Monoids, their boundaries, fractals and C*-algebras

Abstract In this note we establish some connections between the theory of self-similar fractals in the sense of John E. Hutchinson (cf. [3]), and the theory of boundary quotients of C*-algebras associated to monoids. Although we must leave several important questions open, we show that the existence of self-similar ℳ-fractals for a given monoid ℳ, gives rise to examples of C*-algebras (1.9) generalizing the boundary quotients ∂Cλ*(𝒨) \partial C_\lambda ^*(\mathcal{M}) discussed by X. Li in [4, §7, p. 71]. The starting point for our investigations is the observation that the universal boundary of a finitely 1-generated monoid carries naturally two topologies. The fine topology plays a prominent role in the construction of these boundary quotients, while the cone topology can be used to define canonical measures on the attractor of an ℳ-fractal for a finitely 1-generated monoid ℳ.


Introduction
On a monoid M (=semigroup with unit M ) there is naturally de ned a re exive and transitive relation " ", i.e., for ω, τ ∈ M one de nes ω τ if, and only if, there exists σ ∈ M satisfying ω = τ · σ. In particular, one may consider (M , ) as a partially ordered set. Moreover, if M is N -graded, then (M , ) is a (noetherian) partially ordered set (see is a continuous map. The monoid M will be said to be T -regular, if (1.2) is a homeomorphism. E.g., nitely generated free monoids are T -regular (cf. Proposition 3.11, § 4.1). The universal boundary ðM = (∂M , T f (M ) with the ne topology can be identi ed with the Laca boundary E(M ) of the monoid M . This topological space plays an essential role for de ning boundary quotients of C * -algebras associated to monoids (cf. [4, § 7], [5]). Indeed one has the following (cf. Theorem 3.10). On the other hand the induced mapping ϕ E is given by a map (cf. Proposition 3.12). Hence for the purpose of constructing Borel measures the ne topology seems to be inappropriate.
Theorem B can be used to de ne the C * -algebra for every nitely -generated N -graded monoid M , where β(ω) is the mapping induced by left multiplication with ω (cf. § 4.5). We will show by explicit calculation that for the monoid Fn, freely generated by a set of cardinality n, the C * -algebra C * (Fn , µ Fn ) coincides with the Cuntz algebra On (cf. Proposition 4.4), while for the right-angled Artin monoid M Γ associated to the nite graph Γ, C * (M Γ , µ M Γ ) coincides with the boundary quotients introduced by Crisp and Laca in [2] (cf. § 4.6). Nevertheless the following more general question remains unanswered. From now on we will assume that the N -graded monoid M = k∈N M k is nitely -generated. In the context of self-similar fractals in the sense of John E. Hutchinson (cf. [3]) it will turn out to be convenient to endow ∂M with the cone topology Tc(M ). Let (X, d) be a complete metric space with a left M -action α : M −→ C(X, X) by continuous maps. Such a presentation will be said to be contracting, if there exists a positive real number for all x, y ∈ X, s ∈ M (cf. [3, § 2.2]). For such a metric space (X, d) there exists a unique compact subset Obviously, by de nition every map α(t) ∈ C(X, X) is contracting, and thus has a unique xed point x t ∈ X. For short we call K = K(α) ⊂ X the attractor of the representation α. One has the following (cf. Proposition 5.4).
Theorem C. Let M = k∈N M k be a nitely -generated N -graded monoid, let (X, d) be a compact metric space and let α : M −→ C(X, X) be a contracting representation of M . Then for any point x ∈ X, α induces a continuous map κx : ∂M −→ K(α).

Moreover, if M is T -regular, then κx is surjective.
Under the general hypothesis of Theorem C we do not know whether the topological space (∂M , Tc(M )) is necessarily compact (see Question 3). However, in case that it is compact, we call (∂M , Tc(M )) universal attractor of the nitely -generated N -graded T -regular monoid M .
Remark 1.2. Let M be a nitely -generated monoid. Then ∂M carries canonically a probability measure µ M (cf. § 4.5). Thus, by Theorem C, the attractor of the M -fractal ((X, d), α) carries the contact probability measure µx = µ κx • for every point x ∈ X, which is given by By (1), the monoid M is acting on K, and thus also on L (K, C, µx) by bounded linear operators γ(ω), ω ∈ M (cf. § 5.2) This de nes a C * -algebra (cf. § 5.2) (1.9) In case that the equivalence relation∼ generated by on ∂M is di erent from ≈ (cf. (1.1)) the canonical map j : ∂M → ∂M /∼ is not the identity.

Question 2.
Does there exist a nitely -generated τ-regular monoid M for which the mapj is not the identity, and an M -fractal ((X, d), α) such that C * (M , X, dµ X ) is not isomorphic to ∂C λ (M )?

