Introduction Inverse Semigroups vs Groupoids Semigroups P-graphs

Semigroups, P-graphs, and groupoids

Jean Renault

Universite d’Orleans

3 June 2016

“New directions in inverse semigroups”Ottawa

Introduction Inverse Semigroups vs Groupoids Semigroups P-graphs

Introduction

Many constructions of C*-algebras from algebraic or dynamicalsystems can be done through a groupoid.

Let us consider the case of a semigroup P (assumed here to bediscrete countable and with a unit e), is there a semigroupC*-algebra C ∗(P)? G. Murphy has given a natural answer: theuniversal C*-algebra for isometric representations. However, thisC*-algebra is hard to study.

On the other hand, there exist several C*-algebras associated withP which are much more tractable, for example, the left or rightreduced C*-algebras acting on `2(P) or the Wiener-HopfC*-algebra W(P,Q) (assuming, as we shall always do, that P is asubsemigroup of a group Q). The reason that these C*-algebrasare easier to study is that they can be described as groupoidC*-algebras.

Introduction Inverse Semigroups vs Groupoids Semigroups P-graphs

Introduction (cont’d)

I shall concentrate here on the C*-algebra C ∗red(P) of the leftregular representation L on `2(P) of a subsemigroup P of adiscrete group Q containing the unit element e: for s, L(s) is theisometry defined by L(s)εx = εsx for all x ∈ P.

We shall also see that the same procedure applies to topologicalP-graphs and gives a groupoid description of the associatedToeplitz and Cuntz algebras.

In each of these situations, we shall first construct an inversesemigroup.

Introduction Inverse Semigroups vs Groupoids Semigroups P-graphs

Inverse semigroups

There is no need here to recall the definition of an inversesemigroup. I do it anyway to set my notation.

Definition

A semigroup S is called an inverse semigroup if for all s ∈ S , thereexist a unique s∗ ∈ S such that

ss∗s = s s∗ss∗ = s∗

The set of idempotents E = E (S) = e ∈ S : e2 = e is itself acommutative self-adjoint inverse semigroup (equivalently, an infsemi-lattice) and plays an important role.

Introduction Inverse Semigroups vs Groupoids Semigroups P-graphs

Two basic examples

1. Let X be a topological space. Then the set I(X ) ofhomeomorphisms ϕ : U → V where U and V are open subsets ofX is an inverse semigroup.

2. A family of S of partial isometries closed under involution andcomposition (then the range and domain projections commute) isan inverse semigroup.

In these examples, the set of idempotents is contained in aBoolean algebra, hence has a natural spectrum.

Introduction Inverse Semigroups vs Groupoids Semigroups P-graphs

Action of an inverse semigroup on a space

Definition

An action of an inverse semigroup S on a topological space X is aninverse semigroup morphism α : S → I(X ).

An abstract inverse semigroup S admits canonical actions, forexample, it acts on the spectrum E of the commutative C*-algebraC ∗(E ) and also on the tight spectrum closure(Emax) as defined byExel.

Introduction Inverse Semigroups vs Groupoids Semigroups P-graphs

The groupoid of germs of an inverse semigroup action

Given an action (X ,S , α), we define following[Khoshkam-Skandalis 02] and [Exel 08]:

Definition

The groupoid of germs Germ(X , S , α) of the action is the quotientof

S ∗ X = (s, x) : s ∈ S , x ∈ dom(αs)

by the relation [s, x ] = [t, y ] iff x = y and there exists e ∈ E suchthat x ∈ dom(αe) and se = te.It is an etale groupoid.

The choice of X = E gives the universal groupoid G (S) of S ,which gives a groupoid realization of the full and of the reducedC*-algebras of the inverse semigroup S .

Introduction Inverse Semigroups vs Groupoids Semigroups P-graphs

Semigroup actions

Let us define now a semigroup action:

Definition

A left action of a semigroup P on a topological space X is a map

T : (n, x) ∈ P ∗ X 7→ nx ∈ X

where P ∗ X is an open subset of P × X , such that

1 for all x ∈ X , (e, x) ∈ P ∗ X and ex = x ;

2 if (m, x) ∈ P ∗ X , then (n,mx) ∈ P ∗ X iff (nm, x) ∈ P ∗ X ; ifthis holds, we have n(mx) = (nm)x ;

3 for all n ∈ P, the map defined by Tnx = nx is a localhomeomorphism with domainU(n) = x ∈ X : (n, x) ∈ P ∗ X and rangeV (n) = nx : (n, x) ∈ P ∗ X.

