A Spectral Triple for a Solenoid Based on the Sierpinski Gasket
aa r X i v : . [ m a t h . OA ] J un A SPECTRAL TRIPLE FOR A SOLENOID BASED ON THE SIERPINSKIGASKET
VALERIANO AIELLO, DANIELE GUIDO, AND TOMMASO ISOLA
Abstract.
The Sierpinski gasket admits a natural locally isometric ramified self-covering.A semifinite spectral triple is constructed on the resulting solenoidal space, and its maingeometrical features are discussed. Introduction
The topological space known as Sierpinski gasket [17] is considered here, its geometricalproperties being described by the discrete spectral triple considered in [13], cf. also [12].It turns out that this space possesses a ramified self-covering, which is locally isometricwith respect to the Euclidean metric on the base and on the covering, which are induced bysuitable spectral triples.On the one hand such self-covering produces a tower of covering spaces and hence a compactsolenoidal space; on the other hand it gives rise to an endomorphism of the C ∗ -algebra ofcontinuous functions on the gasket, hence to an inductive family of C ∗ -algebras and to aninductive limit C ∗ -algebra (cf. [6]) which is simply the algebra of continuous functions on thesolenoid.Each of the C ∗ -algebras of the inductive family may then be endowed with a spectral triple[5, 9]. The local isometricity of the covering implies that the Lip-norms given by the Diracoperators on the algebras are compatible with the inductive maps.As we did in [1], our aim here is to show that this compatible family of spectral triplesproduces a spectral triple on the solenoidal space. Because of the local isometricity, thelimiting Dirac operator has no longer compact resolvent, but its resolvent is τ -compact withrespect to a natural trace on the C ∗ -algebra of geometric operators. This implies that thespectral triple on the solenoid is semifinite [3].However, in the examples considered in [1], the family of spectral triples had a simple tensorproduct structure, namely the Hilbert spaces were a tensor product of the Hilbert space for thebase space and a finite dimensional Hilbert space, and the Dirac operators could be describedas (a finite sum of) tensor product operators. Then the ambient C ∗ -algebra turned out tobe a product of B ( H ) for the base space and a UHF algebra, allowing a GNS representationw.r.t. a semifinite trace.In the example treated here two problems forbid such simple description. The first is alocal problem, due to the ramification points. This implies that the algebra of a coveringis not a free module on the algebra of the base space; in particular, functions on a coveringspace form a proper sub-algebra of the direct sum of finitely many copies of the algebra forthe base space. Mathematics Subject Classification.
Key words and phrases.
Self-similar fractals, Noncommutative geometry, ramified coverings.This work has been partially supported by GNAMPA, MIUR and the ERC Advanced Grant QUEST .
The second is a non-local problem which concerns the Hilbert spaces, which are ℓ spaceson edges, and the associated operator algebras. Indeed, the Hilbert spaces of the coveringscannot be described as finite sums of copies of the Hilbert space on the base space due toappearance of longer and longer edges on larger and larger coverings.Therefore the construction of the semifinite trace has to be done in a different way, thepresent method being inspired by the treatment in [16, 10, 4].The idea in [16], partly modified in [10], was to replace the von Neumann trace used byAtiyah [2] for his index theorem for covering manifolds, by a trace on the C ∗ -algebra of finitepropagation operators acting on sections of a bundle on an open manifold. Unfortunatelysuch trace is not canonical, since it depends on a generalized limit procedure. However, in,the case of infinite self-similar CW-complexes, it was observed in [4] that such trace becomescanonical when restricted to the C ∗ -algebra of geometrical operators. We adapt these resultsto our present context, namely that of an infinite fractafold [18], which has some featuressimilar to those of an open manifold, but also sufficient self-similarity to obtain a canonicaltrace.We then construct a semifinite spectral triple on the algebra of continuous functions onthe Sierpinski solenoid, where the main role of the semifinite von Neumann algebra is insteadplayed by the C ∗ -algebra of geometric operators together with its semicontinuous semifinitetrace. Finally, we show that the metric dimension of such spectral triple coincides with theHausdorff dimension of the Sierpinski gasket, the noncommutative integral coincides with thenormalized integral w.r.t. the Hausdorff measure on a dense sub-algebra, and the Connesdistance on points recovers exactly the geodesic distance on the open fractafold.We mention that the construction given in the present paper goes in the direction of possiblydefining a C ∗ spectral triple, in which the semifinite von Neumann algebra is replaced by a C ∗ -algebra with a trace to which both the Dirac operator and the ”functions” on the noncommutative space are affiliated, where the compactness of the resolvent of the Dirac operatoris measured by the trace on the C ∗ -algebra, cf. also [11].This paper is divided in five sections. After this introduction, Section 2 describes the ge-ometry of the ramified covering and the corresponding inductive structure, together with itsfunctional counterpart given by a family of compatible spectral triples. Section 3 concernsthe self-similarity structure of the Sierpinski solenoid, whence the description of the inductivefamily of C ∗ -algebras as algebras of bounded functions on the fractafold. The 4th Sectiondescribes the algebra of geometric operators and the construction of a semicontinuous semifi-nite trace on it. Finally, the semifinite spectral triple together with its main features arecontained in Section 5.2. A ramified covering of the Sierpinski gasket
Let us recall that the Sierpinski gasket K may be defined as the compact subset of aclosed equilateral triangle with vertices v , v , v (numbered in a counterclockwise order) inthe Euclidean plane with the property K = [ j =0 , , w j ( K ) , where w j is the dilation around v j with contraction parameter 1 /
