A Spectral Triple for a Solenoid Based on the Sierpinski Gasket

A Spectral Triple for a Solenoid Based
on the Sierpinski Gasket††This paper is a contribution to the Special Issue on
Noncommutative Manifolds and their Symmetries in honour of Giovanni Landi. The
full collection is available at https://www.emis.de/journals/SIGMA/Landi.html

Valeriano AIELLO a, Daniele GUIDO b and Tommaso ISOLA b

V. Aiello, D. Guido and T. Isola

a) Mathematisches Institut, Universität Bern, Alpeneggstrasse 22, 3012 Bern,
Switzerland<EMAIL_ADDRESS>
b) Dipartimento di Matematica, Università di Roma “Tor Vergata”, I–00133 Roma,
Italy<EMAIL_ADDRESS><EMAIL_ADDRESS>
Received June 23, 2020, in final form February 10, 2021; Published online
March 02, 2021

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

self-similar fractals; noncommutative geometry; ramified coverings 58B34;
28A80; 47D07; 46L05

##  1 Introduction

In this note, we introduce a semifinite spectral triple on the $C^{*}$-algebra
of continuous functions on the solenoid associated with a self-covering of the
Sierpinski gasket. Such triple is finitely summable, its metric dimension
coincides with the Hausdorff dimension of the gasket, and the associated non-
commutative integral coincides up to a constant with a Bohr–Følner mean on the
solenoid, hence reproduces the suitably normalized Hausdorff measure on
periodic functions. The open infinite Sierpinski fractafold with a unique
boundary point considered by Teplyaev [51] embeds continuously as a dense
subspace of the solenoid, and the Connes distance restricted to such subspace
reproduces the geodesic distance on such fractafold. On the one hand, this
shows that our spectral triple describes aspects of both local and coarse
geometry [45]. On the other hand, this implies that the topology induced by
the Connes distance, being non compact, does not coincide with the
weak∗-topology on the states of the solenoid algebra, as we call the
$C^{*}$-algebra of continuous functions on the solenoid. This means that the
solenoid, endowed with our spectral triple, is not a quantum metric space in
the sense of Rieffel [41].

Related research concerning projective limits of (possibly quantum) spaces and
the associated solenoids appeared recently in the literature. In the framework
of noncommutative geometry, we mention: [37], where projective families of
compact quantum spaces have been studied, showing their convergence to the
solenoid w.r.t. the Gromov–Hausdorff propinquity distance; [1], where, in the
same spirit as in this note, a semifinite spectral triple has been associated
with the projective limit generated by endomorphisms of $C^{*}$-algebras
associated with commutative and noncommutative spaces; [18], where a spectral
triple on the stable Ruelle algebra for Wieler solenoids has been considered
and its unboundedd KK-theory has been studied, based on the Morita equivalence
between the stable Ruelle algebra and a Cuntz–Pimsner algebra. In the same
paper these techniques are used for the study of limit sets of regular self-
similar groups (cf. [39]).

When fractals are concerned, we mention the projective family of finite
coverings of the octahedron gasket considered in [50], where, as in our
present situation, an intermediate infinite fractafold between the tower of
coverings and the projective limit is considered. Periodic and almost periodic
functions on the infinite fractafold are considered, and a Fourier series
description for the periodic functions is given, based on periodic
eigenfunctions of the Laplacian (cf. also [46] for higher-dimensional
examples). Let us remark that such coverings, as the ones considered in this
paper, are not associated with groups of deck transformations.

The starting point for the construction of this paper is the existence of a
locally isometric ramified three-fold self-covering of the Sierpinski gasket
with trivial group of deck transformations. Such self-covering gives rise to a
projective family of coverings, whose projective limit is by definition a
solenoid. Dually, the algebras of continuous functions on the coverings form
an injective family, whose direct limit (in the category of $C^{*}$-algebras)
is the solenoid algebra. In [1] we already considered various examples of
self-coverings or, dually, of endomorphisms of some $C^{*}$-algebras, most of
which were regular finite self-coverings. There we constructed a spectral
triple on the solenoid algebra as a suitable limit of spectral triples on the
algebras of continuous functions on the coverings. Given a spectral triple on
the base space, attaching a spectral triple to a finite covering is not a
difficult task, and in our present case consists simply in “dilating” the
triple on the base gasket so that the projections are locally isometric.
However, there is no commonly accepted procedure to define a limit of spectral
triples. Since the method used in [1] cannot be used here (see below), we
follow another route, in a sense spatializing the construction, namely showing
that there exists an open fractafold which is intermediate between the
projective family of coverings and the solenoid. More precisely, such
fractafold space turns out to be an infinite covering of each of the finite
coverings of the family, and embeds in a continuous way in the solenoid. In
this way all the algebras (and their direct limit) will act on a suitable
$L^{2}$-space of the open fractafold, as do the Dirac operators of the
associated spectral triples. In this way the limiting Dirac operator is well
defined, but the compact resolvent property will be lost.

Let us notice here that we are not constructing a spectral triple on the open
fractafold, where a weaker compact resolvent property (cf. [16, Chapter IV,
Remark 12]) is retained, namely $f(D^{2}+I)^{-1/2}$ is a compact operator,
where $D$ is the Dirac operator and $f$ is any function with compact support
on the fractafold. Since we are constructing a spectral triple on the
solenoid, which is a compact space, the weaker form does not help.

In order to recover the needed compactness of the resolvent, we use a
procedure first proposed by J. Roe for open manifolds with an amenable
exhaustion in [42, 43], where, based on the observation that the von Neumann
trace used by Atiyah [4] for his index theorem for covering manifolds can be
reformulated in the case of amenable groups via the Følner condition, he
considered amenable exhaustions on open manifolds and constructed a trace for
finite-propagation operators acting on sections of a fiber bundle on the
manifold via a renormalization procedure. 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 [13] that such
trace becomes canonical when restricted to the $C^{*}$-algebra of geometric
operators.

We adapt these results to our present context, namely we replace the usual
trace with a renormalized trace associated with an exhaustion of the infinite
fractafold. Such trace comes together with a noncommutative $C^{*}$-algebra,
the algebra of geometric operators, which is similar in spirit to the Roe
$C^{*}$-algebras of coarse geometry [31, 42, 43, 44, 52]. This algebra
contains the solenoid algebra, and the limiting Dirac operator is affiliated
to it in a suitable sense. Such Dirac operator turns out to be $\tau$-compact
w.r.t. the renormalized trace. We refer to [13, 26] for an analogous
construction of the $C^{*}$-algebra and of a canonical trace based on the
self-similarity structure.

As discussed above, the starting point for the construction of a spectral
triple on the solenoid algebra is the association of a spectral triple to the
fractal known as the Sierpinski gasket [47].

The study of fractal spaces from a spectral, or noncommutative, point of view
has now a long history, starting from the early papers of Kigami and Lapidus
[33, 35, 36]. As for the spectral triples, various constructions have been
considered in the literature, mainly based on “small” triples attached to
specific subsets of the fractal, following a general procedure first
introduced by Connes, then considered in [27, 28], and subsequently abstracted
in [10]. More precisely, the spectral triple on the Cantor set described by
Connes [16] inspired two kinds of spectral triples for various families of
fractals in [27, 28]. These triples were further analysed in [29] for the
class of nested fractals. Such specral triples are obtained as direct sums of
triples on two points (boundary points of an edge in some cases), and we call
them discrete spectral triples. We then mention some spectral triples obtained
as direct sums of spectral triples on 1-dimensional subsets, such as those
considered in [3, 11, 12, 15, 34], where the 1-dimensional subsets are
segments, circles or quasi-circles. Discrete spectral triples give a good
description of metric aspects of the fractal, such as Hausdorff dimension and
measure and geodesic distance, and, as shown in [24], may also reconstruct the
energy functional (Dirichlet form) on the fractal, but are not suited for the
study of K-theoretical properties since the pairing with K-theory is trivial.
Conversely, spectral triples based on segments or circles describe both metric
and K-theoretic properties of the fractal but can’t be used for describing the
Dirichlet form. Finally, the spectral triple based on quasi-circles considered
in [15] describes metric and K-theoretic aspects together with the energy
form, but requires a rather technical approach.

In the present paper, we make use of the simple discrete spectral triple on
the gasket as described in [29], thus obtaining a semifinite spectral triple
on the solenoid algebra which recovers the metric dimension and the
Bohr–Følner mean of the solenoid, and the geodesic distance on the infinite
fractafold. Further analysis on the solenoid is possible, e.g., the
construction of a Dirichlet form via noncommutative geometry or the study of
K-theoretic properties. As explained above, the latter step will require a
different choice of the spectral triple on the base gasket, such as the
triples considered in [11, 12, 15], which admit a non-trivial pairing with the
K-theory of the gasket.

As already mentioned, our aim here is to show that the family of spectral
triples on the finite coverings produces a spectral triple on the solenoidal
space. 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 ${\mathcal{H}}$ 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 ${\mathcal{B}}({\mathcal{H}})$ and a UHF
algebra, allowing a GNS representation w.r.t. a semifinite trace.

In the example treated here we choose a different approach since 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 $\ell^{2}$-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 the appearance of longer and longer edges on larger and larger
coverings.

We conclude this introduction by mentioning two further developments of the
present analysis.

First, the construction of the spectral triple on the solenoid algebra allows
the possibility of lifting a spectral triple from a $C^{*}$-algebra to the
crossed product of the C*-algebra with a single endomorphism [2], thus
generalising the results on crossed products with an automorphism group
considered in [6, 30, 40].

Second, we observe that the construction given in the present paper goes in
the direction of possibly defining 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 non-commutative
space are affiliated, where the compactness of the resolvent of the Dirac
operator is measured by the trace on the $C^{*}$-algebra, cf. also [25].

This paper is divided in six sections. After this introduction, Section 2
contains some preliminary notions on fractals and spectral triples, Section 3
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 4 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 Section 5 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 6.

##  2 Preliminaries

In this section we shall briefly recall various notions that will be used in
the paper. Though these notions are well known among the experts, our note
concerns different themes, namely spectral triples in noncommutative geometry
and nested fractals (the Sierpinski gasket in particular), so that we decided
to write this section with the aim of helping readers with different
background to follow the various arguments, by collecting here the main
notions and results that will be useful in the following.

###  2.1 Spectral triples

The notion of spectral triple plays a key role in Alain Connes’noncommutative
geometry [16, 23]. Basically, it consists of a triple
$({\mathcal{L}},{\mathcal{H}},D)$, where ${\mathcal{L}}$ is a *-algebra acting
faithfully on the Hilbert space ${\mathcal{H}}$, and $D$ is an unbounded self-
adjoint operator on ${\mathcal{H}}$ satisfying the properties

  * $(1)$

$\big{(}1+D^{2}\big{)}^{-1/2}$ is a compact operator,

  * $(2)$

$\pi(a)\mathcal{D}(D)\subset\mathcal{D}(D)$, and $[D,\pi(a)]$ is bounded for
all $a\in{\mathcal{L}}$.

