K-Theory of Approximately Central Projections in the Flip Orbifold
KK-Theory of ApproximatelyCentral Projections in the Flip Orbifold
Samuel G. Walters
University of Northern British Columbia A BSTRACT . For an approximately central (AC) Powers-Rieffel projection e in theirrational Flip orbifold C*-algebra A Φ θ , where Φ is the Flip automorphism of therotation C*-algebra A θ , we compute the Connes-Chern character of the cut-down of any projection by e in terms of K-theoretic invariants of these projec-tions. This result is then applied to computing a complete K-theoretic invari-ant for the projection e with respect to central equivalence (within the orbifold).Thus, in addition to the canonical trace, there is a × K-matrix invariant K ( e ) arising from unbounded traces of the cutdowns of a canonically constructedbasis for K ( A Φ θ ) = Z . Thanks to a theorem of Kishimoto, this enables us to tellwhen AC projections in A Φ θ are Murray-von Neumann equivalent via an approxi-mately central partial isometry (or unitary) in A Φ θ . As additional application, weobtain the K-matrix of canonical SL ( , Z ) -automorphisms of e and show thatthere is a subsequence of e such that e , σ ( e ) , κ ( e ) , κ ( e ) , σ κ ( e ) , σ κ ( e ) – which arethe orbit elements of e under the symmetric group S ⊂ SL ( , Z ) – are pairwisecentrally not equivalent, and that each SL ( , Z ) image of e is centrally equivalentto one of these, where σ , κ are the Fourier and Cubic transform automorphismsof the rotation algebra. C ONTENTS
1. Introduction 22. Background Material 82.1. The Connes-Chern Character 82.2. Continuous field of Rieffel projections 92.3. The K basis 112.4. Kishimoto’s Theorem 132.5. Two Poisson Lemmas 163. The AC Powers-Rieffel Projection 174. The Projection as Rieffel C*-Inner Product 204.1. Computation of (cid:104) f , f (cid:105) D ⊥
23 4.2. Computation of (cid:104) f , f (cid:105) D φ ij (cid:104) fV , f (cid:105) D φ ij (cid:104) fV , f (cid:105) D E ( t ) M q (cid:48) ( C ( T )) Date : Year of the Pandemic, 2020.2000
Mathematics Subject Classification.
Key words and phrases.
C*-algebras, irrational rotation algebra, noncommutative torus,K-theory, Connes Chern character, projections. a r X i v : . [ m a t h . OA ] J a n -Theory of Approximately Central Powers-Rieffel Projections, S. Walters 2
1. I
NTRODUCTION
We study the problem of when a pair of approximately central (AC) projec-tions are Murray-von Neumann equivalent by means of a partial isometry (orunitary) that is approximately central. In doing so, additional K-theoretic in-formation on the AC projections is required. Such information was found byKishimoto [11] (Theorem 2.1) for certain classes of C*-algebras, which includealgebras studied in this paper. The objective of this paper is to formalize thisinformation into a topological K-theory invariant for AC projections, and pro-ceed with computing it explicitly for an AC Flip-invariant Powers-Rieffel pro-jection in the irrational rotation C*-algebra A θ , which will be denoted through-out this paper by e (see equation (1.4)). The projection e is similar to oneconstructed by Elliott and Lin [10], and is essentially the same as the unitprojection in the Elliott-Evans tower construction [9].Our main results are stated in Theorems 1.4, 1.5, and 1.6 of this section.In this paper we will be concerned with the Flip orbifold C*-algebra A Φ θ = { x ∈ A θ : Φ ( x ) = x } the fixed point C*-subalgebra of the irrational rotation algebra A θ under theFlip automorphism Φ defined by Φ ( U ) = U − , Φ ( V ) = V − where U , V are unitaries generating A θ (also called noncommutative torus) sat-isfying the usual Heisenberg commutation relation VU = e π i θ UV . (1.1)Throughout the paper, θ is a fixed irrational number, < θ < . Both A θ andits Flip orbifold have canonical bounded traces which are unique normalizedtraces denote by τ .The approximately central projections studied in this paper depend on in-teger parameters. In our case for example, e = e q (cid:48) , q , p , θ is a Powers-Rieffel pro-jection that depends on a sequence of consecutive convergents p / q , p (cid:48) / q (cid:48) of θ .Since the rotation algebra A θ is generated by the unitaries U , V , a projection e Kishimoto’s main Theorem 2.1 in [11] is stated for separable, nuclear, purely infinite,simple C*-algebras satisfying UCT. In Remark 2.9 of [11], he notes that it also applies to simpleAT C*-algebras of real rank zero, which includes the irrational rotation C*-algebras, knownto be AT from [9] and [10], and also includes their canonical orbifolds under the canonicalautomorphisms of order 2, 3, 4, and 6 as they are known to be AF from [4] [6] [15] [18]. Recallthat AF-algebras are AT-algebras ([13], Corollary 3.2.17). -Theory of Approximately Central Powers-Rieffel Projections, S. Walters 3 is AC in A θ if (cid:107) eU − U e (cid:107) , (cid:107) eV − Ve (cid:107) → as q → ∞ . A projection e is AC in the Fliporbifold A Φ θ if (cid:107) e ( U + U ∗ ) − ( U + U ∗ ) e (cid:107) → , (cid:107) e ( V + V ∗ ) − ( V + V ∗ ) e (cid:107) → as q → ∞ (since it is known from [3] that U + U ∗ , V + V ∗ generate A Φ θ ). We do notknow if AC in A Φ θ implies AC in A θ in general (though for many projections inthe C*-algebra generated by certain powers of U , V this can be checked). Definition 1.1.
Two AC projections are centrally equivalent in an algebra A (or AC-equivalent in A ) if they are Murray-von Neumann equivalent by a partialisometry in A that is approximately central in A (for large enough parameter).Kishimoto’s Theorem 2.1 in [11] (restated in Section 2.4 below), as appliedto the Flip orbifold A Φ θ (which known to be an AF-algebra [4], [15]), implies thattwo AC projections e and f in A Φ θ are centrally equivalent in A Φ θ if and only ifthe cutdown of a given finite generating set of projections [ P j ] for K ( A Φ θ ) by e and f have the same K -class, [ χ ( eP j e )] = [ χ ( f P j f )] ∈ K ( A Φ θ ) (1.2)for each j , where χ is the characteristic function of the interval [ , ∞ ) . Ofcourse, equation (1.2) is understood to hold for large enough integer param-eters which e and f depend on. Since A Φ θ is AF (so its K = ), the K side ofKishimoto’s conditions (see Theorem 2.2 below) are trivially satisfied. In ourparticular case, P j are projections in A Φ θ . In Section 2.3 we construct a specific basis [ P ] , . . . , [ P ] for K ( A Φ θ ) = Z (see(2.9) and (2.10)), with specific projections P s in A Φ θ , with respect to which wecompute the classes [ χ ( eP s e )] – which would therefore determine the centralequivalence class of the projection e in the Flip orbifold. These K -classes willbe identified explicitly by computing their Connes-Chern character T : K ( A Φ θ ) → R , T ( x ) = ( τ ( x ) ; φ ( x ) , φ ( x ) , φ ( x ) , φ ( x )) (1.3)in terms of the canonical trace τ , and four basic unbounded traces φ jk (definedin Section 2.1 below). The map T is known to be a group monomorphismfor irrational θ (see [14], Proposition 3.2). Therefore, in terms of the Connes-Chern character we will calculate the numerical invariants for e τ χ ( eP s e ) , φ jk χ ( eP s e ) ( s = , ..., , jk = , , , ). For ease of notation, let us write χ s : = χ ( eP s ( θ ) e ) It seems reasonable to expect that if [ χ ( ege )] < [ χ ( f g f )] for each g = P j , then there exist ACpartial isometry u such that uu ∗ = e and u ∗ u ≤ f ; but the author has no proof. -Theory of Approximately Central Powers-Rieffel Projections, S. Walters 4 for the cutdown projections by e (for large enough parameter). We also find itconvenient to organize these classes for e into a vector (cid:126) τ ( e ) = (cid:2) τ χ τ χ τ χ τ χ τ χ τ χ (cid:3) consisting of the canonical traces of the cutdowns, together with a topologicalK-matrix involving the unbounded traces which we denote by K ( e ) = φ ( χ ) φ ( χ ) φ ( χ ) φ ( χ ) φ ( χ ) φ ( χ ) φ ( χ ) φ ( χ ) φ ( χ ) φ ( χ ) φ ( χ ) φ ( χ ) φ ( χ ) φ ( χ ) φ ( χ ) φ ( χ ) φ ( χ ) φ ( χ ) φ ( χ ) φ ( χ ) φ ( χ ) φ ( χ ) φ ( χ ) φ ( χ ) wherein the ( jk , s ) -entry consists of the unbounded trace φ jk ( χ s ) . Therefore,in the Flip orbifold, the central equivalence class of e is fully determined bythe pair consisting of the canonical trace vector (cid:126) τ ( e ) , and the × topologicalmatrix K ( e ) . Notation 1.2.
We shall use the divisor delta function δ mn = if n divides m , and δ mn = otherwise. We also use the notation e ( t ) : = e π it . Thus, (cid:80) n − j = e ( m jn ) = n δ mn . Standing Condition 1.3.
Without loss of generality we can assume that thereare infinitely many consecutive convergents pq , p (cid:48) q (cid:48) of θ such that pq < θ < p (cid:48) q (cid:48) , < q (cid:48) ( q θ − p ) < . (See Remark 3.1 for why.) This will be assumed in the hypotheses of Theorems1.4, 1.5, and 1.6.The AC Flip-invariant Powers-Rieffel projection in A Φ θ that will be studiedthroughout this paper is e : = e q (cid:48) , q , p , θ = G τ ( U q (cid:48) ) V − q + F τ ( U q (cid:48) ) + V q G τ ( U q (cid:48) ) (1.4) = ζ q (cid:48) , q , p , θ E ( q (cid:48) ( q θ − p )) (1.5)depending on the convergent parameters p , q , p (cid:48) , q (cid:48) , as stated in the StandingCondition, with trace τ ( e ) = q (cid:48) ( q θ − p ) ∈ ( , ) . The Rieffel functions F τ , G τ in(1.4) and the continuous field E ( t ) in (1.5) are described in Section 2.2 (see(2.3)), and the C*-morphism ζ is defined in (3.1). No confusion should arisewith occasionally denoting the trace of e simply by τ : = q (cid:48) ( q θ − p ) .In light of this background, our results can now be stated in terms of thefollowing three theorems (some of the notation of which is explained later). -Theory of Approximately Central Powers-Rieffel Projections, S. Walters 5 Theorem 1.4.
Let θ be irrational with convergents satisfying the StandingCondition. Let t → P ( t ) be a continuous section, defined in some neighborhoodof θ , of smooth projections of the continuous field { A Φ t } < t < of Flip orbifolds.Then the unbounded trace of the cutdown projection χ ( ePe ) , where P = P ( θ ) , isgiven by φ jk χ ( ePe ) = a − jk C ( P ) + a + jk C ( P ) for jk = , , , , where C ( P ) = φ ( P ) + φ ( P ) , C ( P ) = φ , q (cid:48) ( P ) + ( − ) p (cid:48) φ , q (cid:48) ( P ) (1.6)and a − jk = ( − ) p jk , for even q (cid:48) δ q δ j δ k + ( − ) p jk δ q − for odd q (cid:48) a + jk = ( − ) j + p jk for even q (cid:48) ( − ) p (cid:48) j (cid:104) δ k − + ( − ) p j δ q − k − (cid:105) for odd q (cid:48) . The canonical trace of the cutdown is τ χ ( ePe ) = τ ( e ) τ p (cid:48) / q (cid:48) ( P ( p (cid:48) q (cid:48) )) where τ p (cid:48) / q (cid:48) is the canonical normalized trace of the rational rotation algebra A p (cid:48) / q (cid:48) .With this result established, we can write down the topological K-matrix ofthe projection e according to the following theorem. Theorem 1.5.
Let θ be an irrational number. The Powers-Rieffel projection e has K-matrix given by the × matrices according to parities (as indicated bysubscripts): K ( e q (cid:48) , q , p , θ ) = − −
10 0 ( − ) p ( − ) p + ( − ) p ( − ) p + q (cid:48) even K ( e q (cid:48) , q , p , θ ) = δ p (cid:48) δ p (cid:48) − − δ p (cid:48) − δ p (cid:48) − q even-Theory of Approximately Central Powers-Rieffel Projections, S. Walters 6 K ( e q (cid:48) , q , p , θ ) = + δ p (cid:48) δ p (cid:48) − δ p (cid:48) − − δ p (cid:48) − + δ p (cid:48) δ p (cid:48) − δ p (cid:48) − − δ p (cid:48) − δ p (cid:48) − − δ p (cid:48) − + δ p (cid:48) δ p (cid:48) − ( − ) p δ p (cid:48) − − δ p (cid:48) − ( − ) p ( + δ p (cid:48) ) δ p (cid:48) − q , q (cid:48) both odd . In addition, the canonical trace vector is (cid:126) τ ( e q (cid:48) , q , p , θ ) = ( q θ − p ) (cid:2) q (cid:48) p p (cid:48) p (cid:48) p (cid:48) p (cid:48) (cid:3) where p : = (cid:40) p (cid:48) for < θ < ( q (cid:48) − p (cid:48) ) for < θ < . Application.
One interesting application of the preceding theorem is deter-mination of whether or not α ( e ) and β ( e ) are centrally equivalent for smoothautomorphisms α , β of the Flip orbifold. Or, equivalently, when is α ( e ) cen-trally equivalent to e ? We will answer this for the canonical automorphismsgiven by the Fourier σ and Cubic κ transforms (studied in [2] [5] [16] [17] [18])defined by σ ( U ) = V − , σ ( V ) = U κ ( U ) = e ( − θ ) U − V , κ ( V ) = U − . These have order 2 and 3, respectively, on the Flip orbifold, and they have asimple relations with the unbounded traces: σ swaps φ and φ , fixing φ , φ , while κ induces the cyclic permutation φ → φ → φ , and fixing φ . With thisinformation at hand, one easily calculates the action of these automorphismson the basis [ P ] , . . . , [ P ] of K ( A Φ θ ) mentioned earlier. This would then lead tothe following result. Theorem 1.6.
Considering the Powers-Rieffel AC projection e (given by (1.4))as running over parameters where q is even and p (cid:48) is odd, one has its K-matrixand those of its Fourier and Cubic transforms K ( e ) = −
10 0 0 0 0 00 0 0 0 0 0 , K ( κ ( e )) = − K ( κ ( e )) = − , K ( σ ( e )) = −
10 0 0 0 0 0 -Theory of Approximately Central Powers-Rieffel Projections, S. Walters 7 K ( σ κ ( e )) = − , K ( σ κ ( e )) = − . In particular, as these matrices are pairwise distinct, the AC projections e , σ ( e ) , κ ( e ) , κ ( e ) , σ κ ( e ) , σ κ ( e ) (1.7)are pairwise not centrally equivalent.Note that the six projections in (1.7) all have the same orbifold K class andconstitute the S orbit of e (where the symmetric group S is generated by σ and κ ). Indeed, in the case considered in the preceding theorem where q iseven ( q (cid:48) , p odd), the Connes-Chern character of e can be obtained from (3.5) as T ( e ) = ( τ ( e ) ; 1 , , , ) where the trace of e is also the trace of any automorphism of e (by uniquenessof the normalised canonical trace). In view of what was just noted regardinghow σ , κ act on the φ jk , we see that all the projections in (1.7) have exactly thesame Connes-Chern character as e , so that they are all Murray-von Neumannequivalent (in the Flip orbifold). Theorem 1.6, however, says that they arepairwise centrally inequivalent.In [21] we showed that in fact for any AC projection e in the Flip orbifold,its S orbit (1.7) gives the only possible central classes, in the sense that forany canonical automorphism α arising from SL ( , Z ) , the projection α ( e ) iscentrally equivalent to one of the projections in (1.7). The point of Theorem1.6 is then that all six projections can be pairwise centrally distinct. It ispossible also to give examples where some or all of the projections in (1.7) arecentrally equivalent - see Theorem 1.4 of [21].We have included many details in our calculations in this paper so as tosave the reader from numerous and onerous checking (and hopefully to alsobe clear about our reasoning). We hope this may be of help. Structure Of Paper.