2. For the sake of simplicity,we assume that the length of the side of K is 1. Definition 2.1.
We call cell any element of the family { w i · · · · · w i k ( K ) : k ≥ } . The sizeof a cell is the length of its side; clearly if C = w i · · · · · w i k ( K ), size( C ) = 2 − k . SPECTRAL TRIPLE FOR A SOLENOID BASED ON THE SIERPINSKI GASKET 3
We call (oriented) edge any element of the family E = { w i · · · · · w i k ( e ) : k ≥ , e is one ofthe oriented edges of the triangle } . Clearly if e is one of the oriented edges of the triangleand e = w i · · · · · w i k ( e ), length( e ) = 2 − k .Let us now consider the set K = w − K , the middle point x i,i +1 of the segment ( w − v i , w − v i +1 ), i = 0 , ,
2, the map R i +1 ,i : w − w i K → w − w i +1 K consisting of the rotation of π around thepoint x i,i +1 , i = 0 , ,
2, and observe that(2.1) R i,i +2 ◦ R i +2 ,i +1 ◦ R i +1 ,i = id w − w i K , i = 0 , , . Setting R i,i +1 = R − i +1 ,i , the previous identities may also be written as R i +2 ,i +1 ◦ R i +1 ,i = R i +2 ,i , i = 0 , , p : K → K given by p ( x ) = x, x ∈ K,R , ( x ) , x ∈ w − w K,R , ( x ) , x ∈ w − w K, and observe that this map, which appears to be doubly defined in the points x i,i +1 , i = 0 , , Proposition 2.2.
The map p is a well defined continuous map which is a ramified covering,with ramification points given by { x i,i +1 , i = 0 , , } . Moreover, the covering map is isometricon suitable neighbourhoods of the non-ramification points. Since K and K are homeomorphic, this map may be seen as a self-covering of the gasket.The map p gives rise to an embedding α , : C ( K ) → C ( K ), hence, following [6], to aninductive family of C ∗ -algebras A n = C ( K n ), K n = w − n K , whose inductive limit A ∞ consistsof continuous function on the solenoidal space based on the gasket.Following [12, 13], we build a spectral triple on any of the algebras A n , n ≥
0. Denotingby E n = { w − n e, e ∈ E } the set of oriented edges in K n , we define H n = ℓ ( E n ), F asthe map the changes the orientation of an edge, D n as the map sending an edge e ∈ E n tolength( e ) − F e . As proved in [12, 13], the triple ( A n , H n , D n ) is a spectral triple. Moreover,by [13], Corollary 5.14, the Connes distances induced by these spectral triples recover thegeodesic distances on the points of the gaskets K n , hence the local isometricity of p implies L m + q ( α m + q,m ( f )) = L m ( f ), where L m ( f ) = k [ D, f ] k is the Lip-norm associated with thetriple ( A m , H m , D m ).3. A groupoid of local isometries on the infinite Sierpinski fractafold
Let us consider the infinite fractafold K ∞ = ∪ n ≥ K n [18] endowed with the Hausdorffmeasure µ d of dimension d = log 3log 2 normalized to be 1 on K = K , with the exhaustion { K n } n ≥ , and with the family of local isometries R = { R ni +1 ,i , R ni,i +1 : i = 0 , , , n ≥ } ,where R ni,j = w − n R i,j w n : C nj → C ni , and C ni := w − n − w i K , n ≥ i, j ∈ { , , } . We alsodenote by s ( γ ) and r ( γ ) the domain and range of the local isometry γ . Such local isometriesact on points and on oriented edges of K ∞ .We say that the product of the two local isometries γ , γ ∈ R is defined if γ − ( s ( γ )) ∩ s ( γ ) = ∅ . In this case we consider the product γ · γ : γ − ( s ( γ )) ∩ s ( γ ) → r (cid:0) γ | s ( γ )) ∩ r ( γ ) (cid:1) . VALERIANO AIELLO, DANIELE GUIDO, AND TOMMASO ISOLA
We then consider the family G consisting of all (the well-defined) finite products of isome-tries in R . Clearly, any γ in G is a local isometry, and its domain and range are cells of thesame size. We set G n = { g ∈ G : s ( γ ) & ρ ( γ ) are cells of size 2 n } , n ≥ Proposition 3.1.
For any n ≥ , C , C cells of size n , ∃ ! γ ∈ G n such that s ( γ ) = C , r ( γ ) = C . In particular, if C has size n , the identity map of C belongs to G n , n ≥ .Proof. It is enough to show that for any cell C of size 2 n there exists a unique γ ∈ G n such that γ : C → K n . For any cell C , let m = level( C ) be the minimum number such that C ⊂ K m .We prove the existence: if C has size 2 n and level( C ) = m > n , then C ⊂ C m − i , for some i = 1 ,
2, hence R m − ,i ( C ) ⊂ K m − . Iterating, the result follows. The second statement followsdirectly by equation (2.1).As for the uniqueness, ∀ n ≥
0, we call R ni, ascending, i = 1 , R n ,i descending, i = 1 , R ni,j constant-level, i, j ∈ { , } . Indeed, if C ⊂ s ( R ni, ), then level( C ) ≤ n and level( R ni, ( C )) = n + 1; if C ⊂ s ( R n ,i ), then level( C ) = n + 1, level( R n ,i ( C )) ≤ n and level( R nj,i ( C )) = n + 1, i, j ∈ { , } , n ≥ • The product R nl,k · R mj,i of two constant-level elements R nl,k , R mj,i is defined iff n = m and k = j , therefore any product of constant-level elements in R is either the identitymap on the domain or coincides with a single constant-level element. • Any product of constant level elements in R followed by a descending element coincideswith a single descending element: indeed, if the product of constant level elements isthe identity, the statement is trivially true; if it coincides with a single element, say R ni,j with i, j ∈ { , } , then, by compatibility, the descending element should be R n ,i so that the product is R n ,i , by equation (2.1). • Given a cell C with size( C ) = 2 n and level( C ) > n , the exists a unique descendingelement γ ∈ R such that C ⊂ s ( γ ): indeed, if m = level( C ), then C ⊂ C m − i , for some i ∈ { , } . The only descending element is then γ = R m − ,i . • Any product of an ascending element followed by a descending one is the identityon the domain: indeed if the ascending element is R ni, , then, by compatibility, thedescending element should be R n ,i .Now let size( C ) = 2 n , γ ∈ G n such that γ : C → K n , γ = γ p · γ p − · . . . γ · γ , where γ j ∈ R ,1 ≤ j ≤ p . Since level( C ) ≥ level( K n ) = n , for any possible ascending element γ i thereshould be a j > i such that γ j is descending. If i + q is the minimum among such j ’s, allterms γ j , i < j < i + q , are constant-level, hence the product γ i + q · γ i + q − · · · · · γ i = id s ( γ i ) .Then, we note that γ p can only be descending. As a consequence, γ can be reduced to aproduct of descending elements, and, by the uniqueness of the descending element acting ona given cell, we get the result. (cid:3) Let us observe that each G n , and so also G , is a groupoid under the usual composition rule,namely two local isometries are composable if the domain of the first coincides with the rangeof the latter.We now consider the action on points of the local isometries in G . Proposition 3.2.