Introduction Inverse Semigroups vs Groupoids Semigroups P-graphs

Examples

1 One-sided subshifts of finite type, z 7→ z2 on the circle: themaps Tn : U(n)→ V (n) are not necessarily injective but theyare local homeomorphisms.

2 Rational maps on the sphere: exclude the singular points toget local homeomorphisms.

3 Putnam’s algebra AU : start with a self-homeomorphism T ofX and consider its restriction T|U where U is an open subsetof X .

4 infinite graphs (e.g. Exel-Laca’s Cuntz-Krieger algebras).

5 We shall see that P-graphs lead to such semigroup actions.

Introduction Inverse Semigroups vs Groupoids Semigroups P-graphs

The groupoid of germs of a semigroup action

Assuming that P ⊂ Q, where Q is a group, we now define thegroupoid of germs of the action:

We let S be the sub-inverse semigroup of Q × I(X ) generated bythe elements (n,Tn|U) where n ∈ P and U is an open set on whichTn is injective. Then, the restriction α of the second projection ofthe product Q × I(X ) is an action of the inverse semigroup S onX . We define the groupoid of germs of (X ,P,T ) as the groupoidof germs Germ(X ,S, α) and we denote it by Germ(X ,P,T ).

Introduction Inverse Semigroups vs Groupoids Semigroups P-graphs

directed action

Through the groupoid Germ(X ,P,T ), we can define theC*-algebra (full or reduced) of the semigroup action (X ,P,T ).However, very little can be said about it. The following conditionwill make the groupoid of germs (and its C*-algebra) moretractable.

Definition

We say that a semigroup action (X ,P,T ) is directed if for all pairs(m, n) ∈ P × P such that U(m) ∩ U(n) 6= ∅, there exist(a, b) ∈ P × P such that am = bn and U(am) ⊃ U(m) ∩ U(n).

Introduction Inverse Semigroups vs Groupoids Semigroups P-graphs

The semi-direct product groupoid

Given a semigroup action (X ,P,T ) where P ⊂ Q, we defineG (X ,P,T ) as the set

(x ,m−1n, y) : m, n ∈ P, x ∈ U(m), y ∈ U(n),Tmx = Tny.

When P = N one retrieves the well-known groupoid of anendomorphism studied by V. Deaconu:

G (X ,T ) = (x ,m − n, y) : m, n ∈ N, x , y ∈ X ,Tmx = T ny.

In the general case, G (X ,P,T ) is not necessarily a groupoid (moreprecisely a subgroupoid of X × Q × X ).

Introduction Inverse Semigroups vs Groupoids Semigroups P-graphs

Identification of the groupoid of germs

However, when the action is directed, it is a groupoid and it is infact the groupoid of germs defined earlier

Theorem (R 2015)

Assume that the action (X ,P,T ) is directed. Then

1 G (X ,P,T ) is a locally compact Hausdorff etale groupoid;

2 G (X ,P,T ) is the groupoid of germs of the action.

We shall see shortly that this explains a posteriori why both Oresemigroups (e.g. [R-Exel 07] or [R-Sundar 15]) and quasi-latticeordered semigroups [Nica 92]), rather than more generalsemigroups have been considered.

Introduction Inverse Semigroups vs Groupoids Semigroups P-graphs

The left inverse hull of a semigroup

Let us consider the left action L of P on itself. If P is leftcancellative, the left inverse hull Sl(P) of P is defined as theinverse semigroup generated by the left translations L(s) : P → sP.Equivalently, it is the inverse semigroup of partial isometriesgenerated by the isometries L(s) : `2(P)→ `2(sP).

The inverse semigroup Sl(P) acts on D = C ∗red(P) ∩ `∞(P) by

d 7→ L(s)dL(s)∗. This induces an action α of Sl(P) on X (P) = D.

Introduction Inverse Semigroups vs Groupoids Semigroups P-graphs

The reduced C*-algebra of a semigroup as a groupoidalgebra

Theorem (X. Li 12)

Let P be a left cancellative semigroup. Then C ∗red(P) is thereduced C*-algebra of the groupoid of germs Germ(X ,Sl(P), α) ofthe action of the left inverse hull on X (P).

Moreover, if P is a subsemigroup of a group Q, there is a cocycle

c : Germ(X ,Sl(P), α)→ Q

such that c([L(s), x ]) = s for all s ∈ P and x ∈ X (P).

Introduction Inverse Semigroups vs Groupoids Semigroups P-graphs

The case of Ore and quasi-lattice ordered semigroups

Assume now that P be a left cancellative semigroup. We have twoways to view the left translation of P on itself .