Posets and their boundaries
A poset (or partially ordered set) is a set X together with a re exive and transitive relation : X × X → {t, f } with the property that for all x, y ∈ X satisfying x y and y x follows that x = y. By N = { , , . . .} we denote the set of positive integers, and by N = { , , , . . .} we denote the set of non-negative integers, i.e., N is a commutative monoid.

. Complete posets
For a poset (X, ) let denote the set of decreasing functions which we will -if necessary -identify with the set of decreasing sequences. A poset (X, ) is said to be complete, if for all f ∈ D(N, X, ) there exists an element z ∈ X such that (CP ) f (n) z for all n ∈ N, and (CP ) if y ∈ X satis es f (n) y for all n ∈ N, then z y. Note that -if it existsz ∈ X is the unique element satisfying (i) and (ii) for f ∈ D(N, X, ). As usually, z = min(f ) is called the minimum of f ∈ D(N, X, ).

. The poset completion of a poset
Let (X, ) be a poset. For u, v ∈ D(N, X, ) put and put where cz ∈ D(N, X, ), z ∈ X, is given by cz(n) = z for all n ∈ N. Let ≈ be the equivalence relation generated by ∼ and put X = D(N, X, )/ ≈. Then the following properties hold for (X, ).
Proof. (a) The relation is obviously re exive. Let u, v, w ∈ D(N, X, ), u v, v w. Then for all n ∈ N there exists hn , kn ∈ N such that u(hn) v(kn) w(n). Thus, u w. (b) Let u ∈ D(N, X, ) and let α : N → N be a strictly increasing function. Let m < n, m, n ∈ N. Since α is strictly increasing, α(m) < α(n). Then there exist m , n ∈ N such that m ≤ α(m) < α(n) ≤ n . Then one has u(m ) u(α(m)) u(α(n)) u(n ). Thus u u • α and u • α u, proving that u ≈ u • α. (d) Let {u k } k∈N ∈ D(N, X, ), i.e., u k ∈ X for all k ∈ N. Then one has u u . . . by de nition. Since each u k ∈ D(N, X, ), one has u k (n) u k (m) for all n ≤ m, m, n ∈ N. We de ne v ∈ D(N, X, ) by v(n) = un(n), n ∈ N. Then [v] ∈ X is the minimum of {u k } k∈N . This yields the claim.
Assigning every element x ∈ X the equivalence class containing the constant function cx ∈ D(N, X, ) yields a strictly decreasing mapping of posets ι X : X → X. From now on (X, ) will be considered as a sub-poset of (X, ). The poset (X, ) will be called the poset completion of (X, ). The following fact is straightforward.

. The universal boundary of a poset
For a poset (X, ) the poset ∂X = X \ im(i X ) will be called the universal boundary of the poset (X, ). From now on we use the notation x y as a short form for x y and x ≠ y. A function f : N → X will be said to be strictly decreasing, if f (n + ) ≺ f (n) for all n ∈ N. The following fact will turn out to be useful.

Fact 2.3. Let f ∈ D(N, X, ) be a decreasing function such that [f ] ∈ ∂X. Then there exists a strictly decreasing
Proof. By hypothesis, J = im(f ) is an in nite set. In particular, the set Ω = { min(f − ({j}) | j ∈ J } is an in nite and unbounded subset of N. Let e : N → Ω be the enumeration function of Ω, i.e., e( ) = min(Ω), and recursively one has e(k + ) = min(Ω \ {e( ), . . . , e(k)}). Then, by construction, h = f • e is strictly decreasing, and, by Proposition 2.1(b), one has f ≈ h, and hence the claim. Fact 2.4. Let (X, ) be a noetherian poset, and let (X, ) be its completion. Then for all τ ∈ X one has Cτ(X) ⊆ X. In particular, Cτ(X) = Cτ(X), where the cocones are taken in the respective posets.
where "≤" denotes the natural order relation on Z. Then (X, ) is a poset and its completion is given by X =

. The cone topology
Let (X, ) be a poset, and let (X, ) denote its completion. For τ, ω ∈ X let In particular, is a base of a topology Tc(X) -the cone topology -on X. By construction, the subspace X is discrete and open, and the subspace ∂X is closed.
For ω ∈ X let Nc(ω) denote the set of all open neighborhoods of ω with respect to the cone-topology, and put S (ω) = U∈Nc(ω) U. Then, by construction, one has S (ω) = {ω} for ω ∈ X, and S (ω) = Cω(X) for ω ∈ ∂X. This implies the following. Proposition 2.5. Let (X, ) be a poset, and let (X, ) denote its completion. Then (X, Tc(X)) is a T -space (or Kolmogorov space).