Let’s summarize what we do in this paper.In Section 2 we gather the necessary background material, notation, resultsand introduce our notation for the K-matrix relative to a constructed basis forgroup K ( A Φ θ ) required by our proofs.In Section 3 the approximately central Powers-Rieffel projection e is con-structed from the continuous field constructed in Section 2 and its unboundedtraces are calculated.In Section 4, the projection is realized as a C*-inner product e = (cid:104) f , f (cid:105) D usingRiefflel’s equivalence bimodule paradigm [12]. This realization will facilitate -Theory of Approximately Central Powers-Rieffel Projections, S. Walters 8 computation of the topological invariants of the cutdown χ ( ePe ) of any projec-tion P by e , which is carried out in Section 5. This will prove Theorem 1.4.In Section 6, Theorem 1.4 is applied to the K -basis constructed in Section2.3 in order to obtain the full K-matrix of e (for various parity situations ofinteger parameters that e depends on). This will then prove Theorem 1.5.In Section 7, the unbounded traces of two technical C*-inner products re-quired in Section 6 are computed (see Lemma 7.1).Sections 8 and 9 are appendices involving some basic unbounded trace com-putations used in the paper. (The Acknowledgement paragraph is right beforethe References.)In a forthcoming paper [22] we calculate the K-matrix of Fourier-invariantprojections in the Fourier orbifold A σθ for the Fourier transform σ . The greatercomplication in the Fourier case arises from the fact that Fourier-invariantprojections do not have a Powers-Rieffel form, from additional unboundedtraces on K ( A σθ ) = Z , as well as additional K -basis elements, giving rise to a × K-matrix.
2. B
ACKGROUND M ATERIAL
In this section we write down the relevant Connes-Chern character for theFlip orbifold; define a continuous field of Powers-Rieffel projections, which isa continuous section of the continuous field of Flip orbifolds { A Φ t : 0 < t < } ; construct a canonical basis consisting of continuous fields of projections for K ( A Φ t ) ; state Kishomoto’s Theorem ([11], Theorem 2.1) in the form we require;and state a couple of Poisson Summation formulas used in our calculations.2.1. The Connes-Chern Character.
The Flip automorphism Φ of the rotationC*algebra A θ has four associated unbounded Φ -traces φ θ jk defined on the basicunitaries U m V n , satisfying (1.1), by (see [14] or [15]) φ θ jk ( U m V n ) = e ( − θ mn ) δ m − j δ n − k (2.1)for jk = , , , , m , n ∈ Z (and δ ba was defined in Notation 1.2). Invariably, wemay write φ θ jk simply as φ jk when θ is understood. These are linear functionalsdefined on the canonical smooth dense *-subalgebra A ∞ θ satisfying the Φ -traceproperty φ jk ( xy ) = φ jk ( Φ ( y ) x ) for x , y ∈ A ∞ θ . (Of course, such maps are Φ -invariant). In addition, they areHermitian maps: they are real on Hermitian elements. Clearly, on the smoothorbifold A Φ , ∞ θ – the fixed point *-subalgebra of A ∞ θ under the Flip – they give rise -Theory of Approximately Central Powers-Rieffel Projections, S. Walters 9 to trace functionals that are not continuous in the C*-norm. Together withthe canonical trace τ one has the Connes-Chern character which we write as T : K ( A Φ θ ) → R , T ( x ) = ( τ ( x ) ; φ ( x ) , φ ( x ) , φ ( x ) , φ ( x )) . (2.2)In [14] (Proposition 3.2) this map is known to be injective for irrational θ . Forthe identity element one has T ( ) = (
1; 1 , , , ) . The ranges of the traces φ jk onprojections in A Φ θ are known to be half-integers, while the canonical trace hasrange ( Z + Z θ ) ∩ [ , ] on projections.2.2. Continuous field of Rieffel projections.
There is a natural continuous(section) field E : [ , ) → { A t } of Flip-invariant Powers-Rieffel projections E ( t ) = G t ( U t ) V − t + F t ( U t ) + V t G t ( U t ) (2.3)where F t ( x ) = (cid:88) n ∈ Z f t ( x + n ) , G t ( x ) = (cid:88) n ∈ Z g t ( x + n ) are periodizations of the t -parameterized family ( ≤ t < ) of continuous (orsmooth) functions f t ( x ) , g t ( x ) on R , compactly supported in [ − , ] , as graphedin Figure 1. Figure 1.
Graphs of f t and g t . One requires f t ( x + t ) = − f t ( x ) (2.4)for − ≤ x ≤ − t , sets f t = on − t ≤ x ≤ t − , and f t ( − x ) = f t ( x ) for − ≤ x ≤ . The condition (2.4) also holds for − ≤ x ≤ (trivially for − t < x ≤ ). Since f t is even, we have f t ( x ) + f t ( t − x ) = , for ≤ x ≤ . (2.5) In [14] we worked with the crossed product algebra A θ × Φ Z , but since this algebra isstrongly Morita equivalent to the fixed point algebra, the injectivity follows for the latter andis easy to see. This is a slight modification of the usual constructions; e.g., see proof of Lemma 2.1 [14]. -Theory of Approximately Central Powers-Rieffel Projections, S. Walters 10
Taking x = − t in (2.4) and using the fact that f t is even, gives f t ( t ) = . (2.6)Further, g t is defined by g t ( x ) = (cid:112) f t ( x )( − f t ( x )) for t − ≤ x ≤ , and g t ( x ) = elsewhere. It is easy to check that g t ( t − x ) = g t ( x ) holds for all x (since x → t − x leaves its supporting interval [ t − , ] invariant.Note that the Flip invariance of E ( t ) requires that F t ( U t ) = F t ( U − t ) , whichtranslates into saying f τ ( x ) = f τ ( − x ) , and G t ( U t ) = G t ( e ( t ) U − t ) , i.e. G t ( x ) = G t ( t − x ) for all x . These are easily checked as G t has period 1 and x → t − x leaves theinterval [ t − , ] invariant.To give a specific smooth example of f t (where t ≥ is now fixed), choose any C ∞ function h ( x ) on the closed interval [ − , − t ] such that h ( − ) = , h ( − t ) = , and all derivatives of h vanish at the endpoints − , − t . One can then define f t based on such h by f t ( x ) = h ( x ) − ≤ x ≤ − t − h ( − x − t ) − t ≤ x ≤ − t − t ≤ x ≤ t − − h ( x − t ) t − ≤ x ≤ t h ( − x ) t ≤ x ≤ It is easy now to see that f t is a smooth even function, compactly supported on [ − , ] , and satisfies the condition (2.5). Of course, the corresponding function g t will also be C ∞ and compactly supported in the interval indicated by Figure 1.These ensure that the projection E ( t ) is smooth for each t , which we emphasizeis defined at t = as well. Remark 2.1.
It is worthwhile remembering the conditions which ensure that E ( t ) is a projection. Writing F : = F t , G : = G t , those conditions are: both F , G arenon-negative, of period 1 (as functions of the real variable x ), and satisfy F ( x ) = F ( x ) + G ( x ) + G ( x − t ) G ( x ) (cid:2) − F ( x ) − F ( x + t ) (cid:3) = G ( x ) G ( x + t ) = for all x . These conditions ensure that E ( t ) = E ( t ) = E ( t ) ∗ is a projection.The field E ( t ) has trace τ ( E ( t )) = τ ( F t ( U t )) = (cid:90) − f t ( x ) dx = t -Theory of Approximately Central Powers-Rieffel Projections, S. Walters 11 (in view of Figure 1). In Appendix A (Section 8), the Connes-Chern characterof E ( t ) is computed to be T ( E ( t )) = ( t ; , , , ) for ≤ t < .Now we extend the field E to the interval < s < by setting F ( s ) = − β s E ( − s ) where β s : A − s → A s is the canonical isomorphism β s ( U − s ) = − U s , β s ( V − s ) = − V − s which commutes with the canonical Flip so F ( s ) ∈ A Φ s is also Flip invariant.We have τ ( F ( s )) = s and the unbounded topological invariants of F ( s ) can beobtained from the (easy to check) relation φ − sjk = ( − ) jk + j + k φ sjk β s . (2.7)This gives us the Connes-Chern character of F T ( F ( s )) = ( s ; , , , ) for s ∈ ( , ] - so it conveniently has the same unbounded traces as E .2.3. The K basis. Recall that the torus T acts canonically on the rotationalgebra by mapping a pair ( a , b ) ∈ T to the automorphism U → aU , V → bV . Itis easy to check that the only such toral automorphisms that commute withthe Flip are (aside from the identity) γ ( U ) = − U , γ ( V ) = V γ ( U ) = U , γ ( V ) = − V γ ( U ) = − U , γ ( V ) = − V where γ = γ γ . These give us Flip invariant projection fields γ E ( t ) , γ E ( t ) , γ E ( t ) defined for t ∈ [ , ) as well as fields γ F ( s ) , γ F ( s ) , γ F ( s ) defined for s ∈ ( , ] .It’s easy to check the following relations between the unbounded traces and γ j φ jk γ = ( − ) j φ jk , φ jk γ = ( − ) k φ jk , φ jk γ = ( − ) j + k φ jk . As a result, we obtain the invariants for the E -fields T ( E ( θ )) = ( θ ; , , , ) T ( γ E ( θ )) = ( θ ; , , − , − ) T ( γ E ( θ )) = ( θ ; , − , , − ) T ( γ E ( θ )) = ( θ ; , − , − , ) -Theory of Approximately Central Powers-Rieffel Projections, S. Walters 12 for θ ∈ [ , ) . Similarly, for the F -fields one has T ( F ( θ )) = ( θ ; , , , ) T ( γ F ( θ )) = ( θ ; , , − , − ) T ( γ F ( θ )) = ( θ ; , − , , − ) T ( γ F ( θ )) = ( θ ; , − , − , ) for θ ∈ ( , ] .We shall let P = (the identity projection field), and write P ( θ ) = (cid:40) F ( θ ) for < θ < E ( θ ) for < θ < , P ( θ ) = (cid:40) γ F ( θ ) for < θ < γ E ( θ ) for < θ < P ( θ ) = (cid:40) γ F ( θ ) for < θ < γ E ( θ ) for < θ < P ( θ ) = (cid:40) γ F ( θ ) for < θ < γ E ( θ ) for < θ < . We now construct P ( θ ) as follows. If < θ < , we set P ( θ ) = (cid:40) η F ( θ ) for < θ < η E ( θ ) for < θ < where η : A θ → A θ is the C*-morphism η ( U θ ) = − U θ , η ( V θ ) = V θ . From therelations φ θ k η = φ θ k − φ θ k , φ θ k η = (2.8)one obtains T ( P ( θ )) = ( θ ; 0 , , , ) . Now suppose < θ < . Then < − θ < so that P ( − θ ) ∈ A − θ (as inpreceding case), which we compose with β θ to obtain the projection P ( θ ) = β θ P ( − θ ) ∈ A θ , ( < θ < ) thus giving T ( P ( θ )) = ( − θ ; 0 , , , ) in view of equations (2.7) and the vanishing unbounded traces of P ( − θ ) .From the above it is clear that regardless of whether θ < or < θ , theprojections P j ( θ ) have the same unbounded traces.We now claim that the group K ( A Φ θ ) = Z has the following projections P = , P ( θ ) , P ( θ ) , P ( θ ) , P ( θ ) , P ( θ ) (2.9) -Theory of Approximately Central Powers-Rieffel Projections, S. Walters 13 as basis. As noted above, their Connes-Chern characters are T ( P ) = (
1; 1 , , , ) T ( P ) = (cid:40) ( θ ; 0 , , , ) for < θ < ( − θ ; 0 , , , ) for < θ < T ( P ) = ( θ ; , , , ) (2.10) T ( P ) = ( θ ; , , − , − ) T ( P ) = ( θ ; , − , , − ) T ( P ) = ( θ ; , − , − , ) . It is known from Proposition 3.2 of [14] that a basis for the range of theConnes-Chern character T K ( A Φ θ ) consists of the six vectors (
2; 0 , , , )(
1; 1 , , , ) = T ( )(
1; 0 , , , )(
1; 0 , , , )(
1; 0 , , , ) (cid:40) ( θ ; , − , , − ) = T ( P ( θ )) for < θ < ( θ ; , , − , − ) = T ( P ( θ )) for < θ < . It can be checked that these six vectors have the same integral span as thevectors in (2.10). Therefore, the P j ( θ ) ’s form a basis for K ( A Φ θ ) .We do not actually use the exact form of the basis projections P j ( θ ) in com-puting the traces of their cutdowns by the AC projection e . Their topologicalinvariants, together with those of e , will be sufficient for that purpose. The in-teresting thing we learn here is the manner by which these invariants interactto giving invariants for cutdowns, as in Theorems 1.4, 1.5.2.4. Kishimoto’s Theorem.
We now state Kishimoto’s Theorem 2.1 [11] in aform appropriate for our purposes and specifically for simple AT -algebras ofreal rank zero (satisfying UCT).
Theorem 2.2. (Kishimoto [11], Theorem 2.1.) Let A be a simple AT -C*-algebra A of real rank zero or a separable simple nuclear purely infinite C*-algebrasatisfying the Universal Coefficient Theorem. Assume that K ( A ) is finitelygenerated by classes of projections g , . . . , g k in A , and that K ( A ) is finitelygenerated by classes of unitaries u , . . . , u (cid:96) in A . Then for each ε > and each -Theory of Approximately Central Powers-Rieffel Projections, S. Walters 14 finite subset F ⊂ A , there exists δ > and a finite subset G ⊂ A such that forany pair of projections e , e in A satisfying (cid:107) e x − xe (cid:107) < δ , (cid:107) e x − xe (cid:107) < δ for x ∈ { g , . . . , g k } ∪ { u , . . . , u (cid:96) } ∪ G , and also satisfying [ χ ( e g i e )] = [ χ ( e g i e )] ∈ K ( A ) (2.11)for i = , . . . , k , and [ u j e + ( − e )] = [ u j e + ( − e )] ∈ K ( A ) for j = , . . . , (cid:96) , there exists a partial isometry v in A such that e = v ∗ v , e = vv ∗ , (cid:107) vx − xv (cid:107) < ε for each x ∈ F .The conditions (2.11) on the generators g j ’s imply that e and e have thesame class in K . In our case, the algebras ( A θ and A Φ θ ) have the cancellationproperty so that the projections are Murray von-Neumann equivalent and infact are unitarily equivalent via a unitary in the algebra.In our case, we apply this theorem to the orbifold A Φ θ which has vanishing K (since it is AF) so we are only concerned with the K conditions (2.11) in clas-sifying AC projections with respect to AC Murray von-Neumann equivalence.To this end, we shall use the basis projections (2.9) for K ( A Φ θ ) , and the K classes of their cutdown projections which will be convenient to write as χ i : = χ ( eP i ( θ ) e ) (for large enough integer parameters in e , of course). Since the Connes-Cherncharacter T mentioned in Section 2.1 is injective on K , it follows that the cen-tral equivalence class of e is fully determined by the canonical traces τ ( χ i ) andthe unbounded traces φ jk ( χ i ) for i = , . . . , . We find it convenient to organizethese numerical invariants for e into a 6-dimensional trace vector (cid:126) τ ( e ) = (cid:2) τ χ τ χ τ χ τ χ τ χ τ χ (cid:3) consisting of the canonical traces of the cutdowns, and a topological K-matrixinvolving the unbounded traces which we lay out as a × matrix K ( e ) = φ ( χ ) φ ( χ ) φ ( χ ) φ ( χ ) φ ( χ ) φ ( χ ) φ ( χ ) φ ( χ ) φ ( χ ) φ ( χ ) φ ( χ ) φ ( χ ) φ ( χ ) φ ( χ ) φ ( χ ) φ ( χ ) φ ( χ ) φ ( χ ) φ ( χ ) φ ( χ ) φ ( χ ) φ ( χ ) φ ( χ ) φ ( χ ) and we call it the K-matrix of e . Its entries all lie in Z , where the i -th columnconsists of the unbounded traces of χ i (arranged from top to bottom in the -Theory of Approximately Central Powers-Rieffel Projections, S. Walters 15 same order they appear in the character map T ). Therefore, in the Flip orbifold,the central equivalence class of e is fully determined by the pair (cid:126) τ ( e ) and K ( e ) .For instance, the K-matrix of the identity is K ( ) =
12 12 12 12
12 12 − − −
12 12 − − −
12 12 (2.12)(since χ i = P i ( θ ) ) where the columns are just the (unbounded) topological in-variants of the basis projections P , . . . , P . Its canonical trace vector is (cid:126) τ ( ) = (cid:104) θ θ θ θ θ (cid:105) for < θ < , (cid:104) − θ θ θ θ θ (cid:105) for < θ < . If α is a smooth automorphism, the K-invariant of the AC projection α ( e ) isdetermined by the K classes [ χ ( α ( e ) P α ( e ))] = α ∗ [ χ ( e α − ( P ) e )] where P = P i as in (2.9). These are determined by their canonical traces τ [ χ ( α ( e ) P α ( e ))] = τα ∗ [ χ ( e α − ( P ) e )] = τ [ χ ( e α − ( P ) e )] (since the canonical trace is unique, τα = τ ), and the unbounded traces φ jk [ χ ( α ( e ) P α ( e ))] = φ jk α ∗ [ χ ( e α − ( P ) e )] . (2.13)For the unbounded traces, one needs to calculate φ jk α in terms of φ jk , anddetermine the K class of [ α − ( P )] in terms of the basis. This would then allowfor the calculation of the K-matrix of α ( e ) in terms of that of e . Here is a usefuland relevant case, particularly for Theorem 1.6.For canonical automorphisms, such as those arising from SL ( , Z ) (or even γ , γ , γ ), α − permutes the basis [ P i ] , and φ jk α is a permutation of the φ jk . (The γ j ’s would change the signs of some φ jk ’s.) Therefore, to obtain the K-matrix of α ( e ) from the K-matrix of e , one(1) permutes the columns of K ( e ) according to how α − acts on [ P i ] , (2) permutes the rows of the result in (1) according to how α acts on φ jk .It can be checked, almost by inspection, that applying this procedure to theidentity element, where α is the Fourier and Cubic transforms or γ j ’s, leavesthe K-matrix (2.12) of the identity unchanged (as it should). -Theory of Approximately Central Powers-Rieffel Projections, S. Walters 16 Two Poisson Lemmas.
