A spectral triple for noncommutative compact surfaces
aa r X i v : . [ m a t h . OA ] F e b A SPECTRAL TRIPLE FOR NONCOMMUTATIVE COMPACTSURFACES
FREDY DÍAZ GARCÍA AND ELMAR WAGNER
Dedicated to Paul Baum on the occasion of his 80th birthday who taught me like nobody else thatteaching difficult mathematics doesn’t have to be difficult and can be fun.
Abstract.
A Dirac operator is presented that will yield a + -summable regular evenspectral triple for all noncommutative compact surfaces defined as subalgebras of theToeplitz algebra. Connes’ conditions for noncommutative spin geometries are analyzedand it is argued that the failure of some requirements is mainly due to a wrong choiceof a noncommutative spin bundle. Introduction
One of the most fundamental notions of Connes’ noncommutative geometry [2] is thatof a spectral triple. An extensive amount of research has been dedicated to finding novelexamples. However, proving that a spectral triple satisfies all requirements of a noncom-mutative spin geometry can be a difficult task. Among the noteworthy examples are thenoncommutative torus which satisfies all proposed conditions [4], the isospectral defor-mations of toric manifolds [6], and the standard Podleś sphere [9]. In the latter case, theoriginal conditions had to be modified to conform with the basic principles of a noncom-mutative spin geometry [13]. This seems to be a common feature for examples arising inquantum group theory. In particular, for the natural spectral triple on quantum SU(2) [8],it was proven that the requirement of a real structure demands a modification of originalframework. A proposal for such a modification can be found in [1].In this paper, we present a + -summable regular even spectral triple for the noncommu-tative compact surfaces from [14] and study it in the context of Connes’ noncommutativespin geometry [10]. It turns out that the same Dirac operator can be used for all noncom-mutative compact surfaces including the non-orientable ones. This already indicates thatthese spectral triples are not very useful for the calculation of topological invariants (yetthis could be a welcome effect: if all quantum spaces behaved like the classical ones, therewere no need for a quantization). The reason is that the spectral triples are defined on the“wrong” Hilbert space, namely on the holomorphic functions of the Bergman space on theunit disc rather than on sections of the noncommutative spin bundle. Although this seemsobvious, such effects could easily be disguised in more complicated examples, especiallyif the finiteness and the regularity conditions can be satisfied as in our case. Under morefavorable conditions, the finiteness and the regularity conditions lead the correct modulerelated to the spin bundle. Here, however, as a consequence of working on the “wrong”Hilbert space, the spectrum of the Dirac operator resembles that of a 1-dimensional spec-tral triple and so does a possible real structure. Furthermore, the first order condition can Mathematics Subject Classification.
Primary 58B34; Secondary 46L87.
Key words and phrases.
Noncommutative surfaces, Toeplitz algebra, spectral triple. only be satisfied up to compacts whereas the orientation condition and the Poincaré dual-ity fail. In this sense, our paper is rather a warning that modifying or not fulfilling someaxiomatic conditions may have substantial effects on the noncommutative spin geometry.2.
Noncommutative compact surfaces
The noncommutative compact surfaces of any genus [14] will be defined as subalgebrasof the continuous functions on the quantum disc. As explained in [11], the universal C*-algebra of the quantum disc can be represented by the Toeplitz algebra. For a descriptionof the Toeplitz algebra, consider the open complex unit disc D := { z ∈ C : | z | < } withthe standard Lebesgue measure and let ¯ D := { z ∈ C : | z | ≤ } denote its closure in C .We write L ( D ) for the Hilbert space of square-integrable functions and A ( D ) for theclosed subspace of holomorphic functions on D . Let ˆ P denote the orthogonal projectionfrom L ( D ) onto A ( D ) . The Toeplitz operator ˆ T f ∈ B ( A ( D )) with symbol f ∈ C ( ¯ D ) isdefined by ˆ T f ( ψ ) := ˆ P ( f ψ ) , ψ ∈ A ( D ) ⊂ L ( D ) , and the Toeplitz algebra T is the C*-algebra generated by all ˆ T f in B ( A ( D )) . As iswell known [12], the compact operators K ( ℓ ( N )) ∼ = K ( A ( D )) belong to T and