A spectral triple for a solenoid based on the Sierpinski Gasket

The Sierpinski gasket admits a natural locally isometric ramified self-covering. A semifinite spectral triple is constructed on the resulting solenoidal space, and its main geometrical features are discussed.


Introduction
The topological space known as Sierpinski gasket [16] is considered here, its geometrical properties being described by the discrete spectral triple considered in [12], cf. also [11].
It turns out that this space possesses a ramified self-covering, which is locally isometric with respect to suitable distances on the base and on the covering, which are induced by suitable spectral triples.
On the one hand such self-covering produces a tower of covering spaces and hence a compact solenoidal space; on the other hand it gives rise to an endomorphism of the C * -algebra of continuous functions on the gasket, hence to an inductive family of C * -algebras and to an inductive limit C * -algebra (cf. [6]) which is simply the algebra of continuous functions on the solenoid.
Each of the C * -algebras of the inductive family may then be endowed with a spectral triple [5,10]. The local isometricity of the covering implies that the Lip-norms given by the Dirac operators 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 triples produces a spectral triple on the solenoidal space. Because of the local isometricity, the limiting Dirac operator has no longer compact resolvent, but its resolvent is τ -compact with respect to a natural trace on the inductive limit C * -algebra. This implies that the spectral triple on the solenoid is semifinite [3].
However, in the examples considered in [1], the family of spectral triples had a simple tensor product structure, namely the Hilbert spaces were a tensor product of the Hilbert space for the base space and a finite dimensional Hilbert space, and the Dirac operators could be described as (a finite sum of) tensor product operators. Then the ambient C * -algebra turned out to be a product of B(H) for the base space and a UHF algebra, allowing a GNS representation w.r.t. a semifinite trace.
In the example treated here two problems forbid such simple description. The first is a local problem, due to the ramification points. This implies that the algebra of a covering is not a free module on the algebra of the base space; in particular, functions on a covering space form a proper sub-algebra of the direct sum of finitely many copies of the algebra for the base space.
The second is a non-local problem which concerns the Hilbert spaces, which are ℓ 2 spaces on edges, and the associated operator algebras. Indeed, the Hilbert spaces of the coverings cannot be described as finite sums of copies of the Hilbert space on the base space due to appearance 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, the present method being inspired by the treatment in [15,9,4].
The idea in [15], partly modified in [9], was to replace the von Neumann trace used by Atiyah [2] for his index theorem for covering manifolds, by a trace on the C * -algebra of finite propagation operators acting on sections of a bundle on an open manifold. Unfortunately such 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 becomes canonical when restricted to the C * -algebra of geometrical operators. We adapt these results to our present context, namely that of an infinite fractafold [17], which has some features similar to those of an open manifold, but also sufficient self-similarity to obtain a canonical trace.
We then construct a semifinite spectral triple on the algebra of continuous functions on the Sierpinski solenoid, where the main role of the semifinite von Neumann algebra is instead played by the C * -algebra of geometric operators together with its semicontinuous semifinite trace. Finally, we show that the metric dimension of such spectral triple coincides with the Hausdorff dimension of the Sierpinski gasket, the noncommutative integral coincides with the normalized integral w.r.t. the Hausdorff measure on a dense sub-algebra, and the Connes distance on points recovers exactly the geodesic distance on the open fractafold.
This paper is divided in five sections. After this introduction, Section 2 describes the geometry of the ramified covering and the corresponding inductive structure, together with its functional counterpart given by a family of compatible spectral triples. Section 3 concerns the self-similarity structure of the Sierpinski solenoid, whence the description of the inductive family of C * -algebras as algebras of bounded functions on the fractafold. The 4th Section describes the algebra of geometric operators and the construction of a semicontinuous semifinite trace on it. Finally, the semifinite spectral triple together with its main features are contained in Section 5.

A ramified covering of the Sierpinski gasket
Let us recall that the Sierpinski gasket K may be defined as the compact subset of a closed equilateral triangle with vertices v 0 , v 1 , v 2 (numbered in a counterclockwise order) in the Euclidean plane with the property 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 1 · · · · · w i k (K) : k ≥ 0}. The size of a cell is the length of its side; clearly if C = w i 1 · · · · · w i k (K), size(C) = 2 −k . We call (oriented) edge any element of the family E 0 = {w i 1 · · · · · w i k (e) : k ≥ 0, e is one of the oriented edges of the triangle}. Clearly if e 0 is one of the oriented edges of the triangle and e = w i 1 · · · · · w i k (e 0 ), length(e) = 2 −k .
We then construct the map p : K 1 → K given by 0 w 2 K, and observe that this map, which appears to be doubly defined in the points x i,i+1 , i = 0, 1, 2, is indeed well defined.
The following result is easily verified.
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, 1, 2}. Moreover, the covering map is isometric on suitable neighbourhoods of the non-ramification points.
Since K 1 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 1,0 : C(K) → C(K 1 ), hence, following [6], to an inductive family of C * -algebras A n = C(K n ), K n = w −n 0 K, whose inductive limit A ∞ consists of continuous function on the solenoidal space based on the gasket.
Following [11,12], we build a spectral triple on any of the algebras A n , n ≥ 0. Denoting by E n = {w −n 0 e, e ∈ E 0 } the set of oriented edges in K n , we define H n = ℓ 2 (E n ), F as the map the changes the orientation of an edge, D n as the map sending an edge e ∈ E n to length(e) −1 F e. As proved in [11,12], the triple (A n , H n , D n ) is a spectral triple. Moreover, by [12], Corollary 5.14, the Connes distances induced by these spectral triples recover the geodesic distances on the points of the gaskets K n , hence the local isometricity of p implies