Let us define e A n as the algebra e A n = { f ∈ C b ( K ∞ ) : f ( γ ( x )) = f ( x ) , x ∈ s ( γ ) , γ ∈ G n } . SPECTRAL TRIPLE FOR A SOLENOID BASED ON THE SIERPINSKI GASKET 5
Then, for any n ≥ , the following diagram commutes, (3.1) e A n ⊂ e A n +1 y ι n y ι n +1 A n α n +1 ,n −→ A n +1 where ι n : f ∈ e A n → f | K n ∈ A n are isomorphisms. Hence the inductive limit A ∞ isisomorphic to a C ∗ -subalgebra of C b ( K ∞ ) .Proof. The request in the definition of e A n means that the value of f in any point of K ∞ isdetermined by the value on K n , while such request gives no restrictions on the values of f on K n . The other assertions easily follow. (cid:3) As shown above, we may identify the algebra A n , 0 ≤ n ≤ ∞ , with its isomorphic copy e A n in C b ( K ∞ ), so that the embeddings α k,j become inclusions. Moreover, we may consider theoperator e D n on ℓ ( E ∞ ), with E ∞ = ∪ n ≥ E n , given by e D n e = length( e ) − F e , if length( e ) ≤ n ,and e D n e = 0, if length( e ) > n . Then the spectral triples ( A n , H n , D n ) are isomorphicto the spectral triples ( e A n , H n , e D n ), where C b ( K ∞ ) acts on the space ℓ ( E ∞ ) through therepresentation ρ given by ρ ( f ) e = f ( e + ) e , where e + denotes the target of the oriented edge e . Remark . Because of the isomorphism above, from now on we shall remove the tildes anddenote by A n the subalgebras of C b ( K ∞ ) and by D n the operators acting on ℓ ( E ∞ ).4. The C ∗ -algebra of geometric operators and a tracial weight on it. We now come to the action of local isometries on edges. We shall use the following notation,where in the table below to any subset of edges listed on the left we indicate on the right theprojection on the closed subspace spanned by the same subset:
Table 1.
Edges and projections.
Subsets of E ∞ Projections E n = { e ⊂ K n } , n ≥ P n E k,pn = { e ∈ E n : 2 k ≤ length( e ) ≤ p } , for k ≤ p ≤ n P k,pn E kn = E k,kn = { e ∈ E n : length( e ) = 2 k } , for k ≤ n P kn E k,p = ∪ n E k,pn = { e ∈ E ∞ : 2 k ≤ length( e ) ≤ p } P k,p E k = E k,k = { e ∈ E ∞ : length( e ) = 2 k } P k E C = { e ∈ E ∞ : e ⊂ C } , C being a cell P C .Let us note that any local isometry γ ∈ G , γ : s ( γ ) → r ( γ ), gives rise to a partial isometry V γ defined as V γ e = ( γ ( e ) , e ⊂ s ( γ ) , , elsewhere . In particular, if C is a cell, and γ = id C , V γ = P C . We then consider the subalgebras B n of B ( ℓ ( E ∞ )), B n = { V γ : γ ∈ G m , m ≥ n } ′ , B fin = [ n B n , B ∞ = B fin , VALERIANO AIELLO, DANIELE GUIDO, AND TOMMASO ISOLA and note that the elements of B n commute with the projections P C , for all cells C s.t.size( C ) ≥ n . By definition, the sequence B n is increasing, therefore, since the B n ’s are vonNeumann algebras, B ∞ is a C ∗ -algebra. Let us observe that, ∀ n ≥ ρ ( A n ) ⊂ B n . Definition 4.1.
The elements of the C ∗ -algebra B ∞ are called geometric operators.We now consider the hereditary positive cone(4.1) I +0 = { T ∈ B +fin : ∃ c T ∈ R such that tr( P m T ) ≤ c T µ d ( K m ) , ∀ m ≥ } . Lemma 4.2.
For any T ∈ I +0 , the sequence tr( P m T ) µ d ( K m ) is eventually increasing, hence convergent.In particular (4.2) tr( P pp T ) = 0 ∀ p > m ⇒ τ ( T ) = tr( P m T ) µ d ( K m ) Proof.
