A combinatorial Fredholm module on self-similar sets built on n -cubes
AA COMBINATORIAL FREDHOLM MODULE ON SELF-SIMILARSETS BUILT ON n -CUBES TAKASHI MARUYAMA AND TATSUKI SETO
Abstract.
We construct a Fredholm module on self-similar sets such as theCantor dust, the Sierpinski carpet and the Menger sponge. Our constructionis a higher dimensional analogue of Connes’ combinatorial construction of theFredholm module on the Cantor set. We also calculate the Dixmier trace oftwo operators induced by the Fredholm module.
Introduction
In the 1990s, A. Connes [2, Chapter IV] introduced the quantized calculus, whichis the calculus using the Fredholm modules. The Fredholm module on an involutivealgebra A is a pair ( H, F ), where H is a Hilbert space so that A acts on H and F is a bounded operator on H such that a ( F − F ∗ ) , a ( F − , [ F, a ] ∈ K ( H ) for any a ∈ A . In particular, the commutator [ F, a ] is a quantized differential of a . Thenotion and calculus of Fredholm modules provided us many techniques in studyingvarious spaces. For example, noncompact spaces, foliated spaces, noncommutativespaces, fractal spaces, etc. In the present paper, we study the Fredholm module onsome fractal spaces called self-similar sets.The first study of quantized calculus on self-similar sets is given by Connes [2,Chapter IV]. Connes defined the Fredholm module ( H, F ) on C ( CS ), where CS is the Cantor set realized in the interval [0 , I = ( a, b ) be an open interval and set H I = (cid:96) ( { a } ) ⊕ (cid:96) ( { b } )and F I = (cid:20) (cid:21) on H I . Then ( H, F ) is a direct sum of ( H I , F I ) on all removedopen intervals of the construction of CS and defines an element in K ( C ( CS )).Connes also calculates the non-vanishing Dixmier trace Tr ω ( | [ F, x ] | dim H ( CS ) ), where x is the coordinate function on R (we consider x as a multiplication operator) anddim H ( CS ) is the Hausdorff dimension of the Cantor set CS . Since the commutator[ F, x ] is a quantized differential of x , we call | [ F, x ] | dim H ( CS ) the quantized volumemeasure on CS and Tr ω ( | [ F, x ] | dim H ( CS ) ) the quantized volume on CS in the presentpaper.In this paper, we generalize the quantized volume measure and the quantize vol-ume to higher dimensional self-similar sets. For the generalization, we constructa Fredholm module different from those constructed by other previous researches.We now present what we mean by the generalization of Connes’ quantized volumeTr ω ( | [ F, x ] | dim H ( CS ) ) in the present paper. When we construct the Fredholm mod-ule ( H K , F K ) on a fractal set K in R n such that an algebra of functions on R n acts Mathematics Subject Classification.
Primary 46L87; Secondary 28A80.
Key words and phrases.
Fredholm module, spectral triple, self-similar set, Hausdorff dimension. a r X i v : . [ m a t h . K T ] M a y T. MARUYAMA AND T. SETO on H K , a commutator of operators [ F K , x α ] ( α = 1 , . . . , n ), where x α is the α -th co-ordinate function on R n , is obtained. The commutator [ F K , x α ] is a quantized differ-ential of x α , hence we say the operator | [ F K , x ][ F K , x ] · · · [ F K , x n ] | p is a quantizedvolume measure of the volume measure dx dx · · · dx n on R n , where, p ∈ R is de-fined by using a fractal dimension on K . Then the value Tr ω ( | [ F K , x ][ F K , x ] · · · [ F K , x n ] | p )is a generalization of Connes’ quantized volume on K . We define a Fredholm module( H K , F K ) that is a generalization of Connes’ quantized volume is invariant underthe Euclidean isometries in R n as is the Euclidean volume in R n .Here, let us explain some examples that motivate us to conduct this work. Letus construct the Fredholm module on a self-similar set K built on the square in R . When we refer a standard way to construct Fredholm modules on a self-similarset (see [4, section 2]), it suffices to construct the Fredholm module ( H S , F S ) ona subset S ⊂ K . As constructed in [3], when we choose S = { a, b } (2 points), wehave the Fredholm module ( H S , F S ) same as Connes’ one. The Fredholm moduleon S gives rise to the Fredholm module ( H K , F K ) defined by the direct sum onall steps in the construction of K . Then the commutator [ F K , x ] (resp. [ F K , y ])is essentially given by the length of the projection of a segment ab to the x -axis(resp. y -axis). We can calculate the value Tr ω ( | [ F K , x ][ F K , y ] | p ), however, thevalue may vanish as follows; assume that the vertices of the square are numberedcounterclockwise in the order v , v , v , v . Now, when every edges of the squareis parallel to x -axis or y -axis and K is the Cantor dust (see Figure 5), we have[ F K , x ][ F K , y ] = 0 if S = { v i , v j } is an edge of the square. On the other hand, ifwe take S = { v , v } (a diagonal line of the square), we have the non-trivial valueTr ω ( | [ F K , x ][ F K , y ] | dim H ( K ) / ). So the subset S = { v , v } may be a subset to bechoosen in the case for the Cantor dust.However, the value Tr ω ( | [ F K , x ][ F K , y ] | dim H ( K ) / ) with S = { v , v } is not pre-served under the rotation of the square. In fact, for a self-similar set K that rotatethe above Cantor dust the angle π/ v , we have [ F K , x ][ F K , y ] = 0 with S = { v , v } . Thus it depends on K how to choose a set S so that the valueTr ω ( | [ F K , x ][ F K , y ] | p ) does not vanish. In the present paper, we give a way to con-struct a Fredholm module for K that specifies a unified choice of a subset S ⊂ K and show that the Fredholm module induces non-trivial higher dimensional quan-tized volume measure invariant under the Euclidean isometries in R n .The outline of our construction of the Fredholm module ( H K , F K ) on a self-similar set K is the following. Let γ n = [0 , n be the n -cube and { f s : γ n → γ n } ( s = 1 , , . . . , N ) be similitudes with the similarity ratio 0 < r s <
1, respectively.We do not require the open set condition. Then all f s ◦ · · · ◦ f s j ( γ n ) are smallcopies of the n -cube and we have a decreasing sequence of compact sets K j = (cid:91) ( s ,...,s j ) f s ◦· · ·◦ f s j ( γ n ). Then we get the limiting set K = ∞ (cid:92) j =0 K j . Our constructionis made of 2 steps. The first step is the construction of the Fredholm module( H , F n ) on the n -cube; see subsection 1.1. In our construction, we focus on all vertices (instead of 2 points) of n -cubes, that is, we set H = (cid:96) ( { vertices } ) witha suitable Z -grading. In the definition of F n , we use induction on the dimension n and the resulting Fredholm module represents the Kasparov product ( n -times)of Connes’ Fredholm module on an interval. The second step is taking the directsum of ( H , F n ) on all the copies of n -cubes; see subsection 2.1. Our Fredholmmodule ( H K , F K ) is defined over C ( V K ), where we denote by V K the closure of REDHOLM MODULE ON n -CUBE 3 the vertices of all n -cubes f s ◦ · · · ◦ f s j ( γ n ). Note that V K includes K properlyin general. However, for some important examples such as the Cantor dust, theSierpinski carpet and the Menger sponge, the set V K coincides with K . Divide bythe length of edges of each n -cubes, and we get the Dirac operator D K on K andthe spectral triple on K . The class in K ( C ( V K )) of our Fredholm module is nottrivial in general; see Theorem 2.5.In order to generalize the quantized volume measure, we prove the commutators[ F n , x α ] ( α = 1 , . . . , n ), where x α is the α -th coordinate function on R n , satisfya relation of the Clifford algebra, which is a quantization of the relation of theexterior differential dx α ( α = 1 , . . . , n ). Then the volume element in the Cliffordalgebra induces the non-trivial quantized volume measure and non-trivial quantizedvolume. The calculation of the quantized volume measure and the quantized volumeis given in subsection 3.2.Fredholm modules on self-similar sets are constructed by various researchers.They also investigate spectral triples, which are unbounded picture of the Fredhlmmodule, on some self-similar sets. For example, E. Christensen-C. Ivan-M. L.Lapidus [1] defined a spectral triple on the Sierpinski gasket SG by using the Diracoperator on the circle, which defines an element in K ( C ( SG )). D. Guido-T. Isola[3] defined a spectral triple on self-similar sets with the open set condition in higherdimension by using Connes’ Fredholm module on an interval. Guido-Isola [4] alsodefined a spectral triple on nested fractals by using Connes’ Fredholm module onan interval. See also Introduction in [4] for related researches.Let us compare our spectral triple with Guido-Isola’s triples. First, our Fredholmmodule can not be constructed on self-similar sets on any subset in R n . However,our construction does not required the open set condition. An example of the casefor a self-similar set without the open set condition is in subsection 4.5. Second, ourtriple and the triple in [3] is not constructed on the algebra C ( K ) of the continuousfunctions on K . So the functional defined by f (cid:55)→ Tr ω ( f | D K | − p ) is not defined on C ( K ). However, our algebra C ( V K ) coincides with C ( K ) for some important ex-amples such as the Cantor dust, the Sierpinski carpet and the Menger sponge. Thecalculation of the value Tr ω ( | D K | − p ) for our Dirac operator is given in subsection3.1. The triple in [4] is on C ( K ), however, the Cantor dust, the Sierpinski carpetand the Menger sponge are not in the class of nested fractals.Ours G-I’s [3] G-I’s [4]space self-similar set on n -cube self-similar set on R n nested fractalalgebra C ( V K ) C ( C ) C ( K )Further, we will study more noncommutative geometry of our Fredholm module( H K , F K ) and the corresponding spectral triple ( H K , D K ) in future papers.1. Fredholm module on n -cube Definition of Fredholm module.
