)Therefore, once we have constructed a projection in the inner product form e=〈ξ,ξ〉D to be Flip and Cubic invariant, it will be invariant under the Hexic transform.3

Lattice subgroups, C⁎-inner products, and covariance

Let θ be an irrational number in the class G and let p/q be a rational approximation, where p=p12 is an even perfect square and q≥1 are relatively prime, satisfying0<θ−pq<0.995q2 as in (1.5).Let M=R×Zq×Zq and consider the lattice D in the self-dual Abelian group G=M×M (≅M×Mˆ) with basis(3.1)D:[ε1ε2]=[αp10;000000;αp10], where α=θ−pq. It is clear that D is invariant under the Cubic map C(u,v)=(v,−u−v) on G since Cε1=−ε2 and Cε2=ε1−ε2.Since a fundamental domain for D is [0,α)×Zq×Zq×[0,α)×Zq×Zq, the covolume of D in G is |G/D|=α2q2=q2θ−pq<1. (Recall that the measure of each point of Zq is 1/q, as in Example 2.2 above.) The C⁎-algebra C⁎(D,h) is generated by the two unitaries πε1,πε2 satisfying the Heisenberg commutation relationπε1πε2πε1⁎πε2⁎=h(ε1,ε2)h(ε2,ε1)¯=e(α2+p12q)=e(θ)=λ. Using (2.15), one checks that(3.2)κ(πε1)=πε2−1,κ(πε2)=λ−1/2πε2−1πε1. (In the second of these equalities we used the condition that p=p12 is even. One can modify the current argument if p is odd.) Therefore, on settingU=πε2,V=πε1, we obtain the desired relationsVU=λUV,κ(U)=λ−1/2U−1V,κ(V)=U−1.The complementary lattice D⊥ can easily be checked to be generated by the following basis vectorsD⊥:[δ1δ2δ3δ4]=[β−cp10000000β−cp10001000000001] whereβ:=1qα>1 and c is an integer such that cp≡1 mod q. We haveC(δ1)=−δ2,C(δ2)=δ1−δ2,C(δ3)=−δ4,C(δ4)=δ3−δ4.We consider the associated unitary generatorsVj:=πδj⁎=π−δj for j=1,2,3,4 which can be checked to satisfy the commutation relations(3.3)V1V2=e(θ′)V2V1,V3V4=e(1q)V4V3,VjVk=VkVj,V3q=V4q=I, for j=1,2 and k=3,4, where θ′=β2+c2pq=c2pθ+d′qθ−p (for some integer d′) is in the usual GL(2,Z) orbit of θ. The C⁎-algebra C⁎(D⊥,h¯) is generated by the unitaries V1, V2, V3, V4 and is isomorphic to Mq(Aθ′).In view of (2.16), one checks that the Cubic transform κ′ on C⁎(D⊥,h¯) is given on these unitary generators as follows(3.4)κ′(V1)=V2−1,κ′(V2)=e(−12θ′)V2−1V1,(3.5)κ′(V3)=V4−1,κ′(V4)=−e(−12q)V4−1V3.We now consider the Schwartz function f on M defined byf(x,n,m)=31/8qh(x)φ(n,m),h(x)=e−πz1x2,φ(n,m)=e(nmq), where, z1=12(3−i)=i−1/3=e(−1/12). We have hc=h, though φ is not Cubic invariant but instead satisfies φc=φW0 for some