We shall also say that $({\mathcal{L}},{\mathcal{H}},D)$ is a spectral triple
on the $C^{*}$-algebra ${\mathcal{A}}$ generated by ${\mathcal{L}}$.

Such triple is meant as a generalization of a compact smooth manifold, the
algebra ${\mathcal{L}}$ replacing the algebra of smooth functions, the Hilbert
space describing a vector bundle (a spin bundle indeed) on which the algebra
of functions acts, and the operator $D$ generalizing the notion of Dirac
operator. Further structure may be added to the properties above, allowing
deeper analysis of the geometric features of the noncommutative manifold, but
these are not needed in this paper.

Property $(2)$ above allows the definition of a (possibily infinite) distance
(Connes distance) on the state space of the $C^{*}$-algebra ${\mathcal{A}}$
generated by ${\mathcal{L}}$ , defined as

$\displaystyle d(\varphi,\psi)=\sup\\{|\varphi(a)-\psi(a)|\colon\|[D,a]\|\leq
1,\,a\in{\mathcal{L}}\\}.$

When the Connes distance induces the weak∗-topology on the state space, the
seminorm $\|[D,a]\|$ on ${\mathcal{A}}$ is called a Lip-norm (cf. [41]) and
the algebra ${\mathcal{A}}$ endowed with the Connes distance is a quantum
metric space.

A spectral triple is called finitely summable if $\big{(}1+D^{2}\big{)}^{-s}$
has finite trace for some $s>0$, in this case the abscissa of convergence $d$
of the function $\operatorname{tr}\big{(}1+D^{2}\big{)}^{-s}$ is called the
metric dimension of the triple. Then the logarithmic singular trace introduced
by Dixmier [19] may be used to define a noncommmutative integral on
${\mathcal{A}}$. Let us denote by $\\{\mu_{n}(T)\\}$ the sequence (with
multiplicity) of singular values of the compact operator $T$, arranged in
decreasing order. Then, on the positive compact operators for which the
sequence $\sum_{k=1}^{n}\mu_{n}(T)$, is at most logarithimically divergent, we
may consider the positive functional

$\displaystyle\operatorname{tr}_{\omega}(T)=\operatorname{Lim}_{\omega}\frac{\sum_{k=1}^{n}\mu_{n}(T)}{\log
n},$

where $\operatorname{Lim}_{\omega}$ is a suitable generalized limit. Such
functional extends to a positive trace on ${\mathcal{B}}({\mathcal{H}})$ which
vanishes on trace class operators, and is called Dixmier (logarithmic) trace.

If $\big{(}1+D^{2}\big{)}^{-d}$ is in the domain of the Dixmier trace, one
defines the following noncommutative integral:

$\displaystyle\oint
a=\operatorname{tr}_{\omega}\big{(}a\big{(}I+D^{2}\big{)}^{-d/2}\big{)},\qquad
a\in{\mathcal{A}}.$

When the function $\big{(}1+D^{2}\big{)}^{-s}$ has a finite residue for $s=d$,
such residue turns out to coincide, up to a constant, with the Dixmier trace,
which therefore does not depend on the generalized limit procedure (cf. [16],
and [9, Theorem 3.8]):

$\displaystyle
d\cdot\operatorname{tr}_{\omega}\big{(}a\big{(}I+D^{2}\big{)}^{-d/2}\big{)}={\operatorname{Res}}_{s=d}\operatorname{tr}(a|D|^{-s}).$

We note in passing that spectral triples may also describe non-compact smooth
manifolds, with the algebra ${\mathcal{L}}$ describing smooth functions with
compact support and property (1) replaced by $a\big{(}1+D^{2}\big{)}^{-1/2}$
is a compact operator for any $a\in{\mathcal{L}}$.

###  2.2 Semifinite spectral triples

The notion of spectral triple has been generalized to the semifinite case, by
replacing the ambient algebra ${\mathcal{B}}({\mathcal{H}})$ with a semifinite
von Neumann algebra ${\mathcal{M}}$ endowed with a normal semifinite faithful
trace $\tau$. We recall that an operator $T$ affiliated with
$({\mathcal{M}},\tau)$ is called $\tau$-compact if its generalized s-number
function $\mu_{t}(T)$ is infinitesimal or, equivalently, if
$\tau(e_{(t,\infty)}(T))<\infty$, for any $t>0$ (cf. [21, Section 1.8, p. 34],
[22, Proposition 3.2]).

######  Definition 2.1 ([8]).

An odd semifinite spectral triple $({\mathcal{L}},{\mathcal{M}},D)$ on a
unital C∗-algebra ${\mathcal{A}}$ is given by a unital, norm-dense,
∗-subalgebra ${\mathcal{L}}\subset{\mathcal{A}}$, a semifinite von Neumann
algebra $({\mathcal{M}},\tau)$, acting on a (separable) Hilbert space
${\mathcal{H}}$, a faithful representation
$\pi\colon{\mathcal{A}}\to{\mathcal{B}}({\mathcal{H}})$ such that
$\pi({\mathcal{A}})\subset{\mathcal{M}}$, and an unbounded self-adjoint
operator $D\widehat{\in}{\mathcal{M}}$ such that

  * $(1)$

$\big{(}1+D^{2}\big{)}^{-1/2}$ is a $\tau$-compact operator,

  * $(2)$

$\pi(a)\mathcal{D}(D)\subset\mathcal{D}(D)$, and $[D,\pi(a)]\in{\mathcal{M}}$,
for all $a\in{\mathcal{L}}$.

As in the type $I$ case, such triple is called finitely summable if
$\big{(}1+D^{2}\big{)}^{-s}$ has finite trace for some $s>0$, and $d$ denotes
the abscissa of convergence of the function $\tau\big{(}1+D^{2}\big{)}^{-s}$,
and is called the metric dimension of the triple. The logarithmic Dixmier
trace associated with the normal trace $\tau$ may be defined in this case too,
(cf. [9, 24]) and, when the function $\big{(}1+D^{2}\big{)}^{-s}$ has a finite
residue for $s=d$, the equality
$d\cdot\operatorname{tr}_{\omega}\big{(}a|D|^{-d}\big{)}={\operatorname{Res}}_{s=d}\operatorname{tr}\big{(}a|D|^{-s}\big{)}$
still holds [9, Theorem 3.8].

###  2.3 Self-similar fractals

Let $\Omega:=\\{w_{i}\colon i=1,\ldots,k\\}$ be a family of contracting
similarities of ${\mathbb{R}}^{N}$, with scaling parameters
$\\{\lambda_{i}\\}$. The unique non-empty compact subset $K$ of
${\mathbb{R}}^{N}$ such that $K=\bigcup_{i=1}^{k}w_{i}(K)$ is called the self-
similar fractal defined by $\\{w_{i}\\}_{i=1,\ldots,k}$. For any
$i\in\\{1,\ldots,k\\}$, let $p_{i}\in{\mathbb{R}}^{N}$ be the unique fixed-
point of $w_{i}$, and say that $p_{i}$ is an essential fixed-point of $\Omega$
if there are $i^{\prime},j,j^{\prime}\in\\{1,\ldots,k\\}$ such that
$i^{\prime}\neq i$, and $w_{j}(p_{i})=w_{j^{\prime}}(p_{i^{\prime}})$. Denote
by $V_{0}(K)$ the set of essential fixed-points of $\Omega$, and let
$E_{0}(K):=\\{(p,q)\colon p,q\in V_{0},\ p\neq q\\}$. Observe that
$(V_{0},E_{0})$ is a directed finite graph whose edges are in $1:1$
correspondence with ordered pairs of distinct vertices.

######  Definition 2.2.

We call an element of the family $\\{w_{i_{1}}\cdots w_{i_{k}}(K)\colon k\geq
0\\}$ a cell, and call its diameter the size of the cell. We call an element
of the family $E(K)=\\{w_{i_{1}}\cdots w_{i_{k}}(e)\colon k\geq 0$, $e\in
E_{0}(K)\\}$ an $($oriented$)$ edge of $K$. We denote by $e^{-}$ resp. $e^{+}$
the source, resp. the target of the oriented edge $e$.

As an example, the Sierpinski gasket is the self-similar fractal determined by
3 similarities with scaling parameter 1/2 centered in the vertices of an
equilateral triangle (see Fig. 1).

Figure 1: The first four steps of the construction of the gasket.

Under suitable conditions, the Hausdorff dimension $d_{H}$ of a self-similar
fractal coincides with its scaling dimension, namely with the only positive
number $d$ such that $\sum_{i=1}^{k}\lambda_{i}^{d}=1$, therefore when all
scaling parameters coincide with $\lambda$ we have $d_{H}=\frac{\log
k}{\log(1/\lambda)}$. In particular, the Hausdorff dimension of the Sierpinski
gasket is $\frac{\log 3}{\log 2}$. We note in passing that one of the most
important aspects of the Sierpinski gasket and of more general classes of
fractals is the existence of a self-similar diffusion, associated with a
Dirichlet form, see, e.g., [32]. Even though Dirichlet forms on fractals can
be recovered in the noncommutative geometry framework [29], and in particular
by means of the spectral triples which we use in this paper, we do not analyse
this aspect in the present note.

In [28] discrete spectral triples have been introduced on some classes of
fractals, generalizing an example of Connes in [16, Chapter 4.3, Example 23].
Such triples have been further studied in [29] for nested fractals. On a self-
similar fractal $K$, the triple $({\mathcal{L}},{\mathcal{H}},D)$ on the
$C^{*}$-algebra ${\mathcal{A}}={\mathcal{C}}(K)$ is defined as follows:

######  Definition 2.3.

  * $(a)$

${\mathcal{H}}=\ell^{2}(E(K))$,

  * $(b)$

${\mathcal{A}}$ acts on the Hilbert space as $\rho(f)e=f(e^{+})e$,
$f\in{\mathcal{A}}_{n}$, $e\in E_{n}$,

  * $(c)$

$F$ is the orientation-reversing map on edges,

  * $(d)$

$D$ maps an edge $e\in E(K)$ to $\mathrm{length}(e)^{-1}Fe$,

  * $(e)$

${\mathcal{L}}$ is given by the elements $f\in{\mathcal{A}}$ such that
$\|[D,\rho(f)]\|<\infty$.