We shall have need for the following two Poissonlemmas for our later computations.
Lemma 2.3. (Poisson Summation.) Let f ( x ) be a continuous function on R that is compactly supported. Then for each x , ∞ (cid:88) n = − ∞ (cid:98) f ( n ) e π inx = ∞ (cid:88) n = − ∞ f ( x + n ) where (cid:98) f ( s ) = (cid:82) R f ( t ) e ( − st ) dt is the Fourier transform of f over R . Proof.
The short proof below doesn’t require f to have compact support, onlythat f is integrable and decays at ± ∞ so that h ( x ) : = (cid:80) ∞ n = − ∞ f ( x + n ) is a well-defined integrable periodic function (for instance, for Schwartz functions f ).The Fourier transform of the 1-periodic function h is (cid:98) h ( m ) = (cid:90) h ( x ) e ( − mx ) dx = (cid:90) ∞ (cid:88) n = − ∞ f ( x + n ) e ( − mx ) dx = ∞ (cid:88) n = − ∞ (cid:90) f ( x + n ) e ( − mx ) dx = ∞ (cid:88) n = − ∞ n + (cid:90) n f ( t ) e ( − mt ) dt = ∞ (cid:90) − ∞ f ( t ) e ( − mt ) dt = (cid:98) f ( m ) . Therefore, by Fourier inversion for h ( x ) we get ∞ (cid:88) n = − ∞ f ( x + n ) = h ( x ) = ∞ (cid:88) m = − ∞ (cid:98) h ( m ) e π imx = ∞ (cid:88) m = − ∞ (cid:98) f ( m ) e π imx as required. Lemma 2.4.
For any Schwartz function H ( x ) on the real line we have thefollowing forms of the Poisson Summation (for all real x ), ∞ (cid:88) n = − ∞ (cid:98) H ( n ) e ( nx ) = ∞ (cid:88) n = − ∞ H ( x + n ) + H ( x + + n ) ∞ (cid:88) n = − ∞ (cid:98) H ( n + ) e ( nx ) = e ( − x ) ∞ (cid:88) n = − ∞ H ( x + n ) − H ( x + + n ) Proof.
From Poisson Summation for H , ∞ (cid:88) n = − ∞ (cid:98) H ( n ) e ( nx ) = ∞ (cid:88) n = − ∞ H ( x + n ) -Theory of Approximately Central Powers-Rieffel Projections, S. Walters 17 replace x → x + ∞ (cid:88) n = − ∞ (cid:98) H ( n ) ( − ) n e ( nx ) = ∞ (cid:88) n = − ∞ H ( x + + n ) and add the preceding two equalities to get ∞ (cid:88) n = − ∞ (cid:98) H ( n ) e ( nx )[ + ( − ) n ] = ∞ (cid:88) n = − ∞ H ( x + n ) + H ( x + + n ) ∞ (cid:88) n = − ∞ (cid:98) H ( n ) e ( nx ) = ∞ (cid:88) n = − ∞ H ( x + n ) + H ( x + + n ) replace x by x , ∞ (cid:88) n = − ∞ (cid:98) H ( n ) e ( nx ) = ∞ (cid:88) n = − ∞ H ( x + n ) + H ( x + + n ) . Subtracting we get ∞ (cid:88) n = − ∞ (cid:98) H ( n ) e ( nx )[ − ( − ) n ] = ∞ (cid:88) n = − ∞ H ( x + n ) − H ( x + + n ) which becomes ∞ (cid:88) n = − ∞ (cid:98) H ( n + ) e ( nx ) = e ( − x ) ∞ (cid:88) n = − ∞ H ( x + n ) − H ( x + + n ) or ∞ (cid:88) n = − ∞ (cid:98) H ( n + ) e ( nx ) = e ( − x ) ∞ (cid:88) n = − ∞ H ( x + n ) − H ( x + + n ) as desired.
3. T HE AC P
OWERS -R IEFFEL P ROJECTION
In this section we construct the Powers-Rieffel projection and compute itsunbounded traces.To build approximately central projections from the continuous field E ofSection 2.2, we consider, for given integers q (cid:48) , q , p and irrational θ , the naturalC*-monomorphism ζ = ζ q (cid:48) , q , p , θ defined by ζ : A τ → A θ , ζ ( U τ ) = U q (cid:48) θ = U q (cid:48) , ζ ( V τ ) = V q θ = V q (3.1) -Theory of Approximately Central Powers-Rieffel Projections, S. Walters 18 where we will write τ : = q (cid:48) ( q θ − p ) for brevity (which shan’t be confused withthe trace map τ !). One now uses the field E ( t ) to obtain the Powers-Rieffelprojection e = ζ q (cid:48) , q , p , θ E ( τ ) (3.2) = ζ ( G τ ( U τ ) V − τ + F τ ( U τ ) + V τ G τ ( U τ ))= G τ ( U q (cid:48) ) V − q + F τ ( U q (cid:48) ) + V q G τ ( U q (cid:48) ) (3.3)the K-matrix of which will be computed.It is not hard to see that e is approximately central in the rotation algebra(e.g., it’s easy to see that it approximately commutes with U since (cid:107) V q U − UV q (cid:107) → easily follows).Following [10], we will resurrect this projection as a C*-inner product from aRieffel equivalence bimodule framework (see equation (4.9)). This can certainlybe done for consecutive pairs of convergents pq < θ < p (cid:48) q (cid:48) , where p (cid:48) q − pq (cid:48) = . Remark 3.1.
It is known that there are infinitely many pairs of consecutiverational convergents pq < θ < p (cid:48) q (cid:48) such that q (cid:48) ( q θ − p ) is bounded away from 0and 1. For example, by Lemma 3 of Elliott and Evans [9], for each irrational θ one can show that < q (cid:48) ( q θ − p ) < is satisfied for infinitely many suchconvergent pairs. Therefore, there are infinitely many pairs satisfying one ofthe inequalities < q (cid:48) ( q θ − p ) < or < q (cid:48) ( q θ − p ) < . There is no loss of generality in assuming that the irrational θ conforms to thelatter of these conditions, as we have stipulated in inequality (4.7) below. Onecan reduce the latter case to the former case as follows. Let’s suppose that < q (cid:48) ( q θ − p ) < for infinitely many convergent pairs. One easily converts thisto the former case by looking at the corresponding AC Powers-Rieffel projectionof trace given by the complementary quantity < − q (cid:48) ( q θ − p ) = q ( p (cid:48) − q (cid:48) θ ) = q ( q (cid:48) [ − θ ] − ( q (cid:48) − p (cid:48) )) < . The following lemma shows how the morphism ζ relates the unboundedtraces of A θ and A τ in order to compute the topological invariants of the pro-jection e give by (3.2). Lemma 3.2.
With ζ : A τ → A θ the morphism in (3.1), we have φ θ jk ζ = δ q (cid:48) δ j (cid:104) φ τ k + φ τ k (cid:105) + δ q δ k (cid:104) φ τ j + ( − ) j φ τ j (cid:105) + ( − ) p jk δ q (cid:48) − δ q − φ τ jk where τ = q (cid:48) ( q θ − p ) and φ τ jk are the unbounded Φ -traces for A τ . -Theory of Approximately Central Powers-Rieffel Projections, S. Walters 19 Proof.
We have φ θ jk ζ ( U m τ V n τ ) = φ θ jk ( U q (cid:48) m θ V qn θ ) = e ( − θ q (cid:48) qmn ) δ q (cid:48) m − j δ qn − k = e ( − q (cid:48) [ q θ − p + p ] mn ) δ q (cid:48) m − j δ qn − k = e ( − τ mn )( − ) q (cid:48) pmn δ q (cid:48) m − j δ qn − k = e ( − τ mn )( − ) jpn δ q (cid:48) m − j δ qn − k (since q (cid:48) m in ( − ) q (cid:48) pmn can be replaced by j , in view of the delta function). Usingthe identity δ am − b = δ a δ b + δ a − δ m − b , we have φ jk ζ ( U m τ V n τ ) = e ( − τ mn )( − ) jpn (cid:16) δ q (cid:48) δ j + δ q (cid:48) − δ m − j (cid:17)(cid:16) δ q δ k + δ q − δ n − k (cid:17) = e ( − τ mn )( − ) jpn (cid:16) δ q (cid:48) δ j δ q δ k + δ q (cid:48) δ j δ q − δ n − k + δ q (cid:48) − δ m − j δ q δ k + δ q (cid:48) − δ m − j δ q − δ n − k (cid:17) . The first term δ q (cid:48) δ q = vanishes as q , q (cid:48) are coprime. Also, δ q (cid:48) δ q − = δ q (cid:48) (since if q (cid:48) is even, q has to be odd, and if q (cid:48) is odd both vanish), and similarly δ q (cid:48) − δ q = δ q .Thus we get φ jk ζ ( U m τ V n τ ) = e ( − τ mn )( − ) jpn (cid:16) δ q (cid:48) δ j δ n − k + δ q δ m − j δ k + δ q (cid:48) − δ q − δ m − j δ n − k (cid:17) = e ( − τ mn ) (cid:104) ( − ) jpn δ q (cid:48) δ j δ n − k + ( − ) jpn δ q δ m − j δ k + ( − ) jpn δ q (cid:48) − δ q − δ m − j δ n − k (cid:105) = e ( − τ mn ) (cid:104) δ q (cid:48) δ j δ n − k + ( − ) jn δ q δ m − j δ k + ( − ) p jk δ q (cid:48) − δ q − δ m − j δ n − k (cid:105) (where the middle sign holds since if q is even, p is odd) = δ q (cid:48) δ j e ( − τ mn ) δ n − k + δ q δ k e ( − τ mn )( − ) jn δ m − j + ( − ) p jk δ q (cid:48) − δ q − e ( − τ mn ) δ m − j δ n − k = δ q (cid:48) δ j (cid:104) φ τ k + φ τ k (cid:105) ( U m τ V n τ ) + δ q δ k (cid:104) φ τ j + ( − ) j φ τ j (cid:105) ( U m τ V n τ ) + ( − ) p jk δ q (cid:48) − δ q − φ τ jk ( U m τ V n τ ) therefore we get φ jk ζ = δ q (cid:48) δ j (cid:104) φ τ k + φ τ k (cid:105) + δ q δ k (cid:104) φ τ j + ( − ) j φ τ j (cid:105) + ( − ) p jk δ q (cid:48) − δ q − φ τ jk as claimed.In Appendix A (Section 8) we calculated the unbounded traces of the field E ( t ) to be φ ( E ) = φ ( E ) = φ ( E ) = φ ( E ) = . Combined with Lemma 3.2, we obtain the unbounded traces of our approxi-mately central projection e = ζ E ( τ ) to be φ θ jk ( e ) = φ τ jk ζ ( E ( τ )) = δ q (cid:48) δ j + δ q δ k (cid:104) + ( − ) j (cid:105) + ( − ) p jk δ q (cid:48) − δ q − or φ jk ( e ) = δ q (cid:48) δ j + δ q δ k δ j + ( − ) p jk δ q (cid:48) − δ q − . (3.4) -Theory of Approximately Central Powers-Rieffel Projections, S. Walters 20 Written out, we have φ ( e ) = δ q (cid:48) + δ q + δ q (cid:48) − δ q − , φ ( e ) = δ q (cid:48) + δ q (cid:48) − δ q − , (3.5) φ ( e ) = δ q (cid:48) − δ q − , φ ( e ) = ( − ) p δ q (cid:48) − δ q − .
