From groups to semigroups and groupoids: weakcontainment vs amenability by examples

Claire Anantharaman-DelarocheUniversite d’Orleans

NBFAS Meeting, Lancaster

April 2, 2016

(April 1–2, 2016) Lancaster 1 / 39

SUMMARY

(1) Semigroups

Cancellative semigroups - ExamplesInverse semigroups - Examples

(2) Etale groupoids

ExamplesCancellative semigroups −→ inverse semigroups −→ etale groupoids

(3) C ∗-algebras of semigroups and groupoids

(4) Weak containment and amenability

CANCELLATIVE SEMIGROUPS

A left cancellative semigroup P is a set P equipped with an associativeoperation (a, b) 7→ ab such that ca = cb ⇒ a = b.

Examples :

P = N ⊂ Z additive

P = Nx ⊂ Qx multiplicative

Pn =

n times︷ ︸︸ ︷N ∗ N ∗ · · · ∗ N ⊂ Fn

P = No Nx ⊂ QoQx

P = R o Rx ⊂ Q(R)oQ(R)x where R is an integral domain andQ(R) its field of fractions

......

INVERSE SEMIGROUPS

An inverse semigroup S is a semigroup such that for each s ∈ S thereexists a unique s∗ ∈ S such that ss∗s = s and s∗ss∗ = s∗.

The set E ={s ∈ S : s2 = s

}of idempotents is a commutative

sub-semigroup of S .

Examples :

Discrete groups = inverse semigroups with a unique idempotent

Cuntz and Cuntz-Krieger inverse semigroups

Graph inverse semigroups

Tiling inverse semigroups

Free inverse semigroups

Inverse semigroups of partial isometries in a Hilbert space

Examples :

Inverse semigroup Inv(X ) of partial bijections of a set X .

Inverse hull of a cancellative semigroup.

Let P be a left cancellative semigroup. For p ∈ P, we denote byLp∈ Inv(P) the bijection x 7→ px from P onto pP.The left inverse hull of P is the inverse sub-semigroup S(P) of Inv(P)generated by the partial bijections Lp, p ∈ P.

Remarks :• Every inverse semigroup S is isomorphic to an inverse sub-semigroup ofInv(S).

• Every inverse semigroup S is isomorphic to an inverse semigroup ofpartial isometries in a Hilbert space.

• {Inverse monoids} ∩ {Left cancellative semigroups}= {Discrete groups}.

ETALE GROUPOIDS

A groupoid G is a small category in which every morphism is invertible.

We have a set G(0) ⊂ G of objects (or units), and four maps

r : G → G(0), s : G → G(0), m : G(2) → G, i : G → G,

called source, target, multiplication and inverse where

G(2) ={

(γ, γ′) : s(γ) = r(γ′)}

(composable morphisms). We write m(γ, γ′) = γγ′ and i(γ) = γ−1.

These structure maps are required to obey obvious axioms.

A locally compact groupoid is a groupoid endowed with a locallycompact topology such that the structure maps are continuous. It is saidto be etale if s and r are local homeomorphisms. Then G(0) is a closedand open subset of G and the fibers Gx = r−1(x), Gx = s−1(x) arediscrete for x ∈ G(0).

Examples of etale groupoids :• Locally compact spaces X = G = G(0).

• Discrete groups.

• Bundle of discrete groups. They are etale groupoids such that r = s.Then r−1(x) is a group for each unit x .

• Groupoids of partial actions. A partial action of a discrete group Gon a locally compact space X is a family (θg )g∈G of homeomorphismsθg : Dg → Dg−1 between open subsets of X such that θe = IdX , θghextends θg ◦ θh. Then

G n X = {(g , x) ∈ G × X : x ∈ Dg}

is an etale groupoid, with the induced topology, and

r(g , x) = (e, gx), s(g , x) = (e, x), (g , x)(h, y) = (gh, z),(g , x)−1 = (g−1, gx).

Examples of etale groupoids :• Groupoids of inverse semigroup actions. An action of an inversesemigroup S on a locally compact space X is an homomorphism θ from Sinto the inverse semigroup Inv(X ) such that for every a ∈ S , the domainDa of θa is open and θa is an homeomorphism from Da onto Da∗ .The groupoid G of germs of θ is the quotient of {(a, x) : a ∈ S , x ∈ Da}with respect to the equivalence relation

(a, x) ∼ (b, y)⇔ x = y and ∃e ∈ E with x ∈ De , ae = be.

We have G(0) = X , and

s([a, x ]) = [e, x ] ≡ x , r([a, x ]) = θa(x), [a, x ][b, y ] = [ab, y ], ....