It turns out that ${\mathcal{L}}$ coincides with the algebra of Lipschitz
functions on $K$, hence is dense in ${\mathcal{A}}$, and the seminorm
$L(f):=\|[D,f]\|$ is a Lip-norm. By Theorem 3.3 in [29], see also Remark 2.11
in [28], the triple $({\mathcal{L}},{\mathcal{H}},D)$ is a finitely summable
spectral triple on ${\mathcal{A}}$, its metric dimension coincides with the
Hausdorff dimension, and the noncommutative integral recovers the Hausdorff
measure up to a constant:

$\displaystyle\oint
f=\operatorname{tr}_{\omega}\big{(}f|D|^{-d}\big{)}=\frac{1}{\log k}\sum_{e\in
E_{0}(K)}\ell(e)^{d}\int_{K}f\,{\rm d}H_{d},\qquad f\in C(K),$ (2.1)

where $H_{d}$ denotes the normalized Hausdorff measure on the fractal $K$.
Moreover, in some cases, and in particular for the Sierpinski gasket, the
Connes distance induced by the Lip-norm $L(f):=\|[D,f]\|$ coincides with the
geodesic distance on the points of the gasket $K$, see [29, Corollary 5.14].

###  2.4 Covering fractafolds and solenoids

Generally speaking, a solenoid is the inverse limit of a projective family of
coverings of a given space [38]. Dually, the solenoid algebra is the direct
limit of the family of algebras of continuous functions on the spaces of the
projective family. In this sense the notion of solenoid makes sense for
injective families of $C^{*}$-algebras, cf., e.g., [1] for sequences generated
by a single endomorphism and [37] for sequences of compact quantum spaces.
Other examples of the treatment of solenoids in the recent literature have
been mentioned in the introduction.

The notion of fractafold as a connected Hausdorff topological space such that
every point has a neighborhood homeomorphic to a neighborhood in a given
fractal has been introduced in [49], even though examples of such notion were
already considered before, e.g., in [5, 48, 51]. In some cases projective
families of covering fractafold spaces related to the Sierpinski gasket have
been considered.

Since the gasket does not admit a simply connected covering, one may consider
coverings where more and more cycles are unfolded, in particular consider the
regular infinite abelian covering $S_{n}$ where all the cycles of size at
least $2^{-n}$ are unfolded. Each of those is a closed fractafold (with
boundary) and they form a projective family. The associated solenoid
$S_{\infty}$, i.e., the projective limit, which turns out to be an abelian
counterpart of the Uniform Universal Cover introduced by Berestovskii and
Plaut [7], has been considered in [14], where it is shown that any locally
exact 1-form on the gasket possesses a potential on $S_{\infty}$.

Another projective family of covering fractafolds has been considered in [50],
each element of the family being a compact finite covering of the octahedral
fractafold modeled on the gasket. Any element of the family is covered by the
infinite Sierpinski gasket with a unique boundary point, which we call
$K_{\infty}$ here (see Fig. 2), considered in [51, Lemma 5.11]. The solenoid
associated with the projective family is also mentioned explicitly in [50],
together with the dense embedding of $K_{\infty}$ in it, and also a
Bohr–Følner mean on the solenoid is considered (p. 1199).

Figure 2: The gasket and its infinite blowup.

In the present paper a self-covering of the gasket gives rise to a projective
family of finite ramified coverings, the fractafold $K_{\infty}$ projects onto
each element of the family and embeds densely in the solenoid, and we recover
the Bohr–Følner mean on the solenoid via a noncommutative integral.

##  3 A ramified covering of the Sierpinski gasket

Let us choose an equilateral triangle of side 1 in the Euclidean plane with
vertices $v_{0}$, $v_{1}$, $v_{2}$ (numbered in a counterclockwise order) and
consider the associated Sierpinski gasket as in the previous section, namely
the set $K$ such that

$\displaystyle K=\bigcup_{j=0,1,2}w_{j}(K),$

where $w_{j}$ is the dilation around $v_{j}$ with contraction parameter $1/2$.
Clearly, for the cell $C=w_{i_{1}}\cdots w_{i_{k}}(K)$,
$\mathrm{size}(C)=2^{-k}$ and, if $e_{0}\in E_{0}(K)$ and $e=w_{i_{1}}\cdots
w_{i_{k}}(e_{0})$, $\mathrm{length}(e)=2^{-k}$.

5In the following we shall set $K_{0}:=K$, $E_{0}=E_{0}(K)$,
$K_{n}=w_{0}^{-n}K_{0}$. Let us now consider the middle point $x_{i,i+1}$ of
the segment $\big{(}w_{0}^{-1}v_{i},w_{0}^{-1}v_{i+1}\big{)}$, $i=0,1,2$, the
map $R_{i+1,i}\colon w_{0}^{-1}w_{i}K\to w_{0}^{-1}w_{i+1}K$ consisting of the
rotation of $\frac{4}{3}\pi$ around the point $x_{i,i+1}$, $i=0,1,2$, and
observe that

$\displaystyle R_{i,i+2}\circ R_{i+2,i+1}\circ R_{i+1,i}={\rm
id}_{w_{0}^{-1}w_{i}K},\qquad i=0,1,2.$ (3.1)

Setting $R_{i,i+1}=R_{i+1,i}^{-1}$, the previous identities may also be
written as

$\displaystyle R_{i+2,i+1}\circ R_{i+1,i}=R_{i+2,i},\qquad i=0,1,2.$

We then construct the map $p\colon K_{1}\to K$ given by

$\displaystyle p(x)=\begin{cases}x,&x\in K,\\\ R_{0,1}(x),&x\in
w_{0}^{-1}w_{1}K,\\\ R_{0,2}(x),&x\in w_{0}^{-1}w_{2}K,\end{cases}$

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 (see Fig. 3).

$v_{0}$

$x_{2,0}$

$x_{1,2}$

$w_{0}^{-1}v_{2}$

$x_{0,1}=v_{1}$

$w_{0}^{-1}v_{1}$

$R_{0,2}$

$R_{0,1}$

Figure 3: The covering map $p\colon K_{1}\to K$.

The following result is easily verified.

######  Proposition 3.1.

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
$\alpha_{1,0}\colon{\mathcal{C}}(K)\to{\mathcal{C}}(K_{1})$, hence, following
[17], to an inductive family of C∗-algebras
${\mathcal{A}}_{n}={\mathcal{C}}(K_{n})$, whose inductive limit
${\mathcal{A}}_{\infty}$ consists of continuous function on the solenoidal
space based on the gasket. As in Definition 2.3, we consider the triple
$({\mathcal{L}}_{n},{\mathcal{H}}_{n},D_{n})$ on the $C^{*}$-algebra
${\mathcal{A}}_{n}$, $n\geq 0$, where ${\mathcal{H}}_{n}=\ell^{2}(E_{n})$,
$E_{n}=\\{w_{0}^{-n}e,\,e\in E_{0}\\}$ (the set of oriented edges in $K_{n}$).

Let us also note that, since the covering projections are locally isometric
and any Lip-norm $L_{m}(f)=\|[D_{m},f]\|$ associated with the triple
$({\mathcal{A}}_{m},{\mathcal{H}}_{m},D_{m})$ produces the geodesic distance
on $K_{m}$, we get $L_{m+q}(\alpha_{m+q,m}(f))=L_{m}(f)$, namely we obtain a
seminorm on the algebraic inductive limit of the ${\mathcal{A}}_{n}$’s.

##  4 A groupoid of local isometries on the infinite
Sierpinski fractafold

Let us consider the infinite fractafold $K_{\infty}=\cup_{n\geq 0}K_{n}$ [51]
endowed with the Hausdorff measure $\operatorname{\mu_{d}}$ of dimension
$d=\frac{\log 3}{\log 2}$ normalized to be 1 on $K=K_{0}$, with the exhaustion
$\\{K_{n}\\}_{n\geq 0}$, and with the family of local isometries
$R=\big{\\{}R^{n}_{i+1,i},R^{n}_{i,i+1}\colon i=0,1,2,n\geq 0\big{\\}}$, where
$R^{n}_{i,j}=w_{0}^{-n}R_{i,j}w_{0}^{n}\colon C^{n}_{j}\to C^{n}_{i}$, and
$C^{n}_{i}:=w_{0}^{-n-1}w_{i}K$, $n\geq 0$, $i,j\in\\{0,1,2\\}$. We also
denote by $s(\gamma)$ and $r(\gamma)$ the domain and range of the local
isometry $\gamma$. Such local isometries act on points and on oriented edges
of $K_{\infty}$.

We say that the product of the two local isometries $\gamma_{1}$,
$\gamma_{2}\in R$ is defined if $\gamma_{2}^{-1}(s(\gamma_{1}))\cap$
$s(\gamma_{2})\neq\varnothing$. In this case we consider the product

$\displaystyle\gamma_{1}\cdot\gamma_{2}\colon\
\gamma_{2}^{-1}(s(\gamma_{1}))\cap s(\gamma_{2})\to
r\big{(}\gamma_{1}|_{s(\gamma_{1}))\cap r(\gamma_{2})}\big{)}.$

We then consider the family ${\mathcal{G}}$ consisting of all (the well-
defined) finite products of isometries in $R$. Clearly, any $\gamma$ in
${\mathcal{G}}$ is a local isometry, and its domain and range are cells of the
same size. We set ${\mathcal{G}}_{n}=\\{g\in{\mathcal{G}}\colon
s(\gamma)\,\&\,r(\gamma)$ are cells of size $2^{n}\\}$, $n\geq 0$.

######  Proposition 4.1.

For any $n\geq 0$, $C_{1}$, $C_{2}$ cells of size $2^{n}$,
$\exists!\,\gamma\in{\mathcal{G}}_{n}$ such that $s(\gamma)=C_{1}$,
$r(\gamma)=C_{2}$. In particular, if $C$ has size $2^{n}$, the identity map of
$C$ belongs to ${\mathcal{G}}_{n}$, $n\geq 0$.

###### Proof.

It is enough to show that for any cell $C$ of size $2^{n}$ there exists a
unique $\gamma\in{\mathcal{G}}_{n}$ such that $\gamma\colon C\to K_{n}$. For
any cell $C$, let $m=\mathrm{level}(C)$ be the minimum number such that
$C\subset K_{m}$. We prove the existence: if $C$ has size $2^{n}$ and
$\mathrm{level}(C)=m>n$, then $C\subset C^{m-1}_{i}$, for some $i=1,2$, hence
$R^{m-1}_{0,i}(C)\subset K_{m-1}$. Iterating, the result follows. The second
statement follows directly by equation (3.1).

As for the uniqueness, $\forall\,n\geq 0$, we call $R^{n}_{i,0}$ ascending,
$i=1,2$, $R^{n}_{0,i}$ descending, $i=1,2$, $R^{n}_{i,j}$ constant-level,
$i,j\in\\{1,2\\}$. Indeed, if $C\subset s(R^{n}_{i,0})$, then
$\mathrm{level}(C)\leq n$ and $\mathrm{level}(R^{n}_{i,0}(C))=n+1$; if
$C\subset s(R^{n}_{0,i})$, then $\mathrm{level}(C)=n+1$,
$\mathrm{level}(R^{n}_{0,i}(C))\leq n$ and
$\mathrm{level}(R^{n}_{j,i}(C))=n+1$, $i,j\in\\{1,2\\}$, $n\geq 0$.