4. T HE P ROJECTION AS R IEFFEL
C*-I
NNER P RODUCT
In this section our goal is to express the Powers-Rieffel projection e in (1.4)as a C*-inner product by applying Rieffel’s equivalence bimodule theorem [12].Doing so will help facilitate the interaction that e has with any projection P sothat the topological invariants of the cutdown χ ( ePe ) can be calculated.First, however, we give a quick summary of Rieffel’s bimodule backgroundalong with needed notation.Let G = M × (cid:98) M where (cid:98) M is the Pontryagin dual group of characters on thelocally compact Abelian group M , and h the Heisenberg cocycle on G given by h (( m , s ) , ( m (cid:48) , s (cid:48) )) = (cid:104) m , s (cid:48) (cid:105) for m , m (cid:48) ∈ M and s , s (cid:48) ∈ (cid:98) M . The Heisenberg projective unitary representation π : G → L ( L ( M )) is given by phase multiplication and translation [ π ( m , s ) f ]( n ) = (cid:104) n , s (cid:105) f ( n + m ) for f ∈ L ( M ) , where M is equipped with its Haar-Plancheral measure (whichis unique up to positive scalar multiples). It is projective (with respect to h ) inthe sense that π x π y = h ( x , y ) π x + y , π ∗ x = h ( x , x ) π − x (4.1)for x , y ∈ G . We let S ( M ) denote Schwartz space of M .If D is a discrete lattice subgroup of G (i.e. cocompact), it has the associatedtwisted group C*-algebra C ∗ ( D , h ) of the bounded operators on L ( M ) generatedby the unitaries π x for x ∈ D . It is the universal C*-algebra generated by uni-taries { π x : x ∈ D } satisfying the projective commutation relations (4.1). Fromthe latter relation we have π x π y = h ( x , y ) h ( y , x ) π y π x (4.2)for x , y ∈ D . Doing the same for the complementary lattice D ⊥ = { y ∈ G : h ( x , y ) h ( y , x ) = , ∀ x ∈ D } one obtains the C*-algebra C ∗ ( D ⊥ , h ) generated by the unitaries π ∗ y for y ∈ D ⊥ (which also satisfy the preceding commutation relation with h in place of h ). -Theory of Approximately Central Powers-Rieffel Projections, S. Walters 21 Rieffel’s theorem states that the Schwartz space S ( M ) can be completed toan equivalence (or imprimitivity) C ∗ ( D , h ) - C ∗ ( D ⊥ , h ) bimodule, making these al-gebras strongly Morita equivalent.On the C*-algebras C ∗ ( D , h ) and C ∗ ( D ⊥ , h ) there are canonical Flip automor-phisms defined, respectively, by Φ ( π x ) = π − x , Φ (cid:48) ( π y ) = π − y for x ∈ D , y ∈ D ⊥ . These can easily be shown to be multiplicative with respect tothe C*-inner products in the sense that Φ (cid:104) f , g (cid:105) D = (cid:104) ˜ f , ˜ g (cid:105) D , Φ (cid:48) (cid:104) f , g (cid:105) D ⊥ = (cid:104) ˜ f , ˜ g (cid:105) D ⊥ (4.3)where ˜ f ( t ) = f ( − t ) and for f , g ∈ S ( M ) . It is also easy to see that for the left andright module actions one has (cid:102) a f = Φ ( a ) ˜ f , (cid:102) hb = ˜ h Φ (cid:48) ( b ) for a ∈ C ∗ ( D , h ) and b ∈ C ∗ ( D ⊥ , h ) . The are easy to check by taking, for the firstequation, a = π x , x ∈ D (and likewise for the second).We now apply this construction to the locally compact Abelian group M = R × Z q × Z q (cid:48) and lattice subgroup D = Z ε + Z ε of G = M × M generated by the basis vectors ε = ( α q , p ,
0; 0 , , ) ε = ( , ,
1; 1 , , ) (4.4)where α = q θ − p . From h ( ε , ε ) = e ( α q + pq ) = e ( θ ) , we have associated unitariesgenerating the irrational rotation algebra: V = π ε , U = π ε , VU = e ( θ ) UV (in view of (4.2)) so that the twisted group C*-algebra C ∗ ( D , h ) ∼ = A θ , generatedby π ε , π ε , is just the irrational rotation algebra. The Flip Φ , as defined aboveon the unitaries π x agrees with that originally defined: Φ ( U ) = U − , Φ ( V ) = V − .Recall that the measure of each element of Z q is / √ q , so that its total mea-sure is √ q . Since a fundamental domain of the lattice D in G is [ , α q ) × Z q × Z q (cid:48) × [ , ) × Z q × Z q (cid:48) we obtain the covolume of D in G as the product of measures of each compo-nent | G / D | = α q · √ q · (cid:112) q (cid:48) · · √ q · (cid:112) q (cid:48) = q (cid:48) α = q (cid:48) ( q θ − p ) = : τ which will be the trace of the Powers-Rieffel projection e . -Theory of Approximately Central Powers-Rieffel Projections, S. Walters 22 A straightforward computation gives the complementary lattice of D as D ⊥ = Z δ + Z δ + Z δ with basis vectors δ = ( qq (cid:48) , p ,
0; 0 , , p (cid:48) ) δ = ( , , α , q (cid:48) , ) δ = ( , ,
1; 0 , , ) (4.5)as readily checked. (Note that π ∗ δ j = π − δ j since π ∗ x = h ( x , x ) π − x as in our case h ( δ j , δ j ) = .) We have associated unitaries V = π − δ , V = π − δ , V = π − δ satisfying the commutation relations V V = e ( θ (cid:48) ) V V , V V = e ( p (cid:48) q (cid:48) ) V V , V V = V V , V q (cid:48) = . (4.6)They generate the C*-algebra C ∗ ( D ⊥ , h ) isomorphic to a q (cid:48) × q (cid:48) matrix algebraover some irrational rotation algebra. The Flip Φ (cid:48) ( π y ) = π − y on this algebra iseasily seen to be given by Φ (cid:48) ( V ) = V − , Φ (cid:48) ( V ) = V − , Φ (cid:48) ( V ) = V − . The parameter θ (cid:48) in (4.6) is calculated using (4.2) e ( θ (cid:48) ) = π − δ π − δ π ∗− δ π ∗− δ = h ( δ , δ ) h ( δ , δ ) giving us (modulo the integers) θ (cid:48) : = qq (cid:48) α + pq (cid:48) q = qq (cid:48) α + p (cid:48) q − q ≡ Z qq (cid:48) α − q = − q (cid:48) α qq (cid:48) α = q α (cid:48) qq (cid:48) α = α (cid:48) q (cid:48) α since p (cid:48) q − pq (cid:48) = and q (cid:48) α + q α (cid:48) = , where α (cid:48) = p (cid:48) − q (cid:48) θ , α = q θ − p . We now consider the function (as in [10]) f ( t , r , s ) = c δ rq δ sq (cid:48) (cid:112) f ( t ) , c = √ qq (cid:48) α where c is a normalizing constant, f is continuous and supported on theinterval [ − q (cid:48) , q (cid:48) ] , and f = on [ q (cid:48) − α , α − q (cid:48) ] , and f ( t − α ) = − f ( t ) for α − q (cid:48) ≤ t ≤ q (cid:48) as shown in Figure 2. -Theory of Approximately Central Powers-Rieffel Projections, S. Walters 23 Figure 2.
Graphs of f , g . According to our Standing Condition 1.3 (in the Introduction), we have q α (cid:48) < < q (cid:48) α = τ . (4.7)In terms of the function f τ defined in Section 2.2, we could in fact take f ( t ) = f τ ( q (cid:48) t ) (4.8)where τ = q (cid:48) α .4.1. Computation of (cid:104) f , f (cid:105) D ⊥ . Recall that the D ⊥ -inner product of f in oursetup is (cid:104) f , f (cid:105) D ⊥ = (cid:88) n , n q (cid:48) − (cid:88) n = (cid:104) f , f (cid:105) D ⊥ ( n δ + n δ + n δ ) π ∗ n δ + n δ + n δ where the coefficients will be worked out soon. First, let’s find π ∗ n δ + n δ + n δ .From π u + v = h ( u , v ) π u π v , we get π n δ + n δ + n δ = h ( n δ , n δ + n δ ) π n δ π n δ + n δ = h ( n δ , n δ ) π n δ h ( n δ , n δ ) π n δ π n δ = e ( − n n θ (cid:48) ) π n δ π n δ π n δ so π ∗ n δ + n δ + n δ = e ( n n θ (cid:48) ) π − n δ π − n δ π − n δ = e ( n n θ (cid:48) ) V n V n V n so the inner product becomes (cid:104) f , f (cid:105) D ⊥ = (cid:88) n , n q (cid:48) − (cid:88) n = (cid:104) f , f (cid:105) D ⊥ ( n δ + n δ + n δ ) · e ( n n θ (cid:48) ) V n V n V n -Theory of Approximately Central Powers-Rieffel Projections, S. Walters 24 where the coefficients can be worked out as follows: (cid:104) f , f (cid:105) D ⊥ ( n δ + n δ + n δ ) = (cid:104) f , f (cid:105) D ⊥ ( n qq (cid:48) , n p , n ; n α , n q (cid:48) , n p (cid:48) )= √ qq (cid:48) q − (cid:88) r = q (cid:48) − (cid:88) s = e ( n q (cid:48) rq + n p (cid:48) sq (cid:48) ) (cid:90) R f ( t , r , s ) f ( t + n qq (cid:48) , r + n p , s + n ) e ( t n α ) dt = c √ qq (cid:48) q − (cid:88) r = q (cid:48) − (cid:88) s = e ( n q (cid:48) rq + n p (cid:48) sq (cid:48) ) (cid:90) R δ rq δ sq (cid:48) (cid:112) f ( t ) δ r + n pq δ s + n q (cid:48) (cid:113) f ( t + n qq (cid:48) ) e ( t n α ) dt = α δ n q δ n q (cid:48) (cid:90) R (cid:113) f ( t ) f ( t + n qq (cid:48) ) e ( t n α ) dt . From (4.8) we put f ( t ) = f τ ( x ) where x = q (cid:48) t to get (cid:104) f , f (cid:105) D ⊥ ( n δ + n δ + n δ ) = τ δ n q δ n q (cid:48) (cid:90) R (cid:113) f τ ( x ) f τ ( x + n q ) e ( x n τ ) dx (as τ = q (cid:48) α ). This gives (cid:104) f , f (cid:105) D ⊥ = τ (cid:88) n , n q (cid:48) − (cid:88) n = δ n q δ n q (cid:48) (cid:90) R (cid:113) f τ ( x ) f τ ( x + n q ) e ( x n τ ) dx · e ( n n θ (cid:48) ) V n V n V n setting n = qk , n = (and writing n = m ), = τ (cid:88) k , m (cid:90) R (cid:112) f τ ( x ) f τ ( x + k ) e ( x m τ ) dx · e ( qkm θ (cid:48) ) V m V qk = τ (cid:88) m (cid:90) R f τ ( x ) e ( x m τ ) dx · V m since the integrand here vanishes for k (cid:54) = . The latter integral can be calcu-lated as follows (cf. Figure 1 with t = τ ) (cid:90) R f τ ( x ) e ( x m τ ) dx = − τ (cid:90) − f τ ( x ) e ( x m τ ) dx + τ − (cid:90) − τ f τ ( x ) e ( x m τ ) dx + (cid:90) τ − f τ ( x ) e ( x m τ ) dx = − τ (cid:90) − f τ ( x ) e ( x m τ ) dx + τ − (cid:90) − τ e ( x m τ ) dx + − τ (cid:90) − f τ ( x + τ ) e (( x + τ ) m τ ) dx -Theory of Approximately Central Powers-Rieffel Projections, S. Walters 25 by making the change of variable x → x + τ in the third integral. From f τ ( x ) + f τ ( x + τ ) = for − ≤ x ≤ − τ , we get (cid:90) R f τ ( x ) e ( x m τ ) dx = − τ (cid:90) − e ( x m τ ) dx + τ − (cid:90) − τ e ( x m τ ) dx = τ − (cid:90) − e ( x m τ ) dx = τ δ m , . Therefore the D ⊥ -inner product is (cid:104) f , f (cid:105) D ⊥ = (cid:80) m δ m , · V m = . This means thatthe D -inner product (cid:104) f , f (cid:105) D = e (4.9)is a projection, which we now proceed to calculate and show to be equal to thePowers-Rieffel projection e .4.2. Computation of (cid:104) f , f (cid:105) D . Recall that the D -inner product of Schwartz func-tions f , g on M is given by (cid:104) f , g (cid:105) D = | G / D | (cid:88) w ∈ D (cid:104) f , g (cid:105) D ( w ) π w = τ (cid:88) m , n (cid:104) f , g (cid:105) D ( m ε + n ε ) U n V m since | G / D | = τ and π w = π m ε + n ε = h ( m ε , n ε ) π m ε π n ε = e ( − mn θ ) V m U n = U n V m . The coefficients are (cid:104) f , f (cid:105) D ( m ε + n ε ) = (cid:104) f , f (cid:105) D ( m α q , mp , n ; n , n , )= (cid:90) M f ( t , r , s ) f (( t , r , s ) + ( m α q , mp , n )) · (cid:104) ( t , r , s ) , ( n , n , ) (cid:105) dtdrds = (cid:90) R × Z q × Z q (cid:48) f ( t , r , s ) f ( t + m α q , r + mp , s + n ) e ( − tn ) e ( − rnq ) dt √ q √ q (cid:48) = √ qq (cid:48) q − (cid:88) r = q (cid:48) − (cid:88) s = e ( − rnq ) (cid:90) R f ( t , r , s ) f ( t + m α q , r + mp , s + n ) e ( − tn ) dt = c √ qq (cid:48) q − (cid:88) r = q (cid:48) − (cid:88) s = e ( − rnq ) (cid:90) R δ rq δ sq (cid:48) δ r + mpq δ s + nq (cid:48) (cid:113) f ( t ) f ( t + m α q ) e ( − tn ) dt = α δ mq δ nq (cid:48) (cid:90) R (cid:113) f ( t ) f ( t + m α q ) e ( − tn ) dt -Theory of Approximately Central Powers-Rieffel Projections, S. Walters 26 which, again using the change of variable x = q (cid:48) t and f ( t ) = f τ ( x ) , gives (cid:104) f , f (cid:105) D ( m ε + n ε ) = τ δ mq δ nq (cid:48) (cid:90) R (cid:113) f τ ( x ) f τ ( x + m τ q ) e ( − nxq (cid:48) ) dx . The inner product becomes (cid:104) f , f (cid:105) D = τ (cid:88) m , n (cid:104) f , f (cid:105) D ( m ε + n ε ) U n V m = (cid:88) m , n δ mq δ nq (cid:48) (cid:90) R (cid:113) f τ ( x ) f τ ( x + m τ q ) e ( − nxq (cid:48) ) dx U n V m = (cid:88) k ,(cid:96) (cid:90) R (cid:112) f τ ( x ) f τ ( x + k τ ) e ( − (cid:96) x ) dx U q (cid:48) (cid:96) V qk . (by setting m = qk and n = q (cid:48) (cid:96) ). It is easy to see that from condition < τ (as in(4.7)) the integrand here vanishes for | k | ≥ , so the sum over k is concentratedat k = − , , , hence the inner product can be written (cid:104) f , f (cid:105) D = (cid:88) (cid:96) (cid:90) R (cid:112) f τ ( x ) f τ ( x − τ ) e ( − (cid:96) x ) dx U q (cid:48) (cid:96) V − q + (cid:88) (cid:96) (cid:90) R f τ ( x ) e ( − (cid:96) x ) dx U q (cid:48) (cid:96) + (cid:88) (cid:96) (cid:90) R (cid:112) f τ ( x ) f τ ( x + τ ) e ( − (cid:96) x ) dx U q (cid:48) (cid:96) V q or (cid:104) f , f (cid:105) D = (cid:101) G ( U q (cid:48) ) V − q + (cid:101) F ( U q (cid:48) ) + V q (cid:101) G ( U q (cid:48) ) where (cid:101) F ( z ) = (cid:88) (cid:96) (cid:90) R f τ ( x ) e ( − (cid:96) x ) dx · z (cid:96) = (cid:88) (cid:96) (cid:98) f τ ( (cid:96) ) · z (cid:96) and (cid:101) G ( z ) = (cid:88) (cid:96) (cid:90) R (cid:112) f τ ( x ) f τ ( x − τ ) e ( − (cid:96) x ) dx · z (cid:96) . (4.10)In light of the Poisson Lemma 2.3, we see that (cid:101) F ( z ) = F τ ( z ) and (cid:101) G ( z ) = G τ ( z ) are the same periodization functions mentioned at the beginning of Section2.2. Therefore, (cid:104) f , f (cid:105) D = G τ ( U q (cid:48) ) V − q + F τ ( U q (cid:48) ) + V q G τ ( U q (cid:48) ) = ζ E ( τ ) = e hence the Powers-Rieffel projection e in (3.2) is a C*-inner product. -Theory of Approximately Central Powers-Rieffel Projections, S. Walters 27 The Morita Isomorphism.
Now that the projection e = (cid:104) f , f (cid:105) D is an innerproduct such that (cid:104) f , f (cid:105) D ⊥ = , there is the associated (Morita) isomorphism η : eA θ e −→ C ∗ ( D ⊥ , h ) , η ( x ) = (cid:104) f , x f (cid:105) D ⊥ , η − ( y ) = (cid:104) f y , f (cid:105) D (4.11)where we note (and easy to check) that η is a homomorphism with respectto the opposite multiplication on C ∗ ( D ⊥ , h ) . Further, since the canonical nor-malized traces τ , τ (cid:48) of A θ and C ∗ ( D ⊥ , h ) (respectively) are related by τ (cid:104) g , h (cid:105) D = τ ( e ) τ (cid:48) (cid:104) h , g (cid:105) D ⊥ , one has τ ( x ) = τ ( e ) τ (cid:48) ( η ( x )) (4.12)for x ∈ eA θ e .Further, this Morita isomorphism intertwines the Flip automorphisms η Φ = Φ (cid:48) η . (4.13)Indeed, from (4.3) for any two Schwartz function h , k we have η Φ (cid:104) h , k (cid:105) D = η (cid:104) ˜ h , ˜ k (cid:105) D = (cid:104) f , (cid:104) ˜ h , ˜ k (cid:105) D f (cid:105) D ⊥ which, upon applying the Flip Φ (cid:48) (and using (cid:102) ah = Φ ( a ) ˜ h , recalling ˜ f = f is even),gives Φ (cid:48) η Φ (cid:104) h , k (cid:105) D = Φ (cid:48) (cid:104) f , (cid:104) ˜ h , ˜ k (cid:105) D f (cid:105) D ⊥ = (cid:104) ˜ f , [ (cid:104) ˜ h , ˜ k (cid:105) D f ] ∼ (cid:105) D ⊥ = (cid:104) ˜ f , Φ ( (cid:104) ˜ h , ˜ k (cid:105) D ) · ˜ f (cid:105) D ⊥ = (cid:104) f , (cid:104) h , k (cid:105) D f (cid:105) D ⊥ = η ( (cid:104) h , k (cid:105) D ) .