In this subsection, we construct a “good”Fredholm module on n -cubes γ n . For the simplicity, we set γ n = [0 , e ] n in R n withthe length of edge e >
0. However, we can construct the Fredholm module on any n -cubes by the same way.Let V be the set of vertices of γ n , which is given by V = { ( a , . . . , a n ) ∈ R n ; a i = 0 or e ( i = 1 , , . . . , n ) } . T. MARUYAMA AND T. SETO
We give a number of vertices in V inductively. For n = 1, an interval γ = [0 , e ] hastwo vertices 0 and e . Set v = 0 and v = e . For a general n , we assume that wehave a number of vertices of γ n − . Then a number of vertices of γ n is as follows:(1) Set v i = ( a , . . . , a n − ,
0) = ( a , . . . , a n − ) (0 ≤ i ≤ n − −
1) under theinclusion γ n − → γ n − × { } ⊂ γ n .(2) Set v n − − i = ( a , . . . , a n − , e ) (0 ≤ i ≤ n − −
1) if v i = ( a , . . . , a n − , Example 1.1. (1)
When n = 2 , the numbering of vertices given by v = (0 , , v = ( e, , v = ( e, e ) , v = (0 , e ) ; see Figure 1. (2) When n = 3 , the numbering of vertices is given by v = (0 , , , v = ( e, , , v = ( e, e, , v = (0 , e, ,v = (0 , e, e ) , v = ( e, e, e ) , v = ( e, , e ) , v = (0 , , e ) . See Figure 2. v v v v Figure 1. n = 2. v v v v v v v v Figure 2. n = 3.Set V = { v i ; i = even } and V = { v i ; i = odd } , so we have V = V ∪ V . Set H + = (cid:96) ( V ) = (cid:96) ( v ) ⊕ (cid:96) ( v ) ⊕ · · · ⊕ (cid:96) ( v n − ) , H − = (cid:96) ( V ) = (cid:96) ( v ) ⊕ (cid:96) ( v ) ⊕ · · · ⊕ (cid:96) ( v n − )and H = H + ⊕ H − . The vector space H ( ∼ = C n ) is a Hilbert space of dimension 2 n with an inner product (cid:104) f, g (cid:105) = n − (cid:88) i =0 f ( v i ) g ( v i ) . We assume that H is Z -graded with the grading (cid:15) = ± H ± , respectively. The C ∗ -algebra C ( V ) of continuous functions on V acts on H by multiplication: ρ ( f ) = ( f ( v ) ⊕ f ( v ) ⊕ · · · ⊕ f ( v n − )) ⊕ ( f ( v ) ⊕ f ( v ) ⊕ · · · ⊕ f ( v n − )) . A Fredholm operator F n on H is also defined inductively. Set X = 1 and X n = (cid:20) O X n − X n − O (cid:21) ∈ M n − ( C ) ( n ≥ G = 1, G n = (cid:20) G n − − X n − X n − G n − (cid:21) ∈ M n − ( C ) ( n ≥
2) and U n = 1 √ n G n ( n ≥ Proposition 1.2. U n is a unitary matrix. REDHOLM MODULE ON n -CUBE 5 Proof.
Firstly, we have X n G ∗ n − G n X n = (cid:20) O X n − X n − O (cid:21) (cid:20) G ∗ n − X n − − X n − G ∗ n − (cid:21) − (cid:20) G n − − X n − X n − G n − (cid:21) (cid:20) O X n − X n − O (cid:21) = X ⊗ ( X n − G ∗ n − − G n − X n − )= · · · = X n ⊗ ( X G ∗ − G X ) = O. We prove U n U ∗ n = E n by induction. Clearly, U = 1 is unitary. Assume that U n − is a unitary matrix. Then we have G n − G ∗ n − + X n − = ( n − E n − + E n − = nE n − . Thus we obtain G n G ∗ n = (cid:20) G n − − X n − X n − G n − (cid:21) (cid:20) G ∗ n − X n − − X n − G ∗ n − (cid:21) = (cid:20) G n − G ∗ n − + X n − G n − X n − − X n − G ∗ n − X n − G ∗ n − − G n − X n − X n − + G n − G ∗ n − (cid:21) = (cid:20) G n − G ∗ n − + X n − ( X n − G ∗ n − − G n − X n − ) ∗ X n − G ∗ n − − G n − X n − X n − + G n − G ∗ n − (cid:21) = nE n − . Therefore, U n = 1 √ n G n is a unitary matrix. (cid:3) Set F n = (cid:20) U ∗ n U n (cid:21) ∈ M n ( C ). By Proposition 1.2, we have F n = E n and F ∗ n = F n . We consider that F n is a bounded operator on a finite dimensionalHilbert space H = ( (cid:96) ( v ) ⊕ (cid:96) ( v ) ⊕ · · · ⊕ (cid:96) ( v n − )) ⊕ ( (cid:96) ( v ) ⊕ (cid:96) ( v ) ⊕ · · · ⊕ (cid:96) ( v n − )) ∼ = C n by the multiplication of a matrix F n . Because of F n (cid:15) + (cid:15)F n = O , ( H , F n ) is an evenFredholm module on C ( V ). Example 1.3. (1)
When n = 1 , we have F = (cid:20) (cid:21) , which is introduced by [2,Chapter IV. 3. ε ] . (2) When n = 2 , we have G = (cid:20) −
11 1 (cid:21) , U = 1 √ (cid:20) −
11 1 (cid:21) and F = 1 √ − −
11 1 . (3) When n = 3 , we have G = − −
11 1 − −
11 0 1 1 , U = 1 √ G and F = (cid:20) U ∗ U (cid:21) . Remark 1.4.