Let T ∈ B + n . Then we have, for m ≥ n ,tr( P m +1 T ) = X e ⊂ K m +1 ( e, T e ) = X i =0 , , X e ∈ C mi ( e, T e ) + X e ∈ E m +1 m +1 ( e, T e ) = 3 tr( P m T ) + tr( P m +1 m +1 T ) , hence(4.3) tr( P m +1 T ) µ d ( K m +1 ) = tr( P m T ) µ d ( K m ) + tr( P m +1 m +1 T ) µ d ( K m +1 ) , from which the thesis follows. (cid:3) We then define the weight τ on B + ∞ as follows:(4.4) τ ( T ) = lim m →∞ tr( P m T ) µ d ( K m ) , T ∈ I +0 , elsewhere . The next step is to regularize the weight τ in order to obtain a semicontinuous semifinitetracial weight τ on B ∞ . Lemma 4.3.
For any T ∈ I +0 , A ∈ B fin , it holds AT A ∗ ∈ I +0 , and τ ( AT A ∗ ) ≤ k A k τ ( T ) .Proof. Let A ∈ B n . Then, for any m > n , we havetr( P m AT A ∗ ) = tr( A ∗ AP m T ) ≤ k A ∗ A k tr( P m T ) ≤ k A k c T µ d ( K m ) , and the thesis follows. (cid:3) Proposition 4.4.
For all p ∈ N , recall that P − p, ∞ is the orthogonal projection onto the closedvector space generated by { e ∈ ℓ ( E ∞ ) : length( e ) ≥ − p } , and let ϕ p ( T ) := τ ( P − p, ∞ T P − p, ∞ ) , ∀ T ∈ B + ∞ . Then P − p, ∞ ∈ B , ϕ p is a positive linear functional, and ϕ p ( T ) ≤ ϕ p +1 ( T ) ≤ τ ( T ) , ∀ T ∈ B + ∞ .Proof. We first observe that(4.5) tr( P jn ) = { e ∈ K n : length( e ) = 2 j } = 6 · n − j , j ≤ n. Then it is easy to verify that P − p, ∞ ∈ B . Since(4.6) ϕ p ( I ) = τ ( P − p, ∞ ) = lim n →∞ tr P − p,nn µ d ( K n ) = lim n →∞ − n n X j = − p tr( P jn ) = ∞ X j = − p · − j = 3 p +2 , SPECTRAL TRIPLE FOR A SOLENOID BASED ON THE SIERPINSKI GASKET 7 ϕ p extends by linearity to a positive functional on B ∞ . Moreover, by Lemma 4.3, ϕ p ( T ) ≤ τ ( T ), ∀ T ∈ B + ∞ . Finally, since P − p, ∞ P n = P n P − p, ∞ = P −∞ ,nn , ∀ n ∈ N , we get, for all T ∈ B + ∞ , ϕ p +1 ( T ) − ϕ p ( T ) = τ ( P − ( p +1) , ∞ T P − (+1) p, ∞ ) − τ ( P − p, ∞ T P − p, ∞ )= lim n →∞ tr(( P − ( p +1) ,n ) n − P − p,nn ) T ) µ d ( K n ) = lim n →∞ tr( P − ( p +1)) n T ) µ d ( K n ) ≥ . (cid:3) Proposition 4.5.
Let τ ( T ) := lim p →∞ ϕ p ( T ) , ∀ T ∈ B + ∞ . Then ( i ) τ is a lower semicontinuous weight on B ∞ , ( ii ) τ ( T ) = τ ( T ) , ∀ T ∈ I +0 .Proof. ( i ) Let T ∈ B + ∞ . Since { ϕ p ( T ) } p ∈ N is an increasing sequence, there exists lim p →∞ ϕ p ( T ) =sup p ∈ N ϕ p ( T ). Then τ is a weight on B + ∞ . Since ϕ p is continuous, τ is lower semicontinuous.( ii ) Let us prove that, ∀ T ∈ B + n ,(4.7) tr( P jm T ) µ d ( K m ) = tr( P jn T ) µ d ( K n ) , j ≤ n ≤ m. Indeed,tr( P jm +1 T ) = X e ⊂ K m +1 length( e )=2 j ( e, T e ) = X i =0 X e ⊂ C mi length( e )=2 j ( e, T e ) = X i =0 X e ⊂ K m length( e )=2 j ( V R mi e, T V R mi e )= X i =0 X e ⊂ K m length( e )=2 j ( e, T e ) = 3 tr( P jm T ) , from which (4.7) follows. Let us now prove that(4.8) τ ( T ) = sup p ∈ N ϕ p ( T ) = τ ( T ) , T ∈ I +0 . Let T ∈ B + n ∩ I +0 , and ε >
0. From the definition of τ ( T ), there exists r ∈ N , r > n ,such that tr( P r T ) µ d ( K r ) > τ ( T ) − ε . Since tr( P r T ) µ d ( K r ) = P rj = −∞ tr( P jr T ) µ d ( K r ) , there exists p ∈ N such that P rj = − p tr( P jr T ) µ d ( K r ) > tr( P r T ) µ d ( K r ) − ε > τ ( T ) − ε . Then, for any s ∈ N , s > r , we havetr( P s P − p, ∞ T P − p, ∞ P s ) µ d ( K s ) = s X j = − p tr( P js T ) µ d ( K s ) = r X j = − p tr( P js T ) µ d ( K s ) + s X j = r +1 tr( P js T ) µ d ( K s ) (4.7) = r X j = − p tr( P jr T ) µ d ( K r ) + s X j = r +1 tr( P js T ) µ d ( K s ) > τ ( T ) − ε, and, passing to the limit for s → ∞ , we get ϕ p ( T ) = τ ( P − p, ∞ T P − p, ∞ ) = lim s →∞ tr( P s P − p, ∞ T P − p, ∞ P s ) µ d ( K s ) ≥ τ ( T ) − ε, and equation (4.8) follows. (cid:3) We want to prove that τ is a tracial weight. VALERIANO AIELLO, DANIELE GUIDO, AND TOMMASO ISOLA
Definition 4.6.