1 As a left action L(s)t = st. This action is directed iff P isright reversible, i.e. for all pairs (m, n) ∈ P × P,Pm ∩ Pn 6= ∅. This can be written PP−1 ⊂ P−1P.

2 As a right action of P, by defining L(s)−1(sx) = x . Thisaction is directed iff P satisfies the Clifford condition: ifmP ∩ nP 6= ∅, it is of the form rP for some r ∈ P.

These conditions define respectively the right Ore and thequasi-lattice ordered semigroups.

Introduction Inverse Semigroups vs Groupoids Semigroups P-graphs

The case of Ore and quasi-lattice ordered semigroups(cont’d)

When we pass to the action of P on the spectrum X (P), thedirectedness of the action is preserved. Therefore, in these cases,the groupoid of germs can be described as a semi-direct groupoid.

Examples

Nd ⊂ Zd are both Ore and quasi-lattice ordered, which isequivalent to lattice ordered.

If Q is abelian and P − P = Q, P ⊂ Q is Ore.

SFd ⊂ Fd is quasi-lattice ordered (and not Ore). Graph productsof quasi-lattice ordered semigroups are quasi-lattice ordered[Crisp-Laca 02].

Introduction Inverse Semigroups vs Groupoids Semigroups P-graphs

topological category

Let us turn to topological P-graphs. We first need the definition ofa topological category (or semi-groupoid). It is just like atopological groupoid but without the existence of an inverse.

Definition

A topological category is a small category Λ with set of objectsΛ(0), range and source maps r , s : Λ→ Λ(0), composition map

m : Λ(2) = Λ ∗ Λ→ Λ,

endowed with a topology compatible with its structure.

More precisely, we require that all above maps are continuous andthat s is a local homeomorphism.

Introduction Inverse Semigroups vs Groupoids Semigroups P-graphs

topological P-graphs

Definition

A topological P-graph is a topological category Λ endowed with amap, called the degree map, d : Λ→ P which satisfies thefollowing properties

1 for all m ∈ P, Λm = d−1(m) is open;

2 for all (µ, ν) ∈ Λ(2), d(µν) = d(µ)d(ν) and for all v ∈ Λ(0),d(v) = e;

3 it has the unique factorization property: for all m, n ∈ P, thecomposition map Λm ∗ Λn → Λmn is a homeomorphism.

Introduction Inverse Semigroups vs Groupoids Semigroups P-graphs

Examples

1 Graphs are N-graphs. More accurately, given a graph (E ,V ),we let

Λ = xn . . . x2x1 finite path of the graph

Here, P = N and d(xn . . . x2x1) = n. For example, when Vhas a single element, Λ is the set of all finite words withletters in E .

2 Topological graphs are topological N-graphs. Here E ,V aretopological spaces, r , s : E → V are continuous and s is alocal homeomorphism.

Introduction Inverse Semigroups vs Groupoids Semigroups P-graphs

from P-actions to topological P-graphs

Proposition

Let T : X ∗ P → X be a semigroup action as above. ThenΛ = X ∗ P has a natural structure of a topological P-graph.

It is given by Λ(0) = X , the range and source maps r , s : Λ→ Xare respectively r(x , n) = x and s(x , n) = xn. The composition isgiven by

(x ,m)(xm, n) = (x ,mn)

The degree map d : Λ→ P is simply d(x , n) = n.

Introduction Inverse Semigroups vs Groupoids Semigroups P-graphs

from topological P-graphs to P-actions

One can view a topological P-graph as a generalized P-action. Wecan follow the same pattern as in the construction of the inversehull of a semigroup: given a P-graph Λ, we let P act on Λ by

Λ ∗ P = (µν,m) ∈ Λ× P : d(µ) = m

and T : Λ ∗ P → Λ by T (µν,m) = ν if d(µ) = m.

As in the case of the action of P on itself by translation, the actionof P can be extended to the “path space” X , which is a suitablecompletion of the “finite path space Λ”. Its groupoid of germsdefines the Toeplitz algebra of the P-graph. Its reduction to the“boundary path space” ∂X defines the Cuntz algebra of theP-graph. Here are the details:

Introduction Inverse Semigroups vs Groupoids Semigroups P-graphs

Assumptions

This action of P on Λ is not so interesting! it is proper, i.e. thesemi-direct product Λ o P is a proper groupoid. However, underadditional assumptions, the space Λ admits a naturalcompactification, its order compactification Λ which we are goingto define and things become much more interesting! The ideasgoes back to [Nica 92] who considered the case Λ = P.

Assumptions: We shall assume that

1 P is a quasi-lattice ordered semigroup of a group Q.

2 the map (r , d) : Λ→ Λ(0) × P sending λ to (r(λ), d(λ)) isproper.