The components of G n correspond to the following orientation ofedges, the correspondence is similar to adjacency matrices of oriented graphs. When n = 1 , the orientation of the graph γ = [0 , e ] is from v = 0 to v = e ; we denotesuch an orientation by v → v . Assume that we have the orientation of the edgesof γ n − . T. MARUYAMA AND T. SETO (1)
Assume ≤ i, j ≤ n − − . The orientation in γ n is from v i to v j ; v i → v j ,when the orientation in γ n − is from v i to v j . Here, we consider that γ n − isa subset in γ n under the inclusion γ n − → γ n − × { } ⊂ γ n . (2) v i → v n − − i (0 ≤ i ≤ n − − , which means ( a , . . . , a n − , → ( a , . . . , a n − , e ) . (3) v n − − i ← v n − − j if v i → v j (0 ≤ i, j ≤ n − − . v v (cid:111) (cid:111) v (cid:47) (cid:47) (cid:79) (cid:79) v (cid:79) (cid:79) Figure 3. orientationof edges of γ v (cid:126) (cid:126) (cid:47) (cid:47) v (cid:126) (cid:126) v v (cid:111) (cid:111) v (cid:79) (cid:79) v (cid:111) (cid:111) (cid:79) (cid:79) v (cid:47) (cid:47) (cid:62) (cid:62) (cid:79) (cid:79) v (cid:62) (cid:62) (cid:79) (cid:79) Figure 4. orientationof edges of γ Then the ( i, j ) -component g ij (1 ≤ i, j ≤ n − ) of G n is as follows. (1) g ij = 1 when v j − → v i − . (2) g ij = − when v j − ← v i − . (3) g ij = 0 when v j − and v i − do not connect by an edge. Calculation of quantized differential form.
In this subsection we calcu-late an operator [ F n , x α ] for the coordinate function x α on R n ( α = 1 , , . . . , n ).We also show they satisfy a relation of the Clifford algebra, which is a quantizationof an exterior algebra, on the Euclidean vector space of dimension n .Set d n f = [ F n , f ] = (cid:20) d − n fd + n f (cid:21) , so we have d + n f = U f + − f − Ud − n f = U ∗ f − − f + U ∗ = − ( U ¯ f + − ¯ f − U ) ∗ = − t d + n f, where we set f + = f | V and f − = f | V . Denote by A ◦ B = [ a ij b ij ] the Hadamardproduct of two matrices A = [ a ij ] and B = [ b ij ] of the same size. Proposition 1.5.
For any f ∈ C ( V ) , we set f a,b = f ( v a ) − f ( v b ) and ∆ n f =[ f j, i +1 ] i,j =0 , ,..., n − − ∈ B ( (cid:96) ( V ) , (cid:96) ( V )) ∼ = M n − ( C ) . We have d n f = 1 √ n (cid:20) − t (∆ n f ◦ G n )∆ n f ◦ G n (cid:21) . REDHOLM MODULE ON n -CUBE 7 Proof.
As is in Remark 1.4, we denote G n = [ g ij ]. We have √ nd + n f = G n f ( v ) f ( v ) . . . f ( v n − ) − f ( v ) f ( v ) . . . f ( v n − ) G n = (cid:2) g ij f ( v j ) (cid:3) − (cid:2) f ( v i − ) g ij (cid:3) = (cid:2) f j, i − g ij (cid:3) = ∆ n f ◦ G n . (cid:3) Thus an ( i, j )-component of d + n f is 0 if v i − and v j do not connect by an edge. Proposition 1.6.
For the coordinate function x α on R n ( α = 1 , , . . . , n ), we set e α ( n ) = √ ne d n x α . We have (1) e α ( n ) = √ ne d n x α = E n − α − ⊗ (cid:20) − (cid:21) − E n − α − ⊗ (cid:20) − (cid:21) ⊗ X α . Here, we set E / ⊗ (cid:20) − (cid:21) = 1 .Proof. Firstly, because of ∆ n x n = − eX n , Proposition 1.5 and the definition of G n ,we have e n ( n ) = (cid:20) X n − X n (cid:21) .Next we calculate e n − n ) = √ ne d + n x n − . By the definition of the numbering ofvertices and the orientation of edges of γ n , for 0 ≤ i, j ≤ n −
1, “ v i → v j is positive(resp. negative) with x n − direction” if and only if “ v i +2 n − ← v j +2 n − is negative(resp. positive) with x n − direction”. So we have e n − n ) = (cid:34) e n − n − − e n − n − (cid:35) = (cid:20) − (cid:21) ⊗ ( − X n − ). This implies e n − n ) = E ⊗ (cid:20) − (cid:21) − E ⊗ (cid:20) − (cid:21) ⊗ X n − . We calculate e α ( n ) ( α = 1 , , . . . , n −
2) by induction on n ≥
3. Note that thecalculation of e α ( n ) for n = 1 , e = (cid:20) − (cid:21) , e = − − , e = − − . Assume that equation (1) holds for n −
1. By the definition of the numbering ofvertices and the orientation of edges of γ n , for 1 ≤ α ≤ n −
2, “ i → j is positive T. MARUYAMA AND T. SETO (resp. negative) with x α direction” if and only if “ v i +2 n − → v j +2 n − is positive(resp. negative) with x α direction”. So we have e α +( n ) = (cid:34) e α +( n − e α +( n − (cid:35) = E ⊗ e α +( n − = − E ⊗ (cid:18) E n − − α − ⊗ (cid:20) − (cid:21) ⊗ X α (cid:19) = − E n − α − ⊗ (cid:20) − (cid:21) ⊗ X α . Therefore we have e α ( n ) = E n − α − ⊗ (cid:20) − (cid:21) − E n − α − ⊗ (cid:20) − (cid:21) ⊗ X α ( α = 1 , , . . . , n − . We have equation (1) by the above calculations for any n and α = 1 , , . . . n . (cid:3) By the explicit formula of e α ( n ) in Proposition 1.6, we have a Clifford relation of d n x α . Proposition 1.7.
We have e α ( n ) e β ( n ) = (cid:40) − e β ( n ) e α ( n ) ( α (cid:54) = β ) − E n ( α = β ) . By d n x α = e √ n e α ( n ) , we have d n x α d n x β = − d n x β d n x α ( α (cid:54) = β ) − e n E n ( α = β ) . Proof.
Firstly, we have e α ( n ) e α ( n ) = − E n − α − ⊗ (cid:20) − (cid:21) − E n − α − ⊗ (cid:20) − (cid:21) ⊗ X α = − E n . Set k = α − β >
0, then we have X α = X k +1 ⊗ X β . So we can rewrite thefollowing: e α ( n ) = E n − α − ⊗ (cid:20) X k +1 − X k +1 (cid:21) − E n − α − ⊗ (cid:20) X k +1 − X k +1 (cid:21) ⊗ X β ,e β ( n ) = E n − α − ⊗ E k ⊗ (cid:20) − (cid:21) − E n − α − ⊗ E k ⊗ (cid:20) − (cid:21) ⊗ X β . REDHOLM MODULE ON n -CUBE 9 Now, we set (cid:15) = (cid:20) − (cid:21) , so we have (cid:20) X k +1 − X k +1 (cid:21) (cid:18) E k ⊗ (cid:20) − (cid:21)(cid:19) = (cid:20) X k ⊗ ( X (cid:15) ) − X k ⊗ ( X (cid:15) ) (cid:21) and (cid:18) E k ⊗ (cid:20) − (cid:21)(cid:19) (cid:20) X k +1 − X k +1 (cid:21) = (cid:20) X k ⊗ ( (cid:15) X ) − X k ⊗ ( (cid:15) X ) (cid:21) . Thus the relation e α ( n ) e α ( n ) = − e α ( n ) e α ( n ) ( α (cid:54) = β ) holds since we have X (cid:15) + (cid:15) X = O . (cid:3) Remark 1.8.
When we take the limit as the length of edges tends to , that is e → , we have d n x α d n x α = − e n E n → O. Thus we regard d n x α as a quantization of the ordinal exterior differential dx α on R n . Remark 1.9.
For any unitary matrix U ∈ U (2 n − ) , an odd matrix F = (cid:20) U ∗ U (cid:21) defines an operator on H , so F defines a Fredholm module on C ( V ) . Moreover,since any F is homotopic to F n , it defines a same K -homology class in K ( C ( V )) .However, the general F sometimes does not have good properties. For example,we assume U = E n − , the identity matrix. Then we have [ F, x α ] = O for α =2 , , . . . , n . Thus we cannot regard [ F, x α ] as a quantization of the ordinal exteriordifferential dx α on R n . By Proposition 1.7, we have the volume element ω n = e n ) e n ) · · · e n ( n ) in theClifford algebra. We can easily calculate its absolute value | ω n | . We do not use | ω n | directly, but we use | d n x d n x · · · d n x n | , which is a constant multiple of | ω n | ,in section 3.2. Proposition 1.10.