The following facts hold:

  * •

The product $R^{n}_{l,k}\cdot R^{m}_{j,i}$ 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 $i,j\in\\{1,2\\}$,
then, by compatibility, the descending element should be $R^{n}_{0,i}$ so that
the product is $R^{n}_{0,i},$ by equation (3.1).

  * •

Given a cell $C$ with $\mathrm{size}(C)=2^{n}$ and $\mathrm{level}(C)>n$, the
exists a unique descending element $\gamma\in R$ such that $C\subset
s(\gamma)$: indeed, if $m=\mathrm{level}(C)$, then $C\subset C^{m-1}_{i}$, for
some $i\in\\{1,2\\}$. The only descending element is then
$\gamma=R^{m-1}_{0,i}$.

  * •

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 $\mathrm{size}(C)=2^{n}$, $\gamma\in{\mathcal{G}}_{n}$ such that
$\gamma\colon C\to K_{n}$,
$\gamma=\gamma_{p}\cdot\gamma_{p-1}\cdots\gamma_{2}\cdot\gamma_{1}$, where
$\gamma_{j}\in R$, $1\leq j\leq p$. Since
$\mathrm{level}(C)\geq\mathrm{level}(K_{n})=n$, for any possible ascending
element $\gamma_{i}$ there should be a $j>i$ such that $\gamma_{j}$ is
descending. If $i+q$ is the minimum among such $j$’s, all terms $\gamma_{j}$,
$i<j<i+q$, are constant-level, hence the product
$\gamma_{i+q}\cdot\gamma_{i+q-1}\cdots\gamma_{i}={\rm id}_{s(\gamma_{i})}$.
Then, we note that $\gamma_{p}$ can only be descending. As a consequence,
$\gamma$ 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 ${\mathcal{G}}_{n}$, and so also ${\mathcal{G}}$, 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
${\mathcal{G}}$.

######  Proposition 4.2.

Let us define $\widetilde{\mathcal{A}}_{n}$ as the algebra

$\displaystyle\widetilde{\mathcal{A}}_{n}=\\{f\in{\mathcal{C}}_{b}(K_{\infty})\colon
f(\gamma(x))=f(x),\,x\in s(\gamma),\,\gamma\in{\mathcal{G}}_{n}\\}.$

Then, for any $n\geq 0$, the following diagram commutes,

$\displaystyle\begin{matrix}\widetilde{\mathcal{A}}_{n}&\subset&\widetilde{\mathcal{A}}_{n+1}\\\
\Big{\downarrow}\iota_{n}&&\Big{\downarrow}\iota_{n+1}\\\
{\mathcal{A}}_{n}&\mathop{\longrightarrow}\limits^{\alpha_{n+1,n}}&{\mathcal{A}}_{n+1},\end{matrix}$

where $\iota_{n}\colon f\in\widetilde{\mathcal{A}}_{n}\to
f|_{K_{n}}\in{\mathcal{A}}_{n}$ are isomorphisms. Hence the inductive limit
${\mathcal{A}}_{\infty}$ is isomorphic to a $C^{*}$-subalgebra of
${\mathcal{C}}_{b}(K_{\infty})$.

###### Proof.

The request in the definition of $\widetilde{\mathcal{A}}_{n}$ means that the
value of $f$ in any point of $K_{\infty}$ 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 ${\mathcal{A}}_{n}$, $0\leq
n\leq\infty$, with its isomorphic copy $\widetilde{\mathcal{A}}_{n}$ in
${\mathcal{C}}_{b}(K_{\infty})$, so that the embeddings $\alpha_{k,j}$ become
inclusions. Moreover, we may consider the operator $\widetilde{D}_{n}$ on
$\ell^{2}(E_{\infty})$, with $E_{\infty}=\cup_{n\geq 0}E_{n}$, given by
$\widetilde{D}_{n}e=\mathrm{length}(e)^{-1}Fe$, if $\mathrm{length}(e)\leq
2^{n}$, and $\widetilde{D}_{n}e=0$, if $\mathrm{length}(e)>2^{n}$, where $F$
is defined as in Definition 2.3$(c)$. Then the spectral triples
$({\mathcal{A}}_{n},{\mathcal{H}}_{n},D_{n})$ are isomorphic to the spectral
triples $(\widetilde{\mathcal{A}}_{n},{\mathcal{H}}_{n},\widetilde{D}_{n})$,
where ${\mathcal{C}}_{b}(K_{\infty})$ acts on the space $\ell^{2}(E_{\infty})$
through the representation $\rho$ given by $\rho(f)e=f(e^{+})e$.

######  Remark 4.3.

Because of the isomorphism above, from now on we shall remove the tildes and
denote by ${\mathcal{A}}_{n}$ the subalgebras of
${\mathcal{C}}_{b}(K_{\infty})$ and by $D_{n}$ the operators acting on
$\ell^{2}(E_{\infty})$.

##  5 The $\boldsymbol{C^{*}}$-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. Subsets of $E_{\infty}$ | Projections
---|---
$E_{n}=\\{e\subset K_{n}\\}$, $n\geq 0$ |  $P_{n}$
$E^{k,p}_{n}=\big{\\{}e\in E_{n}\colon 2^{k}\leq\mathrm{length}(e)\leq 2^{p}\big{\\}}$, for $k\leq p\leq n$ |  $P_{n}^{k,p}$
$E^{k}_{n}=E^{k,k}_{n}=\big{\\{}e\in E_{n}\colon\mathrm{length}(e)=2^{k}\big{\\}}$, for $k\leq n$ |  $P^{k}_{n}$
$E^{k,p}=\cup_{n}E^{k,p}_{n}=\big{\\{}e\in E_{\infty}\colon 2^{k}\leq\mathrm{length}(e)\leq 2^{p}\big{\\}}$ |  $P^{k,p}$
$E^{k}=E^{k,k}=\big{\\{}e\in E_{\infty}\colon\mathrm{length}(e)=2^{k}\big{\\}}$ |  $P^{k}$
$E_{C}=\\{e\in E_{\infty}\colon e\subset C\\}$, $C$ being a cell |  $P_{C}$

Let us note that any local isometry $\gamma\in{\mathcal{G}}$, $\gamma\colon
s(\gamma)\to r(\gamma)$, gives rise to a partial isometry $V_{\gamma}$ defined
as

$\displaystyle V_{\gamma}e=\begin{cases}\gamma(e),&e\subset s(\gamma),\\\
0,&\mathrm{elsewhere}.\end{cases}$

In particular, if $C$ is a cell, and $\gamma={\rm id}_{C}$,
$V_{\gamma}=P_{C}$. We then consider the subalgebras ${\mathcal{B}}_{n}$ of
$B(\ell^{2}(E_{\infty}))$,

$\displaystyle{\mathcal{B}}_{n}=\\{V_{\gamma}\colon\gamma\in{\mathcal{G}}_{m},\,m\geq
n\\}^{\prime},\qquad{\mathcal{B}}_{\text{fin}}=\bigcup_{n}{\mathcal{B}}_{n},\qquad{\mathcal{B}}_{\infty}=\overline{{\mathcal{B}}_{\text{fin}}},$

and note that the elements of ${\mathcal{B}}_{n}$ commute with the projections
$P_{C}$, for all cells $C$ s.t. $\mathrm{size}(C)\geq 2^{n}$. By definition,
the sequence ${\mathcal{B}}_{n}$ is increasing, therefore, since the
${\mathcal{B}}_{n}$’s are von Neumann algebras, ${\mathcal{B}}_{\infty}$ is a
$C^{*}$-algebra. Let us observe that, $\forall\,n\geq 0$,
$\rho({\mathcal{A}}_{n})\subset{\mathcal{B}}_{n}$.

######  Definition 5.1.

The elements of the $C^{*}$-algebra ${\mathcal{B}}_{\infty}$ are called
geometric operators.

Now consider the hereditary positive cone

$\displaystyle\mathcal{I}_{0}^{+}=\big{\\{}T\in{\mathcal{B}}_{\text{fin}}^{+}\colon\exists\,c_{T}\in{\mathbb{R}}\mathrm{\
such\ that\ }\operatorname{tr}(P_{m}T)\leq
c_{T}\operatorname{\mu_{d}}(K_{m}),\,\forall\,m\geq 0\big{\\}}.$

######  Lemma 5.2.

For any $T\in\mathcal{I}_{0}^{+}$, the sequence
$\frac{\operatorname{tr}(P_{m}T)}{\operatorname{\mu_{d}}(K_{m})}$ is
eventually increasing, hence convergent. In particular

$\displaystyle\operatorname{tr}(P_{p}^{p}T)=0\qquad\forall\,p>m\Rightarrow\tau_{0}(T)=\frac{\operatorname{tr}(P_{m}T)}{\operatorname{\mu_{d}}(K_{m})}.$
(5.1)

###### Proof.

Let $T\in{\mathcal{B}}_{n}^{+}$. Then we have, for $m\geq n$,

$\displaystyle\operatorname{tr}(P_{{m+1}}T)=\\!\\!\sum_{e\subset
K_{m+1}}(e,Te)=\\!\\!\sum_{i=0,1,2}\sum_{e\in C^{m}_{i}}(e,Te)+\\!\sum_{e\in
E_{m+1}^{m+1}}\\!(e,Te)=3\operatorname{tr}(P_{{m}}T)+\operatorname{tr}(P_{m+1}^{m+1}T),$

hence

$\displaystyle\frac{\operatorname{tr}(P_{m+1}T)}{\operatorname{\mu_{d}}(K_{m+1})}=\frac{\operatorname{tr}(P_{m}T)}{\operatorname{\mu_{d}}(K_{m})}+\frac{\operatorname{tr}(P_{m+1}^{m+1}T)}{\operatorname{\mu_{d}}(K_{m+1})},$

from which the thesis follows. ∎

We then define the weight $\tau_{0}$ on ${\mathcal{B}}_{\infty}^{+}$ as
follows:

$\displaystyle\tau_{0}(T)=\begin{cases}\displaystyle{\lim_{m\to\infty}\frac{\operatorname{tr}(P_{m}T)}{\operatorname{\mu_{d}}(K_{m})}},&T\in{\mathcal{I}_{0}^{+}},\\\
0,&\mathrm{elsewhere}.\end{cases}$

The next step is to regularize the weight $\tau_{0}$ in order to obtain a
semicontinuous semifinite tracial weight $\tau$ on ${\mathcal{B}}_{\infty}$.

######  Lemma 5.3.

For any $T\in\mathcal{I}_{0}^{+}$, $A\in{\mathcal{B}}_{\text{fin}}$, it holds
$ATA^{*}\in\mathcal{I}_{0}^{+}$, and
$\tau_{0}(ATA^{*})\leq\|A\|^{2}\tau_{0}(T)$.