Introduction Inverse Semigroups vs Groupoids Semigroups P-graphs

Quasi-lattice ordered semigroups

Definition

One says that a semigroup P of a group Q is quasi-lattice orderedif P ∩ P−1 = e and as soon as a, b ∈ P have a common upperbound, they have a least common upper bound.

Proposition

1 T is an action of P on Λ by partial local homeomorphisms.

2 If P is quasi-lattice ordered, this action is directed.

Introduction Inverse Semigroups vs Groupoids Semigroups P-graphs

the path space Ω

We define on Λ a pre-order in the same fashion as on P:

µ ≤ ν ⇔ ∃α ∈ Λ : ν = µα.

Let C(Λ) be the space of closed subsets of Λ endowed with theFell’s topology. Define the map F : Λ→ C(Λ) by

F (λ) = µ ∈ Λ : µ ≤ λ.

Then the closure Λ of F (Λ) is the set of hereditary and directedclosed subsets of Λ.

Definition

The path space is Ω = Λ \ ∅.

Introduction Inverse Semigroups vs Groupoids Semigroups P-graphs

the action of P on Ω

The action of P on Λ extends to Ω: we define

Ω ∗ P = (A, n) ∈ Ω× P : ∃λ ∈ A : n ≤ d(λ)

T : Ω ∗ P → Ω : (A, n) 7→ A.n

Proposition

Let Λ be a P-graph. Assume that P is a quasi-lattice orderedsubsemigroup of a group Q and that Λ is (r , d)-proper. Then,

1 T : Ω ∗ P → Ω is an action of P on Ω by partial localhomeomorphisms.

2 The action is directed.

Introduction Inverse Semigroups vs Groupoids Semigroups P-graphs

the Toeplitz algebra of the P-graph

Therefore, we can construct the semi-direct product Ω o P whichis an etale, locally compact and Hausdorff groupoid and we canconstruct its C*-algebra.

Proposition

The C*-algebra C ∗(Ω o P) is the Toeplitz C*-algebra TC ∗(Λ) ofthe P-graph Λ.

The groupoid techniques of R-Williams give that if P is aquasi-lattice ordered subsemigroup of an amenable group Q and ifΛ is (r , d)-proper, then the Toeplitz C*-algebra TC ∗(Λ) is nuclear.

Introduction Inverse Semigroups vs Groupoids Semigroups P-graphs

the boundary action

The definition of the boundary path space ∂Ω, adapted from[Yeend, 2007], is a bit technical: its elements are the subsetsA ∈ Ω whose elements are extendable in A, a notion which I donot want to define here. It is very likely that ∂Ω is the closure ofthe maximal hereditary directed closed subsets.

Proposition

Under the above assumptions, ∂Ω is a closed subset of Ω invariantunder the action of P and with respect to the groupoidG (Ω,P,T ).

The Cuntz P-graph C*-algebra is the reduction of C ∗(Ω o P) to∂Ω.

Introduction Inverse Semigroups vs Groupoids Semigroups P-graphs

the Cuntz algebra of the P-graph

The reduction of the groupoid G (Ω,P,T ) to the boundary pathspace ∂Ω is the semidirect product ∂Ω o P. Again, we canconstruct its C*-algebra.

Proposition

The C*-algebra C ∗(∂Ω o P) is the Cuntz C*-algebra C ∗(Λ) of theP-graph Λ.

The same groupoid techniques as above show that if P is aquasi-lattice ordered subsemigroup of an amenable group Q and ifΛ is (r , d)-proper, then the Cuntz C*-algebra C ∗(Λ) is nuclear.

Introduction Inverse Semigroups vs Groupoids Semigroups P-graphs

application to semigroup actions

We can use this construction to define the C*-algebra of asemigroup action (X ,P,T ) which is not necessarily directed butwhere P quasi-lattice ordered.

We introduce the P-graph Λ = X ∗ P and consider instead thedirected action (∂Ω,P, T ). Then we can define C ∗(X ,P,T ) asC ∗(Λ). When X is reduced to a point, we get Nica’s C ∗(P).

Proposition

If the action (X ,P,T ) is already directed, then there exists aP-equivariant homeomorphism of X onto ∂Ω which implements agroupoid isomorphism from G (X ,P,T ) onto G (∂Ω,P, T ).Therefore C ∗(X ,P,T ) and C ∗(Λ) are isomorphic.

Introduction Inverse Semigroups vs Groupoids Semigroups P-graphs

References

R-Williams, arXiv:1501.03027A. Nica, A. Paterson, R. Exel, X. Li, M. Norling, ...Khoshkam-Skandalis 02