that thequotient T /K ( ℓ ( N )) ∼ = C ( S ) gives rise to the C*-algebra extension(1) / / K ( ℓ ( N )) / / T ˆ σ / / C ( S ) / / , with the symbol map ˆ σ : T −→ C ( S ) given by ˆ σ ( ˆ T f ) = f ↾ S for all f ∈ C ( ¯ D ) .Recall that each closed surface can be constructed from a convex polygon by a suitableidentification of its edges. Instead of edges of a polygon, we will consider arcs on theboundary of the unit disc. In this manner, the C*-algebra of continuous functions on aclosed surface can be viewed as a subalgebra of C ( ¯ D ) . Identifying points on the boundarymeans that the functions belonging to the specified subalgebra must have the same valueson identified points. In correspondence with the presentation of oriented closed surfaces,let g ∈ N and define g arcs on the circle S by a k , a − k : [0 , −→ S , a k ( t ) := e π i k − t g , a − k ( t ) := e π i g + k − t g , k = 1 , . . . , g. By standard operations from algebraic topology, one can readily show that T g := ¯ D / ∼ , z ∼ z and a k ( t ) ∼ a − k ( t ) for all z ∈ D , t ∈ [0 , , k = 1 , . . . , g, is homeomorphic to a closed oriented surface of genus g .Viewing the symbol map ˆ σ : T → C ( S ) as the counterpart of an embedding of thecircle into the quantum disc, we define(2) C ( T g,q ) := { f ∈ T : ˆ σ ( f )( a k ( t )) = ˆ σ ( f )( a − k ( t )) for all t ∈ [0 , , k = 1 , . . . , g } , see [14]. To include the 2-sphere with genus 0, we consider additionally a , a − : [0 , −→ S , a ( t ) := e π i t , a − ( t ) := e − π i t . and define C ( S q ) := C ( T ,q ) as in (2) with k = 0 . Similarly, for non-orientable closedsurfaces, let a k , b k : [0 , −→ S , a k ( t ) := e π i k − tg , b k ( t ) := e π i − k + tg , k = 1 , . . . , g, (3)and set(4) C ( P g,q ) := { f ∈ T : ˆ σ ( f )( a k ( t )) = ˆ σ ( f )( b k ( t )) for all t ∈ [0 , , k = 1 , . . . , g } . SPECTRAL TRIPLE FOR NONCOMMUTATIVE COMPACT SURFACES 3
From the continuity of the symbol map, it follows that C ( T g,q ) and C ( P g,q ) are C*-subalgebras of T .An alternative, albeit less illustrative description of these C*-algebras can be given byconsidering L ( S ) with orthonormal basis { e k := √ π u k : k ∈ Z } , where u (e i t ) = e i t denotes the unitary generator of C ( S ) ⊂ L ( S ) . Let P denote the orthogonal projectionfrom L ( S ) onto the closed subspace ℓ ( N ) := span { e n : n ∈ N } . Then the Toeplitzalgebra T ⊂ B ( ℓ ( N )) is generated by the operators T f , f ∈ C ( S ) , where(5) T f ( φ ) := P ( f φ ) , φ ∈ span { e n : n ∈ N } ⊂ L ( S ) , and the symbol map σ : T → C ( S ) reads σ ( T f ) = f .On the topological side, the C*-algebra of continuous functions on S satisfying theconditions on ˆ σ ( f ) in (2) is isomorphic to C ( g ∨ k =1 S ) , where where g ∨ k =1 S denotes thetopological wedge product of n circles. Indicating the orientation of the identified arcswith a + or a − , we set C ( W - g S ) := { f ∈ C ( S ) : f ( a k ( t )) = f ( a − k ( t )) for all t ∈ [0 , , k = 1 , . . . , g } ,C ( W + g S ) := { f ∈ C ( S ) : f ( a k ( t )) = f ( b k ( t )) for all t ∈ [0 , , k = 1 , . . . , g } . Then the definitions of C ( T g,q ) and C ( P g,q ) can be rewritten as C ( T g,q ) = { f ∈ T : σ ( f ) ∈ C ( W - g S ) } , C ( P g,q ) = { f ∈ T : σ ( f ) ∈ C ( W + g S ) } . By the short exact sequence (1) and the definitions of C ( T g,q ) and C ( P g,q ) , these C*-algebras contain obviously K ( ℓ ( N )) = ker( σ ) . Restricting the symbol map to C ( T g,q ) and C ( P g,q ) yields the C*-extensions / / K ( ℓ ( N )) / / C ( T g,q ) σ / / C ( g ∨ k =1 S ) / / , / / K ( ℓ ( N )) / / C ( P g,q ) σ / / C ( g ∨ k =1 S ) / / . (6)The C*-algebra extensions (6) provide a computational tool for calculating the K -groups of the noncommutative compact surfaces. The result can be found in [14] and isgiven by K ( C ( T g,q )) ∼ = Z ⊕ Z , K ( C ( T g,q )) ∼ = g ⊕ k =1 Z ,K ( C ( P g,q )) ∼ = Z ⊕ Z , K ( C ( P g,q )) ∼ = g − ⊕ k =1 Z . (7)Moreover, a set of free generators for K ( C ( T g,q )) is given by the trivial projection [1] ∈ K ( C ( T g,q )) and [ p e ] ∈ K ( K ( ℓ ( N ))) ֒ → K ( C ( T g,q )) , where p e denotes the 1-dimen-sional projection onto C e . Note that the K -groups coincide with those of the classicalcounterparts but the function algebras C ( T g ) and C ( P g ) do not contain non-trivial × -projections.3. Differential geometry of noncommutative compact surfaces
Spectral triples and regularity.