An operator U ∈ B ( ℓ ( E ∞ )) is called δ -unitary, δ >
0, if k U ∗ U − k < δ ,and k U U ∗ − k < δ .Let us denote with U δ the set of δ -unitaries in B fin and observe that, if δ < U δ consistsof invertible operators, and U ∈ U δ implies U − ∈ U δ/ (1 − δ ) . Proposition 4.7.
The weight τ is is ε -invariant for δ -unitaries in B fin , namely, for any ε ∈ (0 , , there is δ > s.t., for any U ∈ U δ , and T ∈ B + ∞ , (1 − ε ) τ ( T ) ≤ τ ( U T U ∗ ) ≤ (1 + ε ) τ ( T ) . Proof.
We first observe that, if δ ∈ (0 ,
1) and U ∈ U δ , T ∈ I +0 ⇔ U T U ∗ ∈ I +0 . Indeed,choose n such that U, T ∈ B n . Then tr( P n U T U ∗ ) = tr( U ∗ U P n T P n ) ≤ k U ∗ U k tr( P n T ) ≤ (1 + δ ) c T µ d ( K n ), ∀ n ∈ N , so that U T U ∗ ∈ I +0 . Moreover, τ ( U T U ∗ ) = lim n →∞ tr( P n U T U ∗ ) µ d ( K n ) ≤ k U ∗ U k lim n →∞ tr( P n T ) µ d ( K n ) = k U ∗ U k τ ( T ) < (1 + δ ) τ ( T ) . Conversely,
U T U ∗ ∈ I +0 , and U − ∈ U δ/ (1 − d ) = ⇒ T ∈ I +0 . Moreover, τ ( T ) ≤ k ( U − ) ∗ U − k τ ( U T U ∗ ) < − δ τ ( U T U ∗ ) . The result follows by the choice δ = ε . (cid:3) Theorem 4.8.
The lower semicontinuous weight τ in Proposition 4.5 is a trace on B ∞ , thatis, setting J + := { A ∈ B + ∞ : τ ( A ) < ∞} , and extending τ to the vector space J generated by J + , we get ( i ) J is an ideal in B ∞ , ( ii ) τ ( AB ) = τ ( BA ) , for any A ∈ J , B ∈ B ∞ .Proof. ( i ) Let us prove that J + is a unitarily-invariant face in B + ∞ , and suffices it to provethat A ∈ J + implies that U AU ∗ ∈ J + , for any U ∈ U ( B ∞ ), the set of unitaries in B ∞ .To reach a contradiction, assume that there exists U ∈ U ( B ∞ ) such that τ ( U AU ∗ ) = ∞ .Then there is p ∈ N such that ϕ p ( U AU ∗ ) > τ ( A ) + 2. Let δ < V ∈ U δ implies τ ( V AV ∗ ) ≤ τ ( A ), and let U ∈ B fin be such that k U − U k < min { δ , k A kk ϕ p k } . Theinequalities k U U ∗ − k = k U ∗ U U ∗ − U ∗ k ≤ k U ∗ U − kk U ∗ k + k U ∗ − U ∗ k < δ and k U ∗ U − k < δ , prove that U ∈ U δ . Since | ϕ p ( U AU ∗ ) − ϕ p ( U AU ∗ ) | ≤ k ϕ p kk A kk U − U k <
1, we get2 τ ( A ) ≥ τ ( U AU ∗ ) ≥ ϕ p ( U AU ∗ ) ≥ ϕ p ( U AU ∗ ) − ≥ τ ( A ) + 1which is absurd.( ii ) We only need to prove that τ is unitarily-invariant. Let A ∈ J + , U ∈ U ( B ∞ ). For any ε >
0, there is p ∈ N such that ϕ p ( U AU ∗ ) > τ ( U AU ∗ ) − ε , since, by (1), τ ( U AU ∗ ) is finite.Then, arguing as in the proof of (1), we can find U ∈ B fin , so close to U that | ϕ p ( U AU ∗ ) − ϕ p ( U AU ∗ ) | < ε (1 − ε ) τ ( A ) ≤ τ ( U AU ∗ ) ≤ (1 + ε ) τ ( A ) . Then τ ( A ) ≥
11 + ε τ ( U AU ∗ ) ≥
11 + ε ϕ p ( U AU ∗ ) ≥
11 + ε ( ϕ p ( U AU ∗ ) − ε ) ≥
11 + ε ( τ ( U AU ∗ ) − ε ) . SPECTRAL TRIPLE FOR A SOLENOID BASED ON THE SIERPINSKI GASKET 9
By the arbitrariness of ε >
0, we get τ ( A ) ≥ τ ( U AU ∗ ). Exchanging A with U AU ∗ , we getthe thesis. (cid:3) Proposition 4.9.