5. C
UTDOWN A PPROXIMATION
In this section we obtain the cutdown approximations η ( eVe ) ≈ V , η ( eU e ) ≈ V (5.1)needed in the next section. (Recall, a ≈ b means (cid:107) a − b (cid:107) → as q , q (cid:48) → ∞ .)Since V is unitary of order q (cid:48) and V is a unitary with full spectrum, both sat-isfying V V = e ( p (cid:48) q (cid:48) ) V V (as in (4.6)), they generate a C*-subalgebra isomorphicto the circle algebra M q (cid:48) ⊗ C ( T ) ∼ = M q (cid:48) ( C ( T )) which, in view of (5.1), approximatesthe corner algebra eA θ e . This makes e a circle algebra projection.From U = π ε , where ε = ( , ,
1; 1 , , ) , and f ( t , r , s ) = c δ rq δ sq (cid:48) (cid:112) f ( t ) , we get ( U f )( t , k , (cid:96) ) = ( π ε f )( t , k , (cid:96) ) = e ( t + kq ) f ( t , k , (cid:96) + ) = ce ( t + kq ) δ kq δ (cid:96) + q (cid:48) (cid:112) f ( t ) . -Theory of Approximately Central Powers-Rieffel Projections, S. Walters 28 We now calculate the D ⊥ -inner product coefficient (cid:104) f , U f (cid:105) D ⊥ ( n δ + n δ + n δ ) = (cid:104) f , U f (cid:105) D ⊥ ( n qq (cid:48) , n p , n ; n α , n q (cid:48) , n p (cid:48) )= √ qq (cid:48) q − (cid:88) r = q (cid:48) − (cid:88) s = e ( n q (cid:48) rq + n p (cid:48) sq (cid:48) ) (cid:90) R f ( t , r , s )( U f )( t + n qq (cid:48) , r + n p , s + n ) e ( t n α ) dt = c √ qq (cid:48) q − (cid:88) r = q (cid:48) − (cid:88) s = e ( n q (cid:48) rq + n p (cid:48) sq (cid:48) ) (cid:90) R δ rq δ sq (cid:48) (cid:112) f ( t )( U f )( t + n qq (cid:48) , r + n p , s + n ) e ( t n α ) dt = c √ qq (cid:48) (cid:90) R (cid:112) f ( t )( U f )( t + n qq (cid:48) , n p , n ) e ( t n α ) dt = c √ qq (cid:48) (cid:90) R (cid:112) f ( t ) e ( t + n qq (cid:48) + n pq ) δ n pq δ n + q (cid:48) (cid:113) f ( t + n qq (cid:48) ) e ( t n α ) dt since n must be divisible by q (if this is nonzero) we can remove n pq = α δ n q δ n + q (cid:48) e ( n qq (cid:48) ) (cid:90) R (cid:113) f ( t ) f ( t + n qq (cid:48) ) e ( t [ + n α ]) dt which, in view of f ( t ) = f τ ( x ) where x = q (cid:48) t , becomes = τ δ n q δ n + q (cid:48) e ( n qq (cid:48) ) (cid:90) R (cid:113) f τ ( x ) f τ ( x + n q ) e ( xq (cid:48) [ + n α ]) dx as τ = q (cid:48) α in the first factor. Therefore, the C*-inner product now becomes (asin Section 4.1) (cid:104) f , U f (cid:105) D ⊥ = (cid:88) n , n q (cid:48) − (cid:88) n = τ δ n q δ n + q (cid:48) e ( n qq (cid:48) ) (cid:90) R (cid:113) f τ ( x ) f τ ( x + n q ) e ( xq (cid:48) [ + n α ]) dx · e ( n n θ (cid:48) ) V n V n V n = τ V − (cid:88) n , n δ n q e ( n qq (cid:48) ) (cid:90) R (cid:113) f τ ( x ) f τ ( x + n q ) e ( xq (cid:48) [ + n α ]) dx · e ( n n θ (cid:48) ) V n V n now put n = qk (and write n = m ) = τ V − (cid:88) k , m e ( kq (cid:48) ) (cid:90) R (cid:112) f τ ( x ) f τ ( x + k ) e ( xq (cid:48) [ + m α ]) dx · e ( qkm θ (cid:48) ) V m V qk . Since the product f τ ( x ) f τ ( x + k ) = for k (cid:54) = , we get (cid:104) f , U f (cid:105) D ⊥ = V − · τ (cid:88) m (cid:90) R f τ ( x ) e ( xq (cid:48) ) e ( mx τ ) dx · V m . -Theory of Approximately Central Powers-Rieffel Projections, S. Walters 29 We now use the Poisson Lemma 2.3 to evaluate the sum (where V is now re-placed by the function e π it , for t ∈ [ , ] , since it is a unitary with full spectrum).Making the substitution x = τ y , we get C ( t ) : = τ (cid:88) m (cid:90) R f τ ( x ) e ( xq (cid:48) ) e ( mx τ ) dx · e π imt = (cid:88) m (cid:90) R f τ ( τ y ) e ( τ yq (cid:48) ) e ( my ) dy · e π imt or, letting h ( y ) = f τ ( τ y ) e ( τ yq (cid:48) ) , becomes C ( t ) = (cid:88) m (cid:90) R h ( y ) e ( my ) dy · e π imt = (cid:88) m (cid:98) h ( − m ) e π imt = (cid:88) m (cid:98) h ( m ) e − π imt . We will show that C ( t ) converges uniformly in t to 1 for large q , q (cid:48) , where we cantake ≤ t ≤ . By Lemma 2.3 we have C ( t ) = (cid:88) n h ( n − t ) = (cid:88) n f τ ( τ n − τ t ) e ( τ n − τ tq (cid:48) ) . Since f τ is supported on [ − , ] , the only n ’s that contribute to this sum arethose satisfying τ | n − t | < . Since τ > , we have | n − t | ≤ τ | n − t | < which gives − < n − t < . Adding this inequality to ≤ t ≤ gives − < n < so that n = , .Using the fact that f τ is even, we obtain C ( t ) = f τ ( τ t ) e ( − τ tq (cid:48) ) + f τ ( τ − τ t ) e ( τ − τ tq (cid:48) ) ≈ f τ ( τ t ) + f τ ( τ − τ t ) = uniformly in t for large q (cid:48) . The last equality here can be seen by looking atthe cases ≤ τ t ≤ and < τ t ≤ τ separately. The former case follows fromour observation in (2.5), and in the latter case we have ≤ τ − τ t < τ − where f τ ( τ − τ t ) = and f τ ( τ t ) = .This gives us the cutdown approximation η ( eU e ) = (cid:104) f , U f (cid:105) D ⊥ ≈ V − . We next calculate (cid:104) f , V f (cid:105) D ⊥ where V = π ε = π ( α q , p ,
0; 0 , , ) . We have ( V f )( t , r , s ) = f ( t + α q , r + p , s ) = c δ r + pq δ sq (cid:48) (cid:113) f ( t + α q ) . The C*-inner product coefficients are (cid:104) f , V f (cid:105) D ⊥ ( n δ + n δ + n δ )= √ qq (cid:48) q − (cid:88) r = q (cid:48) − (cid:88) s = e ( n q (cid:48) rq + n p (cid:48) sq (cid:48) ) (cid:90) R f ( t , r , s ) V f ( t + n qq (cid:48) , r + n p , s + n ) e ( t n α ) dt = c √ qq (cid:48) q − (cid:88) r = q (cid:48) − (cid:88) s = e ( n q (cid:48) rq + n p (cid:48) sq (cid:48) ) (cid:90) R δ rq δ sq (cid:48) (cid:112) f ( t ) · V f ( t + n qq (cid:48) , r + n p , s + n ) e ( n t α ) dt -Theory of Approximately Central Powers-Rieffel Projections, S. Walters 30 in which both r , s have to be 0 = c √ qq (cid:48) (cid:90) R (cid:112) f ( t ) V f ( t + n qq (cid:48) , n p , n ) e ( n t α ) dt = α δ n + q δ n q (cid:48) (cid:90) R (cid:113) f ( t ) f ( t + n qq (cid:48) + α q ) e ( n t α ) dt = τ δ n + q δ n q (cid:48) (cid:90) R (cid:113) f τ ( x ) f τ ( x + n q + τ q ) e ( n x τ ) dx (again using the substitution x = q (cid:48) t and f ( t ) = f τ ( x ) ), therefore (cid:104) f , V f (cid:105) D ⊥ = τ (cid:88) n , n q (cid:48) − (cid:88) n = δ n + q δ n q (cid:48) (cid:90) R (cid:113) f τ ( x ) f τ ( x + n + τ q ) e ( n x τ ) dx · e ( n n θ (cid:48) ) V n V n V n = τ (cid:88) n , n δ n + q (cid:90) R (cid:113) f τ ( x ) f τ ( x + n + τ q ) e ( n x τ ) dx · V n V n here we put n = − + qn (and n = m ) = V − τ (cid:88) n , m (cid:90) R (cid:112) f τ ( x ) f τ ( x − α (cid:48) + n ) e ( mx τ ) dx · V nq V m since − τ q = α (cid:48) . It is easy to see that the integrand here vanishes for n (cid:54) = , .Thus, (cid:104) f , V f (cid:105) D ⊥ = V − τ (cid:88) m (cid:90) R (cid:112) f τ ( x ) f τ ( x − α (cid:48) ) e ( mx τ ) dx · V m + V − V q τ (cid:88) m (cid:90) R (cid:112) f τ ( x ) f τ ( x − α (cid:48) + ) e ( mx τ ) dx · V m making the substitution y = x / τ , (cid:104) f , V f (cid:105) D ⊥ = V − (cid:88) m (cid:90) R (cid:112) f τ ( τ y ) f τ ( τ y − α (cid:48) ) e ( my ) dy · V m + V − V q (cid:88) m (cid:90) R (cid:112) f τ ( τ y ) f τ ( τ y − α (cid:48) + ) e ( my ) dy · V m = V − (cid:88) m (cid:98) h ( − m ) · V m + V − V q (cid:88) m (cid:98) h ( − m ) · V m -Theory of Approximately Central Powers-Rieffel Projections, S. Walters 31 or (cid:104) f , V f (cid:105) D ⊥ = V − (cid:88) m (cid:98) h ( m ) · V − m + V − V q (cid:88) m (cid:98) h ( m ) · V − m (5.2)where we have written h ( y ) = (cid:112) f τ ( τ y ) f τ ( τ y − α (cid:48) ) , h ( y ) = (cid:112) f τ ( τ y ) f τ ( τ y − α (cid:48) + ) . By Lemma 2.3, the first sum is (as done earlier) (cid:88) m (cid:98) h ( m ) e − π imt = (cid:88) m h ( m − t ) = h ( − t ) + h ( − t ) since h ( m − t ) = for m (cid:54) = , and ≤ t ≤ . (Note f τ ( τ ( m − t )) = for m (cid:54) = , since | m − t | ≤ τ | m − t | < gives − < m − t < , and adding ≤ t ≤ gives − < m < hence m = , .) Since α (cid:48) → for large q , q (cid:48) , it follows that h ( − t ) → f τ ( τ t ) and h ( − t ) → f τ ( τ − τ t ) (both uniformly) hence their sum h ( − t ) + h ( − t ) → f τ ( τ t ) + f τ ( τ − τ t ) = by equation (2.5) for f τ . This shows the the first term in (5.2) for (cid:104) f , V f (cid:105) D ⊥ converges to V − in norm. It remains now to check that the second termconverges to 0 for large q , q (cid:48) . For the second term we likewise have only the m = , terms (cid:88) m (cid:98) h ( m ) e − π imt = (cid:88) m h ( m − t ) = h ( − t ) + h ( − t ) . The fact that this converges uniformly to 0 follows from f τ ( s ) f τ ( s + − α (cid:48) ) → uniformly in s . Since f τ is supported on [ − , ] , this product is 0 unless − ≤ s ≤ − + α (cid:48) , and on this interval (which shrinks to − as α (cid:48) → ) one has f τ ( s ) → f τ ( − ) = . Therefore, we have obtained the norm approximation η ( eVe ) = (cid:104) f , V f (cid:105) D ⊥ ≈ V − for large q , q (cid:48) .