A groupoid of local isometries on the infinite Sierpinski fractafold
Let us consider the infinite fractafold K ∞ = ∪ n≥0 K n [17] endowed with the Hausdorff measure µ d of dimension d = log 3 log 2 normalized to be 1 on K = K 0 , with the exhaustion {K n } n≥0 , and with the family of local isometries We also denote by s(γ) and r(γ) the domain and range of the local isometry γ. Such local isometries act on points and on oriented edges of K ∞ .
We say that the product of the two local isometries In this case we consider the product . We then consider the family G consisting of all (the well-defined) finite products of isometries in R. Clearly, any γ in G is a local isometry, and its domain and range are cells of the same size. We set G n = {g ∈ G : s(γ) & ρ(γ) are cells of size 2 n }, n ≥ 0.
Proposition 3.1. For any n ≥ 0, C 1 , C 2 cells of size 2 n , ∃!γ ∈ G n such that s(γ) = C 1 , r(γ) = C 2 . In particular, if C has size 2 n , the identity map of C belongs to G n , n ≥ 0.
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 Iterating, the result follows. The second statement follows directly by equation (2.1). As for the uniqueness, ∀n ≥ 0, we call R n i,0 ascending, The following facts hold: • The product R n l,k · R m j,i of two constant-level elements R n l,k , R m j,i is defined iff n = m and k = j, therefore any product of constant-level elements in R is either the identity map on the domain or coincides with a single constant-level element. • Any product of constant level elements in R followed by a descending element coincides with a single descending element: indeed, if the product of constant level elements is the identity, the statement is trivially true; if it coincides with a single element, say R n i,j with i, j ∈ {1, 2}, then, by compatibility, the descending element should be R n 0,i so that the product is R n 0,i , by equation (2.1). • Given a cell C with size(C) = 2 n and level(C) > n, the exists a unique descending The only descending element is then γ = R m−1 0,i . • Any product of an ascending element followed by a descending one is the identity on the domain: indeed if the ascending element is R n i,0 , then, by compatibility, the descending element should be R n 0,i . Now let size(C) = 2 n , γ ∈ G n such that γ : Since level(C) ≥ level(K n ) = n, for any possible ascending element γ i there should be a j > i such that γ j is descending. If i + q is the minimum among such j's, all terms γ j , i < j < i + q, are constant-level, hence the product γ i+q · γ i+q−1 · · · · · γ i = id s(γ i ) . Then, we note that γ p can only be descending. As a consequence, γ can be reduced to a product of descending elements, and, by the uniqueness of the descending element acting on a given cell, we get the result.
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 range of the latter.
We now consider the action on points of the local isometries in G.
Proposition 3.2. Let us define A n as the algebra Then, for any n ≥ 0, the following diagram commutes, Proof. The request in the definition of A n means that the value of f in any point of K ∞ is determined 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.
As shown above, we may identify the algebra A n , 0 ≤ n ≤ ∞, with its isomorphic copy A n in C b (K ∞ ), so that the embeddings α k,j become inclusions. Moreover, we may consider the operator D n on ℓ 2 (E ∞ ), with E ∞ = ∪ n≥0 E n , given by D n e = length(e) −1 F e, if length(e) ≤ 2 n , and D n e = 0, if length(e) > 2 n . Then the spectral triples (A n , H n , D n ) are isomorphic to the spectral triples ( A n , H n , D n ), where C b (K ∞ ) acts on the space ℓ 2 (E ∞ ) through the representation ρ given by ρ(f )e = f (e + )e, where e + denotes the target of the oriented edge e.
Remark 3.3. Because of the isomorphism above, from now on we shall remove the tildes and denote by A n the subalgebras of C b (K ∞ ) and by D n the operators acting on ℓ 2 (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 the projection on the closed subspace spanned by the same subset: Table 1. Edges and projections.
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 2 (E ∞ )), and note that the elements of B n commute with the projections P C , for all cells C s.t. size(C) ≥ 2 n . By definition, the sequence B n is increasing, therefore, since the B n 's are von Neumann algebras, B ∞ is a C * -algebra. Let us observe that, ∀n ≥ 0, ρ(A n ) ⊂ B n . We then define the weight τ 0 on B + ∞ as follows: The next step is to regularize the weight τ 0 in order to obtain a semicontinuous semifinite tracial weight τ on B ∞ .   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 , Indeed, Let T ∈ B + n ∩ I + 0 , and ε > 0. From the definition of τ 0 (T ), there exists r ∈ N, r > n, such that tr(PrT ) Then, for any s ∈ N, s > r, we have tr(P s P −p,∞ T P −p,∞ P s ) and, passing to the limit for s → ∞, we get and equation (4.5) follows.
We want to prove that τ is a tracial weight.
Proof. (i) Let us prove that J + is a unitarily-invariant face in B + ∞ , and suffices it to prove that A ∈ J + implies that UAU * ∈ 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 (UAU * ) = ∞.