Since surfaces are even dimensional, we are look-ing for even spectral triples ( A , H , D, γ ) for our noncommutative compact surfaces, i.e., adense *-subalgebra A of C ( T g,q ) (or C ( P g,q ) ) which is stable under holomorphic functionalcalculus, a faithful representation π : A → B ( H ) , a self-adjoint operator D on H with FREDY DÍAZ GARCÍA AND ELMAR WAGNER compact resolvent and a self-adjoint grading operator γ satisfying γ = id , γD = − Dγ , γπ ( a ) = π ( a ) γ and [ D, π ( a )] := Dπ ( a ) − π ( a ) D ∈ B ( H ) for all a ∈ A .We say that a spectral triple is n + -summable if (1 + | D | ) − ( n + ǫ ) yields a trace classoperator for all ǫ > but (1 + | D | ) − n does not. In this case, one refers to the number n ∈ [0 , ∞ ) as the metric dimension in analogy to Weyl’s formula for the asymptoticbehavior of the eigenvalues of the Laplacian on a compact Riemannian manifold.A spectral triple ( A , H , D ) is called regular, if π ( a ) , [ D, π ( a )]) ∈ ∩ n ∈ N dom( δ n | D | ) for all a ∈ A , where δ | D | ( x ) := [ | D | , x ] and dom( δ | D | ) := { x ∈ B ( H ) : δ | D | ( x ) ∈ B ( H ) } . Let D = F | D | denote the polar decomposition of D . For regular even spectral triples, one canshow that F = (cid:18) F + − F − + (cid:19) provides an even Fredholm module for A and one defines ind( D ) := ind( F + − ) . This Fredholm module is called the fundamental class of D and wesay that the fundamental class is non-trivial if it gives rise to non-trivial index pairings.Before turning our attention to spectral triples for C ( T g,q ) , we will describe moreexplicitly the action of T ⊂ B ( ℓ ( N )) on ℓ ( N ) . As in the previous section, we use theorthonormal basis { e k = √ π u k : k ∈ Z } of L ( S ) . Then T u n = S n and T u − n = S ∗ n ,where S ∈ B ( ℓ ( N )) denotes the shift operator given by Se k = e k +1 , k ∈ N . Expanding f ∈ C ( S ) ⊂ L ( S ) in its Fourier series f = P k ∈ Z f k u k , f k = √ π h e k , f i , andsetting S k := S k , k ≥ , S k := S ∗| k | , k < , we can write(8) T f = P k ∈ Z f k S k , which means that T f e n = P k ∈ N f k − n e k . Identifying C ( m ) ( S ) with the space of m -timescontinuously differentiable π -periodic functions on R , one shows by partial integrationthat(9) T f ′ = i P k ∈ Z kf k S k , f ∈ C (1) ( S ) , where f ′ = dd t f for t ∈ [0 , π ] . In particular, f = P k ∈ Z f k u k ∈ C ∞ ( S ) implies that { f k } k ∈ Z is a sequence of rapid decay.The condition that A should be stable under holomorphic functional calculus is oftenignored because the proof that a chosen subalgebra has this property might be somewhatinvolved. Our choice is presented in the next proposition. The proof follows the argumentsof [5, Proposition 1]. Proposition 3.1.
Let K S ⊂ K ( ℓ ( N )) denote the ideal of matrices of rapid decay, i.e.,operators A given by Ae n = P k ∈ N a kn e k such that lim ( k,n ) →∞ | k α a kn n β | = 0 for all α, β ∈ N . Set C ∞ ( W - g S ) := C ∞ ( S ) ∩ C ( W - g S ) and let A be the *-algebra generated by theelements of K S and C ∞ ( T g,q ) := { T f ∈ C ( T g,q ) : f ∈ C ∞ ( W - g S ) } . Then A is dense in C ( T g,q ) and stable under holomorphic functional calculus.Proof. Note that k T f − T g k ≤ k P k k f − g k ∞ ≤ k f − g k ∞ and x = x − T σ ( x ) + T σ ( x ) with x − T σ ( x ) ∈ K ( ℓ ( N )) . Since each g ∈ C ( W - g S ) can be uniformly approximated by a byfunctions from C ∞ ( W - g S ) , and each compact operator can be approximated by matricesof rapid decay, A is dense in C ( T g,q ) . SPECTRAL TRIPLE FOR NONCOMMUTATIVE COMPACT SURFACES 5
An elementary calculation shows that S m + k ) − S m S k = (1 − S m S ∗ m ) S ∗ ( | k |− m ) (1 − S | k | S ∗| k | ) , k < , < m < | k | ,S m + k ) − S m S k = (1 − S m S ∗ m ) S ( m −| k | ) (1 − S | k | S ∗| k | ) , k < , m ≥ | k | , (10)and 0 otherwise, where the operators in (10) are finite matrices. Let f, g ∈ C ∞ ( W - g S ) .Using the representation (8) together with Equation (10) and the fact that the coefficientsare sequences of rapid decay, one proves that(11) T fg − T f T g = P m,k ∈ Z f m g k ( S m + k − S m S k ) ∈ K S . Thus [ T f , T g ] ∈ K S , so the algebra generated by C ∞ ( T g,q ) is commutative modulo theideal K S . Since K S is an two-sided ideal in A , we get the exact sequence / / K S / / A σ / / C ∞ ( W - g S ) / / . Moreover, K S and C ∞ ( W - g S ) with the usual (semi-)norms for sequences of rapid decayand C ∞ -functions, respectively, can be turned into Fréchet algebras which are stable underholomorphic functional calculus. To prove the proposition, it thus suffices to show that,for all a ∈ A such that a − ∈ C ( T g,q ) , one has a − ∈ A .Now, if a ∈ A such that a − ∈ C ( T g,q ) , then clearly σ ( a − ) = σ ( a ) − ∈ C ∞ ( W - g S ) .Furthermore, − aT σ ( a − ) ∈ A and σ (1 − aT σ ( a − ) ) = 0 , hence − aT σ ( a − ) ∈ K S . As K S is a two-sided ideal in C ( T g,q ) , we obtain a − = a − (1 − aT σ ( a − ) ) + T σ ( a − ) ∈ A . Thisfinishes the proof. (cid:3) As also in the classical case not all continuous functions are differentiable, the lastproposition provides us with a preferred choice of a dense *-subalgebra of C ( T g,q ) for theconstruction of a spectral triple. In [15], a spectral triple for the noncommutative torus C ( T ,q ) was obtained by starting with the action of the first order differential operator ∂∂z on the Bergman space A ( D ) . It turns out that essentially the same Dirac operator worksfor all the noncommutative compact surfaces from Section 2. In the following, we will firstgive the definition of these spectral triples and then prove their fundamental propertiesin a theorem.The simplest choice of a Z -graded Hilbert space with a faithful representation of C ( T g,q ) ⊂ B ( ℓ ( N )) is H := ℓ ( N ) ⊕ ℓ ( N ) . Then the assignment π : A → B ( ℓ ( N ) ⊕ ℓ ( N )) , π ( a )( v + ⊕ v − ) := av + ⊕ av − , evidently defines a faithful *-representation for all *-subalge-bras A ⊂ B ( ℓ ( N )) . The representation commutes with obvious grading operator γ givenby γ ( v + ⊕ v − ) = v + ⊕ ( − v − ) . Consider now the self-adjoint number operator N definedby(12) N e n = ne n , dom( N ) = n P n ∈ N α n e n ∈ ℓ ( N ) : P n ∈ N n | α n | < ∞ o , and let(13) D := (cid:18) S ∗ NN S (cid:19) = (cid:18) S ∗ NS ( N + 1) 0 (cid:19) , dom( D ) = dom( N ) ⊕ dom( N ) ⊂ H . Since ( S ∗ N ) ∗ = N S = S ( N + 1) , D is self-adjoint. Defining F and | D | by the polardecomposition D = F | D | , we get from (13)(14) F = (cid:18) S ∗ S (cid:19) , | D | = (cid:18) N + 1 00 N (cid:19) . Clearly, γD = − Dγ . FREDY DÍAZ GARCÍA AND ELMAR WAGNER
Theorem 3.2.