We have | [ F n , x ] · · · [ F n , x n ] | = e n n n/ E n . By the definition of e α ( n ) , we also have | ω n | = E n .Proof. Because of [ F n , x α ] ∗ [ F n , x α ] = e n e α ∗ ( n ) e α ( n ) = − e n ( e α ( n ) ) = e n E n , we have | [ F n , x ] · · · [ F n , x n ] | = ([ F n , x ] · · · [ F n , x n ]) ∗ [ F n , x ] · · · [ F n , x n ]= [ F n , x n ] ∗ · · · [ F n , x ] ∗ [ F n , x ] · · · [ F n , x n ]= (cid:18) e n (cid:19) n E n . This implies | [ F n , x ] · · · [ F n , x n ] | = e n n n/ E n . (cid:3) Fredholm module on self-similar sets built on n -cubes Fredholm module and spectral triple.
In this subsection, we constructa Fredholm module and a spectral triple on self-similar sets built on any n -cubes γ n . For the simplicity, we assume that the length of edges of γ n equals 1. Let f s : γ n → γ n ( s = 1 , . . . N ) be similitudes. Denote by r s = (cid:107) f s ( x ) − f s ( y ) (cid:107) R n (cid:107) x − y (cid:107) R n ( <
1) ( x (cid:54) = y )the similarity ratio of f s . An iterated function system (IFS) ( γ n , S = { , . . . , N } , { f s } s ∈ S )defines the unique non-empty compact set K = K ( γ n , S = { , . . . , N } , { f s } s ∈ S )called the self-similar set such that K = (cid:83) Ns =1 f s ( K ). Denote by dim S ( K ) thesimilarity dimension of K , that is, the number s that satisfies N (cid:88) s =1 r ss = 1 . If an IFS ( γ n , S, { f s } s ∈ S ) satisfies the open set condition, we have dim H ( K ) =dim S ( K ), where we denote by dim H ( K ) the Hausdorff dimension of K .Set f s = f s ◦ · · · ◦ f s j for s = ( s , . . . , s j ) ∈ S ∞ = (cid:83) ∞ j =0 S × j and f ∅ = id. Forthe simplicity, we will denote by i the vertex f s ( v i ) of an n -cube f s ( γ n ). We alsodenote by V s the vertices of an n -cube f s ( γ n ). Denote by e s the length of edge of f s ( γ n ), which equals (cid:81) jk =1 r s k . As introduced in subsection 1.1, we set the Hilbertspace H s = (cid:96) ( V s ) on an n -cube of the length e s , which splits the positive part H + s and the negative part H − s . Taking direct sum on all n -cubes, we set as follows: H K = (cid:77) s ∈ S ∞ H s , F K = (cid:77) s ∈ S ∞ F n , D K = (cid:77) s ∈ S ∞ e s F n . Denote by V K ⊂ R n the closure of the set of vertices of all n -cubes f s ( γ n ). Thatis, V K is the closure of (cid:91) s ∈ S ∞ V s . If V ⊂ N (cid:91) s =1 f s ( V ) holds, we have V K = K . Ifnot, V K equals the union of (cid:91) s ∈ S ∞ V s and K . Denote by A K the Banach algebra ofLipschitz functions Lip( V K ) on V K with the norm (cid:107) a (cid:107) A K = (cid:107) a (cid:107) ∞ + Lip( a ), wherethe second term is the Lipschitz constant of a Lipschitz function a . The Banachalgebra A K acts on H K by ρ K : A K → B ( H K ) ; ρ K ( a )( ⊕ ξ s ) = ⊕ ( a | V s ) · ξ s . Lemma 2.1.
Set H K = (cid:40) (cid:77) s ∈ S ∞ ξ s ∈ H K ; (cid:107) ⊕ ξ s (cid:107) H K = (cid:88) s ∈ S ∞ e s n − (cid:88) i =0 | ξ s ( i ) | < ∞ (cid:41) , then an operator D K is a self-adjoint operator of dom ( D K ) = H K .Proof. By inclusions {⊕ ξ s ∈ H K ; ξ s = 0 except finite s } ⊂ H K ⊂ H K , H K is adense subset in H K .On each n -cubes f s ( γ n ), we have (cid:107) F n ξ s (cid:107) (cid:96) = (cid:107) U n ξ + s (cid:107) (cid:96) + (cid:107) U ∗ n ξ − s (cid:107) (cid:96) = (cid:107) ξ + s (cid:107) (cid:96) + (cid:107) ξ − s (cid:107) (cid:96) = n − (cid:88) i =0 | ξ s ( i ) | n -CUBE 11 for any function ξ s on V s , where ξ ± s denotes the H ± s part of ξ s , respectively. Sowe have (cid:107) D K ( ⊕ ξ s ) (cid:107) H K = (cid:88) s ∈ S ∞ e s n − (cid:88) i =0 | ξ s ( i ) | = (cid:107) ⊕ ξ s (cid:107) H K for ⊕ ξ s ∈ H K . Thus we have D K ( H K ) ⊂ H K . Therefore D K is a symmetricoperator with domain H K .On the other hand, set ⊕ η s = ⊕ e s F n ξ s for any ⊕ ξ s ∈ H K . Then we have ⊕ η s ∈ H K since we have (cid:107) ⊕ η s (cid:107) H K = (cid:88) s ∈ S ∞ (cid:107) F n ξ s (cid:107) (cid:96) = (cid:88) s ∈ S ∞ (cid:107) ξ s (cid:107) (cid:96) = (cid:107) ⊕ ξ s (cid:107) H K < ∞ . This implies D K ( H K ) ⊃ H K . Thus we have D K ( H K ) = H K . Therefore D K is aself-adjoint operator ofdom( D K ) = (cid:40) (cid:77) s ∈ S ∞ ξ s ∈ H K ; (cid:88) s ∈ S ∞ e s n − (cid:88) i =0 | ξ s ( i ) | < ∞ (cid:41) . (cid:3) Note that we have ρ K ( A K )( H K ) ⊂ H K and F K = D K | D K | − . We prove someregularity of F K and D K . Lemma 2.2.