6. K-
THEOR Y OF P OWERS -R IEFFEL P ROJECTIONS
In this section we prove Theorem 1.4. Theorem 1.5 is then proved from itand Lemma 7.1 (which is proved in Section 7). All norm approximations “ ≈ ”here are understood to hold for large enough integer parameters q , q (cid:48) .We will denote by Ξ the linear *-anti-automorphism of the continuous fieldof rotation algebras { A t } defined by Ξ ( U mt V nt ) = U nt V mt -Theory of Approximately Central Powers-Rieffel Projections, S. Walters 32 on the canonical unitary basis. It is a vector space linear transformationsatisfying Ξ ( ab ) = Ξ ( b ) Ξ ( a ) , Ξ ( a ∗ ) = Ξ ( a ) ∗ for a , b ∈ A t . It follows that we also have Ξ ( V rt U st ) = V st U rt . (We simply write Ξ instead of Ξ t since it will present no confusion.) Further, Ξ commutes with theFlip ΞΦ = ΦΞ so it leaves the Flip orbifild invariant.Let P ( θ ) denote any continuous field of Flip-invariant smooth projections.For example, P could be any of the Powers-Rieffel projection fields forming thebasis for K ( A Φ θ ) = Z given in (2.9). For convenience we write P as a continuoussection P ( t ) = (cid:88) m , n c m , n ( t ) U mt V nt of the continuous field of rotation C*-algebras { A t } , where c m , n ( t ) are rapidlydecreasing coefficients; and from its Flip-invariance one has c − m , − n = c m , n .For large q , the cut down eP ( θ ) e is close to the Flip-invariant projection χ ( eP ( θ ) e ) ≈ eP ( θ ) e . where χ is the characteristic function of the interval [ , ∞ ] .Let A p (cid:48) / q (cid:48) denote the rational rotation algebra generated by the unitaries U (cid:48) = U p (cid:48) / q (cid:48) and V (cid:48) = V p (cid:48) / q (cid:48) satisfying V (cid:48) U (cid:48) = e ( p (cid:48) q (cid:48) ) U (cid:48) V (cid:48) . Let π denote the canonical representation π : A p (cid:48) / q (cid:48) → C ∗ ( V , V ) , π ( U (cid:48) ) = V , π ( V (cid:48) ) = V which exists since V V = e ( p (cid:48) q (cid:48) ) V V from (4.6).We will use the well-known result of Elliott ([7]) that all normalized traceson a rational rotation algebra agree on projections. In particular, τ (cid:48) π andthe canonical trace τ p (cid:48) / q (cid:48) of A p (cid:48) / q (cid:48) are equal on projections. The canonicaltrace of χ ( eP ( θ ) e ) is therefore obtainable from the approximations η ( eU e ) ≈ V − , η ( eVe ) ≈ V − , as follows. Since P ( t ) is Flip-invariant we can write it as -Theory of Approximately Central Powers-Rieffel Projections, S. Walters 33 P ( t ) = (cid:80) m , n c m , n ( t ) U − mt V − nt , hence for sufficiently large q , q (cid:48) , we have η ( χ ( eP ( θ ) e )) ≈ η ( eP ( θ ) e ) = η (cid:88) m , n c m , n ( θ ) eU − m θ V − n θ e ≈ η (cid:88) m , n c m , n ( θ )( eU θ e ) − m ( eV θ e ) − n = (cid:88) m , n c m , n ( θ ) η ( eV θ e ) − n η ( eU θ e ) − m (opposite multiplication) ≈ (cid:88) m , n c m , n ( θ ) V n V m = π (cid:88) m , n c m , n ( θ ) U (cid:48) n V (cid:48) m ≈ π (cid:88) m , n c m , n ( p (cid:48) q (cid:48) ) U (cid:48) n V (cid:48) m ≈ π Ξ (cid:88) m , n c m , n ( p (cid:48) q (cid:48) ) U (cid:48) m V (cid:48) n = π Ξ P ( p (cid:48) q (cid:48) ) . This shows that the projections χ ( eP ( θ ) e ) and η − π Ξ P ( p (cid:48) q (cid:48) ) , being close, aretherefore unitarily equivalent in the Flip orbifold A Φ θ , and in particular theyhave the same canonical and unbounded trace invariants. Thus τ (cid:48) ( η ( χ ( eP ( θ ) e ))) = τ (cid:48) π Ξ P ( p (cid:48) q (cid:48) ) = τ p (cid:48) / q (cid:48) ( Ξ P ( p (cid:48) q (cid:48) )) = τ p (cid:48) / q (cid:48) ( P ( p (cid:48) q (cid:48) )) . where the last equality holds since τ p (cid:48) / q (cid:48) Ξ is a normalized trace on the rationalrotation algebra A p (cid:48) / q (cid:48) so it agrees with τ p (cid:48) / q (cid:48) on the projections.Hence from (4.12), we get τ ( χ ( eP ( θ ) e )) = τ ( e ) τ (cid:48) ( η ( χ ( eP ( θ ) e ))) = q (cid:48) ( q θ − p ) τ p (cid:48) / q (cid:48) ( P ( p (cid:48) q (cid:48) )) . (6.1)We now compute the unbounded traces φ jk of the cutdown projection χ ( eP ( θ ) e ) (which requires more work). We have φ jk ( χ ( eP ( θ ) e )) = φ jk ( η − π Ξ P ( p (cid:48) q (cid:48) )) = ( φ jk η − ) π Ξ P ( p (cid:48) q (cid:48) ) . Here, it is easily checked that φ jk η − is a Φ (cid:48) -trace when restricted to the C*-algebra generated by V , V since φ jk are Φ -traces and using the intertwiningrelation (4.13). In Section 9 (see equations (9.1)) we showed that the vectorspace of Φ (cid:48) -traces on C ∗ ( V , V ) is 2-dimensional with basis the Φ (cid:48) -traces ψ ( V n V m ) = e ( p (cid:48) mn q (cid:48) ) δ m , ψ ( V n V m ) = e ( p (cid:48) mn q (cid:48) )( − ) p (cid:48) n δ m − q (cid:48) . Therefore, on C ∗ ( V , V ) (the range of π ) there are scalars a − jk , a + jk such that φ jk η − = a − jk ψ + a + jk ψ . -Theory of Approximately Central Powers-Rieffel Projections, S. Walters 34 We then have φ jk ( χ ( eP ( θ ) e )) = a − jk ψ π Ξ P ( p (cid:48) q (cid:48) ) + a + jk ψ π Ξ P ( p (cid:48) q (cid:48) ) . The maps ψ j π Ξ ( j = , ) are readily found on the basis elements as follows ψ π Ξ ( U (cid:48) m V (cid:48) n ) = ψ π ( U (cid:48) n V (cid:48) m ) = ψ ( V n V m ) = e ( − p (cid:48) mnq (cid:48) ) ψ ( V m V n )= e ( − p (cid:48) mn q (cid:48) ) e ( p (cid:48) mn q (cid:48) ) δ n = e ( − p (cid:48) mn q (cid:48) ) δ n = ( φ + φ )( U (cid:48) m V (cid:48) n ) hence ψ π Ξ = φ + φ on A p (cid:48) / q (cid:48) , where here φ ≡ φ p (cid:48) / q (cid:48) , φ ≡ φ p (cid:48) / q (cid:48) – two of the four basic unboundedtraces on the rotation algebra, namely φ i j ( U (cid:48) m V (cid:48) n ) = e ( − p (cid:48) mn q (cid:48) ) δ m − i δ n − j . Similarly, ψ π Ξ ( U (cid:48) m V (cid:48) n ) = ψ ( V n V m ) = e ( − p (cid:48) mnq (cid:48) ) ψ ( V m V n ) = e ( − p (cid:48) mn q (cid:48) )( − ) p (cid:48) m δ n − q (cid:48) = ( φ , q (cid:48) + ( − ) p (cid:48) φ , q (cid:48) )( U (cid:48) m V (cid:48) n ) since ( − ) p (cid:48) m = δ m + ( − ) p (cid:48) δ m − – where, of course, φ , s is φ or φ depending onparity of s . Therefore, ψ π Ξ = φ , q (cid:48) + ( − ) p (cid:48) φ , q (cid:48) . We have therefore obtained φ jk ( χ ( eP ( θ ) e )) = a − jk C ( P ) + a + jk C ( P ) (6.2)where C ( P ) = φ ( P ) + φ ( P ) , C ( P ) = φ , q (cid:48) ( P ) + ( − ) p (cid:48) φ , q (cid:48) ( P ) (6.3)where P = P ( p (cid:48) q (cid:48) ) on the right sides. The invariants φ jk P ( p (cid:48) q (cid:48) ) depend only on thefield P ( θ ) and do not depend specifically on θ , p (cid:48) , q (cid:48) - for instance, this can beseen from unbounded trace values of the basis fields listed in (2.10). Notehowever, how C depends on the parities of p (cid:48) and q (cid:48) in (6.3).It now remains to find the coefficients a − jk , a + jk which depend only on the ACprojection e . First, evaluate the equation φ jk η − = a − jk ψ + a + jk ψ at the identity η ( e ) = to get φ jk ( e ) = a − jk + a + jk δ q (cid:48) . (6.4)Next, evaluate it at V , where ψ ( V ) = , ψ ( V ) = ( − ) p (cid:48) δ q (cid:48) = − δ q (cid:48) (since p (cid:48) , q (cid:48) arecoprime), to get φ jk η − ( V ) = a − jk − a + jk δ q (cid:48) . (6.5)Now evaluate it at V (noting ψ ( V ) = , ψ ( V ) = δ q (cid:48) − ) φ jk η − ( V ) = a + jk δ q (cid:48) − . (6.6) -Theory of Approximately Central Powers-Rieffel Projections, S. Walters 35 We now consider the two parity cases for q (cid:48) . CASE: q (cid:48) is even. Then p (cid:48) is odd and equations (6.4) and (6.5) become φ jk ( e ) = a − jk + a + jk , φ jk η − ( V ) = a − jk − a + jk which give a − jk = [ φ jk ( e ) + φ jk η − ( V )] , a + jk = [ φ jk ( e ) − φ jk η − ( V )] ( q (cid:48) even ) . From (3.4) we have φ jk ( e ) = δ j ( q (cid:48) even, q odd), and from Lemma 7.1, and itsconsequent equation (7.2) in this case, we have φ jk η − ( V ) = φ jk (cid:104) f V , f (cid:105) D = ( − ) pk δ j − . We then get a − jk = [ δ j + ( − ) pk δ j − ] , a + jk = [ δ j − ( − ) pk δ j − ] ( q (cid:48) even ) which simplify to a − jk = ( − ) p jk , a + jk = ( − ) j + p jk ( q (cid:48) even ) (6.7)This, together with (6.3), give us the results indicated in Theorem 1.4 for thecase where q (cid:48) is even. CASE: q (cid:48) is odd. In this case, equation (6.6) gives a + jk = φ jk η − ( V ) and (6.4) gives a − jk = φ jk ( e ) = δ q δ j δ k + ( − ) p jk δ q − (6.8)from (3.4) (since q (cid:48) is odd). By Lemma 7.1, and consequent equation (7.1), wehave a + jk = φ jk η − ( V ) = φ jk (cid:104) f V , f (cid:105) D = ( − ) p (cid:48) j [ δ k − + ( − ) p j δ q − k − ] (6.9)which are the values given in Theorem 1.4 in the case that q (cid:48) is odd. Thiscompletes the proof of Theorem 1.4 (the canonical traces having already beenobtained above).We now proceed to prove Theorem 1.5 by calculating the K-matrix of theprojection e , which we do for three parity cases. As stated in the Introduction,for simplicity we let χ i : = χ ( eP i ( θ ) e ) denote the cutdown projection of the i -thbasis generator P i by e . The trace vector of e consists of the traces of thesecutdowns (cid:126) τ ( e ) = (cid:2) τ χ τ χ τ χ τ χ τ χ τ χ (cid:3) . In view of (6.1), they are τ χ i = q (cid:48) ( q θ − p ) τ p (cid:48) / q (cid:48) ( P i ( p (cid:48) q (cid:48) )) . -Theory of Approximately Central Powers-Rieffel Projections, S. Walters 36 Inserting the traces of the six fields P i as indicated in (2.10), we get the tracevector (cid:126) τ ( e q ) = (cid:2) q (cid:48) ( q θ − p ) τ χ p (cid:48) ( q θ − p ) p (cid:48) ( q θ − p ) p (cid:48) ( q θ − p ) p (cid:48) ( q θ − p ) (cid:3) where τ χ = (cid:40) p (cid:48) ( q θ − p ) for < θ < ( q (cid:48) − p (cid:48) )( q θ − p ) for < θ < . This gives the canonical traces side of the K -theory of the AC projection e asstated in Theorem 1.5.In view of equation (6.2), it is convenient to write the full K-matrix K ( e ) =[ φ jk ( χ i )] jk , i of e (relative to the ordered K basis (2.9)) as a matrix product K ( e ) = φ ( χ ) φ ( χ ) φ ( χ ) φ ( χ ) φ ( χ ) φ ( χ ) φ ( χ ) φ ( χ ) φ ( χ ) φ ( χ ) φ ( χ ) φ ( χ ) φ ( χ ) φ ( χ ) φ ( χ ) φ ( χ ) φ ( χ ) φ ( χ ) φ ( χ ) φ ( χ ) φ ( χ ) φ ( χ ) φ ( χ ) φ ( χ ) = AC where A = a − a + a − a + a − a + a − a + , C = (cid:20) C ( P ) C ( P ) C ( P ) C ( P ) C ( P ) C ( P ) C ( P ) C ( P ) C ( P ) C ( P ) C ( P ) C ( P ) (cid:21) . Of course, A depends only the AC projection e , and C is a matrix of topologicalinvariants of the ordered basis (given in (2.9)).It is more convenient to consider the following three cases separately:(i) q (cid:48) even,(ii) q (cid:48) odd and q even, and(iii) q (cid:48) and q both odd. Case (i): q (cid:48) even. From (6.7) we have a − jk = ( − ) p jk , a + jk = ( − ) j + p jk , so A = − ( − ) p ( − ) p + Equations (6.3) in the even q (cid:48) case (so p (cid:48) is odd) become C ( P ) = φ ( P ) + φ ( P ) , C ( P ) = φ ( P ) − φ ( P ) which, in view of the unbounded traces in (2.10), give C = (cid:20) (cid:21) -Theory of Approximately Central Powers-Rieffel Projections, S. Walters 37 Therefore we obtain the K-matrix in the even q (cid:48) case to be K ( e ) = − ( − ) p ( − ) p + (cid:20) (cid:21) or K ( e ) = − −
10 0 ( − ) p ( − ) p + ( − ) p ( − ) p + q (cid:48) even (where we have subscripted the matrix with the parity case to which it applies). Case (ii): q even. In this case p and q (cid:48) are odd and equations (6.8) and (6.9)become a − jk = δ j δ k , a + jk = δ j δ k − since a + jk = ( − ) p (cid:48) j [ δ k − + ( − ) p j δ q − k − ] = ( − ) p (cid:48) j [ + ( − ) j ] δ k − = ( − ) p (cid:48) j δ j δ k − = δ j δ k − . This gives A = and (6.3) becomes C ( P ) = φ ( P ) + φ ( P ) , C ( P ) = φ ( P ) + ( − ) p (cid:48) φ ( P ) which lead to C = (cid:34) δ p (cid:48) δ p (cid:48) − − δ p (cid:48) − δ p (cid:48) − (cid:35) where we made use of ( + ( − ) p (cid:48) ) = δ p (cid:48) and ( − ( − ) p (cid:48) ) = δ p (cid:48) − . Therefore, K ( e ) = (cid:34) δ p (cid:48) δ p (cid:48) − − δ p (cid:48) − δ p (cid:48) − (cid:35) or K ( e ) = δ p (cid:48) δ p (cid:48) − − δ p (cid:48) − δ p (cid:48) − q even-Theory of Approximately Central Powers-Rieffel Projections, S. Walters 38 Case (iii): q (cid:48) , q both odd. Here, equations (6.8) and (6.9) give a − jk = ( − ) p jk , a + jk = ( − ) j + p jk since a + jk = ( − ) p (cid:48) j [ δ k − + ( − ) p j δ k ] = ( − ) p (cid:48) j ( − ) p j ( k + ) = ( − ) j ( p (cid:48) + p + pk ) = ( − ) j ( + pk ) where the last equality holds because one of p , p (cid:48) will be even and the otherodd (from qp (cid:48) − q (cid:48) p = , where q , q (cid:48) are both odd). Therefore, A = − ( − ) p ( − ) p − . The matrix C is the same as in the previous case (ii) (where q (cid:48) is odd), thus K ( e ) = − ( − ) p − ( − ) p (cid:34) δ p (cid:48) δ p (cid:48) − − δ p (cid:48) − δ p (cid:48) − (cid:35) = + δ p (cid:48) δ p (cid:48) − − δ p (cid:48) − δ p (cid:48) − + δ p (cid:48) δ p (cid:48) − − δ p (cid:48) − δ p (cid:48) − δ p (cid:48) − − δ p (cid:48) − + δ p (cid:48) δ p (cid:48) − ( − ) p ( − ) p δ p (cid:48) − − ( − ) p δ p (cid:48) − ( − ) p ( + δ p (cid:48) ) ( − ) p δ p (cid:48) − q (cid:48) , q both odd Since q (cid:48) , q are both odd, one of p or p (cid:48) will be even which would simplify thematrix a bit. (E.g., ( − ) p δ p (cid:48) − = δ p (cid:48) − since if p (cid:48) is odd p must be even. Also, − δ p (cid:48) = δ p (cid:48) − .) With this in mind, the preceding K-matrix becomes K ( e ) = + δ p (cid:48) δ p (cid:48) − δ p (cid:48) − − δ p (cid:48) − + δ p (cid:48) δ p (cid:48) − δ p (cid:48) − − δ p (cid:48) − δ p (cid:48) − − δ p (cid:48) − + δ p (cid:48) δ p (cid:48) − ( − ) p δ p (cid:48) − − δ p (cid:48) − ( − ) p ( + δ p (cid:48) ) δ p (cid:48) − q (cid:48) , q both odd These all give us the matrices in Theorem 1.5 and therefore complete its proof.
7. U
NBOUNDED T RACES OF
C*-I
NNER P RODUCTS
In this section we calculate the unbounded traces of the inner products (cid:104) f V , f (cid:105) D and (cid:104) f V , f (cid:105) D given by the following lemma. These quantities areneeded for the calculations in Section 6 in computing the coefficients a − , a + . -Theory of Approximately Central Powers-Rieffel Projections, S. Walters 39 Lemma 7.1.
For i j = , , , , we have φ i j (cid:104) f V , f (cid:105) D = δ j − (cid:104) δ i + δ q (cid:48) − i ( − ) p (cid:48) (cid:105) + δ q − j − (cid:104) δ i + δ q (cid:48) − i ( − ) pq (cid:48) + p (cid:48) (cid:105) and φ i j (cid:104) f V , f (cid:105) D = ( δ i − + δ q (cid:48) − − i )( δ j + ( − ) pi δ q − j ) . Remark 7.2.