unitary matrix W0, as shown by Lemma 4.2 below. It will then follow that fc=fW0.We now compute the inner product 〈f,f〉D. Its series expansion coefficients are〈f,f〉D(mε1+nε2)=〈f,f〉D(mα,mp1,0;nα,np1,0)=1q∑r,s=0q−1∫Rf(x,r,s)f(x+mα,r+mp1,s)¯e(−xnα)e(−rnp1q)dx=1q231/4∑r,s=0q−1∫Rh(x)φ(r,s)h(x+mα)φ(r+mp1,s)¯e(−xnα)e(−rnp1q)dx=1q231/4∫Rh(x)h(x+mα)¯e(−xnα)dx⋅∑r,s=0q−1φ(r,s)φ(r+mp1,s)¯e(−rnp1q)=1q231/4H(mα,nα)¯⋅Ω where(3.6)H(a,b)=∫Rh(x)¯h(x+a)e(bx)dx=3−1/4e−π(13+i)abe−π3(a2+b2) (note: 13+i=2i3z1) andΩ=∑r,s=0q−1φ(r,s)φ(r+mp1,s)¯e(−rnp1q)=∑r,s=0q−1e(−smp1q)e(−rnp1q)=q2δqmδqn. Thus,〈f,f〉D(mε1+nε2)=31/4H(mα,nα)¯δqmδqn and (since |G/D|=q2α2),〈f,f〉D=|G/D|∑m,n〈f,f〉D(mε1+nε2)πmε1+nε2=31/4q2α2∑m,nH(mα,nα)¯δqmδqnπmε1+nε2=31/4q2α2∑m,nH(mqα,nqα)¯πmqε1+nqε2. From πmqε1+nqε2=UqnVqm and inserting the expression for H from (3.6), we obtain〈f,f〉D=q2α2∑m,ne−π(13−i)q2α2mne−π3q2α2(m2+n2)UqnVqm. We now explain that the positive element 〈f,f〉D is both invariant and covariant.To show that the element 〈f,f〉D has covariant form, consider the continuous cross section X of the continuous field of rotation C⁎-algebras {At}0<t<1 (obtained by replacing q2α2 by a free parameter t) defined byX(t)=t∑m,ne−π(13−i)tmne−π3t(m2+n2)UtnVtm (for 0<t<1) which in fact consists of smooth elements in At∞. (Basically, X(t) is the above expression for 〈f,f〉D after setting p=0, q=1, and q2α2=t.) Letting θq:=q2θ−pq=q2α2, and defining the homomorphism ζ=ζq,θ:Aθq→Aθ by(3.7)ζ(Uθq)=Uθq,ζ(Vθq)=Vθq, one immediately gets the covariance relation(3.8)〈f,f〉D=ζ(X(θq)). (For our purposes, the covariance of an element simply means that it arises from a continuous section of the field of C⁎-algebras {At} in this manner.)Further, with κ0 denoting the Cubic transform of Aθq, one has the commutative diagram so that(3.9)κζ=ζκ0. It is straightforward to check that X(t) is Cubic invariant (for each t), therefore it immediately follows that 〈f,f〉D is invariant under the Cubic transform κ.Another way to see the invariance is using Proposition 2.4 and Lemma 4.2 below (which gives fc=fW0):κ〈f,f〉D=〈fc,fc〉D=〈fW0,fW0〉D=〈f,f〉D. Further, by inspection X(t) and 〈f,f〉D are invariant under the Flip ϕ (since f is an even function), therefore they are invariant under the Hexic transform ρ=ϕκ2. It now follows that the support projections of X(t) and 〈f,f〉D (after they are shown to exist below) are ρ invariant as well.4