Let
A ⊂ C ( T g,q ) denote the pre-C*-algebra from Proposition 3.1, andlet H , D , γ and the representation π : A → B ( H ) be given as in the previous paragraph.Then ( A , H , D, γ ) yields a + -summable regular even spectral triple for C ( T g,q ) . The Diracoperator D has discrete spectrum spec( D ) = Z , each eigenvalue k ∈ Z has multiplicity 1and a complete orthonormal basis of associated eigenvectors is given by (15) b k := √ ( e k − ⊕ e k ) , b − k := √ ( − e k − ⊕ e k ) , k > , b := 0 ⊕ e . Moreover, the fundamental class of D is non-trivial.Proof. It was already discussed above that γ , D and π ( a ) , a ∈ A , satisfy the commutationrelations of a Z -graded spectral triple. To prove that commutators the [ D, π ( a )] arebounded, it suffices to consider the generators from K S and C ∞ ( T g,q ) since, by the Leibnizrule for commutators, [ D, π ( ab )] = [ D, π ( a )] π ( b ) + π ( a )[ D, π ( b )] .If the operator A ∈ K S is given by a matrix of rapid decay ( a kn ) k,n ∈ N , then thematrices ( na kn ) k,n ∈ N and ( ka kn ) k,n ∈ N corresponding to N A and AN , respectively, areagain matrices of rapid decay and thus define bounded operators belonging to K S . Usingthe relations S ∗ N = ( N + 1) S ∗ and S ( N + 1) = N S , we get(16) [ D, π ( A )] = (cid:18) S ∗ N A − AN S ∗ − AS ∗ SN A − AN S + SA (cid:19) which is bounded because all the operators A , S , S ∗ , AN and N A belong to B ( ℓ ( N )) .We can conclude even more, namely that the entries of [ D, A ] belong to K S since, as easilyseen, the product of a matrix of rapid decay with the shift operator or its adjoint yieldsagain a matrix of rapid decay.Next, let f ∈ C ∞ ( W - g S ) so that T f ∈ C ∞ ( T g,q ) . Then, by Equations (8) and (9), [ S ∗ N, T f ] e n = P k ∈ N kf k − n e k − − P k ∈ N nf k − n +1 e k = P k ∈ N ( k − n + 1) f k − n +1 e k = P k ∈ Z kf k S k +1 e n = − i T ¯ uf ′ e n , (17)hence(18) [ S ∗ N, T f ] = − i T ¯ uf ′ ∈ T , [ N S, T f ] = ( − [ S ∗ N, T ¯ f ]) ∗ = − i T uf ′ ∈ T . Therefore [ D, π ( T f )] ∈ B ( H ) since the entries of this × -matrix belong to the Toeplitzalgebra T ⊂ B ( ℓ ( N )) . This finishes the proof that [ D, π ( a )] ∈ B ( H ) for all a ∈ A .Furthermore, one readily verifies that the vectors b k , k ∈ Z , in Equation (15) form abasis of eigenvectors for H and that Db k = kb k . Since each eigenvalue k ∈ Z = spec( D ) hasmultiplicity 1, the resolvent ( D + i) − is compact and the spectral triple is + -summable.To prove the regularity of the spectral triple, first note that [ | D | , A ] = (cid:18) [ N, A ] [ N, A ] + A [ N, A ] − A [ N, A ] (cid:19) , A = (cid:18) A A A A (cid:19) ∈ B ( ℓ ( N ) ⊕ ℓ ( N )) . Hence it suffice to show that all elements a ∈ A and all entries of [ D, π ( a )] belong to ∩ n ∈ N dom( δ nN ) , where δ N ( x ) := [ N, x ] for x ∈ B ( ℓ ( N )) .If A ∈ K S is given by a rapid decay matrix, then N n A ∈ K s and AN n ∈ K s for all n ∈ N by the very definition of a rapid decay matrix. This implies K s ⊂ ∩ n ∈ N dom( δ nN ) .As concluded below Equation (16), all entries of [ D, π ( A )] belong again to K S . Thereforewe have π ( A ) , [ D, π ( A )] ∈ ∩ n ∈ N dom( δ n | D | ) for all A ∈ K S .Let now f ∈ C ∞ ( S ) . As in (17), we compute [ N, T f ] e n = P k ∈ N ( k − n ) f k − n e k , thus(19) δ nN ( T f ) = ( − i) n T f ( n ) SPECTRAL TRIPLE FOR NONCOMMUTATIVE COMPACT SURFACES 7 by Equation (9). Consequently, T f ∈ ∩ n ∈ N dom( δ nN ) . As uf ′ and ¯ uf ′ also belong to C ∞ ( S ) , we conclude from the definition of C ∞ ( T g,q ) together with Equations (18) and(19) that π ( T f ) , [ D, π ( T f )] ∈ ∩ n ∈ N dom( δ n | D | ) for all T f ∈ C ∞ ( T g,q ) . Since the elementsfrom K S and C ∞ ( T g,q ) generate A , the regularity follows from the Leibniz rule for com-mutators.Having a regular, even spectral triple, we know that its fundamental class F from (14)defines an even Fredholm module. Since ind( D ) = ind( S ∗ ) = 1 = 0 , the fundamentalclass is non-trivial. (cid:3) The restriction to C ( T g,q ) was made only for notational convenience. Clearly, we have K S ⊂ C ( P g,q ) without any modification. Analyzing the proofs of Proposition 3.1 andTheorem 3.2, one easily realizes that only the differentiability of the function f in thedefinition of T f ∈ C ∞ ( T g,q ) was used. So, if we define(20) C ∞ ( P g,q ) := { T f ∈ C ( P g,q ) : f ∈ C ( W + g S ) ∪ C ∞ ( S ) } , then all arguments in the proofs remain valid. Therefore we can state the analogousresults of Theorem 3.2 for C ( P g,q ) . Corollary 3.3.