In Section 6 the computation for φ i j (cid:104) f V , f (cid:105) D is needed for thecase that q (cid:48) is odd (see equations (6.8) and (6.9)), so in this case it simplifies to φ i j (cid:104) f V , f (cid:105) D = ( − ) p (cid:48) i (cid:104) δ j − + ( − ) pi δ q − j − (cid:105) . ( q (cid:48) odd ) (7.1)We have also used the computation for φ i j (cid:104) f V , f (cid:105) D for when q (cid:48) is even (so q , p (cid:48) are odd), which simplifies it to φ i j (cid:104) f V , f (cid:105) D = δ i − ( − ) p j ( q (cid:48) even ) (7.2)7.1. Computation of φ i j (cid:104) f V , f (cid:105) D . We will first need to compute η − ( V ) = (cid:104) f V , f (cid:105) D where (as in Section 4) V = π − δ , δ = ( qq (cid:48) , p ,
0; 0 , , p (cid:48) ) . Recalling that f ( t , r , s ) = c δ rq δ sq (cid:48) (cid:112) f ( t ) , where c = √ qq (cid:48) α , we have f V ( t , r , s ) = f π ( − qq (cid:48) , − p ,
0; 0 , , − p (cid:48) ) ( t , r , s ) = e ( − p (cid:48) sq (cid:48) ) f ( t − qq (cid:48) , r − p , s ) = ce ( − p (cid:48) sq (cid:48) ) δ r − pq δ sq (cid:48) (cid:113) f ( t − qq (cid:48) ) . From the delta factor δ sq (cid:48) the exponential appearing here can be replaced by 1,thus f V ( t , r , s ) = c δ r − pq δ sq (cid:48) (cid:113) f ( t − qq (cid:48) ) . We therefore get the D -inner product coefficients (cid:104) f V , f (cid:105) D ( m ε + n ε ) = √ qq (cid:48) q − (cid:88) r = q (cid:48) − (cid:88) s = e ( − rnq ) (cid:90) R f V ( t , r , s ) f ( t + m α q , r + mp , s + n ) e ( − tn ) dt = c √ qq (cid:48) q − (cid:88) r = q (cid:48) − (cid:88) s = e ( − rnq ) δ r − pq δ sq (cid:48) δ r + mpq δ s + nq (cid:48) (cid:90) R (cid:113) f ( t − qq (cid:48) ) f ( t + m α q ) e ( − tn ) dt which, in view of the first two delta functions, we set r = p and s = = α e ( − pnq ) δ m + q δ nq (cid:48) (cid:90) R (cid:113) f ( t − qq (cid:48) ) f ( t + m α q ) e ( − tn ) dt . -Theory of Approximately Central Powers-Rieffel Projections, S. Walters 40 Using the equality f ( t ) = f τ ( q (cid:48) t ) , by (4.8), and making the change of variable x = q (cid:48) t , this becomes (using q (cid:48) α = τ ) = τ e ( − pnq ) δ m + q δ nq (cid:48) (cid:90) R (cid:113) f τ ( x − q ) f τ ( x + m τ q ) e ( − nxq (cid:48) ) dx . This gives the D -inner product (noting that | G / D | = τ = q (cid:48) α ) (cid:104) f V , f (cid:105) D = τ (cid:88) m , n (cid:104) f V , f (cid:105) D ( m ε + n ε ) U n V m = (cid:88) m , n e ( − pnq ) δ m + q δ nq (cid:48) (cid:90) R (cid:113) f τ ( x − q ) f τ ( x + m τ q ) e ( − nxq (cid:48) ) dx · U n V m . Now we set m = qk − and n = q (cid:48) (cid:96), = (cid:88) k ,(cid:96) e ( − pq (cid:48) (cid:96) q ) (cid:90) R (cid:113) f τ ( x − q ) f τ ( x + k τ − τ q ) e ( − (cid:96) x ) dx · U q (cid:48) (cid:96) V qk − . Making the translation x → x + q in the integral gives = (cid:88) k ,(cid:96) e ( − pq (cid:48) (cid:96) q ) (cid:90) R (cid:113) f τ ( x ) f τ ( x + k τ + − τ q ) e ( − (cid:96) [ x + q ]) dx · U q (cid:48) (cid:96) V qk − = (cid:88) k ,(cid:96) e ( − pq (cid:48) (cid:96) q ) e ( − (cid:96) q ) (cid:90) R (cid:113) f τ ( x ) f τ ( x + k τ + − τ q ) e ( − (cid:96) x ) dx · U q (cid:48) (cid:96) V qk − . From p (cid:48) q − pq (cid:48) = we have e ( − pq (cid:48) (cid:96) q ) e ( − (cid:96) q ) = e ( − ( pq (cid:48) + ) (cid:96) q ) = e ( − p (cid:48) q (cid:96) q ) = e ( − p (cid:48) (cid:96) ) = , andfrom = q (cid:48) α + q α (cid:48) = τ + q α (cid:48) we have − τ q = α (cid:48) , hence = (cid:88) k ,(cid:96) (cid:90) R (cid:112) f τ ( x ) f τ ( x + k τ + α (cid:48) ) e ( − (cid:96) x ) dx · U q (cid:48) (cid:96) V qk − . Since the function f τ is supported on the interval − < x < , we must also have − < x + k τ + α (cid:48) < (otherwise the integrand vanishes). From these inequalities,we have k τ < k τ + α (cid:48) < − x < which implies k < τ < since τ > (in viewof Standing Condition 1.3). Further, these inequalities also imply k τ + α (cid:48) > − − x > − , so k τ > − − α (cid:48) . But as α (cid:48) < q (since from τ > and = τ + q α (cid:48) onehas q α (cid:48) < ), we have k τ > − − q ≥ − for q ≥ and hence k > − τ > − (from τ > ). This shows that the preceding sum runs only over k = − , − , , : (cid:104) f V , f (cid:105) D = (cid:88) k = − (cid:88) (cid:96) (cid:90) R H k ( x ) e ( − (cid:96) x ) dx · U q (cid:48) (cid:96) V qk − = (cid:88) k = − (cid:88) (cid:96) (cid:98) H k ( (cid:96) ) · U q (cid:48) (cid:96) V qk − . -Theory of Approximately Central Powers-Rieffel Projections, S. Walters 41 where we have written H k ( x ) = (cid:112) f τ ( x ) f τ ( x + k τ + α (cid:48) ) for simplicity and used itsFourier transform. Applying the unbounded trace φ i j ( U m V n ) = e ( − θ mn ) δ m − i δ n − j we get φ i j (cid:104) f V , f (cid:105) D = (cid:88) k = − (cid:88) (cid:96) (cid:98) H k ( (cid:96) ) φ i j ( U q (cid:48) (cid:96) V qk − ) = (cid:88) k = − δ qk − − j (cid:88) (cid:96) (cid:98) H k ( (cid:96) ) · e ( − θ q (cid:48) (cid:96) ( qk − )) δ q (cid:48) (cid:96) − i = δ j − ( Ω + Ω − ) + δ q − − j ( Ω − + Ω ) where Ω k = (cid:88) (cid:96) (cid:98) H k ( (cid:96) ) · e ( − θ q (cid:48) (cid:96) ( qk − )) δ q (cid:48) (cid:96) − i . Writing Ω k with respect to even and odd (cid:96) = n , n + , gives Ω k = δ i (cid:88) n (cid:98) H k ( n ) · e ( − q (cid:48) θ ( qk − ) n ) + δ q (cid:48) − i e ( − θ q (cid:48) ( qk − )) (cid:88) n (cid:98) H k ( n + ) · e ( − q (cid:48) θ ( qk − ) n )= δ i (cid:88) n (cid:98) H k ( n ) · e ( nx (cid:48) ) + δ q (cid:48) − i e ( x (cid:48) ) (cid:88) n (cid:98) H k ( n + ) · e ( nx (cid:48) ) where x (cid:48) = − q (cid:48) θ ( qk − ) . By Lemma 2.4 this becomes = δ i (cid:88) n H k ( x (cid:48) + n ) + H k ( x (cid:48) + + n ) + δ q (cid:48) − i (cid:88) n H ( x (cid:48) + n ) − H ( x (cid:48) + + n ) so that Ω k = ( δ i + δ q (cid:48) − i ) (cid:88) n H k ( x (cid:48) + n ) + ( δ i − δ q (cid:48) − i ) (cid:88) n H k ( x (cid:48) + + n ) . We work out the two sums in Ω k by working out the sum (cid:88) n H k ( x (cid:48) + ε + n ) = (cid:88) n (cid:113) f τ ( x (cid:48) + ε + n ) f τ ( x (cid:48) + ε + n + k τ + α (cid:48) ) where ε = , . (This Gossamer of a monster is mostly fur!) It is convenient towrite x (cid:48) (using q α (cid:48) = − q (cid:48) α = − τ ) as follows x (cid:48) = − q (cid:48) θ ( qk − ) = − k pq (cid:48) + p (cid:48) − k τ − α (cid:48) which is easily checked.The function values f τ ( x (cid:48) + ε + n ) and f τ ( x (cid:48) + ε + n + k τ + α (cid:48) ) are nonzero whenboth arguments lie in ( − , ) , i.e. when the inequalities − < x (cid:48) + ε + n < , − < x (cid:48) + ε + n + k τ + α (cid:48) < hold. Using the above form for x (cid:48) , these inequalities become − < − k pq (cid:48) + p (cid:48) − k τ − α (cid:48) + ε + n < , − < − k pq (cid:48) + p (cid:48) − k τ − α (cid:48) + ε + n + k τ + α (cid:48) < . -Theory of Approximately Central Powers-Rieffel Projections, S. Walters 42 Adding these inequalities gives − < [ − k pq (cid:48) + p (cid:48) + ε + n ] < , or − < − k pq (cid:48) + p (cid:48) + ε + n < . Since middle number is an integer we get − k pq (cid:48) + p (cid:48) + ε + n = . For suchinteger n to exist, the integer − k pq (cid:48) + p (cid:48) + ε must be even and the following suminvolves only the term with n = ( k pq (cid:48) − p (cid:48) − ε ) , thus we have (cid:88) n H k ( x (cid:48) + ε + n ) = δ − kpq (cid:48) + p (cid:48) + ε H k ( x (cid:48) + ε + k pq (cid:48) − p (cid:48) − ε )= δ kpq (cid:48) + p (cid:48) + ε H k ( − kpq (cid:48) + p (cid:48) − k τ − α (cid:48) + k pq (cid:48) − p (cid:48) )= δ kpq (cid:48) + p (cid:48) + ε f τ ( k τ + α (cid:48) ) using fact that f τ is even in the last equality. This yields Ω k = ( δ i + δ q (cid:48) − i ) δ kpq (cid:48) + p (cid:48) f τ ( k τ + α (cid:48) ) + ( δ i − δ q (cid:48) − i ) δ kpq (cid:48) + p (cid:48) + f τ ( k τ + α (cid:48) )= (cid:104) ( δ i + δ q (cid:48) − i ) δ kpq (cid:48) + p (cid:48) + ( δ i − δ q (cid:48) − i ) δ kpq (cid:48) + p (cid:48) + (cid:105) f τ ( k τ + α (cid:48) )= (cid:104) δ i δ kpq (cid:48) + p (cid:48) + δ q (cid:48) − i δ kpq (cid:48) + p (cid:48) + δ i δ kpq (cid:48) + p (cid:48) + − δ q (cid:48) − i δ kpq (cid:48) + p (cid:48) + (cid:105) f τ ( k τ + α (cid:48) )= (cid:104) δ i + δ q (cid:48) − i ( δ kpq (cid:48) + p (cid:48) − δ kpq (cid:48) + p (cid:48) + ) (cid:105) f τ ( k τ + α (cid:48) ) therefore Ω k = (cid:104) δ i + δ q (cid:48) − i ( − ) kpq (cid:48) + p (cid:48) (cid:105) f τ ( k τ + α (cid:48) ) . Setting k = and k = − gives Ω = (cid:104) δ i + δ q (cid:48) − i ( − ) p (cid:48) (cid:105) f τ ( α (cid:48) ) , Ω − = (cid:104) δ i + δ q (cid:48) − i ( − ) p (cid:48) (cid:105) f τ ( α (cid:48) − τ ) which sum to Ω + Ω − = (cid:104) δ i + δ q (cid:48) − i ( − ) p (cid:48) (cid:105) since f τ ( α (cid:48) ) + f τ ( α (cid:48) − τ ) = (which follows from (2.5) by taking x = α (cid:48) and t = τ there). Similarly, for k = − , we have Ω − = (cid:104) δ i + δ q (cid:48) − i ( − ) pq (cid:48) + p (cid:48) (cid:105) f τ ( − τ + α (cid:48) ) , Ω = (cid:104) δ i + δ q (cid:48) − i ( − ) pq (cid:48) + p (cid:48) (cid:105) f τ ( τ + α (cid:48) ) which sum to Ω + Ω − = (cid:104) δ i + δ q (cid:48) − i ( − ) pq (cid:48) + p (cid:48) (cid:105) since f τ ( − τ + α (cid:48) ) + f τ ( τ + α (cid:48) ) = (which follows from (2.4) by taking x = − τ + α (cid:48) ∈ ( − , ) and t = τ ). We have therefore obtained φ i j (cid:104) f V , f (cid:105) D = δ j − ( Ω + Ω − ) + δ q − j − ( Ω − + Ω )= δ j − (cid:104) δ i + δ q (cid:48) − i ( − ) p (cid:48) (cid:105) + δ q − j − (cid:104) δ i + δ q (cid:48) − i ( − ) pq (cid:48) + p (cid:48) (cid:105) -Theory of Approximately Central Powers-Rieffel Projections, S. Walters 43 which establishes the equation for φ i j (cid:104) f V , f (cid:105) D in the statement of Lemma 7.1.Now if q (cid:48) is odd, as in fact is needed for the computation in Section 6, thisresult simplifies to φ i j (cid:104) f V , f (cid:105) D = δ q − j − (cid:104) δ i + δ i − ( − ) p + p (cid:48) (cid:105) + δ j − (cid:104) δ i + δ i − ( − ) p (cid:48) (cid:105) = (cid:104) δ q − j − ( − ) pi + p (cid:48) i + δ j − ( − ) p (cid:48) i (cid:105) = ( − ) p (cid:48) i (cid:104) δ q − j − ( − ) pi + δ j − (cid:105) ( q (cid:48) odd ) as noted in the remark following Lemma 7.1.7.2. Computation of φ i j (cid:104) f V , f (cid:105) D . Here we compute the unbounded traces φ i j of η − ( V ) = (cid:104) f V , f (cid:105) D . We have, using V = π − δ and δ = ( , ,
1; 0 , , ) , f V ( t , r , s ) = π − δ ( f )( t , r , s ) = π ( , , −
1; 0 , , ) ( f )( t , r , s ) = f ( t , r , s − ) so (cid:104) f V , f (cid:105) D ( m ε + n ε ) = √ qq (cid:48) q − (cid:88) r = q (cid:48) − (cid:88) s = e ( − rnq ) (cid:90) R f ( t , r , s − ) f ( t + m α q , r + mp , s + n ) e ( − tn ) dt = c √ qq (cid:48) q − (cid:88) r = q (cid:48) − (cid:88) s = e ( − rnq ) (cid:90) R δ rq δ s − q (cid:48) δ r + mpq δ s + nq (cid:48) (cid:113) f ( t ) f ( t + m α q ) e ( − tn ) dt = α δ mq δ n + q (cid:48) (cid:90) R (cid:113) f ( t ) f ( t + m α q ) e ( − tn ) dt in view of (4.8), and using a change of variable x = q (cid:48) t , this becomes = τ δ mq δ n + q (cid:48) (cid:90) R (cid:113) f τ ( x ) f τ ( x + m τ q ) e ( − nxq (cid:48) ) dx . This gives inner product (noting that | G / D | = τ = q (cid:48) α ) (cid:104) f V , f (cid:105) D = τ (cid:88) m , n (cid:104) f V , f (cid:105) D ( m ε + n ε ) U n V m = (cid:88) m , n δ mq δ n + q (cid:48) (cid:90) R (cid:113) f τ ( x ) f τ ( x + m τ q ) e ( − nxq (cid:48) ) dx U n V m setting m = qk and n = q (cid:48) (cid:96) − = (cid:88) k ,(cid:96) (cid:90) R (cid:112) f τ ( x ) f τ ( x + k τ ) e ( − (cid:96) x ) e ( xq (cid:48) ) dx U q (cid:48) (cid:96) − V qk -Theory of Approximately Central Powers-Rieffel Projections, S. Walters 44 and noting that the integrand vanishes for | k | ≥ , = (cid:88) k = − (cid:88) (cid:96) (cid:90) R L k ( x ) e ( − (cid:96) x ) dx U q (cid:48) (cid:96) − V qk = (cid:88) k = − (cid:88) (cid:96) (cid:98) L k ( (cid:96) ) U q (cid:48) (cid:96) − V qk where we put L k ( x ) = e ( xq (cid:48) ) (cid:112) f τ ( x ) f τ ( x + k τ ) . Applying φ i j ( U m V n ) = e ( − θ mn ) δ m − i δ n − j one gets φ i j (cid:104) f V , f (cid:105) D = (cid:88) k = − δ qk − j (cid:88) (cid:96) (cid:98) L k ( (cid:96) ) · e ( − θ ( q (cid:48) (cid:96) − ) qk ) δ q (cid:48) (cid:96) − − i = (cid:88) k = − δ qk − j e ( θ qk ) Λ k where Λ k = (cid:88) (cid:96) (cid:98) L k ( (cid:96) ) · e ( − θ qq (cid:48) k (cid:96) ) δ q (cid:48) (cid:96) − − i . Thus, φ i j (cid:104) f V , f (cid:105) D = δ j Λ + δ q − j (cid:2) e ( θ q ) Λ + e ( − θ q ) Λ − (cid:3) . We have Λ k = δ i − (cid:88) n (cid:98) L k ( n ) · e ( − θ qq (cid:48) kn ) + δ q (cid:48) − − i e ( − θ qq (cid:48) k ) (cid:88) n (cid:98) L k ( n + ) · e ( − θ qq (cid:48) kn ) which by the Poisson formulas in Lemma 2.4 becomes Λ k = δ i − ( A k + B k ) + δ q (cid:48) − − i ( A k − B k ) where A k = (cid:88) n L k ( − θ qq (cid:48) k + n ) , B k = (cid:88) n L k ( − θ qq (cid:48) k + + n ) . First, consider k = : A = (cid:88) n L ( n ) = (cid:88) n e ( nq (cid:48) ) f τ ( n ) = since f τ ( n ) = for n = and 0 otherwise, and B = (cid:88) n L ( + n ) = (cid:88) n e ( + nq (cid:48) ) f τ ( + n ) = since f τ ( + n ) = for all integers n . This gives Λ = ( δ i − + δ q (cid:48) − − i ) . To compute Λ and Λ − , it will suffice to find A = (cid:88) n L ( − θ qq (cid:48) + n ) , B = (cid:88) n L ( − θ qq (cid:48) + + n ) . -Theory of Approximately Central Powers-Rieffel Projections, S. Walters 45 First, it is easy to see that L − ( x ) = L ( − x ) (since f τ is even), and consequently A − = A and B − = B , hence Λ − = Λ . For example, to see B − = B , we have B − = (cid:88) n L − ( θ qq (cid:48) + + n ) = (cid:88) n L ( − θ qq (cid:48) − − n ) = (cid:88) n L ( − θ qq (cid:48) + + n ) = B using the substitution n → − − n . We now show that A = δ q (cid:48) p e ( − τ q (cid:48) ) , B = δ q (cid:48) p − e ( − τ q (cid:48) ) . For the first, write A = (cid:88) n L ( − τ − q (cid:48) p + n ) . Letting ε = when q (cid:48) p is even, and ε = when q (cid:48) p is odd, the preceding sumbecomes, after appropriate translation of the index n , A = (cid:88) n L ( − τ + ε + n ) = (cid:88) n e (cid:18) − τ + ε + nq (cid:48) (cid:19) (cid:113) f τ ( − τ + ε + n ) f τ ( τ + ε + n ) . The function values under the square-root here is nonzero when both of itsarguments lie in the open interval ( − , ) , so their sum ε + n (which is aninteger) is in the open interval ( − , ) , so ε + n = . This means that if q (cid:48) p isodd ( ε = ) then no such n exists and hence A = . And if q (cid:48) p is even, so ε = , then n = : A = e ( − τ q (cid:48) ) (cid:113) f τ ( − τ ) f τ ( τ ) = e ( − τ q (cid:48) ) since f τ ( τ ) = from (2.6). We may then write A = δ q (cid:48) p e ( − τ q (cid:48) ) in either parity case.Likewise (with ε as before), we have B = (cid:88) n L ( − τ − q (cid:48) p + + n ) = (cid:88) n L ( − τ + ε + + n )= (cid:88) n e (cid:18) − τ + ε + + nq (cid:48) (cid:19) (cid:113) f τ ( − τ + ε + + n ) f τ ( τ + ε + + n ) . By the same argument as before, the function values under the square-rootare nonzero when their arguments are in ( − , ) , so their sum + ε + n = .Therefore, B = when q (cid:48) p is even ( ε = ). And when q (cid:48) p is odd we must have n = − and ε = which gives B = e ( − τ q (cid:48) ) (cid:113) f τ ( − τ ) f τ ( τ ) = e ( − τ q (cid:48) ) , so that in eitherparity case we have B = δ q (cid:48) p − e ( − τ q (cid:48) ) . -Theory of Approximately Central Powers-Rieffel Projections, S. Walters 46 We thus get Λ = δ i − ( A + B ) + δ q (cid:48) − − i ( A − B )= δ i − (cid:104) δ q (cid:48) p + δ q (cid:48) p − (cid:105) e ( − τ q (cid:48) ) + δ q (cid:48) − − i (cid:104) δ q (cid:48) p − δ q (cid:48) p − (cid:105) e ( − τ q (cid:48) )= δ i − e ( − τ q (cid:48) ) + δ q (cid:48) − − i ( − ) q (cid:48) p e ( − τ q (cid:48) ) and since e ( θ q ) e ( − τ q (cid:48) ) = ( − ) p we have e ( θ q ) Λ = δ i − ( − ) p + δ q (cid:48) − − i ( − ) q (cid:48) p ( − ) p = δ i − ( − ) p + δ q (cid:48) − − i ( − ) pi (where the last term holds in view of the δ q (cid:48) − − i factor) which is real, and asseen above Λ − = Λ , we get φ i j (cid:104) f V , f (cid:105) D = δ j Λ + δ q − j (cid:2) e ( θ q ) Λ + e ( − θ q ) Λ − (cid:3) = δ j ( δ i − + δ q (cid:48) − − i ) + δ q − j · (cid:104) δ i − ( − ) p + δ q (cid:48) − − i ( − ) pi (cid:105) = δ j ( δ i − + δ q (cid:48) − − i ) + δ q − j · (cid:104) δ i − + δ q (cid:48) − − i (cid:105) ( − ) pi = ( δ i − + δ q (cid:48) − − i )( δ j + ( − ) pi δ q − j ) which is the expression in the statement of Lemma 7.1, the proof of which isnow complete.