. The ne topology
For a partially ordered set (X, ) let denote the set of all subsets of X of cardinality 1, all cones and their complements in X. Then S is a subbasis of a topology T f (X) on X which we will call the ne topology on X. In particular, the set Ω = { X = ≤j≤r X j | X , . . . , Xr ∈ S } is a base of the topology T f (X). By de nition, this topology has the following properties. Fact 2.6. Let (X, ) be a partially ordered set. Then

. The ∼-boundary
There is another type of boundary for a poset, the ∼-boundary, which seems to be relevant for the study of fractals (see (5.6)). Let (X, ) be a noetherian poset, and let (X, ) denote its completion. Put and let∼ denote the equivalence relation on X generated by the relation Ω. Then one has a canonical map π : X → X, (2.13) where X = X/∼. By construction, π| X is injective. The set ∂X = X \ π(X) will be called the ∼-boundary of the poset (X, ). We put 14) The set X carries the quotient topology Tq( X) with respect to the mapping π and the topological space (X, Tc(X)). In particular, the subspace π(X) ⊆ X is discrete and open, and ∂X ⊆ X is closed. For ω ∈ X we put Cω = π(Cω(X)). The space X will be considered merely as topological space. It has the following property.
This yields the claim.

Monoids and their boundaries
A monoid (or semigroup with unit) M is a set with an associative multiplication · : M × M → M and a distinguished element ∈ M satisfying · x = x · = x for all x ∈ M . For a monoid M we denote by .

N -graded monoids
The set of non-negative integers N = { , , , . . .} together with addition is a monoid. A monoid M together with a homomorphism of monoids | | : M → N is called an N -graded monoid. For k ∈ N one de nes One has the following straightforward fact.  satisfying ϕ M , = id M . The N -graded monoid M is said to be -generated, if ϕ M is surjective. In particular, such a monoid is connected. By de nition, free monoids are -generated. Moreover, M is said to be nitely -generated, if it is -generated and M is a nite set. The following important question remains unanswered in this paper.   Proof. Suppose xM = yM , for x, y ∈ M . Then there exist z, w ∈ M such that x = yz and y = xw, so |x| = |y| + |z| and |y| = |x| + |w|. Thus |z| = = |w|. Since M is connected, this implies z = = w.

. Abelian semigroups generated by idempotents
Let E be an abelian semigroup being generated by a set of elements Σ ⊆ E satisfying σ = σ for all σ ∈ Σ, i.e., all elements of Σ are idempotents. Then every element u ∈ E is an idempotent, and one may de ne a partial order " " on E by where F ab (Σ) is the free abelian semigroup over the set Σ, and R is the relation i.e., E = F ab (Σ)/R ∼ , where R ∼ is the equivalence relation on F ab (Σ) generated by the set R. Let Then E coincides with the set of characters of the C * -algebra C * (E) generated by E (satisfying e * = e for all e ∈ E), and hence carries naturally the structure of a compact topological space. By construction, E can be identi ed with a subset of F(Σ, { , }) -the set of functions from Σ to { , }. In more detail, Thus identifying F (Σ, { , }) with { , } Σ , one obtains that

. The semigroup of idempotents generated by a set of subsets of a set
Let X be a set, and let S ⊆ P(X) be a set of subsets of X. Then S generates an algebra of sets A (S) ⊆ P(X), i.e., the sets of A (S) consist of the nite intersections of sets in S. Then is an abelian semigroup being generated by the set of idempotents Moreover, by (3.14), one has
The following theorem shows that for a -generated N -graded monoid M its universal boundary ∂M with the ne topology is a totally-disconnected compact space. Proof. It is well known that χ is surjective (cf. [5,Lemma 2.3]), and thus χ is surjective. By Proposition 3.9, χ is injective, and hence χ is a bijection. The sets In contrast to Proposition 3.2 one has the following property for the Laca boundary of monoids. and thus a map If ϕ is surjective, then ϕ E is surjective, and ϕ • E restricts to a map It is straightforward to verify that ϕ E is continuous and injective.

Free monoids and trees
Let Fn = F x , . . . , xn be the free monoid on n generators. Let S = {x , . . . , xn} be the set of generators, and let | | : Fn → N be the grading morphisms, i.e., |y| = if and only if y ∈ S. The Cayley graph Γ(Fn , S) of Fn with respect to S is the graph de ned by The origin and terminus maps o, t : E → V are given by the projection onto the rst and second coordinate, respectively. Then Γ(Fn , S) is an n-regular tree with root and all edges pointing away from . The graph Γ(Fn , S) coincides with an orientation of the n-regular tree Tn.