Let
A ⊂ C ( P g,q ) denote the *-algebra generated by the elements of K S and C ∞ ( P g,q ) . With the same definitions of H , D , γ and π : A → B ( H ) as in Theo-rem 3.2, ( A , D, H , γ ) yields a + -summable regular even spectral triple for C ( P g,q ) . Thespectrum, the eigenvalues and the orthonormal basis of eigenvectors from Theorem 3.2remain unchanged for the Dirac operator D and its fundamental class is non-trivial. At the first glance, it might be surprising that there exist Dirac operators on non-commutative versions of a non-orientable manifolds. However, the Dirac operator of aspectral triple should rather be viewed as an analog of an elliptic first order differentialoperator and not necessarily as the Dirac operator on a spin manifold. Nevertheless,since the Dirac operator is related to topological invariants via the index theorem, it isa strange effect of noncommutative geometry that the same operator can be used fordifferent noncommutative spaces among which some have K -groups with torsion.3.2. Real structure and first order condition.
In the context of spin geometry, areal structure singles out those manifolds that admit a real spin structure. The resultsbelow will show that our spectral triples cannot be equipped with a real structure in theexact sense. Since this result would not be surprising for a quantum space for which theclassical counterpart does not admit a real spin structure, we restrict ourselves in thissection to the quantized orientable surfaces C ( T g,q ) .A real structure for a spectral triple ( A , H , D ) is given by an anti-unitary operator J satisfying J = ± id and JD = ± DJ . For an even spectral triple ( A , H , D, γ ) , onerequires additionally Jγ = ± γJ . The signs depend on the dimension of the underlyingquantum space. For instance,(21) J = − id , JD = DJ, Jγ = − γJ in dimension 2, and(22) J = id , JD = − DJ in dimension 1.Given a real structure J , one says that the spectral triple satisfies the first order con-dition if for all a, b ∈ A (23) [ π ( a ) , Jπ ( b ) J − ] = 0 , [[ D, π ( a )] , Jπ ( b ) J − ] = 0 for all a, b ∈ A . FREDY DÍAZ GARCÍA AND ELMAR WAGNER
It was observed in [7] and [8] that, under certain circumstances, a real structure mightnot exist for quantized real spin manifolds and it was proposed to modify the first ordercondition (23) by requiring only(24) [ π ( a ) , Jπ ( b ) J − ] ∈ K S ( H ) , [[ D, π ( a )] , Jπ ( b ) J − ] ∈ K S ( H ) for all a, b ∈ A , where K S ( H ) ⊂ B ( H ) denotes the ideal of matrices of rapid decay (associated to an or-thonormal basis). Here we assume that H is separable. In the context of noncommutativegeometry, the matrices of rapid decay are considered as infinitesimals of arbitrary highorder.The first result of this section shows that the spectral triples from Theorem 3.2 do notadmit a real structure in the exact sense. Proposition 3.4.
Let ( A , H , D, γ ) denote the spectral triple described in Theorem 3.2.Then there does not exist an anti-unitary operator J on H satisfying (21) .Proof. From J − DJe k = De k = ke k , it follows that DJe k = kJe k , hence Je k = α k e k ,where α k ∈ C . Since J is anti-unitary, we have necessarily | α k | = 1 , and thus J e n = J ( α k e k ) = ¯ α k Je k = ¯ α k α k e k = e k , which contradicts the first equation of (21). (cid:3) Ignoring for a moment the commutation relations (21) and (22), it is also impossibleto find an anti-unitary operator satisfying (23).
Proposition 3.5.