###### Proof.

Let $A\in{\mathcal{B}}_{n}$. Then, for any $m>n$, we have

$\displaystyle\operatorname{tr}(P_{m}ATA^{*})=\operatorname{tr}(A^{*}AP_{m}T)\leq\|A^{*}A\|\operatorname{tr}(P_{m}T)\leq\|A\|^{2}c_{T}\operatorname{\mu_{d}}(K_{m}),$

and the thesis follows. ∎

######  Proposition 5.4.

For all $p\in\mathbb{N}$, recall that $P^{-p,\infty}$ is the orthogonal
projection onto the closed vector space generated by
$\big{\\{}e\in\ell^{2}(E_{\infty})\colon\mathrm{length}(e)\geq
2^{-p}\big{\\}}$, and let
$\varphi_{p}(T):=\tau_{0}(P^{-p,\infty}TP^{-p,\infty})$,
$\forall\,T\in{\mathcal{B}}_{\infty}^{+}$. Then
$P^{-p,\infty}\in{\mathcal{B}}_{0}$, $\varphi_{p}$ is a positive linear
functional, and $\varphi_{p}(T)\leq\varphi_{p+1}(T)\leq\tau_{0}(T)$,
$\forall\,T\in{\mathcal{B}}_{\infty}^{+}$.

###### Proof.

We first observe that

$\displaystyle\operatorname{tr}(P^{j}_{n})=\\#\big{\\{}e\in
K_{n}\colon\mathrm{length}(e)=2^{j}\big{\\}}=6\cdot 3^{n-j},\qquad j\leq n.$
(5.2)

Then it is easy to verify that $P^{-p,\infty}\in{\mathcal{B}}_{0}$. Since

$\displaystyle\varphi_{p}(I)=\tau_{0}(P^{-p,\infty})=\lim_{n\to\infty}\frac{\operatorname{tr}P^{-p,n}_{n}}{\mu_{d}(K_{n})}=\lim_{n\to\infty}3^{-n}\sum_{j=-p}^{n}\operatorname{tr}(P^{j}_{n})=\sum_{j=-p}^{\infty}6\cdot
3^{-j}=3^{p+2},$ (5.3)

$\varphi_{p}$ extends by linearity to a positive functional on
${\mathcal{B}}_{\infty}$. Moreover, by Lemma 5.3,
$\varphi_{p}(T)\leq\tau_{0}(T)$, $\forall\,T\in{\mathcal{B}}_{\infty}^{+}$.
Finally, since $P^{-p,\infty}P_{n}=P_{n}P^{-p,\infty}=P^{-\infty,n}_{n}$,
$\forall\,n\in\mathbb{N}$, we get, for all $T\in{\mathcal{B}}_{\infty}^{+}$,

$\displaystyle\varphi_{p+1}(T)-\varphi_{p}(T)$
$\displaystyle=\tau_{0}(P^{-(p+1),\infty}TP^{-(+1)p,\infty})-\tau_{0}(P^{-p,\infty}TP^{-p,\infty})$
$\displaystyle=\lim_{n\to\infty}\frac{\operatorname{tr}((P^{-(p+1),n)}_{n}-P^{-p,n}_{n})T)}{\mu_{d}(K_{n})}=\lim_{n\to\infty}\frac{\operatorname{tr}(P^{-(p+1))}_{n}T)}{\mu_{d}(K_{n})}\geq
0.$ ∎

######  Proposition 5.5.

Let $\tau(T):=\lim\limits_{p\to\infty}\varphi_{p}(T)$,
$\forall\,T\in{\mathcal{B}}_{\infty}^{+}$. Then

  * $(i)$

$\tau$ is a lower semicontinuous weight on ${\mathcal{B}}_{\infty}$,

  * $(ii)$

$\tau(T)=\tau_{0}(T)$, $\forall\,T\in\mathcal{I}_{0}^{+}$.

###### Proof.

$(i)$ Let $T\in{\mathcal{B}}_{\infty}^{+}$. Since
$\\{\varphi_{p}(T)\\}_{p\in\mathbb{N}}$ is an increasing sequence, there
exists
$\lim\limits_{p\to\infty}\varphi_{p}(T)=\sup\limits_{p\in\mathbb{N}}\varphi_{p}(T)$.
Then $\tau$ is a weight on ${\mathcal{B}}_{\infty}^{+}$. Since $\varphi_{p}$
is continuous, $\tau$ is lower semicontinuous.

$(ii)$ Let us prove that, $\forall\,T\in{\mathcal{B}}_{n}^{+}$,

$\displaystyle\frac{\operatorname{tr}(P^{j}_{m}T)}{\mu_{d}(K_{m})}=\frac{\operatorname{tr}(P^{j}_{n}T)}{\mu_{d}(K_{n})},\qquad
j\leq n\leq m.$ (5.4)

Indeed,

$\displaystyle\operatorname{tr}\big{(}P^{j}_{m+1}T\big{)}$
$\displaystyle=\sum_{\begin{subarray}{c}e\subset K_{m+1}\\\
\mathrm{length}(e)=2^{j}\end{subarray}}(e,Te)=\sum_{i=0}^{2}\sum_{\begin{subarray}{c}e\subset
C^{m}_{i}\\\
\mathrm{length}(e)=2^{j}\end{subarray}}(e,Te)=\sum_{i=0}^{2}\sum_{\begin{subarray}{c}e\subset
K_{m}\\\
\mathrm{length}(e)=2^{j}\end{subarray}}(V_{R^{m}_{i}}e,TV_{R^{m}_{i}}e)$
$\displaystyle=\sum_{i=0}^{2}\sum_{\begin{subarray}{c}e\subset K_{m}\\\
\mathrm{length}(e)=2^{j}\end{subarray}}(e,Te)=3\operatorname{tr}\big{(}P^{j}_{m}T\big{)},$

from which (5.4) follows. Let us now prove that

$\displaystyle\tau(T)=\sup_{p\in\mathbb{N}}\varphi_{p}(T)=\tau_{0}(T),\qquad
T\in\mathcal{I}_{0}^{+}.$ (5.5)

Let $T\in{\mathcal{B}}_{n}^{+}\cap\mathcal{I}_{0}^{+}$, and $\varepsilon>0$.
From the definition of $\tau_{0}(T)$, there exists $r\in\mathbb{N}$, $r>n$,
such that
$\frac{\operatorname{tr}(P_{r}T)}{\mu_{d}(K_{r})}>\tau_{0}(T)-\varepsilon$.
Since
$\frac{\operatorname{tr}(P_{r}T)}{\mu_{d}(K_{r})}=\sum_{j=-\infty}^{r}\frac{\operatorname{tr}(P^{j}_{r}T)}{\mu_{d}(K_{r})}$,
there exists $p\in\mathbb{N}$ such that
$\sum_{j=-p}^{r}\frac{\operatorname{tr}(P^{j}_{r}T)}{\mu_{d}(K_{r})}>\frac{\operatorname{tr}(P_{r}T)}{\mu_{d}(K_{r})}-\varepsilon>\tau_{0}(T)-2\varepsilon$.
Then, for any $s\in\mathbb{N}$, $s>r$, we have

$\displaystyle\frac{\operatorname{tr}(P_{s}P^{-p,\infty}TP^{-p,\infty}P_{s})}{\mu_{d}(K_{s})}$
$\displaystyle=\sum_{j=-p}^{s}\frac{\operatorname{tr}(P^{j}_{s}T)}{\mu_{d}(K_{s})}=\sum_{j=-p}^{r}\frac{\operatorname{tr}(P^{j}_{s}T)}{\mu_{d}(K_{s})}+\sum_{j=r+1}^{s}\frac{\operatorname{tr}(P^{j}_{s}T)}{\mu_{d}(K_{s})}$
$\displaystyle\stackrel{{\scriptstyle\eqref{uguaglianza}}}{{=}}\sum_{j=-p}^{r}\frac{\operatorname{tr}(P^{j}_{r}T)}{\mu_{d}(K_{r})}+\sum_{j=r+1}^{s}\frac{\operatorname{tr}(P^{j}_{s}T)}{\mu_{d}(K_{s})}>\tau_{0}(T)-2\varepsilon,$

and, passing to the limit for $s\to\infty$, we get

$\displaystyle\varphi_{p}(T)=\tau_{0}(P^{-p,\infty}TP^{-p,\infty})=\lim_{s\to\infty}\frac{\operatorname{tr}(P_{s}P^{-p,\infty}TP^{-p,\infty}P_{s})}{\mu_{d}(K_{s})}\geq\tau_{0}(T)-\varepsilon,$

and equation (5.5) follows. ∎

We want to prove that $\tau$ is a tracial weight.

######  Definition 5.6.

An operator $U\in B\big{(}\ell^{2}(E_{\infty})\big{)}$ is called
$\delta$-unitary, $\delta>0$, if $\|U^{*}U-1\|<\delta$, and
$\|UU^{*}-1\|<\delta$.

Let us denote with ${\mathcal{U}}_{\delta}$ the set of $\delta$-unitaries in
${\mathcal{B}}_{\text{fin}}$ and observe that, if $\delta<1$,
${\mathcal{U}}_{\delta}$ consists of invertible operators, and
$U\in{\mathcal{U}}_{\delta}$ implies
$U^{-1}\in{\mathcal{U}}_{\delta/(1-\delta)}$.

######  Proposition 5.7.

The weight $\tau_{0}$ is $\varepsilon$-invariant for $\delta$-unitaries in
${\mathcal{B}}_{\text{fin}}$, namely, for any $\varepsilon\in(0,1)$, there is
$\delta>0$ s.t., for any $U\in{\mathcal{U}}_{\delta}$, and
$T\in{\mathcal{B}}^{+}_{\infty}$,

$\displaystyle(1-\varepsilon)\tau_{0}(T)\leq\tau_{0}(UTU^{*})\leq(1+\varepsilon)\tau_{0}(T).$

###### Proof.

We first observe that, if $\delta\in(0,1)$ and $U\in{\mathcal{U}}_{\delta}$,
$T\in\mathcal{I}_{0}^{+}\Leftrightarrow UTU^{*}\in\mathcal{I}_{0}^{+}$.
Indeed, choose $n$ such that $U,T\in{\mathcal{B}}_{n}$. Then
$\operatorname{tr}(P_{n}UTU^{*})=\operatorname{tr}(U^{*}UP_{n}TP_{n})\leq\|U^{*}U\|\operatorname{tr}(P_{n}T)\leq(1+\delta)c_{T}\operatorname{\mu_{d}}(K_{n})$,
$\forall\,n\in\mathbb{N}$, so that $UTU^{*}\in\mathcal{I}_{0}^{+}$. Moreover,

$\displaystyle\tau_{0}(UTU^{*})=\lim_{n\to\infty}\frac{\operatorname{tr}(P_{n}UTU^{*})}{\operatorname{\mu_{d}}(K_{n})}\leq\|U^{*}U\|\lim_{n\to\infty}\frac{\operatorname{tr}(P_{n}T)}{\operatorname{\mu_{d}}(K_{n})}=\|U^{*}U\|\tau_{0}(T)<(1+\delta)\tau_{0}(T).$

Conversely, $UTU^{*}\in\mathcal{I}_{0}^{+}$, and
$U^{-1}\in{\mathcal{U}}_{\delta/(1-d)}\implies T\in\mathcal{I}_{0}^{+}$.
Moreover,

$\displaystyle\tau_{0}(T)\leq\big{\|}\big{(}U^{-1}\big{)}^{*}U^{-1}\big{\|}\tau_{0}(UTU^{*})<\frac{1}{1-\delta}\tau_{0}(UTU^{*}).$

The result follows by the choice $\delta=\varepsilon$. ∎

######  Theorem 5.8.

The lower semicontinuous weight $\tau$ in Proposition 5.5 is a trace on
${\mathcal{B}}_{\infty}$, that is, setting
${\mathcal{J}}^{+}:=\\{A\in{\mathcal{B}}_{\infty}^{+}\colon\tau(A)<\infty\\}$,
and extending $\tau$ to the vector space ${\mathcal{J}}$ generated by
${\mathcal{J}}^{+}$, we get

  * $(i)$

${\mathcal{J}}$ is an ideal in ${\mathcal{B}}_{\infty}$,

  * $(ii)$

$\tau(AB)=\tau(BA)$, for any $A\in{\mathcal{J}}$,
$B\in{\mathcal{B}}_{\infty}$.