We have the following. (1) [ F K , a ] ∈ K ( H K ) for any a ∈ C ( V K ) . (2) [ D K , a ] ∈ B ( H K ) for any a ∈ A K . (3) | D K | − ∈ K ( H K ) . (4) ( D K + 1) − / ∈ K ( H K ) . (5) | D K | − p ∈ L ( H K ) ⇐⇒ p > dim S ( K ) , where L ( H K ) is the set of trace classoperators on H K . (6) ( D K + 1) − p/ ∈ L ( H K ) ⇐⇒ p > dim S ( K ) .Proof. (1) First, we take a ∈ A K . For any s ∈ S × j , we have[ F K , a ] | H s = 1 √ n (cid:20) − t (∆ n a ◦ G n )∆ n a ◦ G n (cid:21) . So the operator norm (cid:107) [ F K , a ] | H s (cid:107) is less thanLip( a ) · e s = Lip( a ) · j (cid:89) k =1 r s k . Thus [ F K , a ] is compact for a ∈ A K since we have j (cid:89) k =1 r s k ≤ max s ∈ S r js → j → ∞ . The case for any continuous function is proved by the density of A K in C ( V K ).(2) For any s ∈ S × j , we have[ D K , a ] | H s = 1 √ n (cid:32) j (cid:89) k =1 r s k (cid:33) − (cid:20) − t (∆ n a ◦ G n )∆ n a ◦ G n (cid:21) . So the operator norm (cid:107) [ D K , a ] | H s (cid:107) is less than Lip( a ), which is independentof j . Therefore [ D K , a ] is bounded on H K . (3) Because of | D K | = (cid:77) s ∈ S ∞ e s E n , we have | D K | − = ∞ (cid:77) j =0 (cid:77) s ∈ S × j (cid:32) j (cid:89) k =1 r s k (cid:33) E n .Thus | D K | − is compact since we have j (cid:89) k =1 r s k → j → ∞ .(4) Because of D K + 1 = (cid:77) s ∈ S ∞ (cid:18) e s + 1 (cid:19) E n , we have( D K + 1) − / = ∞ (cid:77) j =0 (cid:77) s ∈ S × j (cid:32) j (cid:89) k =1 r − s k + 1 (cid:33) − / E n . Thus ( D K + 1) − / is a compact operator.(5) Because of | D K | − p = ∞ (cid:77) j =0 (cid:77) s ∈ S × j (cid:32) j (cid:89) k =1 r ps k (cid:33) E n , we haveTr( | D K | − p ) = ∞ (cid:88) j =0 (cid:88) s ∈ S × j n j (cid:89) k =1 r ps k = 2 n ∞ (cid:88) j =0 (cid:32) N (cid:88) s =1 r ps (cid:33) j . Thus we have | D K | − p ∈ L ( H K ) ⇐⇒ N (cid:88) s =1 r ps < . This implies | D K | − p ∈ L ( H K ) ⇐⇒ p > dim S ( K ).(6) Because of ( D K + 1) − p/ = ∞ (cid:77) j =0 (cid:77) s ∈ S × j (cid:32) j (cid:89) k =1 r − s k + 1 (cid:33) − p/ E n , we have Tr(( D K + 1) − p/ ) = ∞ (cid:88) j =0 (cid:88) s ∈ S × j n (cid:32) j (cid:89) k =1 r − s k + 1 (cid:33) − p/ . Thus we have ∞ (cid:88) j =0 (cid:88) s ∈ S × j n − p/ j (cid:89) k =1 r ps k ≤ Tr(( D K + 1) − p/ ) ≤ ∞ (cid:88) j =0 (cid:88) s ∈ S × j n j (cid:89) k =1 r ps k , that is we have2 n − p/ ∞ (cid:88) j =0 (cid:32) N (cid:88) s =1 r ps k (cid:33) j ≤ Tr(( D K + 1) − p/ ) ≤ n ∞ (cid:88) j =0 (cid:32) N (cid:88) s =1 r ps k (cid:33) j . This implies( D K + 1) − p/ ∈ L ( H K ) ⇐⇒ N (cid:88) s =1 r ps k < ⇐⇒ p > dim S ( K ) . (cid:3) REDHOLM MODULE ON n -CUBE 13 Theorem 2.3.
The pair ( H K , F K ) is an even Fredholm module over C ( V K ) withthe Z -grading (cid:15) K = (cid:76) s ∈ S ∞ (cid:15) . The pair ( H K , F K ) is a ([dim S ( K )] + 1) -summableeven Fredholm module over A K . In particular, if we have dim S ( K ) < n , an operator [ F K , a ][ F K , a ] · · · [ F K , a n ] is of trace class for any a , a , . . . , a n ∈ A K .Proof. By the definition of F K , we have F K = 1, F ∗ K = F K and F K (cid:15) K + (cid:15) K F K =0. On the other hand, [ F K , a ] is a compact operator by Lemma 2.2. Therefore,( H K , F K ) is an even Fredholm module over C ( V K ).Next we prove summability of the Fredholm module ( H K , F K ) over A K . Now,since [ D K , a ] is a bounded operator for a ∈ A K and | D K | − ([dim S ( K )]+1) is of traceclass, we have[ F K , a ][ F K , a ] · · · [ F K , a [dim S ( K )]+1 ]=[ D K , a ] | D K | − [ D K , a ] | D K | − · · · [ D K , a [dim S ( K )]+1 ] | D K | − =[ D K , a ][ D K , a ] · · · [ D K , a [dim S ( K )]+1 ] | D K | − ([dim S ( K )]+1) ∈ L ( H K )for a , a , . . . , a [dim S ( K )]+1 ∈ A K . Here, we have [ | D K | − , T ] = 0 if T ∈ B ( H K )is a direct sum of operators on each n -cubes f s ( γ n ). Therefore, ( H K , F K ) is a([dim S ( K )] + 1)-summable even Fredholm module. (cid:3) Theorem 2.4.
The tirple ( A K , H K , D K ) is an even QC ∞ -spectral triple of spectraldimension dim S ( K ) .Proof. By the definition of D K and Lemma 2.2, ( A K , H K , D K ) is an even spectraltriple of spectral dimension dim S ( K ). On the other hand, the spectral triple is of QC ∞ -class since we have [ | D K | , T ] = 0 for an operator T ∈ B ( H K ) of a direct sumof operators on each n -cubes f s ( γ n ). (cid:3) We next prove a nonvanishing property of the K -class of the Fredholm module( H K , F K ). Theorem 2.5.
Denote by X , . . . , X k the all connected components of V ∪ (cid:91) s ∈ S f s ( γ n ) .If there is X i such that (cid:93) ( V ∩ X i ) (cid:54) = (cid:93) ( V ∩ X i ) , the Connes-Chern character Ch ∗ ( H K , F K ) ∈ H even λ ( A K ) induces a non-zero ad-ditive map K ( C ( V K )) ∼ = K ( A K ) → C by the Connes pairing. In particular, [ H K , F K ] ∈ K ( C ( V K )) is not trivial.Proof. Set d = (cid:93) ( V ∩ X i ) , d = (cid:93) ( V ∩ X i )and p ( x ) = (cid:40) x ∈ X i x ∈ V K . Then p is a continuous function and we haveindex( pF + K p : p H + K → p H − K ) = index( pU n p : p(cid:96) ( V ) → p(cid:96) ( V ))= d − d (cid:54) = 0 . Therefore we have Ch ∗ ( H K , F K ) (cid:54) = 0 on K ( C ( V K )). (cid:3) Remark 2.6.
For example, the Sierpinski carpet (see subsection 4.3) and the n -cube γ n do not hold the assumption in Theorem 2.5. In these cases, the Connes-Chern character induces the -map on K ( A K ) . Remark 2.7.
As remarked in Remark 1.9, we can define a Fredholm module on C ( V ) by using any unitary matrix U instead of U n . All properties in subsection 2.1holds without changing proofs in such a situation. Quantized differential form on self-similar sets.
Note that all similitudeson γ n forms f s ( x ) = r s T s x + b s for an orthogonal matrix T s ∈ O ( n ) and b s ∈ R n .It is easy that we calculate the quantum differential form [ F K , x α ] in the case for γ n = [0 , n and T s = E n (for any s ∈ S ), which is the direct sum of the matrix d n x α ; see Proposition 1.6. We can also express [ F K , x α ] explicitly for the generalcase and show that they satisfy “a variation” of the Clifford relation. Proposition 2.8.
We have [ F K , x α ][ F K , x β ] = − [ F K , x β ][ F K , x α ] α (cid:54) = β − (cid:77) s ∈ S ∞ e s n E n α = β . Proof.
Take an orthogonal matrix T s = [ t ij ] i,j ∈ O ( n ) and a vector bs ∈ R n suchthat the image of the affine transformation g s ( x ) = e s T sx + bs of [0 , n equals f s ( γ n ) and g s ( x ) preserves the numbering the vertices of [0 , n and f s ( γ n ). If weassume γ n = [0 , n , we have f s = g s . Note that we have[ F K , x α ] | H s = 1 √ n (cid:20) − t (∆ n x α ◦ G n )∆ n x α ◦ G n (cid:21) . Because of v j − v i − = ± e s T se k if g − s ( v j ) is connecting g − s ( v i − ) by an edgeof the n -cube [0 , n parallel with x k -direction and T se k = (cid:80) nα =1 t αk e α , we have[ F K , x α ] | H s = e s √ n n (cid:88) j =1 t αj e j ( n ) . Thus we have[ F K , x α ][ F K , x β ] | H s = e s n n (cid:88) j =1 t αj e j ( n ) n (cid:88) j =1 t βk e k ( n ) = e s n (cid:88) j,k t αj t βk e j ( n ) e k ( n ) = e s n (cid:88) j (cid:54) = k t αj t βk e j ( n ) e k ( n ) − e s n n (cid:88) j =1 t αj t βj = e s n (cid:88) j (cid:54) = k t αj t βk e j ( n ) e k ( n ) ( α (cid:54) = β ) − e s n E n ( α = β ) . Therefore, we have[ F K , x α ][ F K , x β ] = − [ F K , x β ][ F K , x α ] ( α (cid:54) = β ) − (cid:77) s ∈ S ∞ e s n E n ( α = β ) . (cid:3) REDHOLM MODULE ON n -CUBE 15 By Proposition 2.8, we get an explicit formula for an operator | [ F K , x ] · · · [ F K , x n ] | . Proposition 2.9.