The matrix projection

In this section we carry out C⁎-inner product computations that show:

(i)〈f,f〉D⊥ is invertible, so that the projection e of Theorem 1.2 exists,

(ii)that e is approximately central,

(iii)that the cut downs eUe, eVe are approximated by unitary matrices in a matrix algebra that is ρ invariant and whose identity is e.

(The approximations hold for large q.)To this end, we start by computing the inner product 〈f,Urf〉D⊥=〈f,πε2rf〉D⊥ where r=0,1 is fixed. With r=0 this will give existence and approximate centrality of the projection, and with r=1 we get the matrix approximations for eUe, eVe.Let g=πε2rf=πrε2f so thatg(y,j,k)=(πrε2f)(y,j,k)=(π(0,0,0;rα,rp1,0)f)(y,j,k)=e(rαy)e(rp1jq)f(y,j,k)=31/8qe(rαy)e(rp1jq)h(y)φ(j,k) and〈f,πε2rf〉D⊥(Σjnjδj)=〈f,g〉D⊥(n1β,−n1cp1,n3;n2β,−n2cp1,n4)=∫R×Zq×Zqf(x,n,m)¯g(x+n1β,n−n1cp1,m+n3)e(xn2β+−n2cp1n+mn4q)dxdndm=1q∑m,n=0q−1e(−n2cp1n+mn4q)∫Rf(x,n,m)¯g(x+n1β,n−n1cp1,m+n3)e(xn2β)dx=31/8qq∑m,n=0q−1e(−n2cp1nq)e(mn4q)∫Rh(x)φ(n,m)¯g(x+n1β,n−n1cp1,m+n3)e(xn2β)dx The integral here is31/8q∫Rh(x)φ(n,m)¯e(rα(x+n1β))e(rp1(n−n1cp1)q)h(x+n1β)φ(n−n1cp1,m+n3)e(xn2β)dx=31/8q∫Rh(x)¯e(−nmq)e(rα(x+n1β))e(rp1(n−n1cp1)q)h(x+n1β)e((n−n1cp1)(m+n3)q)e(xn2β)dx=31/8qe((nn3−n1cp1m−n1cp1n3)q)e(rp1(n−n1cp1)q)e(rαn1β)∫Rh(x)¯h(x+n1β)e(x(n2β+rα))dx=31/8qe((nn3−n1cp1m−n1cp1n3)q)e(rp1n−rn1cpq)e(rn1q)H(n1β,n2β+rα) and since cp≡1 mod q we get=31/8qe((nn3−n1cp1m−n1cp1n3)q)e(rp1nq)H(n1β,n2β+rα)=31/8qe(−n1cp1n3q)e(−n1cp1mq)e((n3+rp1)nq)H(n1β,n2β+rα) Hence〈f,πε2rf〉D⊥(Σjnjδj)=31/4q2H(n1β,n2β+rα)e(−cp1n3n1q)∑m,n=0q−1e(−n2cp1nq)e(mn4q)e(−n1cp1mq)e((n3+rp1)nq)=31/4q2H(n1β,n2β+rα)e(−cp1n3n1q)∑m,n=0q−1e((n4−n1cp1)qm)e((n3+rp1−n2cp1)qn)=31/4H(n1β,n2β+rα)e(−cp1n3n1q)δqn4−n1cp1δqn3+rp1−n2cp1.This gives the C⁎-inner product〈f,πε2rf〉D⊥=∑n1n2n3n4〈f,πε2rf〉D⊥(Σnjδj)⋅πΣnjδj⁎=31/4∑n1,n2H(n1β,n2β+rα)∑n3,n4=0q−1e(−cp1n3n1q)δqn4−n1cp1δqn3+rp1−n2cp1⋅πΣnjδj⁎ and here we can just substitute n3=n2cp1−rp1 in the complex exponential and pull it out of the summation because of the delta functions:=31/4∑n1,n2H(n1β,n2β+rα)e(−cp1n1(n2cp1−rp1)q)∑n3,n4=0q−1δqn4−n1cp1δqn3+rp1−n2cp1⋅πΣnjδj⁎. Working out πΣnjδj⁎, we getπΣnjδj=π(n2δ2+n4δ4)+(n1δ1+n3δ3)=πn2δ2+n4δ4πn1δ1+n3δ3=πn2δ2πn4δ4πn1δ1πn3δ3 andπΣnjδj⁎=V3n3V1n1V4n4V2n2 so the inner product becomes〈f,πε2rf〉D⊥=31/4∑n1,n2H(n1β,n2β+rα)e(−cp12n1(n2c−r)q)∑n3,n4=0q−1δqn4−n1cp1δqn3+rp1−n2cp1⋅V3n3V4n4V1n1V2n2=31/4∑n1,n2H(n1β,n2β+rα)e(−c2pn1n2q)e(cpn1rq)e(θ′n1n2)∑n3,n4=0q−1δqn4−n1cp1δqn3+rp1−n2cp1⋅V3n3V2n2V4n4V1n1 inserting θ′=β2+c2pq and using cp≡1 mod q gives=31/4∑n1,n2H(n1β,n2β+rα)e(rn1q)e(β2n1n2)∑n3,n4=0q−1δqn4−n1cp1δqn3+rp1−n2cp1⋅V3n3V2n2V4n4V1n1=31/4∑n1,n2H(n1β,n2β+rα)e(rn1q)e(β2n1n2)⋅V3n2cp1−rp1V2n2V4n1cp1V1n1=31/4V3−rp1∑n1,n2H(n1β,n2β+rα)e(rn1q)e(β2n1n2)⋅(V3cp1V2)n2(V4cp1V1)n1=31/4V3−rp1∑n1,n2H(n1β,n2β+rα)e(rn1q)e(β2n1n2)⋅W2n2W1n1 where the unitariesW1=V4cp1V1,W2=V3cp1V2 satisfy the unitary Heisenberg