###### Proof.

$(i)$ Let us prove that ${\mathcal{J}}^{+}$ is a unitarily-invariant face in
${\mathcal{B}}_{\infty}^{+}$, and suffices it to prove that
$A\in{\mathcal{J}}^{+}$ implies that $UAU^{*}\in{\mathcal{J}}^{+}$, for any
$U\in{\mathcal{U}}({\mathcal{B}}_{\infty})$, the set of unitaries in
${\mathcal{B}}_{\infty}$. To reach a contradiction, assume that there exists
$U\in{\mathcal{U}}({\mathcal{B}}_{\infty})$ such that $\tau(UAU^{*})=\infty$.
Then there is $p\in\mathbb{N}$ such that $\varphi_{p}(UAU^{*})>2\tau(A)+2$.
Let $\delta<3$ be such that $V\in{\mathcal{U}}_{\delta}$ implies
$\tau(VAV^{*})\leq 2\tau(A)$, and let $U_{0}\in{\mathcal{B}}_{\text{fin}}$ be
such that
$\|U-U_{0}\|<\min\big{\\{}\frac{\delta}{3},\frac{1}{3\|A\|\|\varphi_{p}\|}\big{\\}}$.
The inequalities

$\displaystyle\|U_{0}U_{0}^{*}-1\|=\|U^{*}U_{0}U_{0}^{*}-U^{*}\|\leq\|U^{*}U_{0}-1\|\|U_{0}^{*}\|+\|U_{0}^{*}-U^{*}\|<\delta$

and $\|U_{0}^{*}U_{0}-1\|<\delta$, prove that
$U_{0}\in{\mathcal{U}}_{\delta}$. Since

$|\varphi_{p}(U_{0}AU_{0}^{*})-\varphi_{p}(UAU^{*})|\leq
3\|\varphi_{p}\|\|A\|\|U-U_{0}\|<1,$

we get

$\displaystyle
2\tau(A)\geq\tau(U_{0}AU_{0}^{*})\geq\varphi_{p}(U_{0}AU_{0}^{*})\geq\varphi_{p}(UAU^{*})-1\geq
2\tau(A)+1$

which is absurd.

$(ii)$ We only need to prove that $\tau$ is unitarily-invariant. Let
$A\in{\mathcal{J}}^{+}$, $U\in{\mathcal{U}}({\mathcal{B}}_{\infty})$. For any
$\varepsilon>0$, there is $p\in\mathbb{N}$ such that
$\varphi_{p}(UAU^{*})>\tau(UAU^{*})-\varepsilon$, since, by $(1)$,
$\tau(UAU^{*})$ is finite. Then, arguing as in the proof of $(1)$, we can find
$U_{0}\in{\mathcal{B}}_{\text{fin}}$, so close to $U$ that

$\displaystyle|\varphi_{p}(U_{0}AU_{0}^{*})-\varphi_{p}(UAU^{*})|<\varepsilon,$
$\displaystyle(1-\varepsilon)\tau(A)\leq\tau(U_{0}AU_{0}^{*})\leq(1+\varepsilon)\tau(A).$

Then

$\displaystyle\tau(A)$ $\displaystyle\geq\frac{1}{1+\varepsilon}\
\tau(U_{0}AU_{0}^{*})\geq\frac{1}{1+\varepsilon}\
\varphi_{p}(U_{0}AU_{0}^{*})\geq\frac{1}{1+\varepsilon}\
(\varphi_{p}(UAU^{*})-\varepsilon)$ $\displaystyle\geq\frac{1}{1+\varepsilon}\
(\tau(UAU^{*})-2\varepsilon).$

By the arbitrariness of $\varepsilon>0$, we get $\tau(A)\geq\tau(UAU^{*})$.
Exchanging $A$ with $UAU^{*}$, we get the thesis. ∎

######  Proposition 5.9.

The lower semicontinuous tracial weight $\tau$ defined in Proposition 5.5 is
semifinite and faithful.

###### Proof.

Let us recall that, for any $p\in\mathbb{N}$,
$P^{-p,\infty}\in\mathcal{I}_{0}^{+}$ by Proposition 5.4. From Proposition 5.5
follows that $\tau(P^{-p,\infty})=\tau_{0}(P^{-p,\infty})<\infty$, hence
$P^{-p,\infty}\in{\mathcal{J}}^{+}$. Then, for any
$T\in{\mathcal{B}}_{\infty}^{+}$,
$S_{p}:=T^{1/2}P^{-p,\infty}T^{1/2}\in{\mathcal{J}}^{+}$, and $0\leq S_{p}\leq
T$. Moreover,

$\displaystyle\tau(S_{p})$
$\displaystyle=\tau\big{(}T^{1/2}P^{-p,\infty}T^{1/2}\big{)}=\tau(P^{-p,\infty}TP^{-p,\infty})=\sup_{q\in\mathbb{N}}\tau_{0}(Q_{q}P^{-p,\infty}TP^{-p,\infty}Q_{q})$
$\displaystyle=\tau_{0}(P^{-p,\infty}TP^{-p,\infty})=\varphi_{p}(T),$

so that $\sup\limits_{p\in\mathbb{N}}\tau(S_{p})=\tau(T)$, and $\tau$ is
semifinite. Finally, if $T\in{\mathcal{B}}_{\infty}^{+}$ is such that
$\tau(T)=0$, then $\sup\limits_{p\in\mathbb{N}}\varphi_{p}(T)=0$. Since
$\\{\varphi_{p}(T)\\}_{p\in\mathbb{N}}$ is an increasing sequence,
$\varphi_{p}(T)=0$, $\forall\,p\in\mathbb{N}$. Then, for a fixed
$p\in\mathbb{N}$, we get
$0=\tau_{0}(P^{-p,\infty}TP^{-p,\infty})=\lim\limits_{n\to\infty}\frac{\operatorname{tr}(P_{n}P^{-p,\infty}TP^{-p,\infty}P_{n})}{\mu_{d}(K_{n})}$.
Since the sequence
$\big{\\{}\frac{\operatorname{tr}(P_{n}P^{-p,\infty}TP^{-p,\infty}P_{n})}{\mu_{d}(K_{n})}\big{\\}}_{n\in\mathbb{N}}$
is definitely increasing, we get
$\operatorname{tr}(P_{n}P^{-p,\infty}TP^{-p,\infty}P_{n})=0$ definitely, that
is $TP^{-p,\infty}P_{n}=0$ definitely, so that $TP^{-p,\infty}=0$. By the
arbitrariness of $p\in\mathbb{N}$, we get $T=0$. ∎

##  6 A semifinite spectral triple on the inductive limit
$\boldsymbol{{\mathcal{A}}_{\infty}}$

Since the covering we are studying is ramified, the family
$\\{{\mathcal{A}}_{n},{\mathcal{H}}_{n},D_{n}\\}$ 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
${\mathcal{A}}_{\infty}$: our construction is indeed based on the pair
$({\mathcal{B}}_{\infty},\tau)$ of the $C^{*}$-algebra of geometric operators
and the semicontinuous semifinite weight on it.

The Dirac operator will be defined below (Definition 6.4) through its phase
and the functional calculi of its modulus with continuous functions vanishing
at $\infty$. More precisely we shall use the following

######  Definition 6.1.

Let $(\mathfrak{C},\tau)$ be a $C^{*}$-algebra with unit endowed with a
semicontinuous semifinite faithful trace. A selfadjoint operator $T$
affiliated to $(\mathfrak{C},\tau)$ is defined as a pair given by a closed
subset $\sigma(T)$ in ${\mathbb{R}}$ and a $*$ homomorphism
$\phi\colon{\mathcal{C}}_{0}(\sigma(T))\to\mathfrak{C}$,
$f(T)\mathop{=}\limits^{\mathrm{def}}\phi(f)$, provided that the support of
such homomorphism is the identity in the GNS representation $\pi_{\tau}$
induced by the trace $\tau$.

The previous definition was inspired by that in [20] appendix A, and should
not be confused with that of Woronowicz for $C^{*}$-algebras without identity.

######  Remark 6.2.

The $*$-homomorphism $\phi_{\tau}=\pi_{\tau}\circ\phi$ extends to bounded
Borel functions on ${\mathbb{R}}$ and
$e_{(-\infty,t]}\mathop{=}\limits^{\mathrm{def}}\phi_{\tau}(\chi_{(-\infty,t]})$
tends strongly to the identity when $t\to\infty$, hence it is a spectral
family. We shall denote by $\pi_{\tau}(T)$ the selfadjoint operator affiliated
to $\pi_{\tau}(\mathfrak{C})^{\prime\prime}$ given by

$\displaystyle\pi_{\tau}(T)\mathop{=}\limits^{\mathrm{def}}\int_{\mathbb{R}}t\,{\rm
d}e_{(-\infty,t]}.$

######  Proposition 6.3.

Let $T$ be a selfadjoint operator affiliated to $(\mathfrak{C},\tau)$ as
above.

  * $(a)$

Assume that for any $n\in\mathbb{N}$, there is
$\varphi_{n}\in{\mathcal{C}}({\mathbb{R}})\colon 0\leq\varphi_{n}\leq
1,\varphi_{n}=1$ for $|t|\leq a_{n}$, $\varphi_{n}(t)=0$ for $|t|\geq b_{n}$
with $0<a_{n}<b_{n}$ and $\\{a_{n}\\}$, $\\{b_{n}\\}$ increasing to $\infty$.
Then, for any $A\in\mathfrak{C}$, if
$\sup\limits_{n}\|[T\cdot\varphi_{n}(T),A]\|=C<\infty$ then
$[\pi_{t}(T),\pi_{\tau}(A)]$ is bounded and
$\|[\pi_{t}(T),\pi_{\tau}(A)]\|=C$.