We have | [ F K , x ] · · · [ F K , x n ] | = (cid:77) s ∈ S ∞ e n s n n/ E n . Proof.
Similar to the proof of Proposition 1.10. (cid:3)
Remark 2.10.
Set e αK = (cid:77) s ∈ S ∞ e α ( n ) . Then we have the Clifford relation e αK e βK = (cid:40) − e βK e αK ( α (cid:54) = β ) − id H K ( α (cid:54) = β ) . Thus we can regard e αK as a - Q -form in the sense of [6] . Dixmier traces
In this section, we calculate the Dixmier trace of two operators. The valueof second one changes in general if the Fredholm operator F n changes to otherFredholm operator.3.1. Dixmier trace of | D K | − p . In this subsection, we calculate the Dixmier traceof | D K | − p , which is given by the residue at the pole of the zeta function ζ D K ( s ) =Tr( | D K | − s ). Theorem 3.1.
For any p ≥ dim S ( K ) , we have | D K | − p ∈ L (1 , ∞ ) ( H K ) and we have Tr ω ( | D K | − p ) = − n (cid:32) dim S ( K ) N (cid:88) s =1 r dim S ( K ) s log r s (cid:33) − ( p = dim S ( K ))0 ( p > dim S ( K )) . Thus we have Tr ω ( f | D K | − dim S ( K ) ) = − n (cid:32) dim S ( K ) N (cid:88) s =1 r dim S ( K ) s log r s (cid:33) − (cid:90) K f | K d Λ for any f ∈ C ( V K ) by the Riesz-Markov-Kakutani representation theorem. Here, Λ is the dim S ( K ) -dimensional Hausdorff probability measure of K .In particular, if all similarity ratios r s are equal, we have Tr ω ( | D K | − dim S ( K ) ) = 2 n log N .
Proof.
By proof of Lemma 2.2, we haveTr( | D K | − p ) = 2 n ∞ (cid:88) j =0 (cid:32) N (cid:88) s =1 r ps (cid:33) j = 2 n (cid:32) − N (cid:88) s =1 r ps (cid:33) − . Thus we have( z − | D K | − zp ) = 2 n z − − (cid:80) Ns =1 r zps = 2 n z − (cid:80) Ns =1 (cid:16) r dim S ( K ) s − r zps (cid:17) = 2 n (cid:32) N (cid:88) s =1 r dim S ( K ) s − r zps z − (cid:33) − . Thus the value Tr ω ( | D K | − p ) = lim z → +1 ( z − | D K | − zp )= 2 n (cid:32) N (cid:88) s =1 lim z → +1 r dim S ( K ) s − r zps z − (cid:33) − converges for p ≥ dim S ( K ) and we haveTr ω ( | D K | − p ) = − n (cid:32) N (cid:88) s =1 ddz (cid:12)(cid:12)(cid:12)(cid:12) z =1 r z dim S ( K ) s (cid:33) − = − n (cid:32) dim S ( K ) N (cid:88) s =1 r dim S ( K ) s log r s (cid:33) − for p = dim S ( K ) and Tr ω ( | D K | − p ) = 0for p > dim S ( K ). (cid:3) Dixmier trace of | [ F K , x ] · · · [ F K , x n ] | p . In this subsection, we calculate theDixmier trace of | [ F K , x ] · · · [ F K , x n ] | p by using Proposition 2.9. Theorem 3.2.
For any p ≥ n dim S ( K ) , we have | [ F K , x ][ F K , x ] · · · [ F K , x n ] | p ∈L (1 , ∞ ) ( H K ) and we have Tr ω ( | [ F K , x ][ F K , x ] · · · [ F K , x n ] | p ) = 1 n dim S ( K ) / Tr ω ( | D K | − np )= − n n dim S ( K ) / (cid:32) dim S ( K ) N (cid:88) s =1 r dim S ( K ) s log r s (cid:33) − ( p = 1 n dim S ( K ))0 ( p > n dim S ( K )) . Thus we have Tr ω ( f | [ F K , x ][ F K , x ] · · · [ F K , x n ] | n dim S ( K ) )= − n n dim S ( K ) / (cid:32) dim S ( K ) N (cid:88) s =1 r dim S ( K ) s log r s (cid:33) − (cid:90) K f | K d Λ= 1 n dim S ( K ) / Tr ω ( | D K | − dim S ( K ) ) (cid:90) K f | K d Λ for any f ∈ C ( V K ) by the Riesz-Markov-Kakutani representation theorem. Here, Λ is the dim H ( K ) -dimensional Hausdorff probability measure of K .Proof. By Proposition 2.9, we have | [ F K , x ][ F K , x ] · · · [ F K , x n ] | p = (cid:77) s ∈ S ∞ e np s n np/ E n . REDHOLM MODULE ON n -CUBE 17 So we haveTr( | [ F K , x ][ F K , x ] · · · [ F K , x n ] | p ) = 2 n ∞ (cid:88) j =0 (cid:88) ( s ,...s j ) ∈ S j n np/ j (cid:89) k =1 r nps k = 2 n n np/ (cid:32) N (cid:88) s =1 r nps (cid:33) j . Thus we haveTr( | [ F K , x ][ F K , x ] · · · [ F K , x n ] | p ) < ∞ ⇐⇒ p > n dim S ( K )and then we haveTr( | [ F K , x ][ F K , x ] · · · [ F K , x n ] | p ) = 2 n n np/ (cid:32) − N (cid:88) s =1 r nps (cid:33) − . Therefore, the similar proof of Theorem 3.1 implies | [ F K , x ][ F K , x ] · · · [ F K , x n ] | p ∈L (1 , ∞ ) ( H K ) for p ≥ n dim S ( K ). Moreover, we haveTr ω ( | [ F K , x ][ F K , x ] · · · [ F K , x n ] | p ) = lim z → +1 ( z − | [ F K , x ][ F K , x ] · · · [ F K , x n ] | zp )= 2 n n dim S ( K ) / (cid:32) N (cid:88) s =1 lim z → +1 r dim S ( K ) s − r z dim S ( K ) s z − (cid:33) − = − n n dim S ( K ) / (cid:32) dim S ( K ) N (cid:88) s =1 r dim S ( K ) s log r s (cid:33) − for p = 1 n dim S ( K ) andTr ω ( | [ F K , x ][ F K , x ] · · · [ F K , x n ] | p ) = 0for p > n dim S ( K ). (cid:3) Examples
In this section, we apply arguments in Section 2 and 3 to some examples.4.1.
Cantor dust.
The Cantor dust is a generalization of the middle third Cantorset to higher dimension. Let CD n be the Cantor dust defined on γ n = [0 , n , whichis a self-similar set defined by similitudes f s ( x ) = 13 x + 23 n (cid:88) α =1 a α e α ( x ∈ γ n , s = 0 , , , . . . , n − . Here, we express a number s by a n a n − · · · a a in binary and e α is the standardbasis of R n . Since the Cantor dust CD n satisfies the open set condition, we have dim H ( CD n ) = dim S ( CD n ) = n log
2. We have V CD n = CD n since we have V ⊂ n − (cid:91) s =0 f s ( V ). So we have A CD n = Lip( CD n ) and C ( V CD n ) = C ( CD n ) . Figure 5.
The first 3 steps of construction of CD .Since all f s ( γ n ) are disconnecting each other and we have (cid:93) ( V ∩ f ( γ n )) = 1 and (cid:93) ( V ∩ f ( γ n )) = 0, the K -class of ( H CD n , F CD n ) in K ( C ( CD n )) does not vanishby Theorem 2.5. Theorem 4.1.