The lower semicontinuous tracial weight τ defined in Proposition 4.5 issemifinite and faithful.Proof. Let us recall that, for any p ∈ N , P − p, ∞ ∈ I +0 by Proposition 4.4. From Proposition4.5 follows that τ ( P − p, ∞ ) = τ ( P − p, ∞ ) < ∞ , hence P − p, ∞ ∈ J + . Then, for any T ∈ B + ∞ , S p := T / P − p, ∞ T / ∈ J + , and 0 ≤ S p ≤ T . Moreover, τ ( S p ) = τ ( T / P − p, ∞ T / ) = τ ( P − p, ∞ T P − p, ∞ ) = sup q ∈ N τ ( Q q P − p, ∞ T P − p, ∞ Q q )= τ ( P − p, ∞ T P − p, ∞ ) = ϕ p ( T ) , so that sup p ∈ N τ ( S p ) = τ ( T ), and τ is semifinite. Finally, if T ∈ B + ∞ is such that τ ( T ) = 0,then sup p ∈ N ϕ p ( T ) = 0. Since { ϕ p ( T ) } p ∈ N is an increasing sequence, ϕ p ( T ) = 0, ∀ p ∈ N .Then, for a fixed p ∈ N , we get 0 = τ ( P − p, ∞ T P − p, ∞ ) = lim n →∞ tr( P n P − p, ∞ T P − p, ∞ P n ) µ d ( K n ) . Since thesequence { tr( P n P − p, ∞ T P − p, ∞ P n ) µ d ( K n ) } n ∈ N is definitely increasing, we get tr( P n P − p, ∞ T P − p, ∞ P n ) = 0definitely, that is T P − p, ∞ P n = 0 definitely, so that T P − p, ∞ = 0. By the arbitrariness of p ∈ N , we get T = 0. (cid:3) A semifinite spectral triple on the inductive limit A ∞ Since the covering we are studying is ramified, the family { A n , H n , D n } does not havea simple tensor product structure, contrary to what happened in [1]. We therefore use adifferent approach to construct a semifinite spectral triple on A ∞ : our construction is indeedbased on the pair ( B ∞ , τ ) of the C ∗ -algebra of geometric operators and the semicontinuoussemifinite weight on it.The Dirac operator will be defined through its phase F defined above and the functionalcalculi of its modulus with continuous functions vanishing at ∞ . More precisely we shall usethe following Definition 5.1.
Let ( C , τ ) be a C ∗ -algebra with unit endowed with a semicontinuous semifi-nite faithful trace. A selfadjoint operator T affiliated to ( C , τ ) is defined as a pair given bya closed subset σ ( T ) in R and a ∗ homomorphism φ : C ( σ ( T )) → C , f ( T ) def = φ ( f ), providedthat the support of such homomorphism is the identity in the GNS representation π τ inducedby the trace τ .The previous definition was inspired by that in [7] appendix A, and should not be confusedwith that of Woronowicz for C ∗ -algebras without identity. Remark . Let us observe that the ∗ -homomorphism φ τ = π τ ◦ φ extends to bounded Borelfunctions on R and e ( −∞ ,t ] def = φ τ ( χ ( −∞ ,t ] ) tends strongly to the identity when t → ∞ , henceit is a spectral family. We shall denote by π τ ( T ) the selfadjoint operator affiliated to π τ ( C ) ′′ given by π τ ( T ) def = Z R t de ( −∞ ,t ] . Proposition 5.3.
Let T be a selfadjoint operator affiliated to ( C , τ ) as above. (a) Assume that for any n ∈ N , there is ϕ n ∈ C ( R ) : 0 ≤ ϕ n ≤ , ϕ n = 1 for | t | ≤ a n , ϕ n ( t ) = 0 for | t | ≥ b n with < a n < b n and { a n } , { b n } increasing to ∞ . Then, for any A ∈ C , if sup n k [ T · ϕ n ( T ) , A ] k = C < ∞ then [ π t ( T ) , π τ ( A )] is bounded and k [ π t ( T ) , π τ ( A )] k = C . (b) If τ ( f ( T )) < ∞ for any positive function f with compact support on the spectrum of T then π τ ( T ) has τ -compact resolvent.Proof. ( a ). Let D be the domain of π τ ( T ), D the space of vectors in D with boundedsupport w.r.t. to π τ ( T ), and consider the sesquilinear form F ( y, x ) = ( π τ ( T ) y, π τ ( A ) x ) − ( y, π τ ( A ) π τ ( T ) x ) defined on D . By hypothesis, for any x, y ∈ D there exists n such that π τ ( ϕ n ( T )) x = x and π τ (( ϕ n ( T )) y = y , hence F ( y, x ) = ( y, π τ ([ T · ϕ n ( T ) , A ]) x ) ≤ C k x k k y k .By the density of D in D w.r.t.˜the graph norm of π τ ( T ), the same bound holds on D . Then for y, x ∈ D , | ( π τ ( T ) y, π τ ( A ) x ) | ≤ | ( y, π τ ( A ) π τ ( T ) x ) | + | F ( y, x ) | ≤ ( k π τ ( A ) π τ ( T ) x k + C k x k ) k y k which implies π τ ( A ) x belongs to the domain of π τ ( T ) ∗ = π τ ( T ). Therefore π τ ( T ) π τ ( A ) − π τ ( A ) π τ ( T ) is defined on D and its norm is bounded by C . Since C is the optimal bound forthe sesquilinear form F it is indeed the norm of the commutator.( b ) Let λ be in the resolvent of | T | . We then note that for any f positive and zero on aneighbourhood of the origin there is a g positive and with compact support such that f (( | T | − λI ) − ) = g ( | T | ). Therefore τ ( f (( | T | − λI ) − )) < ∞ , hence τ ( e ( t, + ∞ ) ( π τ (( | T | − λI ) − ))) < ∞ for any t >
0, which is the definition of τ -compactess (cf. [8] Proposition 3.2). (cid:3) We then consider the Dirac operator D = F | D | on ℓ ( E ∞ ), where F is the orientationreversing operator on edges and | D | = X n ∈ Z − n P n , σ ( | D | ) = { − n , n ∈ Z } ∪ { } . Proposition 5.4.