. The boundary of the n-regular tree
The boundary ∂Tn of Tn is the set of equivalence classes of in nite paths without backtracking under the relation ∼ de ned by the shift, We denote by [v, w) the unique path starting at v in the class ω and de ne the interval of ∂Tn starting at v. Then ∂Tn is compact with respect to the topology T I generated by {Iv} v∈V . For any [ρ] ∈ ∂Tn there exists a unique ray ρ = (e k ) k∈N , o(ρ) = o(e ) = . One can assign to ρ the decreasing function ωρ ∈ D(N, Fn , ) given by ωρ(k) = t(e k ). The map φ : ∂Tn → ∂Fn given by is a bijection. Hence one can identify ∂Tn with ∂Fn.
. The space (F n , T c (F n )) Every cone Cτ(Fn) de nes a rooted subtree Tτ of Tn satisfying ∂Tτ = ∂Fn ∩ Cτ(Fn). Thus every covering τ∈U Cτ(Fn) ∩ ∂Fn of the boundary of ∂Fn by cones de nes a forest F = τ∈U Tτ. Let F = i∈I F i be the decomposition of F in connected components. Then ∂Tn = ∂F = i∈I ∂F i , where denotes disjoint union. Hence the compactness of ∂Tn implies |I| < ∞.
As ∂F i ⊆ ∂Tn is closed, and hence compact, a similar argument shows that there exist nitely many cones Cτ i,j , ≤ j ≤ r i , such that  . The canonical probability measure on the boundary of a regular tree By Carathéodory's extension theorem the assignment de nes a unique probability measure µ : Bor(∂Tn) → R + . Hence the corresponding probability measure µ : Bor(∂Fn) → R + satis es µ(∂Fn ∩ Cτ(Fn)) = n −|τ| for τ ∈ Fn . (4.7) De nition. Let · : Fn × ∂Fn → ∂Fn be the map given by where xω : N → Fn is given by (xω)(n) = xω(n) Note that this action is well de ned, since ω ∼ ω implies that xω ∼ xω .

. Finitely -generated monoids
Let M be a nitely -generated N -graded monoid. Then one has a canonical surjective graded homomorphism ϕ M : F → M , where F is a nitely generated free monoid (cf. .

Right-angled Artin monoids
Let Γ = (V , E) be a nite undirected graph, i.e. |V| = n < ∞ and E ⊆ P (V), where P (V) denotes the set of subsets of cardinality of V. The right-angled Artin monoid associated to Γ is the monoid M Γ de ned by where µ is the measure de ned on ∂F V by (4.7).
De nition. Let Γ = (V , E) be a graph, and let Γ = (V , E ) and Γ = (V , E ) be subgraphs of Γ. We say that Γ is bipartitly decomposed by Γ and Γ , if V = V V and In this case we will write Γ = Γ ∨ Γ . If no such decomposition exists, Γ will be called coconnected. and the adjoint operators are given by (4.52) where f ∈ L (∂M Γ , C, µ Γ ). It remains to prove that it also satis es (iv). Let In order to show that e i (f ) = for all f ∈ L (∂M Γ , C, µ Γ ) it su ces to show that e i (f ) = for f = f ⊗ · · · ⊗ fr, Hence e i (f ) = and this yields the claim.

Fractals
Let M be a nitely -generated monoid. By an M -fractal we will understand a compact metric space (X, d) with a contracting left M -action α : M −→ C(X, X), i.e., there exists a real number δ < such that for all x, y ∈ X and all ω ∈ M \ { } one has The real number δ will be called the contraction constant. To the authors knowledge the following important question has not been discussed in the literature yet. As α is contracting, one concludes that α(f (n))(x) is a Cauchy sequence for every strictly decreasing sequence f ∈ D(N, M , ) and thus has a limit point α(f )(x) = limn→∞ α(f (n))(x) . In more detail, if α has contracting constant δ < , one has for n, m ∈ N, m > n, that where diam(X) = max{ d(y, z) | y, z ∈ X }. Thus one has a map · : D(N, M , ≺) × X −→ X (5.3) given by [f ] · x = α(f )(x). This map has the following property. Proof. We may assume that f (n) h(n) for all n ∈ N, i.e., there exists yn ∈ M such that f (n) = h(n) · yn. Then, by the same argument which was used for (5.2), one concludes that d(α(f (n))(x), α(h(n)(x)) ≤ δ |h(n)| diam(X) ≤ δ n diam(X). The following property suggest to think of ( ∂M , Tc) as the universal attractor of an M -fractal. . The C * -algebra associated to an M -fractals for a nitely -generated monoid M Let M be a nitely -generated monoid, and let ((X, d), α) be an M -fractal with attractor K. For x ∈ X there exists a continuous mapping κx : ∂M → K (cf. Theorem C). Let µx : Bor(K) → R + be the probability measure given by (1.8). Then M acts on K, and thus also on L (K, C, µx).