The Connes-Chern character Ch ∗ ( H CD n , F CD n ) ∈ H even λ (Lip( CD n )) induces a non-zero additive map K ( C ( CD n )) → C . In particular, [ H CD n , F CD n ] ∈ K ( C ( CD n )) is not trivial. By dim S ( CD n ) = n log
2, we have the following.
Corollary 4.2. (1) ( H CD n , F CD n ) is a ([ n log
2] + 1) -summable even Fredholmmodule over
Lip( CD n ) . (2) (Lip( CD n ) , H CD n , D CD n ) is a QC ∞ -spectral triple of spectral dimension n log . Corollary 4.3.
We have the following. (1) Tr( | D CD n | − p ) = 2 n · p p − n for any p > n log . (2) Tr ω ( | D CD n | − n log ) = 2 n n log 2 . (3) Tr ω ( f | D CD n | − n log ) = 2 n n log 2 (cid:90) CD n f d Λ for any f ∈ C ( CD n ) . Here, Λ is the ( n log -dimensional Hausdorff probability measure of CD n . Corollary 4.4.
An operator | [ F CD n , x ][ F CD n , x ] · · · [ F CD n , x n ] | log is of L (1 , ∞ ) -class and we have Tr ω ( | [ F CD n , x ][ F CD n , x ] · · · [ F CD n , x n ] | log ) = 2 n n (2+ n log / log 2 . Thus we have Tr ω ( f | [ F CD n , x ][ F CD n , x ] · · · [ F CD n , x n ] | log ) = 2 n n (2+ n log / log 2 (cid:90) CD n f d Λ for any f ∈ C ( CD n ) . Here, Λ is the ( n log -dimensional Hausdorff probabilitymeasure of CD n . REDHOLM MODULE ON n -CUBE 19 Middle third Cantor set, revisited.
In this subsection, we focus on themiddle third Cantor set CS = CD .First, we see a relationship between our Fredholm module and Connes’ Fredholmmodule defined in [2, Chapter IV. 3. ε ]. We recall that Connes’ Fredholm module( H, F ) on C ( CS ). Let I i,j = ( a i,j , b i,j ) ( i ∈ N , j = 1 , , . . . , i ) be open intervals in[0 ,
1] which are defined to be I , = (cid:18) , (cid:19) and I i +1 ,j = (cid:18) b i,j − + a i,j , b i,j − + 2 a i,j (cid:19) , where we set b i, = 0 and a i,i +1 = 1. The middle third Cantor set satisfies CS =[0 , \ (cid:91) i,j I i,j . Connes defined H = (cid:77) i,j (cid:96) ( { a i,j , b i,j } ) and F = (cid:77) i,j F . Note that H ⊕ (cid:96) ( { , } ) ∼ = H CS as Hilbert spaces. Lemma 4.5.
Let a < b < c be real numbers. We assume [ (cid:96) ( { a, b } ) , F ] , [ (cid:96) ( { b, c } ) , F ] , [ (cid:96) ( { a, c } ) , F ] ∈ K ( C ( { a, b, c } )) under homomorphisms K ( C ( { a, b } )) → K ( C ( { a, b, c } )) , K ( C ( { b, c } )) → K ( C ( { a, b, c } )) and K ( C ( { a, c } )) → K ( C ( { a, b, c } )) defined by inclusions { a, b } → { a, b, c } , { b, c } → { a, b, c } and { a, c } → { a, b, c } , respectively. Then we have [ (cid:96) ( { a, b } ) , F ] + [ (cid:96) ( { b, c } ) , F ] = [ (cid:96) ( { a, c } ) , F ] in K ( C ( { a, b, c } )) . Proof.
Denote by b = b = b , { a, b } = { a, b } and { b, c } = { b , c } . We have[ (cid:96) ( { a, b } ) , F ] + [ (cid:96) ( { b , c } ) , F ] = [ (cid:96) ( { a, b } ) ⊕ (cid:96) ( { b , c } ) , F ⊕ F ]= (cid:20) (cid:96) ( { a, c } ) ⊕ (cid:96) ( { b , b } ) , (cid:20) E E (cid:21)(cid:21) . Here the Z -grading operator of the last Fredholm module is defined by ˜ (cid:15) = (cid:15) ⊕ ( − (cid:15) ).Set T t = (cid:20) F cos t sin t sin t − F cos t (cid:21) on (cid:96) ( { a, c } ) ⊕ (cid:96) ( { b , b } ). Then we have T t ˜ (cid:15) + ˜ (cid:15)T t = 0, T = F ⊕ ( − F ) and T π/ = (cid:20) E E (cid:21) . Thus we have[ (cid:96) ( { a, b } ) , F ] + [ (cid:96) ( { b , c } ) , F ] = (cid:2) (cid:96) ( { a, c } ) ⊕ (cid:96) ( { b , b } ) , F ⊕ ( − F ) (cid:3) = [ (cid:96) ( { a, c } ) , F ] − [ (cid:96) ( { b , b } ) , F ]= [ (cid:96) ( { a, c } ) , F ] . Here, the last equality is given by b = b = b . (cid:3) By Lemma 4.5, we have[
H, F ] + [ H CS , F CS ] = [ H CS , F CS ] + [ (cid:96) ( { , } ) , F ] . Therefore we have [
H, F ] = [ (cid:96) ( { , } ) , F ] in K ( C ( CS )). On the other hand, weset p k ( x ) = (cid:40) x ∈ [0 , / k ] ∩ CS for x ∈ CS . Then we have (cid:104) [ H CS , F CS ] , [ p k ] (cid:105) = k and (cid:104) [ (cid:96) ( { , } ) , F ] , [ p k ] (cid:105) = 1 bythe index pairing between K -homology and K -theory. Thus { [ H CS , F CS ] , [ H, F ] } islinearly independent on Z in K ( C ( CS )).Second, we set similitudes f ( x ) = 13 x , f ( x ) = 13 x + 23 e for x ∈ γ and denote by K the self-similar set defined by the IFS ( γ , { f , f } ).Then we have K = CS × { } as sets. So the Fredholm module ( H K , F K ) is newFredholm module of the middle third Cantor set. Note that we have V K (cid:54) = K and (cid:0)(cid:83) s ∈ S ∞ V s (cid:1) ∩ K (cid:54) = ∅ in this case. Figure 6.
The first 3 steps of construction of K .4.3. Sierpinski carpet and its higher dimensional analogue.
The Sierpinskicarpet is another generalization of the middle third Cantor set to dimension 2.The Menger sponge is an analogue of the Sierpinski carpet in 3 dimension. In thissubsection, we see such self-similar sets in n dimension ( n ≥ S n ⊂ N ∪ { } be the index set defined by S n = { s ∈ N ∪{ } ; 0 ≤ s ≤ n − s } . For example, we have S = { , , , , , , , } and S = S ∪ { , , , , , , , , , , , } . Define similitudes f s : γ n → γ n for s ∈ S n by f s ( x ) = 13 x + 13 n (cid:88) α =1 a α e α . Here, we express a number s by a n a n − · · · a a in ternary. Denote by SC n theself-similar set on the IFS ( γ n , S n , { f s } s ∈ S n ). For example, SC is called the Sier-pinski carpet and SC is called the Menger sponge. Since the self-similar set CD n satisfies the open set condition, we have dim H ( CD n ) = dim S ( CD n ) = log ( (cid:93)S n ) =log (2 n − ( n + 2)). We have V SC n = SC n since we have V ⊂ (cid:91) s ∈ S n f s ( V ). So we have A SC n = Lip( SC n ) , C ( V SC n ) = C ( SC n ) . Since X = V ∪ (cid:83) s ∈ S n f s ( γ n ) is connected, we have (cid:93) ( V ∩ X ) = (cid:93) ( V ∩ X ). Sothe assumption in Theorem 2.5 does not hold. REDHOLM MODULE ON n -CUBE 21 Figure 7.
The first 3 steps of construction of SC . Remark 4.6.