commutation relationW1W2=e(β2)W2W1. We have obtained〈f,πε2rf〉D⊥=V3−rp1∑m,n31/4H(mβ,nβ+rα)e(rmq)W1mW2n where31/4H(mβ,nβ+rα)=e−π(13+i)β2m(n+rαβ−1)e−π3β2[m2+(n+rαβ−1)2]. Taking r=0,1 one gets(4.1)〈f,f〉D⊥=∑m,ne−π(13+i)β2mne−π3β2(m2+n2)W1mW2n and(4.2)〈f,πε2f〉D⊥=〈f,Uf〉D⊥=V3−p1Y˜ where(4.3)Y˜:=∑m,ne−π(13+i)β2m(n+αβ−1)e−π3β2[m2+(n+αβ−1)2]e(mq)W1mW2n.Using techniques similar to those used in Section 6 of [24] one checks thatY:=〈f,f〉D⊥=∑m,ne−π(13+i)β2mne−π3β2(m2+n2)W1mW2n is invertible in C⁎(D⊥,h¯) for any β2>1. We do however wish to present a much shorter and quicker proof that 〈f,f〉D⊥ is invertible for β2>1.005 (which suffices for the purposes of Theorem 1.2) as shown by the following lemma.Lemma 4.1For any pair of unitaries W1, W2 in a unital C⁎-algebra, the elementY(x)=∑m,ne−π(13+i)xmne−π3x(m2+n2)W1mW2n is invertible for x>1.005.ProofIt suffices to check that ‖Y(x)−I‖<1 for x>1.005. We haveY(x)−I=∑(m,n)≠(0,0)e(−x2mn)e−π3x(m2+mn+n2)W1mW2n and‖Y(x)−I‖≤∑(m,n)≠(0,0)e−π3x(m2+mn+n2)=∑m,ne−π3x(m2+mn+n2)−1=Θ(x)−1 whereΘ(x)=∑me−π3xm2∑ne−π3x(n2+mn)=∑me−π3xm2eπ43xm2∑ne−π3x(n+m2)2=∑me−π34xm2∑ne−π3x(n+m2)2 (here break the sum over m according to parity, m→2m and m→2m+1)=∑me−π3xm2∑ne−π3xn2+∑me−π3x(m+12)2∑ne−π3x(n+12)2=ϑ3(0,i3x)ϑ3(0,i3x)+ϑ2(0,i3x)ϑ2(0,i3x). (Here we used the classical Theta functions which we recall at the beginning of Section 5.) Note that Θ(x) is a decreasing function as each series is clearly a decreasing function. One checks by direct computation that Θ(1.005)=1.9987… so that for x>1.005 one has ‖Y(x)−I‖≤Θ(x)−1<C0<1 where C0:=Θ(1.005)−1=0.9987… , hence the result. □We therefore conclude that the inner product Y=〈f,f〉D⊥ is positive, invertible, and(4.4)‖Y−I‖<C0<1 whenever β2>1.005, where C0=Θ(1.005)−1=0.9987… is the constant stated in the preceding proof. As θ is in the Gδ class G, so that 0<q(qθ−p)=1β2<11.005=0.99502… , Eq. (4.4) is satisfied and therefore the above inner product 〈f,f〉D⊥ is invertible.The projection. Normalizing f by the positive invertible elementb:=〈f,f〉D⊥−1/2 by forming the Schwartz function ξ:=fb, so that 〈ξ,ξ〉D⊥=1, we obtain the desired projection(4.5)e=〈ξ,ξ〉D of Theorem 1.2. Further, this projection is the support projection of 〈f,f〉D as can be seen quickly by checking that 〈f,f〉D〈fb2,fb2〉D=e.As a result, we have the isomorphism(4.6)η:eAθe→C⁎(D⊥,h¯) of unital C⁎-algebras defined byη(x)=〈ξ,xξ〉D⊥,η−1(y)=〈ξy,ξ〉D for x∈eAθe and y∈C⁎(D⊥,h¯). (It is worthwhile remembering here that when operators acting on the right come out of the inner product 〈,〉D⊥, they do so with the opposite multiplication (opposite of the usual multiplication of operators on the Hilbert space L2(M)). Thus, 〈g1,g2a〉D⊥=〈g1,g2〉D⊥•a=a〈g1,g2〉D⊥.)In view of the fact that 〈f,f〉D is ρ-invariant (discussed near the end of the previous section), it follows that e is ρ-invariant. In view of the trace equation (2.10), the projection e has trace