The following hold: (a)
The elements D and | D | are affiliated to ( B ∞ , τ ) . (b) The following formulas hold: τ ( P n ) = 6 · − n , τ ( P − p, ∞ ) = 3 p +2 , as a consequence theoperator D has τ -compact resolvents (c) The trace τ ( I + D ) − s/ < ∞ if and only if s > d = log 3log 2 and Res s = d τ ( I + D ) − s/ = 6log 2 . Proof. (a) We first observe that the ∗ -homomorphisms for D and | D | have the same supportprojection, then note that since F and P n belong to B (which is a von Neumann algebra)for any n ∈ N , then f ( D ) and f ( | D | ) belong to B for any f ∈ C ( R ); therefore it is enoughto show that the support of f f ( | D | ) is the identity in the representation π τ .In order to prove this, it is enough to show that π τ ( e | D | [0 , p ]) tends to the identity stronglywhen p → ∞ , that is to say that π τ ( e | D | (2 p , ∞ )) tends to 0 strongly when p → ∞ .We consider then the projection P −∞ , which projects on the space generated by the edgeswith length( e ) ≤
1. Clearly, such projection belongs to B , we now show that it is in-deed central there. In fact, if c is a cell with size( c ) = 1, P c commutes with B . Since P −∞ , = P size( c )=1 P c , then P −∞ , commutes with B . On the one hand, the von Neumannalgebra P −∞ , B is isomorphic to B ( ℓ ( K )) and the restriction of τ to P −∞ , B coincideswith the usual trace on B ( ℓ ( K )), therefore the representation π τ is normal when restrictedto P −∞ , B . On the other hand, since e | D | (2 p , ∞ ) = P −∞ , − p − is, for − p ≤
1, a sub-projectionof P −∞ , , and P −∞ , − p − tends to 0 strongly in the given representation, the same holds ofthe representation π τ . SPECTRAL TRIPLE FOR A SOLENOID BASED ON THE SIERPINSKI GASKET 11 (b) We prove the first equation. Indeed τ ( P n ) = lim m tr P nm µ d ( K m ) = tr P n + lim m m X j =1 tr P jj P n µ d ( K j ) . The first summand is non-zero iff n ≤
0, while the second vanishes exactly for such n . Sincelim m m X j =1 tr P jj P n µ d ( K j ) = tr P nn µ d ( K n ) , the result in (4.5) shows that in both cases we obtain 6 · − n . We already proved in (4.6) that τ ( P − p, ∞ ) = 3 p +2 . Since P − p, ∞ ∈ B , the same holds for τ by Proposition 4.5 ( ii ). Then thethesis follows by condition (b) in Proposition 5.3.(c) We have τ ( I + D ∞ ) − s/ = τ ( P −∞ , ( I + D ∞ ) − s/ )+ τ ( P , + ∞ ( I + D ∞ ) − s/ ). A straightforwardcomputation and (4.2) give τ ( P −∞ , ( I + D ∞ ) − s/ ) = tr( P ( I + D ∞ ) − s/ ) = 6 X n ≥ (1 + 2 n ) − s/ n , which converges iff s > d . As for the second summand, we have τ ( P , + ∞ ( I + D ∞ ) − s/ ) = τ ( P , + ∞ ( I + D ∞ ) − s/ ) = lim m tr( P ,mm ( I + D ∞ ) − s/ ) µ d ( K m )= lim m m X j =1 − m tr( P ,mm ( I + D ∞ ) − s/ ) = 6 ∞ X j =1 − j (1 + 2 − j ) − s/ which converges for any s hence does not contribute to the residue. FinallyRes s = d τ ( I + D ∞ ) − s/ = lim s → d + ( s − d ) τ ( I + D ∞ ) − s/ = lim s → d + ( s − log 3log 2 )6 X n ≥ (1 + 2 − n ) − s/ e n (log 3 − s log 2) = 6log 2 lim s → d + s log 2 − log 31 − e − ( s log 2 − log 3) = 6log 2 (cid:3) Proposition 5.5.
For any f ∈ A n sup t> k [ e [ − t,t ] ( D ) · D, ρ ( f )] k = k [ D n , ρ ( f | K n )] k Proof.
We observe that | D | is a multiplication operator on ℓ ( E ∞ ), therefore it commuteswith ρ ( f ). Hence, k [ D e [ − p , p ] ( D ) , ρ ( f )] k = k| D | e [0 , p ] ( | D | ) ( ρ ( f ) − F ρ ( f ) F ) k = sup length( e ) ≥ − p | f ( e + ) − f ( e − ) | length( e ) . As a consequence, sup p ∈ Z k [ D e [ − p , p ] ( D ) , ρ ( f )] k = sup e ∈ E ∞ | f ( e + ) − f ( e − ) | length( e ) . Recall now that any edge e of length 2 n +1 is the union of two adjacent edges e and e oflength 2 n such that e +1 = e − , therefore | f ( e + ) − f ( e − ) | n +1 ≤ (cid:16) | f ( e +1 ) − f ( e − ) | n + | f ( e +2 ) − f ( e − ) | n (cid:17) ≤ sup length( e )=2 n | f ( e + ) − f ( e − ) | length( e ) . Iterating, we get sup e ∈ E ∞ | f ( e + ) − f ( e − ) | length( e ) = sup length( e ) ≤ n | f ( e + ) − f ( e − ) | length( e ) . Since f ∈ A n , sup length( e ) ≤ n | f ( e + ) − f ( e − ) | length( e ) = sup e ∈ K n | f ( e + ) − f ( e − ) | length( e ) = k [ D n , ρ ( f | K n )] k . (cid:3) Definition 5.6 ([3]) . An odd semifinite spectral triple ( L , M , D ) on a unital C ∗ -algebra A is given by a unital, norm-dense, ∗ -subalgebra L ⊂ A , a semifinite von Neumann algebra( M , τ ), acting on a (separable) Hilbert space H , a faithful representation π : A → B ( H ) suchthat π ( A ) ⊂ M , and an unbounded self-adjoint operator D b ∈ M such that(1) (1 + D ) − s/ is a τ -compact operator,(2) π ( a ) D ( D ) ⊂ D ( D ), and [ D, π ( a )] ∈ M , for all a ∈ L . Theorem 5.7.