We assume n = 2 , so SC is a compact set in R . We have K ( C ( SC )) = Z , which is generated by (matrix valued) constant functions on SC . So the index pairing between K -theory and K -homology induces the -map K ( C ( SC )) → Z . Therefore we have [ H SC , F SC ] = 0 in K ( C ( SC )) by [5,Theorem 7.5.5] .On the other hand, we can construct a non-trivial Fredholm module correspondingto the Sierpinski carpet, which is a similar construction to the last of subsection 4.2.Define z : γ → γ by z ( t ) = 13 t . Set ˜ f s = ( f s , z ) : γ → γ for s ∈ S . Then we geta new IFS ( γ , S , { ˜ f s } s ∈ S ) . Denote by (cid:102) SC the self-similar set on the new IFS,so we have (cid:102) SC = SC × { } . The corresponding Fredholm module ( H (cid:102) SC , F (cid:102) SC ) represents a non-trivial element in K ( C ( V (cid:102) SC )) . Remark 4.7.
The construction of IFS in Remark 4.6 can be extended to generalcase. Namely, let ( γ n , S, { f s } s ∈ S ) be an IFS and denote by K the self-similar seton the IFS. Then ( γ n +1 , S, { ( f s , z ) } s ∈ S ) is a new IFS and the corresponding self-similar set denote by (cid:101) K satisfies (cid:101) K = K × { } and [ H (cid:101) K , F (cid:101) K ] (cid:54) = 0 in K ( C ( V (cid:101) K )) . By dim S ( SC n ) = log (2 n − ( n + 2)), we have the following. Corollary 4.8. (1) ( H SC n , F SC n ) is a ([log (2 n − ( n + 2))] + 1) -summable evenFredholm module over Lip( SC n ) . (2) (Lip( SC n ) , H SC n , D SC n ) is a QC ∞ -spectral triple of spectral dimension log (2 n − ( n +2)) . Corollary 4.9.
We have the following. (1) Tr( | D SC n | − p ) = 2 n · p p − n − ( n + 2) for any p > log (2 n − ( n + 2)) . (2) Tr ω ( | D SC n | − log (2 n − ( n +2)) ) = 2 n log(2 n − ( n + 2)) . (3) Tr ω ( f | D SC n | − log (2 n − ( n +2)) ) = 2 n log(2 n − ( n + 2)) (cid:90) SC n f d Λ for any f ∈ C ( SC n ) .Here, Λ is the (log (2 n − ( n + 2))) -dimensional Hausdorff probability measureof SC n . Corollary 4.10.
Set d = 1 n log (2 n − ( n +2)) . An operator | [ F SC n , x ][ F SC n , x ] · · · [ F SC n , x n ] | d is of L (1 , ∞ ) -class and we have Tr ω ( | [ F SC n , x ][ F SC n , x ] · · · [ F SC n , x n ] | d ) = 2 n n nd/ log(2 n − ( n + 2)) . Thus we have Tr ω ( f | [ F CD n , x ][ F CD n , x ] · · · [ F CD n , x n ] | d ) = 2 n n nd/ log(2 n − ( n + 2)) (cid:90) SC n f d Λ for any f ∈ C ( SC n ) . Here, Λ is the (log (2 n − ( n + 2))) -dimensional Hausdorffprobability measure of SC n . With rotations.
Denote by R = (cid:20) cos θ − sin θ sin θ cos θ (cid:21) the rotation matrix. Let f , f , f , f be four similitudes defined by f s ( x ) = 12 √ R (cid:18) x − (cid:20) (cid:21)(cid:19) + b s . Here, we set b = 14 (cid:20) (cid:21) , b = 14 (cid:20) (cid:21) , b = 14 (cid:20) (cid:21) , b = 14 (cid:20) (cid:21) . The IFS ( γ , { f , f , f , f } ) is defined by using a rotation of angle θ . Denote by K the self-similar set on the IFS ( γ , { f , f , f , f } ) that satisfies the open setcondition. Then we have V K (cid:54) = K and (cid:16)(cid:83) s ∈{ , , , } ∞ V s (cid:17) ∩ K = ∅ . Since { (0 , } is a connected component of V ∪ (cid:91) s ∈{ , , , } f s ( γ ), the Fredholm module ( H K , F K )defines a non-trivial element in K ( C ( V K )). Figure 8.
The first 3 steps of construction of K .By dim S ( K ) = log √ Corollary 4.11. (1) ( H K , F K ) is a -summable even Fredholm module over A K . (2) ( A K , H K , D K ) is a QC ∞ -spectral triple of spectral dimension . Corollary 4.12.
We have the following. (1) Tr( | D K | − p ) = 42 p/ − for any p > . (2) Tr ω ( | D K | − / ) = 2log 2 . (3) Tr ω ( f | D K | − / ) = 2log 2 (cid:90) K f | K d Λ for any f ∈ C ( V K ) . Here, Λ is the / -dimensional Hausdorff probability measure of K . REDHOLM MODULE ON n -CUBE 23 The quantized differential form [ F K , x α ] ( α = 1 ,
2) is given by[ F K , x ] = ∞ (cid:77) j =0 (cid:77) s ∈ S × j e s √ jθ − sin jθ − sin jθ − cos jθ − cos jθ sin jθ jθ cos jθ , [ F K , x ] = ∞ (cid:77) j =0 (cid:77) s ∈ S × j e s √ jθ cos jθ jθ − sin jθ − sin jθ − cos jθ − cos jθ sin jθ . by Proposition 1.5. So we have | [ F K , x ][ F K , x ] | = (cid:77) s ∈ S ∞ e s E . This implies
Corollary 4.13.
An operator | [ F K , x ][ F K , x ] | / is of L (1 , ∞ ) -class and we have Tr ω ( | [ F K , x ][ F K , x ] | / ) = √ . Thus we have Tr ω ( f | [ F K , x ][ F K , x ] | / ) = √ (cid:90) K f | K d Λ for any f ∈ C ( V K ) . Here, Λ is the -dimensional Hausdorff probability measure of K . Without the open set condition.
In this subsection, we present an exampleof a self-similar set that does not satisfy the open set condition. In this case, wecan detect the similarity dimension by using our Fredholm module but not detectthe Hausdorff dimension explicitly.Let ( γ , S = { , , , , } , { f s } s ∈ S ) be a IFS defined to be f ( x ) = 13 x , f ( x ) = 13 x + 23 e , f ( x ) = 13 x + 23 e ,f ( x ) = 13 x + 23 e + 23 e , f ( x ) = 23 x + 16 e + 16 e . Note that the IFS doese not satisfies the open set condition. Denote by K theself-similar set on the IFS. Since we have V ⊂ (cid:91) s =1 f s ( V ), we have V K = K . Thesimilarity dimension s = dim S ( K ) of K is given by the identity4 · (cid:18) (cid:19) s + (cid:18) (cid:19) s = 1 . We can easily check 1 < s < Corollary 4.14. (1) ( H K , F K ) is a -summable even Fredholm module over Lip( K ) . (2) (Lip( K ) , H K , D K ) is a QC ∞ -spectral triple of spectral dimension s . Corollary 4.15.
We have the following. (1) Tr( | D K | − p ) = 4 · p p − p − for any p > s . (2) Tr ω ( | D K | − dim S ( K ) ) = 4 · s log 34 s + 2 s s log 2 . (3) Tr ω ( f | D K | − dim S ( K ) ) = 4 · s log 34 s + 2 s s log 2 (cid:90) K f d Λ for any f ∈ C ( K ) . Here, Λ isthe dim H ( K ) -dimensional Hausdorff probability measure of K . Corollary 4.16.
An operator | [ F K , x ][ F K , x ] | s/ is of L (1 , ∞ ) -class and we have Tr ω ( | [ F K , x ][ F K , x ] | d ) = 4 · s log 32 s/ s + 2 s/ s log 2 . Thus we have Tr ω ( f | [ F K , x ][ F K , x ] | d ) = 4 · s log 32 s/ s + 2 s/ s log 2 (cid:90) K f d Λ for any f ∈ C ( K ) . Here, Λ is the dim H ( K ) -dimensional Hausdorff probabilitymeasure of K . References [1] Erik Christensen, Cristina Ivan, and Michel L. Lapidus. Dirac operators and spectral triplesfor some fractal sets built on curves.
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