Let A denote the *-algebra from Proposition 3.1 and π : A → B ( H ) the representation from Theorem 3.2. Then there does not exist a anti-unitary operator J on H satisfying the first order condition (23) .Proof. Recall that H = ℓ ( N ) ⊕ ℓ ( N ) and π ( a )( v + ⊕ v − ) := av + ⊕ av − . If an operator A = (cid:18) A A A A (cid:19) ∈ B ( ℓ ( N ) ⊕ ℓ ( N )) commutes with π ( a ) for all a ∈ A , then we havenecessarily [ k, A ij ] = 0 for all k ∈ K S and i, j = 1 , . Since K S is dense in K ( ℓ ( N )) , itfollows that A ij = c ij id with c ij ∈ C . Therefore we get for any anti-unitary operator J satisfying (23) dim { A ∈ B ( H ) : [ π ( a ) , A ] = 0 for all a ∈ A} = 4 ≥ dim { Jπ ( b ) J − : b ∈ A} = ∞ , which is a contradiction. (cid:3) If one wants to allow for the existence of a real structure despite the negative result ofProposition 3.5, one may consider to weaken the first order condition (23) by requiringonly the modified version (24). But then the problem of Proposition 3.4 still persists.However, the problem was caused by taking the commutation relations (21) of a spectraltriple of dimension 2. On the other hand, our spectral triples are + -summable, thus theirmetric dimension is 1 instead of 2. Under the requirements of Equations (22) and (24),we can prove the following positive result. Proposition 3.6.
Let ( A , H , D, γ ) denote the spectral triple given in Theorem 3.2, andlet { b k : k ∈ Z } be the orthonormal basis defined in (15) . Then the anti-unitary operator J given by (25) Jb k = b − k , k ∈ Z , satisfies the conditions (22) and (24) . SPECTRAL TRIPLE FOR NONCOMMUTATIVE COMPACT SURFACES 9
Proof.
Since b k is an eigenvector of D corresponding to the eigenvalue k ∈ Z , we have JDJ − b k = − kb k = − Db k , thus JD = − DJ . Obviously, J = 1 , so (22) is satisfied.If b ∈ K S , then Jπ ( b ) J − ∈ K S ( H ) and therefore (24) holds for all a ∈ A since K S ( H ) is an ideal in B ( H ) . Similarly, if a ∈ K S , then, as in the proof of Theorem 3.2, N a, aN ∈ K S . Therefore [ D, π ( a )] ∈ K S ( H ) , and since also π ( a ) ∈ K S ( H ) , the condition(24) is now fulfilled for all b ∈ A .It remains to show that (24) holds for a, b ∈ C ∞ ( T g,q ) . So, let a = T g and b = T f ,where g, f ∈ C ∞ ( W - g S ) . From (15) and (25), we get J ( e k ⊕ e n ) = ( − e k ) ⊕ e n for all k, n ∈ N . Writing T f = P k ∈ Z f k S k as in (8) and setting e + n := e n ⊕ and e − n := 0 ⊕ e n ,we compute(26) Jπ ( T f ) J − e ± n = P k ∈ N ¯ f k − n e ± k = X k ∈ Z ¯ f k S k e ± n = π ( T ˆ f ) e ± n , where ˆ f ∈ C ∞ ( S ) is defined by ˆ f ( z ) := f (¯ z ) . Therefore, by (13), (18) and (26), [ π ( T g ) , Jπ ( T f ) J − ] = π ([ T g , T ˆ f ]) , [[ D, π ( T g )] , Jπ ( T f ) J − ] = − i (cid:18) T ¯ ug ′ , T ˆ f ][ T ug ′ , T ˆ f ] 0 (cid:19) . Since g, ˆ f , ug ′ , ¯ ug ′ ∈ C ∞ ( S ) , and hence its Fourier coefficients are sequences of rapiddecay, we conclude as in the paragraph with Equation (11) that the commutators [ T g , T ˆ f ] , [ T ¯ ug ′ , T ˆ f ] and [ T ug ′ , T ˆ f ] belong to K S . From this, the result follows. (cid:3) Finiteness.
We say that a spectral triple ( A , H , D ) satisfies the finiteness condition,if there exists a dense subset of “smooth” vectors H ∞ ⊂ ∩ n ∈ N dom( D n ) which is isomorphicto a finitely generated left A -projective module. This finitely generated projective moduleis considered as a module of smooth sections of the vector bundle on which the Diracoperator acts (e.g. spin bundle). If H ∞ is a core for the self-adjoint operator D , then theDirac operator D is uniquely determined by its restriction to H ∞ .Stable isomorphism classes of finitely generated projective modules of a (pre-)C*-alge-bra are classified by the associated K -group. Recall from Section 2 that a non-trivialgenerator of K ( C ( T g,q )) is given by the 1-dimensional projection p e = 1 − SS ∗ ∈ K S ⊂A . Consider now the finitely generated left A -projective module A ∞ := A (1 − SS ∗ ) ⊂ B ( ℓ ( N )) together with the vector state and the non-negative sesquilinear form(27) ψ ( a ) := h e , ae i , hh b, a ii := ψ ( b ∗ a ) = h be , ae i a, b ∈ A (1 − SS ∗ ) . The next proposition shows that ∩ n ∈ N dom( D n ) ∼ = A (1 − SS ∗ ) ⊕ A (1 − SS ∗ ) satisfyingthus the finiteness condition. We will give the proof for C ( T g,q ) but the statement holdsalso for A ⊂ C ( P g,q ) . Proposition 3.7.