  * $(b)$

If $\tau(f(T))<\infty$ for any positive function $f$ with compact support on
the spectrum of $T$ then $\pi_{\tau}(T)$ has $\tau$-compact resolvent.

###### Proof.

$(a)$ Let ${\mathcal{D}}$ be the domain of $\pi_{\tau}(T)$,
${\mathcal{D}}_{0}$ the space of vectors in ${\mathcal{D}}$ with bounded
support w.r.t. to $\pi_{\tau}(T)$, and consider the sesquilinear form
$F(y,x)=(\pi_{\tau}(T)y,\pi_{\tau}(A)x)-(y,\pi_{\tau}(A)\pi_{\tau}(T)x)$
defined on ${\mathcal{D}}$. By hypothesis, for any $x,y\in{\mathcal{D}}_{0}$
there exists $n$ such that $\pi_{\tau}(\varphi_{n}(T))x=x$ and
$\pi_{\tau}((\varphi_{n}(T))y=y$, hence
$F(y,x)=(y,\pi_{\tau}([T\cdot\varphi_{n}(T),A])x)\leq C\|x\|\ \|y\|$. By the
density of ${\mathcal{D}}_{0}$ in ${\mathcal{D}}$ w.r.t. the graph norm of
$\pi_{\tau}(T)$, the same bound holds on ${\mathcal{D}}$. Then for
$y,x\in{\mathcal{D}}$,
$|(\pi_{\tau}(T)y,\pi_{\tau}(A)x)|\leq|(y,\pi_{\tau}(A)\pi_{\tau}(T)x)|+|F(y,x)|\leq(\|\pi_{\tau}(A)\pi_{\tau}(T)x\|+C\|x\|)\|y\|$
which implies $\pi_{\tau}(A)x$ belongs to the domain of
$\pi_{\tau}(T)^{*}=\pi_{\tau}(T)$. Therefore
$\pi_{\tau}(T)\pi_{\tau}(A)-\pi_{\tau}(A)\pi_{\tau}(T)$ is defined on
${\mathcal{D}}$ and its norm is bounded by $C$. Since $C$ is the optimal bound
for the sesquilinear form $F$ it is indeed the norm of the commutator.

$(b)$ Let $\lambda$ 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 $f\big{(}(|T|-\lambda
I)^{-1}\big{)}=g(|T|)$. Therefore $\tau\big{(}f\big{(}(|T|-\lambda
I)^{-1}\big{)}\big{)}<\infty$, hence
$\tau\big{(}e_{(t,+\infty)}\big{(}\pi_{\tau}\big{(}(|T|-\lambda
I)^{-1}\big{)}\big{)}\big{)}<\infty$ for any $t>0$, i.e.,
$\pi_{\tau}\big{(}(|T|-\lambda I)^{-1}\big{)}$ is $\tau$-compact (cf. Section
2.2). ∎

######  Definition 6.4.

We consider the Dirac operator $D=F|D|$ on $\ell^{2}(E_{\infty})$, where $F$
is the orientation reversing operator on edges and

$\displaystyle|D|=\sum_{n\in{\mathbb{Z}}}2^{-n}P^{n},\qquad\sigma(|D|)=\\{2^{-n},\
n\in{\mathbb{Z}}\\}\cup\\{0\\}.$

######  Proposition 6.5.

The following hold:

  * $(a)$

The elements $D$ and $|D|$ are affiliated to $({\mathcal{B}}_{\infty},\tau)$.

  * $(b)$

The following formulas hold: $\tau(P^{n})=6\cdot 3^{-n}$,
$\tau(P^{-p,\infty})=3^{p+2}$, as a consequence the operator $D$ has
$\tau$-compact resolvents

  * $(c)$

The trace $\tau(I+D^{2})^{-s/2}<\infty$ if and only if $s>d=\frac{\log 3}{\log
2}$ and

$\displaystyle\operatorname{Res}_{s=d}\tau\big{(}I+D^{2}\big{)}^{-s/2}=\frac{6}{\log
2}.$

###### 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
${\mathcal{B}}_{0}$ (which is a von Neumann algebra) for any $n\in\mathbb{N}$,
then $f(D)$ and $f(|D|)$ belong to ${\mathcal{B}}_{0}$ for any
$f\in{\mathcal{C}}_{0}({\mathbb{R}})$; therefore it is enough to show that the
support of $f\mapsto f(|D|)$ is the identity in the representation
$\pi_{\tau}$.

In order to prove this, it is enough to show that
$\pi_{\tau}(e_{|D|}[0,2^{p}])$ tends to the identity strongly when
$p\to\infty$, that is to say that $\pi_{\tau}(e_{|D|}(2^{p},\infty))$ tends to
0 strongly when $p\to\infty$.

We consider then the projection $P^{-\infty,0}$ which projects on the space
generated by the edges with $\mathrm{length}(e)\leq 1$. Clearly, such
projection belongs to ${\mathcal{B}}_{0}$, we now show that it is indeed
central there. In fact, if $c$ is a cell with $\mathrm{size}(c)=1$, $P_{c}$
commutes with ${\mathcal{B}}_{0}$. Since
$P^{-\infty,0}=\sum_{\mathrm{size}(c)=1}P_{c}$, then $P^{-\infty,0}$ commutes
with ${\mathcal{B}}_{0}$. On the one hand, the von Neumann algebra
$P^{-\infty,0}{\mathcal{B}}_{0}$ is isomorphic to
${\mathcal{B}}\big{(}\ell^{2}(K)\big{)}$ and the restriction of $\tau$ to
$P^{-\infty,0}{\mathcal{B}}_{0}$ coincides with the usual trace on
${\mathcal{B}}\big{(}\ell^{2}(K)\big{)}$, therefore the representation
$\pi_{\tau}$ is normal when restricted to $P^{-\infty,0}{\mathcal{B}}_{0}$. On
the other hand, since $e_{|D|}(2^{p},\infty)=P^{-\infty,-p-1}$ is, for $-p\leq
1$, a sub-projection of $P^{-\infty,0}$, and $P^{-\infty,-p-1}$ tends to 0
strongly in the given representation, the same holds of the representation
$\pi_{\tau}$.

($b$) We prove the first equation. Indeed

$\displaystyle\tau(P^{n})=\lim_{m}\frac{\operatorname{tr}P_{m}^{n}}{\operatorname{\mu_{d}}(K_{m})}=\operatorname{tr}P_{0}^{n}+\lim_{m}\sum_{j=1}^{m}\frac{\operatorname{tr}P_{j}^{j}P^{n}}{\operatorname{\mu_{d}}(K_{j})}.$

The first summand is non-zero iff $n\leq 0$, while the second vanishes exactly
for such $n$. Since

$\displaystyle\lim_{m}\sum_{j=1}^{m}\frac{\operatorname{tr}P_{j}^{j}P^{n}}{\operatorname{\mu_{d}}(K_{j})}=\frac{\operatorname{tr}P_{n}^{n}}{\operatorname{\mu_{d}}(K_{n})},$

the result in (5.2) shows that in both cases we obtain $6\cdot 3^{-n}$. We
already proved in (5.3) that $\tau_{0}(P^{-p,\infty})=3^{p+2}$. Since
$P^{-p,\infty}\in{\mathcal{B}}_{0}$, the same holds for $\tau$ by Proposition
5.5$(ii)$. Then the thesis follows by condition $(b)$ in Proposition 6.3.

($c$) We have
$\tau\big{(}I+D^{2}\big{)}^{-s/2}=\tau\big{(}P^{-\infty,0}\big{(}I+D^{2}\big{)}^{-s/2}\big{)}+\tau\big{(}P^{1,+\infty}\big{(}I+D^{2}\big{)}^{-s/2}\big{)}$.
A straightforward computation and (5.1) give

$\displaystyle\tau\big{(}P^{-\infty,0}\big{(}I+D^{2}\big{)}^{-s/2}\big{)}=\operatorname{tr}\big{(}P_{0}\big{(}I+D^{2}\big{)}^{-s/2}\big{)}=6\sum_{n\geq
0}\big{(}1+2^{2n}\big{)}^{-s/2}3^{n},$

which converges iff $s>d$. As for the second summand, we have

$\displaystyle\tau\big{(}P^{1,+\infty}\big{(}I+D^{2}\big{)}^{-s/2}\big{)}$
$\displaystyle=\tau_{0}\big{(}P^{1,+\infty}\big{(}I+D^{2}\big{)}^{-s/2}\big{)}=\lim_{m}\frac{\operatorname{tr}\big{(}P^{1,m}_{m}\big{(}I+D^{2}\big{)}^{-s/2}\big{)}}{\mu_{d}(K_{m})}$
$\displaystyle=\lim_{m}\sum_{j=1}^{m}3^{-m}\operatorname{tr}\big{(}P^{1,m}_{m}\big{(}I+D^{2}\big{)}^{-s/2}\big{)}=6\sum_{j=1}^{\infty}3^{-j}\big{(}1+2^{-2j}\big{)}^{-s/2},$

which converges for any $s$ hence does not contribute to the residue. Finally

$\displaystyle\operatorname{Res}_{s=d}\tau\big{(}I+D^{2}\big{)}^{-s/2}$
$\displaystyle=\lim_{s\to d^{+}}(s-d)\tau\big{(}I+D^{2}\big{)}^{-s/2}$
$\displaystyle=\lim_{s\to d^{+}}\bigg{(}s-\frac{\log 3}{\log
2}\bigg{)}6\sum_{n\geq 0}\big{(}1+2^{-2n}\big{)}^{-s/2}e^{n(\log 3-s\log 2)}$
$\displaystyle=\frac{6}{\log 2}\lim_{s\to d^{+}}\frac{s\log 2-\log
3}{1-e^{-(s\log 2-\log 3)}}=\frac{6}{\log 2}.$ ∎

######  Proposition 6.6.

For any $f\in{\mathcal{A}}_{n}$
$\sup\limits_{t>0}\big{\|}\big{[}e_{[-t,t]}(D)\,D,\rho(f)\big{]}\big{\|}=\|[D_{n},\rho(f|_{K_{n}})]\|$.