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 have a simple tensor product structure, contrary to what happened in [1]. We therefore use a different approach to construct a semifinite spectral triple on A ∞ : our construction is indeed based on the pair (B ∞ , τ ) of the C * -algebra of geometric operators and the semicontinuous semifinite weight on it.
The Dirac operator will be defined through its phase F defined above and the functional calculi of its modulus with continuous functions vanishing at ∞. More precisely we shall use the following Definition 5.1. Let (C, τ ) be a C * -algebra with unit endowed with a semicontinuous semifinite faithful trace. A selfadjoint operator T affiliated to (C, τ ) is defined as a pair given by a closed subset σ(T ) in R and a * homomorphism φ : C(σ(T )) → C, f (T ) def = φ(f ), provided that the support of such homomorphism is the identity in the GNS representation π τ induced by the trace τ .
The previous definition was inspired by that in [7] appendix A, and should not be confused with that of Woronowicz for C * -algebras without identity.
(b) Let λ be in the resolvent of |T |. We then note that for any f positive and zero on a neighbourhood of the origin there is a g positive and with compact support such that f ((|T |− λI) −1 ) = g(|T |). Therefore τ (f ((|T | − λI) −1 )) < ∞, hence τ (e (t,+∞) (π τ ((|T | − λI) −1 ))) < ∞ for any t > 0, which is the definition of τ -compactess (cf. Proof. (a) We first observe that the * -homomorphisms for D and |D| have the same support projection, then note that since F and P n belong to B 0 (which is a von Neumann algebra) for any n ∈ N, then f (D) and f (|D|) belong to B 0 for any f ∈ C 0 (R); therefore it is enough to 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, 2 p ]) tends to the identity strongly when p → ∞, that is to say that π τ (e |D| (2 p , ∞)) tends to 0 strongly when p → ∞. We consider then the projection P −∞,0 which projects on the space generated by the edges with length(e) ≤ 1. Clearly, such projection belongs to B 0 , we now show that it is indeed central there. In fact, if c is a cell with size(c) = 1, P c commutes with B 0 . Since P −∞,0 = size(c)=1 P c , then P −∞,0 commutes with B 0 . On the one hand, the von Neumann algebra P −∞,0 B 0 is isomorphic to B(ℓ 2 (K)) and the restriction of τ to P −∞,0 B 0 coincides with the usual trace on B(ℓ 2 (K)), therefore the representation π τ is normal when restricted to P −∞,0 B 0 . On the other hand, since e |D| (2 p , ∞) = P −∞,−p−1 is, for −p ≤ 1, a sub-projection of P −∞,0 , and P −∞,−p−1 tends to 0 strongly in the given representation, the same holds of the representation π τ .
(b) Observe that the projection P n ∈ B 0 on the edges of length 2 n has finite τ -trace: This implies that τ (e |D| [0, 2 p ]) = τ (P −p,∞ ) < ∞: Recall now that any edge e of length 2 n+1 is the union of two adjacent edges e 1 and e 2 of length 2 n such that e + 1 = e − 2 , therefore Iterating, we get  Theorem 5.7. The triple (L, π τ (B ∞ ) ′′ , π τ (D)) on the unital C * -algebra A ∞ is an odd semifinite spectral triple, where L = ∪ n {f ∈ A n , f Lipschitz}. The spectral triple has metric dimension d = log 3 log 2 , the functional is a finite trace on A ∞ where τ ω is the logarithmic Dixmier trace associated with τ , and Kn f dµ d µ d (K n ) by Theorem 3.3 in [12]. 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 , [π τ (D), π τ • ρ(f )] = [D m , ρ(f | Km )] , and, by Theorem 5.2 and Corollary 5.14 in [12], sup{|f (x) − f (y)| : f ∈ A m , [D m , ρ(f | Kn )] = 1} = d geo (x, y), m ≥ n. Remark 5.8. The last statement in Theorem 5.7 shows that the triple (L, M, D ∞ ) recovers two incompatible aspects of the space A ∞ : on the one hand the compact space given by the spectrum of the unital algebra A ∞ , with the corresponding finite integral, and on the other hand the open fractafold K ∞ with its geodesic distance. In particular, the functional L(f ) = [D, ρ(f )] , f ∈ L, is not a Lip-norm (cf. [14]), since it does not recover the weak * topology on S(A ∞ ).