Consider the finitely generated left A -module A (1 − SS ∗ ) together withthe sesquilinear form given in (27) . Then the left A -modules H ∞ := ∩ n ∈ N dom( D n ) and A (1 − SS ∗ ) ⊕ A (1 − SS ∗ ) are isometrically isomorphic.Proof. Since ∩ n ∈ N dom( D n ) = ∩ n ∈ N dom( N n ) ⊕ ∩ n ∈ N dom( N n ) with the diagonal A -action, it suffices to show that A (1 − SS ∗ ) ∼ = ∩ n ∈ N dom( N n ) . Note that ∩ n ∈ N dom( N n ) = { P k ∈ N α k e k ∈ ℓ ( N ) : P n ∈ N | α k | k n < ∞} = { P k ∈ N α k e k ∈ ℓ ( N ) : ( α k ) k ∈ N ∈ S ( N ) } (28)where S ( N ) denotes the space of sequences of rapid decay. We claim that(29) Φ : A (1 − SS ∗ ) ⊂ B ( ℓ ( N )) −→ ∩ n ∈ N dom( N n ) , Φ( x ) := xe , defines an isometric isomorphism.To see that Φ is well defined, suppose that T f ∈ C ∞ ( T g,q ) , where f ∈ C ∞ ( W - g S ) ,and write f = P k ∈ Z f k u k in its Fourier series expansion. From Equation (8), we get Φ( T f (1 − SS ∗ )) = P k ∈ N f k e k . Since f ∈ C ∞ ( S ) , the sequence ( f k ) n ∈ N belongs to S ( N ) (cf. Section 3.1). Therefore, by (28), Φ( T f (1 − SS ∗ )) ∈ ∩ n ∈ N dom( N n ) . If a ∈ K S and ( a kj ) k,j ∈ N denotes the corresponding matrix of rapid decay, then ( a k ) k ∈ N ∈ S ( N ) andtherefore Φ( a (1 − SS ∗ )) = P k ∈ N a k e k ∈ ∩ n ∈ N dom( N n ) . Since each element from a ∈ A can be written as a = ( a − T σ ( a ) ) + T σ ( a ) ∈ K S + C ∞ ( T g,q ) it follows that Φ( A (1 − SS ∗ )) ⊂∩ n ∈ N dom( N n ) .Clearly, Φ is left A -linear since A ⊂ B ( ℓ ( N )) . To show the surjectivity of Φ , let v = P k ∈ N α k e k ∈ ∩ n ∈ N dom( N n ) . Then a := ( α k δ ,j ) k,j ∈ N ∈ K S , where δ i,j denotesthe Kronecker delta, and Φ( a (1 − SS ∗ )) = v . Moreover, Φ is injective. To see this, assumethat Φ( a (1 − SS ∗ )) = 0 . Then a (1 − SS ∗ ) e = Φ( a (1 − SS ∗ )) = 0 and a (1 − SS ∗ ) e k = 0 for all k = 0 since − SS ∗ is the orthogonal projection onto C e . Thus a (1 − SS ∗ ) = 0 in A (1 − SS ∗ ) ⊂ B ( ℓ ( N )) . So we just proved that the map Φ in (29) defines an isomorphismof left A -modules.It remains to show that Φ is actually an isometry. From (27) and (29), we get h Φ( a (1 − SS ∗ )) , Φ( a (1 − SS ∗ )) i = h ae , ae i = hh a, a ii for all a ∈ A . This completes the proof. (cid:3) Applying the previous proposition to the noncommutative torus C ( T ,q ) , we see thatthe “spin bundle” is given by A (1 − SS ∗ ) ⊕ A (1 − SS ∗ ) rather than A ⊕ A as it shouldhave been in analogy to the classical case. Moreover, by extending the isomorphism Φ in (29) to its closure, the Hilbert space A (1 − SS ∗ ) ∼ = ℓ ( N ) ⊂ L ( S ) looks rather likefunctions on the circle S than on the disc D . This observation is in line with the metricdimension 1. It seem that the interior of the quantum disc K ( ℓ ( N )) = ker( σ ) has thedimension of a (fuzzy) point. Also Equation (18) indicates that the action of the Diracoperator is essentially given by a derivation on the circle.3.4. Existence of a volume form or orientation.
By a volume form we mean aHochschild n -cycle ω , i.e., ω = P j a j ⊗ b j ⊗ a j ⊗ · · · ⊗ a nj ∈ A ⊗ A op ⊗ A ⊗ · · · ⊗ A , δ n ( ω ) := P j a j ⊗ ( b j a j ) ⊗ a j ⊗ · · · ⊗ a nj + n − P k =1 P j ( − k a j ⊗ b j ⊗ · · · ⊗ a kj a k +1 ,j ⊗ · · · ⊗ a nj ± P j a nj a j ⊗ b j ⊗ · · · ⊗ a n − ,j , satisfying(30) γ = π D ( ω ) := P j a j Jb ∗ j J − [ D, a j ] · · · [ D, a nj ] , where n depends on the (metric) dimension of the spectral triple.As pointed out in the paragraph preceding Proposition 3.6, the metric dimension ofour spectral triples is 1. However, for a Hochschild -cycle ω , the expression π D ( ω ) in(30) is an odd operator whereas γ is a diagonal operator, so Equation (30) cannot hold.Assuming that our quantum surfaces have dimension 2, we face the problem of the non-existence of the real structure J , see Proposition 3.4. Unfortunately the problem goesdeeper and cannot be resolved in any other way. SPECTRAL TRIPLE FOR NONCOMMUTATIVE COMPACT SURFACES 11
Proposition 3.8.