(e)=

〈ξ,ξ〉D=|G/D|

′〈ξ,ξ〉D⊥=q2θ−pq. In addition, the covariance relation (3.8) immediately gives the covariance equation (1.6) for e.Approximate centrality. We now prove that the projection e is approximately central in Aθ – and for this it suffices to check that e approximately commutes with U since upon applying κ (or ρ) to the approximation Ue≈eU (noting that e is ρ and κ invariant) one sees that e also approximately commutes with V.As −‖H‖I≤H≤‖H‖I for any Hermitian operator H, taking H=Y−I and using ‖Y−I‖<C0 as we just obtained in (4.4), we get1111

Note that ‖Y‖>1 and ‖Y−1‖>1. For if ‖Y‖≤1, then Y≤I and I−Y would be a positive element in an irrational rotation C⁎-algebra of zero trace, so Y=I, which is not the case. The same contradiction arises if ‖Y−1‖≤1.

(4.7)−C0I≤Y−I≤C0Ior(1−C0)I≤Y≤(1+C0)I.Taking the “η−1” of this and noting that η(e)=I and η(〈f,f〉D)=〈f,f〉D⊥ we get(1−C0)e≤〈f,f〉D≤(1+C0)e.Writing x:=〈f,f〉D, so that x−1 denotes its inverse in eAθe, one has11+C0e≤x−1≤11−C0e,‖x−1‖≤11−C0<834 whenever β2>1.005 (or, equivalently, whenever 0<q(qθ−p)<0.99502… as noted above).In virtually the same way we have done in [24] (see Section 7) one checks ‖xU−Ux‖→0 as q→∞, hence writingeU−Ue=x−1(xU−Ux)+(xU−Ux)x−1+x−1(Ux2−x2U)x−1 we get‖eU−Ue‖≤2‖x−1‖‖xU−Ux‖+‖x−1‖2‖Ux2−x2U‖≤1668‖xU−Ux‖+8342‖Ux2−x2U‖ which goes to 0 for large q. One concludes that the projection e is therefore approximately central: the commutator norms ‖eU−Ue‖ and ‖eV−Ve‖ are arbitrarily small for large q. This proves that e is approximately central as per Theorem 1.2.We now work out the cut down approximation in Theorem 1.2. For this, we need a lemma.Lemma 4.2There exists a unitary matrix W0 in C⁎(V3,V4)≅Mq such that fc=fW0 and ξc=ξW0. Further, W0 satisfies κ′2(W0)κ′(W0)W0=I.ProofMuch as we have done in Section 4 of [24], one removes the first and fourth columns of the group G and from the lattice subgroup D in (3.1), and correspondingly removes the first and fourth columns as well as the δ1,δ2 row vectors in D⊥. These result in lattice subgroups D0,D0⊥ of the finite group G0=M0×M0 where M0=Zq×Zq. With this in mind, for the discrete factor φ of f, one computes that 〈φ,φ〉D0=qI and 〈φ,φ〉D0⊥=qI are scalars. Therefore, by normalizing φ0:=1qφ and taking(4.8)W0=〈φ0,φ0c〉D0⊥ which is a unitary matrix in C⁎(V3,V4)≅Mq, one hasφ0c=〈φ0,φ0〉D0φ0c=φ0〈φ0,φ0c〉D0⊥=φ0W0 whence fc=fW0 since f=h⊗φ and hc=h. Using Proposition 3.9 we getκ′(b−2)=κ′〈f,f〉D⊥=〈fc,fc〉D⊥=〈fW0,fW0〉D⊥=W0⁎•〈f,f〉D⊥•W0=W0b−2W0⁎ whence κ′(b)=W0bW0⁎. Therefore, by Proposition 2.3 this givesξc=(fb)c=fcκ′(b)=(fW0)κ′(b)=f(κ′(b)W0)=f(W0b)=(fb)W0=ξW0 as claimed. The equation κ′2(W0)κ′(W0)W0=I is easy to verify by taking the Cubic transform on the equation φ0c=φ0W0 twice, thereby obtainingφ0=(((φ0c)c)c=φ0[W0•κ′(W0)•κ′2(W0)], which, upon taking the inner product via 〈φ0,⋅〉D0⊥, gives the result. □Next we check that(4.9)ηκ=κ″η where κ″=W0⁎κ′()W0. Note that the matrix algebra Mq=C⁎(V3,V4) is κ″-invariant since it is κ′-invariant (in view of (3.5)) and W0 is a unitary in Mq.For each x∈eAθe we have, using ξ=ξcW0⁎ by Lemma 4.2,ηκ(x)=〈ξ,κ(x)ξ〉D⊥=〈ξcW0⁎,κ(x)ξcW0⁎〉D⊥=W0⁎〈ξc,κ(x)ξc〉D⊥W0=W0⁎〈ξc,(xξ)c〉D⊥W0=W0⁎κ′(〈ξ,xξ〉D⊥)W0=κ″η(x).Cut down approximation. From Eq. (4.2) (and noting that eξ=ξ=fb), the first cut down approximation isη(eUe)=〈ξ,eUeξ〉D⊥=〈ξ,Uξ〉D⊥=b〈f,Uf〉D⊥b=bV3−p1Y˜b=Y−1/2V3−p1Y˜Y−1/2. Here we use the limit approximation‖V3p1YV3−p1−Y‖→0 as q→∞, which can be checked exactly as in [24, (Section 8)]. As the norm of Y is bounded above and below (as we saw above in (4.7)) we get‖V3p1Y−1/2−Y−1/2V3p1‖→0. Further, since ‖Y˜−Y‖→0 for large q, so that ‖Y−1/2Y˜Y−1/2−I‖→0 (since the norms of Y,Y−1/2,Y−1 are all bounded), we obtain the desired cut down approximationη(eUe)≈V3−p1Y−1/2Y˜Y−1/2≈V3−p1 by a unitary in the matrix algebra (in fact, one of its unitary generators).The cut down approximation for η(eVe) by a unitary matrix in Mq now follows from Eq. (4.9) sinceκ″η(eVe)=ηκ(eVe)=η(eU⁎e)≈V3p1 henceη(eVe)≈κ″−1(V3p1)∈Mq. Hence, η(eVe) is approximately in κ″−1(Mq)=Mq.It's not hard to check that the isomorphism η commutes with the Flip:ηϕ=ϕ′η since f and ξ are even functions. As the Hexic transform is ρ=ϕκ2, it follows that the approximating matrix C⁎-algebra Mq is invariant under the Hexic transform also – and, as we noted earlier, that the point projection e=〈ξ,ξ〉D is invariant under ρ as well.This completes the proof of Theorem 1.2.5