8. A
PPENDIX
A: U
NBOUNDED T RACES OF E ( t ) In this section we show that the Connes-Chern character of the continuousfield E ( t ) is T ( E ( t )) = ( t ; , , , ) for ≤ t < .Recall that the continuous field E : [ , ) → { A t } of Flip-invariant Powers-Rieffel projections is given by E ( t ) = G t ( U t ) V − t + F t ( U t ) + V t G t ( U t ) (8.1)where F t , G t are smooth functions, as in Section 2.2. Fix j , k and write φ : = φ tjk , which is defined on A ∞ t by φ tjk ( U mt V nt ) = e ( − t mn ) δ m − j δ n − k . (8.2)By the Φ -trace property of φ , we have φ ( G t ( U t ) V − t ) = φ ( Φ ( V − t ) G t ( U t )) = φ ( V t G t ( U t )) so that φ ( E ( t )) = φ ( F t ( U t )) + φ ( G t ( U t ) V − t ) . -Theory of Approximately Central Powers-Rieffel Projections, S. Walters 47 Expressing F t in terms of its Fourier transform F t ( U t ) = (cid:88) n ∈ Z (cid:98) F t ( n ) U nt where (cid:98) F t ( n ) = (cid:98) f t ( n ) (see proof of Lemma 2.3), we obtain φ ( F t ( U t )) = (cid:88) n ∈ Z (cid:98) F t ( n ) φ ( U nt ) = (cid:88) n ∈ Z (cid:98) F t ( n ) δ n − j δ − k = δ k (cid:88) n ∈ Z (cid:98) F t ( n ) δ n − j . Expanding this series into its even and odd indices, one has (cid:88) n ∈ Z (cid:98) F t ( n ) δ n − j = δ j (cid:88) n ∈ Z (cid:98) F t ( n ) + δ j − (cid:88) n ∈ Z (cid:98) F t ( n + ) = δ j (cid:88) n ∈ Z (cid:98) f t ( n ) + δ j − (cid:88) n ∈ Z (cid:98) f t ( n + ) which, in view of the Poisson Lemma 2.4, become = δ j (cid:32) ∞ (cid:88) n = − ∞ f t ( n ) + f t ( + n ) (cid:33) + δ j − (cid:32) ∞ (cid:88) n = − ∞ f t ( n ) − f t ( + n ) . (cid:33) The sums here are (see Figure 1) (cid:88) n f t ( n ) = f t ( ) = , (cid:88) n f t ( + n ) = so that φ ( F t ( U t )) = δ k ( δ j + δ j − ) = δ k . We similarly compute φ ( G t ( U t ) V − t ) using the Fourier series for G t G t ( U t ) = (cid:88) n ∈ Z (cid:98) G t ( n ) U nt . We have φ ( G t ( U t ) V − t ) = (cid:88) n ∈ Z (cid:98) G t ( n ) φ ( U nt V − t ) = δ − − k (cid:88) n ∈ Z (cid:98) G t ( n ) e ( t n ) δ n − j = δ k − δ j (cid:88) n ∈ Z (cid:98) G t ( n ) e ( tn ) + δ k − δ j − e ( t ) (cid:88) n ∈ Z (cid:98) G t ( n + ) e ( tn ) which again by Lemma 2.4 (with H = g t and using (cid:98) G t = (cid:98) g t ) is = δ k − δ j (cid:32) (cid:88) n g t ( t + n ) + g t ( t + + n ) (cid:33) + δ k − δ j − (cid:32) (cid:88) n g t ( t + n ) − g t ( t + + n ) (cid:33) . The individual sums here are (note f t ( t ) = = g t ( t ) from Section 2.2) (cid:88) n g t ( t + n ) = g t ( t ) = , (cid:88) n g t ( t + + n ) = so φ ( G t ( U t ) V − t ) = δ k − ( δ j + δ j − ) = δ k − . -Theory of Approximately Central Powers-Rieffel Projections, S. Walters 48 Therefore we obtain the desired unbounded traces φ tjk ( E ( t )) = δ k + δ k − = which gives the Connes-Chern character of E ( t ) as T ( E ( t )) = ( t ; , , , ) for ≤ t < (the parameter range over which the field E is defined).We note that the unbounded traces of the projection depend on the followingvalues of the underlying function: f t ( ) = , f t ( ) = , f t ( t ) = , f t ( t + ) = (where the last of these holds since f t is compactly supported on [ − , ] , whereit is an even function). So any homotopic deformation of f t which preservesthese boundary conditions (and of course maintaining the equations between F , G that make E is a projection) would still give us the same unbounded traces.
9. A
PPENDIX
B: F
LIP -T RACES ON M q (cid:48) ( C ( T )) In this section we show that the two functionals ψ ( V n V m ) = e ( p (cid:48) mn q (cid:48) ) δ m (9.1) ψ ( V n V m ) = e ( p (cid:48) mn q (cid:48) )( − ) p (cid:48) n δ m − q (cid:48) form a basis for all Φ (cid:48) -traces on the circle algebra B = C ∗ ( V , V ) ∼ = M q (cid:48) ⊗ C ( T ) generated by unitaries V , V satisfying V V = e ( p (cid:48) q (cid:48) ) V V , V q (cid:48) = I as in (4.6), with V of full spectrum, where Φ (cid:48) is the Flip automorphism of B defined by Φ (cid:48) ( V ) = V − , Φ (cid:48) ( V ) = V − . Proof.
We begin with the canonical epimorphism of the rational rotation alge-bra onto B , π : A p (cid:48) / q (cid:48) → C ∗ ( V , V ) , π ( U (cid:48) ) = V , π ( V (cid:48) ) = V where U (cid:48) = U p (cid:48) / q (cid:48) , V (cid:48) = V p (cid:48) / q (cid:48) satisfy V (cid:48) U (cid:48) = e ( p (cid:48) q (cid:48) ) U (cid:48) V (cid:48) . This surjection intertwinesthe two Flips π Φ = Φ (cid:48) π which is easy to verify, where Φ ( U (cid:48) ) = U (cid:48)− , Φ ( V (cid:48) ) = V (cid:48)− is the Flip on A p (cid:48) / q (cid:48) .Fix a Φ (cid:48) -trace ψ on B . Since ψπ is a Φ -trace on A p (cid:48) / q (cid:48) , it is a linear combinationof the four basic unbounded Φ -traces defined on the basic unitaries U (cid:48) m V (cid:48) n by φ i j ( U (cid:48) m V (cid:48) n ) = e ( − p (cid:48) mn q (cid:48) ) δ m − i δ n − j (9.2) -Theory of Approximately Central Powers-Rieffel Projections, S. Walters 49 ( i j = , , , ). Thus ψπ = a φ + b φ + c φ + d φ for some constants a , b , c , d . Evaluating this gives on the basic unitaries, ψ ( V m V n ) = ψπ ( U (cid:48) m V (cid:48) n ) = a φ ( U (cid:48) m V (cid:48) n ) + b φ ( U (cid:48) m V (cid:48) n ) + c φ ( U (cid:48) m V (cid:48) n ) + d φ ( U (cid:48) m V (cid:48) n )= e ( − p (cid:48) mn q (cid:48) ) (cid:16) a δ m δ n + b δ m δ n − + c δ m − δ n + d δ m − δ n − (cid:17) from e ( p (cid:48) mnq (cid:48) ) V m V n = V n V m , this gives ψ ( V n V m ) = e ( p (cid:48) mn q (cid:48) ) (cid:16) a δ m δ n + b δ m δ n − + c δ m − δ n + d δ m − δ n − (cid:17) . (9.3)Since this expression should be invariant under the translation n → n + q (cid:48) (as V has order q (cid:48) ), we get a δ m δ n + b δ m δ n − + c δ m − δ n + d δ m − δ n − = ( − ) p (cid:48) m a δ m δ n + q (cid:48) + ( − ) p (cid:48) m b δ m δ n + q (cid:48) − + ( − ) p (cid:48) m c δ m − δ n + q (cid:48) + ( − ) p (cid:48) m d δ m − δ n + q (cid:48) − or a δ m δ n + b δ m δ n − + c δ m − δ n + d δ m − δ n − = a δ m δ n + q (cid:48) + b δ m δ n + q (cid:48) − + ( − ) p (cid:48) c δ m − δ n + q (cid:48) + ( − ) p (cid:48) d δ m − δ n + q (cid:48) − which must be satisfied for all four parities of m , n . Setting m = n = , it gives a = a δ q (cid:48) + b δ q (cid:48) − and for m = , n = : b = a δ q (cid:48) − + b δ q (cid:48) for m = , n = : c = ( − ) p (cid:48) c δ q (cid:48) + ( − ) p (cid:48) d δ q (cid:48) − and for m = n = : d = ( − ) p (cid:48) c δ q (cid:48) − + ( − ) p (cid:48) d δ q (cid:48) We consider separately two parity cases for q (cid:48) .If q (cid:48) is even (so p (cid:48) is odd), these become a = a , b = b , c = − c , d = − d , so c = d = , and equation (9.3) becomes ψ ( V n V m ) = e ( p (cid:48) mn q (cid:48) ) (cid:16) a δ n + b δ n − (cid:17) δ m , ( q (cid:48) even ) where a , b are arbitrary scalars. Taking a = b = and a = , b = − , gives us thetwo basic unbounded traces ψ ( V n V m ) = e ( p (cid:48) mn q (cid:48) ) δ m , ψ ( V n V m ) = e ( p (cid:48) mn q (cid:48) )( − ) n δ m ( q (cid:48) even ) . These are easily verified to be well-defined under n → n + q (cid:48) , where q (cid:48) is evenhere, and that both are Φ (cid:48) -traces. -Theory of Approximately Central Powers-Rieffel Projections, S. Walters 50 If q (cid:48) is odd, we get a = b , d = ( − ) p (cid:48) c , from which we can take a and c to beindependent parameters, and equation (9.3) in this case becomes ψ ( V n V m ) = e ( p (cid:48) mn q (cid:48) ) (cid:16) a δ m δ n + a δ m δ n − + c δ m − δ n + c ( − ) p (cid:48) δ m − δ n − (cid:17) = ae ( p (cid:48) mn q (cid:48) )( δ n + δ n − ) δ m + ce ( p (cid:48) mn q (cid:48) )( δ n + ( − ) p (cid:48) δ n − ) δ m − = ae ( p (cid:48) mn q (cid:48) ) δ m + ce ( p (cid:48) mn q (cid:48) )( − ) p (cid:48) n δ m − giving us the two basic unbounded traces in the odd q (cid:48) case ψ ( V n V m ) = e ( p (cid:48) mn q (cid:48) ) δ m , ψ ( V n V m ) = e ( p (cid:48) mn q (cid:48) )( − ) p (cid:48) n δ m − . Combining the two parity cases for q (cid:48) , we can write the two basic Φ (cid:48) -traces asin (9.1) above. (Note that ψ here agrees with the odd q (cid:48) case, and when q (cid:48) iseven, p (cid:48) has to be odd so ( − ) p (cid:48) n = ( − ) n as in ψ in the even case.)Notice that by contrast with the unbounded Φ -trace functionals φ jk for thesmooth rotation algebra, the Φ (cid:48) -traces ψ , ψ are continuous linear functionalson the circle algebra M q (cid:48) ( C ( T )) . Thus, the “unbounded traces” here turn out tobe bounded. Acknowledgement.
This paper and [21] were written at about the time theauthor retires. He is therefore most grateful to his home institution of 26years, the University of Northern British Columbia, for many years of researchand so much other support. The author expresses his nontrivial gratitude tothe many referees who made helpful review reports over the years (includingcritical ones). Thank you. R
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EPAR TMENT OF M ATHEMATICS AND S TATISTICS , U
NIVERSITY OF N OR THERN
B.C., P
RINCE G EORGE , B.C. V2N 4Z9, C
ANADA . Email address : [email protected] or [email protected] URL ::