For all spectral triples from Section 3.1 and any anti-unitary operator J , there does not exist a Hochschild n -cycle satisfying (30) .Proof. For n odd, the same reasoning as in the case n = 1 applies: Since D is odd, theright hand side of (30) would yield an odd operator whereas γ is even, a contradiction.Assume now that there exist an anti-unitary operator J and a Hochschild k -cycle ω = P j a j ⊗ b j ⊗ a j ⊗ · · · ⊗ a k,j satisfying (30). Let ˆ σ : B ( ℓ ( N )) → B ( ℓ ( N )) /K ( ℓ ( N )) denote the cannonical projection and observe that the restriction of ˆ σ to the Toeplitzalgebra yields the symbol map. Recall that N k, kN ∈ K S for all k ∈ K S . From (18), weget for all a ∈ A ˆ σ ([ S ∗ N, a ]) = ˆ σ ([ S ∗ N, a − T σ ( a ) + T σ ( a ) ]) = ˆ σ ([ S ∗ N, T σ ( a ) ]) = ˆ σ ( − i T ¯ uσ ( a ) ′ ) = − i¯ uσ ( a ) ′ , and similarly ˆ σ ([ N S, a ]) = − i uσ ( a ) ′ . Using the facts that the operators σ ( t ) , t ∈ T ,commute and that u ¯ u = 1 , and applying ˆ σ to two consecutive commutators in (30), weobtain ˆ σ ([ D, a i,j ][ D, a i +1 ,j ]) = (cid:18) ˆ σ ([ S ∗ N, a i,j ][ N S, a i +1 ,j ]) 00 ˆ σ ([ N S, a i,j ][ S ∗ N, a i +1 ,j ]) (cid:19) = (cid:18) − σ ( a i,j ) ′ σ ( a i +1 ,j ) ′ − σ ( a i,j ) ′ σ ( a i +1 ,j ) ′ (cid:19) Note that the diagonal entries coincide. This remains true for the right hand side of (30)if n is even. On the other hand, the diagonal entries of γ differ by a minus sign. Thereforeapplying ˆ σ to the diagonal elements of (30) and equating gives P j ( − k ˆ σ ( a j Jb ∗ j J − ) σ ( a ,j ) ′ σ ( a ,j ) ′ · · · σ ( a k − ,j ) ′ σ ( a k,j ) ′ = − , which is a contradiction. (cid:3) Observe that the problem of Proposition 3.8 cannot be solved by changing γ as in [13]as long as the diagonal entries of γ do not coincide.3.5. Poincaré duality.
We mentioned already in Section 3.1 that regular spectral triplesgive rise to index pairings. To be more precise, one can show that, for an even regularspectral triple ( A , H , D, γ ) and a projection P = P = P ∗ ∈ Mat n × n ( A ) , the map π ( P ) D + − π ( P ) : π ( P ) H n − → π ( P ) H n + yields a Fredholm operator and its index does notdepend on the K -class of P . Here, D + − denotes the upper right corner of the oddoperator D . Unfortunately, the formulation of Poincaré duality involves a real structure J . One says that an even spectral triple with real structure J satisfies Poincaré duality [3],if(31) K ( A ) × K ( A ) ∋ ([ P ] , [ Q ]) ind(( π ( P ) ⊗ Jπ ( Q ) J − ) D + − ( π ( P ) ⊗ Jπ ( Q ) J − )) ∈ Z defines a non-degenerate pairing, where P ∈ Mat n × n ( A ) and Q ∈ Mat k × k ( A ) are projec-tions and the operator in (31) acts between the Hilbert spaces ( π ( P ) ⊗ Jπ ( Q ) J − ) H nk − and ( π ( P ) ⊗ Jπ ( Q ) J − ) H nk + .The aim of this section is to explain that the Poincaré duality fails for our spectral triple.The statement doesn’t seem to make much sense because of the non-existence of a realstructure proven in Section 3.2. However, the main problem is not the real structure. Thenext proposition will show that the fundamental class of the Dirac operator only detectsthe rank of the trivial K -classes and leads to a zero pairing with K -classes representedby compact operators. Let us recall here from [14] (see also the end Section 2) that K ( C ( T g,q )) ∼ = Z ⊕ Z ∼ = K ( K ( ℓ ( N ))) ⊕ K ( C ) . Proposition 3.9.
Let ( A , H , D, γ ) denote the spectral triple from Theorem 3.2. For anyodd or even anti-unitary operator J on H = ℓ ( N ) ⊕ ℓ ( N ) , the index pairing (31) (if welldefined) is degenerate.Proof. Let p e denote the projection 1-dimensional projection onto C e ⊂ H − = ℓ ( N ) .Then [ p e ] ∈ K ( C ( T g,q )) , and for any projection Q := π ( Q ) with [ Q ] ∈ K ( C ( T g,q )) ,we have dim(( p e ⊗ JQJ − ) H n − )) = dim( JQJ − (( p e ℓ ( N )) ⊗ C n )) ≤ dim( C ⊗ C n ) = n < ∞ . Hence the operator ( p e ⊗ JQJ − ) S ∗ N ( p e ⊗ JQJ − ) : ( p e ⊗ JQJ − ) ℓ ( N ) n → ( p e ⊗ JQJ − ) ℓ ( N ) n acts between finite dimensional spaces and therefore its index is always 0. (cid:3) Note that the problem arises because the compact operators K ( ℓ ( N )) ⊂ C ( T g,q ) acton H − = ℓ ( N ) by the identity and thus dim( p e H − ) = 1 < ∞ . So we arrive again at theconclusion that the Hilbert space H = ℓ ( N ) ⊕ ℓ ( N ) is “too small” for a non-trivial indexpairing. Acknowledgment.
This work was supported by CIC-UMSNH and is part of the inter-national project “Quantum Dynamics” supported by EU grant H2020-MSCA-RISE-2015-691246 and co-financed by Polish Government grant 3542/H2020/2016/2 awarded for theyears 2016-2019.
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SPECTRAL TRIPLE FOR NONCOMMUTATIVE COMPACT SURFACES 13
Instituto de Física y Matemáticas, Universidad Michoacana de San Nicolás de Hidalgo,Cd. Universitaria, Edificio C-3, 58040 Morelia, Michoacán, México
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