Computation of topological invariants

To facilitate the computation of the topological invariants mentioned in Theorem 1.1 based on the continuous field method that we outlined in the Introduction, we recall the classical Theta functions(5.1)ϑ2(z,s)=∑neπis(n+12)2ei2z(n+12),(5.2)ϑ3(z,s)=∑neπisn2ei2zn,(5.3)ϑ4(z,s)=∑n(−1)neπisn2ei2zn, for z,s∈C and Im(s)>0, where all summations range over the integers. (For a wonderful treatment see [29], whose definitions we adopt.) We further recall the following inversion formulas for Theta functions which we will use:(5.4)ϑ3(z,s)=(−is)−1/2ez2/(πis)ϑ3(zs,−1s),ϑ2(z,s)=(−is)−1/2ez2/(πis)ϑ4(zs,−1s).We begin with the computation of φ1 on the projection E(t) by working out φ1t(X(t)) and then taking its limit as t→0+. For this we first need a lemma.Lemma 5.1For any complex numbers A,B with Re(B)>0, one haslimt→0+⁡tϑk(tA,itB)=1B for k=2,3.ProofThe proof is a simple consequence of the inversion formula for Theta functions. Applying (5.4) one getstϑ3(tA,itB)=t1tBe−tA2/(πB)ϑ3(AiB,itB)=1Be−tA2/(πB)ϑ3(AiB,itB). As t→0+, the ϑ3 factor here goes to 1, hence the result. The same proof holds for the limit involving ϑ2, the only difference is that the inversion formula converts ϑ2 into ϑ4, and the limit of ϑ4(−iAB,itB) is still 1. □Let us now proceed to compute φ1t(X(t)). We haveφ1t(X(t))=t∑m,ne−π3t(m2+n2)e−π(13−i)tmnφ1t(UtnVtm)=t∑m,ne−π3t(m2+n2)e−π(13−i)tmneiπt(m2+n2)=t∑m,neiπ(1+i3)t(m2+n2)e−π(13−i)tmn=t∑neiπ(1+i3)tn2Sn where the sum over m here isSn=∑meiπ(1+i3)tm2e−π(13−i)tmn=ϑ3(iπ2(13−i)tn,(1+i3)t) using the inversion formula here gives=1t113−ieπ4(13−i)tn2ϑ3(π2n,−1t(1+i3)). This givesφ1t(X(t))=t13−i∑neiπ(1+i3)tn2eπ4(13−i)tn2ϑ3(π2n,−1t(1+i3))=t13−i∑neiπ4(3+i3)tn2ϑ3(π2n,−1t(1+i3)) here, the ϑ3 factor depends only on the parity of n (as ϑ3 has period π in the first variable), so the sum breaks down according to parity as=t13−i{∑neiπ(3+i3)tn2ϑ3(0,−1t(1+i3))+∑neiπ(3+i3)t(n+12)2ϑ3(π2,−1t(1+i3))}=t13−i{ϑ3(0,(3+i3)t)ϑ3(0,−1t(1+i3))+ϑ2(0,(3+i3)t)ϑ3(π2,−1t(1+i3))}. Using Lemma 5.1 we obtainφ1(E)=limt→0+⁡φ1t(X(t))=213−i13−3i=1+i32=3ω−1 which gives the value stated in Theorem 1.1.Remark 5.2We carried out computations with the computer algebra software Maple to verify this and later results for these limits.We next compute and show that ψk(E)=ω for k=0,1,2. We haveψkt(X(t))=t∑m,ne−πt(13−i)mne−πt3(m2+n2)ψkt(UtnVtm)=t∑m,ne−πt(13−i)mne−πt3(m2+n2)e(t6(m−n)2)δ3m−n−k the replacement m→n+k+3m gives=t∑m,ne−πt(13−i)(n+k+3m)ne−πt3((n+k+3m)2+n2)e(t6(k+3m)2)=t∑me−πt3(k+3m)2e(t6(k+3m)2)N(m) where (after simplifying