The triple ( L , π τ ( B ∞ ) ′′ , π τ ( D )) on the unital C ∗ -algebra A ∞ is an odd semifi-nite spectral triple, where L = ∪ n { f ∈ A n , f Lipschitz } . The spectral triple has metricdimension d = log 3log 2 , the functional (5.1) I f = τ ω ( ρ ( f )( I + D ∞ ) − d/ ) , is a finite trace on A ∞ where τ ω is the logarithmic Dixmier trace associated with τ , and (5.2) I f = 6log 3 R K n f d µ d µ d ( K n ) , f ∈ A n where µ d is the Hausdorff measure of dimension d normalized as above. The Connes distance d ( ϕ, ψ ) = sup {| ϕ ( f ) − ψ ( f ) | : f ∈ L , k [ π τ ( D ) , π τ ◦ ρ ( f )] k = 1 } , ϕ, ψ ∈ S ( A ∞ ) between states on A ∞ verifies (5.3) d ( δ x , δ y ) = d geo ( x, y ) , x, y ∈ K ∞ , where d geo is the geodesic distance on K ∞ .Proof. The properties of a semifinite spectral triple follow by the properties proved above, inparticular property (1) of Definition 5.6 follows by Proposition 5.3 (a) and Proposition 5.5,while property (2) follows by Proposition 5.4 (b).The functional in equality (5.1) is a finite trace because of Proposition 5.4 ( c ) and the fact thatthe Dixmier trace coincides with the residue when the latter exists and is finite. Equations(5.2) and (5.3) only remain to be proved. We observe that ( I + D ) − d/ − D − dn have finitetrace. Indeed (cid:0) ( I + D ) − d/ − D − dn (cid:1) e = ( (1 + 4 − k ) − d/ e length( e ) = 2 k , k > n, (1 + 4 − k ) − d/ − dk length( e ) = 2 k , k ≤ n. hence, makig use of a formula in Theorem 5.4 (b), we get | τ (( I + D ) − d/ − D − dn ) | ≤ X k>n (1 + 4 − k ) − d/ τ ( P k ) + X k ≤ n (cid:12)(cid:12) (1 + 4 − k ) − d/ − k (cid:12)(cid:12) τ ( P k ) ≤ − ( n +1) ) − d/ X k>n − k + 6 X k ≤ n (cid:12)(cid:12) (1 + 4 k ) − d/ − (cid:12)(cid:12) SPECTRAL TRIPLE FOR A SOLENOID BASED ON THE SIERPINSKI GASKET 13 and both series are convergent. Since the Dixmier trace vanishes on trace class operators,this implies that τ ω ( ρ ( f )( I + D ) − d/ ) = τ ω ( ρ ( f ) D − dn ) = Res s = d τ ( ρ ( f ) D − sn ) , therefore, if f ∈ A n , I f = Res s = d τ ( ρ ( f ) D − sn ) = Res s = d tr( ρ ( f K n ) D − sn ) µ d ( K n ) = tr ω ( ρ ( f ) D − dn ) µ d ( K n ) = 6log 3 R K n f dµ d µ d ( K n )by Theorem 3.3 in [13]. As for equation (5.3), given x, y ∈ K ∞ let n such that x, y ∈ K n , m ≥ n . Then, combining Propositions 5.3 (a) and 5.5, we have, for f ∈ A m , k [ π τ ( D ) , π τ ◦ ρ ( f )] k = k [ D m , ρ ( f | K m )] k , and, by Theorem 5.2 and Corollary 5.14 in [13],sup {| f ( x ) − f ( y ) | : f ∈ A m , k [ D m , ρ ( f | K n )] k = 1 } = d geo ( x, y ) , m ≥ n. Therefore d ( δ x , δ y ) = sup {| f ( x ) − f ( y ) | : f ∈ L , k [ π τ ( D ) , π τ ◦ ρ ( f )] k = 1 } = lim m sup {| f ( x ) − f ( y ) | : f ∈ A m , k [ π τ ( D ) , π τ ◦ ρ ( f )] k = 1 } = lim m sup {| f ( x ) − f ( y ) | : f ∈ A m , k [ D m , ρ ( f | K n )] k = 1 } = d geo ( x, y ) . (cid:3) Remark . The last statement in Theorem 5.7 shows that the triple ( L , M , D ∞ ) recoverstwo incompatible aspects of the space A ∞ : on the one hand the compact space given bythe spectrum of the unital algebra A ∞ , with the corresponding finite integral, and on theother hand the open fractafold K ∞ with its geodesic distance. In particular, the functional L ( f ) = k [ D, ρ ( f )] k , f ∈ L , is not a Lip-norm (cf. [15]), since it does not recover the weak ∗ topology on S ( A ∞ ). Acknowledgement .
V.A. is supported by the Swiss National Science Foundation. D.G. andT.I. acknowledge the MIUR Excellence Department Project awarded to the Department ofMathematics, University of Rome Tor Vergata, CUP E83C18000100006.
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E-mail address : [email protected] (D.G.) Dipartimento di Matematica, Universit`a di Roma “Tor Vergata”, I–00133 Roma,Italy E-mail address : [email protected] (T.I.) Dipartimento di Matematica, Universit`a di Roma “Tor Vergata”, I–00133 Roma, Italy E-mail address ::