###### Proof.

We observe that $|D|$ is a multiplication operator on $\ell^{2}(E_{\infty})$,
therefore it commutes with $\rho(f)$. Hence,

$\displaystyle\big{\|}\big{[}D\,e_{[-2^{p},2^{p}]}(D),\rho(f)\big{]}\big{\|}=\big{\|}|D|\,e_{[0,2^{p}]}(|D|)\,(\rho(f)-F\rho(f)F)\big{\|}\\!=\\!\\!\sup_{\mathrm{length}(e)\geq
2^{-p}}\\!\\!\\!\frac{|f(e^{+})-f(e^{-})|}{\mathrm{length}(e)}.$

As a consequence,

$\displaystyle\sup_{p\in{\mathbb{Z}}}\big{\|}\big{[}D\,e_{[-2^{p},2^{p}]}(D),\rho(f)\big{]}\big{\|}=\sup_{e\in
E_{\infty}}\frac{|f(e^{+})-f(e^{-})|}{\mathrm{length}(e)}.$

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

$\displaystyle\frac{|f(e^{+})-f(e^{-})|}{2^{n+1}}\leq\frac{1}{2}\bigg{(}\frac{|f(e_{1}^{+})-f(e_{1}^{-})|}{2^{n}}+\frac{|f(e_{2}^{+})-f(e_{2}^{-})|}{2^{n}}\bigg{)}\leq\sup_{\mathrm{length}(e)=2^{n}}\frac{|f(e^{+})-f(e^{-})|}{\mathrm{length}(e)}.$

Iterating, we get

$\displaystyle\sup_{e\in
E_{\infty}}\frac{|f(e^{+})-f(e^{-})|}{\mathrm{length}(e)}=\sup_{\mathrm{length}(e)\leq
2^{n}}\frac{|f(e^{+})-f(e^{-})|}{\mathrm{length}(e)}.$

Since $f\in{\mathcal{A}}_{n}$,

$\displaystyle\sup_{\mathrm{length}(e)\leq
2^{n}}\frac{|f(e^{+})-f(e^{-})|}{\mathrm{length}(e)}=\sup_{e\in
K_{n}}\frac{|f(e^{+})-f(e^{-})|}{\mathrm{length}(e)}=\|[D_{n},\rho(f|_{K_{n}})]\|.$
∎

In the following Theorem we identify ${\mathcal{B}}_{\infty}$ with
$\pi_{\tau}({\mathcal{B}}_{\infty})$, the trace $\tau$ on
$\pi_{\tau}({\mathcal{B}}_{\infty})$ with its extension to
$\pi_{\tau}({\mathcal{B}}_{\infty})^{\prime\prime}$, and $D_{n}$ and $D$ as
unbounded operators affiliated with $({\mathcal{B}}_{\infty},\tau)$ with
$\pi_{\tau}(D_{n})$ and $\pi_{\tau}(D)$ as unbounded operators affiliated with
$(\pi_{\tau}({\mathcal{B}}_{\infty})^{\prime\prime},\tau)$.

######  Theorem 6.7.

The triple $({\mathcal{L}},\pi_{\tau}(B_{\infty})^{\prime\prime},D)$ on the
unital C∗-algebra ${\mathcal{A}}_{\infty}$ is an odd semifinite spectral
triple, where
${\mathcal{L}}=\cup_{n}\\{f\in{\mathcal{A}}_{n},f\text{\,Lipschitz}\\}$. The
spectral triple has metric dimension $d=\frac{\log 3}{\log 2}$, the functional

$\displaystyle\oint
f=\tau_{\omega}\big{(}\rho(f)\big{(}I+D^{2}\big{)}^{-d/2}\big{)},$ (6.1)

is a finite trace on ${\mathcal{A}}_{\infty}$ where $\tau_{\omega}$ is the
logarithmic Dixmier trace associated with $\tau$, and

$\displaystyle\oint f=\frac{6}{\log 3}\frac{\int_{K_{n}}f\,{\rm
d}\operatorname{\mu_{d}}}{\operatorname{\mu_{d}}(K_{n})},\qquad
f\in{\mathcal{A}}_{n},$ (6.2)

where $\operatorname{\mu_{d}}$ is the Hausdorff measure of dimension $d$
normalized as above. As a consequence, $\oint f$ is a Bohr–Følner mean on the
solenoid:

$\displaystyle\oint f=\frac{6}{\log
3}\,\lim_{n\in\mathbb{N}}\frac{\int_{K_{n}}f\,{\rm
d}\operatorname{\mu_{d}}}{\operatorname{\mu_{d}}(K_{n})},\qquad
f\in{\mathcal{A}}_{\infty}.$

The Connes distance

$\displaystyle d(\varphi,\psi)=\sup\\{|\varphi(f)-\psi(f)|\colon
f\in{\mathcal{L}},\,\|[D,\rho(f)]\|=1\\},\qquad\varphi,\psi\in{\mathcal{S}}({\mathcal{A}}_{\infty})$

between states on ${\mathcal{A}}_{\infty}$ verifies

$\displaystyle d(\delta_{x},\delta_{y})=d_{\rm geo}(x,y),\qquad x,y\in
K_{\infty},$ (6.3)

where $d_{\rm geo}$ is the geodesic distance on $K_{\infty}$.

###### Proof.

The properties of a semifinite spectral triple follow by the properties proved
above, in particular property $(1)$ of Definition 2.1 follows by Propositions
6.3$(a)$ and 6.6, while property $(2)$ follows by Proposition 6.5$(b)$. The
functional in equality (6.1) is a finite trace by Proposition 6.5$(c)$.
Equations (6.2) and (6.3) only remain to be proved. We observe that
$(I+D^{2})^{-d/2}-|D_{n}|^{-d}$ have finite trace. Indeed

$\displaystyle\big{(}(I+D^{2})^{-d/2}-|D_{n}|^{-d}\big{)}e=\begin{cases}\big{(}1+4^{-k}\big{)}^{-d/2}e,&\mathrm{length}(e)=2^{k},\quad
k>n,\\\
\big{(}\big{(}1+4^{-k}\big{)}^{-d/2}-2^{dk}\big{)}e,&\mathrm{length}(e)=2^{k},\quad
k>nk\leq n,\end{cases}$

hence, makig use of a formula in Theorem 6.5$(b)$, we get

$\displaystyle\big{|}\tau\big{(}\big{(}I+D^{2}\big{)}^{-d/2}-|D_{n}|^{-d}\big{)}\big{|}$
$\displaystyle\leq\sum_{k>n}(1+4^{-k})^{-d/2}\tau\big{(}P^{k}\big{)}+\sum_{k\leq
n}\big{|}\big{(}1+4^{-k}\big{)}^{-d/2}-3^{k}\big{|}\tau\big{(}P^{k}\big{)}$
$\displaystyle\leq
6\big{(}1+4^{-(n+1)}\big{)}^{-d/2}\sum_{k>n}3^{-k}+6\sum_{k\leq
n}\big{|}\big{(}1+4^{k}\big{)}^{-d/2}-1\big{|}$

and both series are convergent. Since the Dixmier trace vanishes on trace
class operators, this implies that

$\displaystyle\tau_{\omega}(\rho(f)\big{(}I+D^{2}\big{)}^{-d/2})=\tau_{\omega}\big{(}\rho(f)|D_{n}|^{-d}\big{)}=\frac{1}{d}\operatorname{Res}_{s=d}\tau\big{(}\rho(f)|D_{n}|^{-s}\big{)},$

therefore, if $f\in{\mathcal{A}}_{n}$,

$\displaystyle\oint
f=\frac{1}{d}\operatorname{Res}_{s=d}\tau\big{(}\rho(f)|D_{n}|^{-s}\big{)}=\frac{1}{d}\operatorname{Res}_{s=d}\frac{\operatorname{tr}(\rho(f_{K_{n}})|D_{n}|^{-s})}{\operatorname{\mu_{d}}(K_{n})}=\frac{\operatorname{tr}_{\omega}\big{(}\rho(f)|D_{n}|^{-d}\big{)}}{\operatorname{\mu_{d}}(K_{n})}.$

Now, by formula (2.1) applied to $K_{n}$,
$\operatorname{tr}_{\omega}(\rho(f)|D_{n}|^{-d})=\frac{6\cdot\ell(e)^{d}}{\log
3}\int_{K_{n}}f\,{\rm d}H_{d}$, where $H_{d}$ is the Hausdorff measure
normalized on $K_{n}$, hence
$H_{d}=(\mu_{d}(K_{d}))^{-1}\mu_{d}=3^{-n}\mu_{d}$, and $e\in E_{0}(K_{n})$,
hence $\ell(e)^{d}=3^{n}$. Therefore
$\operatorname{tr}_{\omega}\big{(}\rho(f)|D_{n}|^{-d}\big{)}=\frac{6}{\log
3}\int_{K_{n}}f\,{\rm d}\mu_{d}$ and formula (6.2) follows. As for equation
(6.3), given $x,y\in K_{\infty}$ let $n$ such that $x,y\in K_{n}$, $m\geq n$.
Then, combining Propositions 6.3$(a)$ and 6.6, we have, for
$f\in{\mathcal{A}}_{m}$,

$\displaystyle\|[D,\rho(f)]\|=\|[D_{m},\rho(f|_{K_{m}})]\|,$

and, by Theorem 5.2 and Corollary 5.14 in [29],

$\displaystyle\sup\\{|f(x)-f(y)|\colon
f\in{\mathcal{A}}_{m},\|[D_{m},\rho(f|_{K_{n}})]\|=1\\}=d_{\rm
geo}(x,y),\qquad m\geq n.$

Therefore

$\displaystyle d(\delta_{x},\delta_{y})$
$\displaystyle=\sup\\{|f(x)-f(y)|\colon
f\in{\mathcal{L}},\,\|[D,\rho(f)]\|=1\\}$ $\displaystyle=\lim_{m}\
\sup\\{|f(x)-f(y)|\colon f\in{\mathcal{A}}_{m},\,\|[D,\rho(f)]\|=1\\}$
$\displaystyle=\lim_{m}\ \sup\\{|f(x)-f(y)|\colon
f\in{\mathcal{A}}_{m},\,\|[D_{m},\rho(f|_{K_{n}})]\|=1\\}=d_{\rm geo}(x,y).$ ∎

######  Remark 6.8.

The last statement in Theorem 6.7 shows that the triple
$({\mathcal{L}},{\mathcal{M}},D_{\infty})$ recovers two incompatible aspects
of the space ${\mathcal{A}}_{\infty}$: on the one hand the compact space given
by the spectrum of the unital algebra ${\mathcal{A}}_{\infty}$, with the
corresponding finite integral, and on the other hand the open fractafold
$K_{\infty}$ with its geodesic distance. In particular, the functional on
${\mathcal{L}}$ given by $L(f)=\|[D,\rho(f)]\|$ is not a Lip-norm in the sense
of Rieffel [41] because it does not give rise to the weak∗ topology on
${\mathcal{S}}({\mathcal{A}}_{\infty})$. In fact, such seminorm produces a
distance which is unbounded on points, therefore the induced topology cannot
be compact.

### Acknowledgements

We thank the referees of this paper for many interesting observations and
suggestions. V.A. is supported by the Swiss National Science Foundation. D.G.
and T.I. are supported in part by GNAMPA-INdAM and the ERC Advanced Grant
669240 QUEST “Quantum Algebraic Structures and Models”, and acknowledge the
MIUR Excellence Department Project awarded to the Department of Mathematics,
University of Rome Tor Vergata, CUP E83C18000100006.

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