exponents)N(m)=∑ne−πt(13−i)[n2+(k+3m)n]e−πt3[2n2+2n(k+3m)]=∑ne−πtcn2e−πtc(k+3m)nwhere c=3−i (has positive real part)=ϑ3(iπ2tc(k+3m),itc) which, under inversion (5.4), becomesN(m)=1tceπ4tc(k+3m)2ϑ3(π2k+3π2m,itc)=1tceπ4tc(k+3m)2ϑ3(π2k+π2m,itc) since ϑ3 has period π in the first variable and, as we noted earlier, that the ϑ3 factor here depends only on the parity of m. Therefore, we obtainψkt(X(t))=ttc∑me−πt3(k+3m)2e(t6(k+3m)2)eπ4tc(k+3m)2ϑ3(π2k+π2m,itc)=tc∑me−πt12(3+3i)(k+3m)2e(t6(k+3m)2)ϑ3(π2k+π2m,itc)=tce−πt12(3+3i)k2e(t6k2)∑me−πt12(3+3i)(9m2+6km)e(t6(9m2+6km))ϑ3(π2k+π2m,itc)=tce−πt12(3+3i)k2e(t6k2)∑me−πt4(3+3i)(3m2+2km)e(t2(3m2+2km))ϑ3(π2k+π2m,itc)=tce−πt12(3+3i)k2e(t6k2)∑meiπ3btm2e2i(πbkt)mϑ3(π2k+π2m,itc) where we wrote b:=14(1+i3)=i4c; summing according to parity of m gives=tce−πt12(3+3i)k2e(t6k2){∑meiπ12btm2e2i(2πbkt)mϑ3(π2k,itc)+∑meiπ12bt(m+12)2e2i(2πbkt)(m+12)ϑ4(π2k,itc)}=tce−πt12(3+3i)k2e(t6k2){ϑ3(2πbkt,12bt)ϑ3(π2k,itc)+ϑ2(2πbkt,12bt)ϑ4(π2k,itc)}. Taking the limit of this in view of Lemma 5.1 we getψk(E)=limt→0+⁡ψkt(X(t))=21c13c=2c3=ω. (Here, both ϑ3(π2k,itc),ϑ4(π2k,itc) have limit 1 as t→0+.) Thus, we can write down the Connes–Chern κ-topological invariant for the projection field E(t) asT3(E(t))=(t;ω,ω,ω). Applying the automorphism γ to this projection we get, in view of Eqs. (1.13), its invariants as wellT3(γE(t))=(t;ω,e(13)ω,e(−13)ω)=(t;ω,ω−1,1−2ω)T3(γ2E(t))=(t;ω,e(23)ω,e(−23)ω)=(t;ω,1−2ω,ω−1)From Eq. (1.12) we therefore obtainφ20(E)=ω,φ21(E)=3ω.We now compute φ3k(E):φ3kt(X(t))=t∑m,ne−π(13−i)tmne−π3t(m2+n2)φ3kt(UtnVtm). When k=0 we getφ30t(X(t))=t∑m,ne−π3tmne−π3t(m2+n2)δ2mδ2n=t∑m,ne−4π3tmne−4π3t(m2+n2)=t∑m,ne−4π3t(m2+mn+n2)=t∑m,ne−4π3t[(m+n2)2+34n2]=t∑ne−3πt3n2∑me−4π3t(m+n2)2 which can be written as two sums depending on the parity of n as follows=t{∑ne−12πt3n2∑me−4πt3m2+∑ne−12πt3(n+12)2∑me−4πt3(m+12)2}=t{ϑ3(0,12it3)ϑ3(0,4it3)+ϑ2(0,12it3)ϑ2(0,4it3)}. Now apply Lemma 5.1 to obtain the limit of thisφ30(E)=21123143=12.For k=1 we similarly getφ31t(X(t))=t∑m,ne−πt3(m2+mn+n2)=t∑m,ne−πt3[(m+n2)2+34n2]=t∑ne−3πt43n2∑me−πt3(m+n2)2=t{∑ne−3πt3n2∑me−πt3m2+∑ne−3πt3(n+12)2∑me−πt3(m+12)2}=t{ϑ3(0,it3)ϑ3(0,it3)+ϑ2(0,it3)ϑ2(0,it3)}. Therefore,φ31(E)=21311/3=2. These give all the topological numbers in Theorem 1.1 which we can summarize in terms of the Connes–Chern ρ-topological invariant for the field E(t) asT6(E(t))=(t;3ω−1,ω,3ω,12,2). The proof of Theorem 1.2 is now complete.We end this section with computation of the topological invariants of the matrix projection of Theorem 1.2.Topological invariants of the matrix projection e of

Theorem 1.2

. To compute the topological invariants of the matrix projection e, we use the values in Theorem 1.1, its covariant form (1.6), together with the following relations between the κ-traces ψkθ of Aθ and the κ0-traces ψkθq of Aθq under the condition that p is even (which we assumed in Eq. (3.2))(5.5)ψ0θζ=ψ0θq+δ3qψ1θq+δ3qψ2θq(5.6)ψ1θζ=e(−p3)δ3q−1ψ1θq+e(p3)δ3q−2ψ2θq(5.7)ψ2θζ=e(p3)δ3q−2ψ1θq+e(−p3)δ3q−1ψ2θq where ζ=ζq,θ was defined by (3.7). (These are quickly verified by evaluating both sides on the basis elements UθqmVθqn.) One therefore obtains the κ-topological invariants of e to be(5.8)ψ0θ(e)=(1+2δ3q)ω,ψ1θ(e)=ψ2θ(e)=[e(−p3)δ3q−1+e(p3)δ3q−2]ω.To obtains its ρ-topological invariants, we similarly check (based on the assumption that p is even, as θ was taken in the class G)φ1θζ=φ1θq,φ30θζ=φ30θq,φ31θζ=φ31θqφ20θζ=(1−δ3q)φ20θq+δ3qφ21θq,φ21θζ=(1−e(−pq3))φ20θq+e(−pq3)φ21θq. These give us the valuesφ1θ(e)=3ω−1,φ20θ(e)=(1+2δ3q)ω,φ21θ(e)=(1+2e(−pq3))ω,φ30θ(e)=12,φ31θ(e)=2. (We note that the values for φ20 and φ21 here are consistent with Eq. (1.12).)

Acknowledgements

The author thanks the referee for helpful comments and questions. This research work has been partially supported by a grant from the Natural Sciences and Engineering Research Council of Canada (NSERC).

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