(where e is any idempotent such that x ∈ De). A basis of the (not alwaysHausdorff) topology of G is given by the

Θ(a,U) = {[a, x ], x ∈ U}

for a ∈ S and U open subset of Da.

Particular case of inverse semigroup actions and their groupoids :

• Case where the inverse semigroup is a group G . Then G acts byhomeomorphisms on the locally compact space X , and the correspondinggroupoid is the transformation groupoid G n X .

• Case where the inverse semigroup S acts on a singleton X . Thecorresponding groupoid G is the quotient of S with respect to theequivalence relation

a ∼σb

if there exists an idempotent e such that ae = be.Then G is a group, denoted S/σ, called the maximal grouphomomorphism image of S (every homomorphism from S in a groupfactors through S/σ).

• Canonical action of S on the dual E of its set of idempotents.

GROUPOID GS OF AN INVERSE SEMIGROUP S .

Let E be the sub-semigroup of idempotents in S , and let E ⊂ {0, 1}E bethe locally compact totally disconnected set of nonzero elements χsatisfying χ(ef ) = χ(e)χ(f ) for all e, f ∈ E .

S acts on E as follows. For a ∈ S ,

Da = {χ : χ(a∗a) = 1}, θa(χ)(e) = χ(a∗ea).

GS is the groupoid associated to this action.

In many case, GS is the groupoid of a partial action (an thereforeHausdorff). A result of Khoshkam-Skandalis shows that for many inversesemigroups, GS is even equivalent to a group action.

C ∗-ALGEBRAS OF AN INVERSE SEMIGROUP S

`1(S) is a Banach ∗-algebra with respect to the operations

(f ∗ g)(s) =∑uv=s

f (u)g(v), f ∗(s) = f (s∗)

The full C ∗-algebra C ∗(S) of S is the enveloping C ∗-algebra of `1(S). It isthe universal C ∗-algebra for the representations of S by partial isometries.

The left regular representation π2 : S → B(`2(S)) is defined by

π2(s)δt = δst if (s∗s)t = t, π2(s)δt = 0 otherwise

The reduced C ∗-algebra C ∗r (S) of S is the C ∗-algebra generated byπ2(S).π2 is faithful on `1(S), and so S is isomorphic to an inverse semigroup ofpartial isometries (Wordingham, 1982).

C ∗-ALGEBRAS OF A SEMIGROUP P embeddable in a group G

The reduced or Toeplitz C ∗-algebra C ∗r (P) is the sub-C ∗-algebra ofB(`2(P)) generated by the isometries Vp : δq 7→ δpq, q ∈ P.

The full C ∗-algebra of P may be defined as C ∗(S(P)).

The Wiener-Hopf C ∗-algebra W (P,G ) is the sub-C ∗-algebra ofB(`2(P)) generated by the operators Wg = EPλgEP , g ∈ G , where λ isthe left regular representation of G , and EP : `2(G )→ `2(P) is theorthogonal projection.

We have

C ∗(P) = C ∗(S(P))−� C ∗r (S(P))h−� C ∗r (P)

κ↪→W (P,G )

where h(π2(Lp)) = Vp for p ∈ P.

C ∗-ALGEBRAS OF A GROUPOID 1 G

• Groupoid ∗-algebra Cc(G) : it is the ∗-algebra of continuous compactlysupported functions on G with product and ∗-operation given by

(f ∗ g)(γ) =∑

γ1γ2=γ

f (γ1)g(γ2), f ∗(γ) = f (γ−1).

• Full C ∗-algebra C ∗(G) : it is the universal completion of Cc(G) withrespect to its representations.

• Reduced C ∗-algebra C ∗r (G). For x ∈ X = G(0), define therepresentation πx of Cc(G) in `2(Gx) by A

∀ξ ∈ `2(Gx), γ ∈ Gx , (πx(f )ξ)(γ) =∑

s(γ1)=s(γ)

f (γγ−11 )ξ(γ1)

C ∗r (G) is the completion of Cc(G) with respect to the norm

‖f ‖r = supx∈X‖πx(f )‖

1. assuming that G is Hausdorff

Paterson, Khoshkam-Skandalis : If GS is the groupoid associated to aninverse semigroup S we have

C ∗(S) = C ∗(GS), C ∗r (S) = C ∗r (GS).

In case where P is a sub-semigroup of a group, we have therefore

C ∗(P) = C ∗(S(P)) = C ∗(GS(P))−� C ∗r (GS(P)) = C ∗r (S(P))h−� C ∗r (P)

When S , P or an etale groupoid G is a discrete group G , the definitions ofthe full and reduced C ∗-algebras that we gave are the classical ones for G .

CANDIDATES TO DEFINE AMENABILITY

Let G be a discrete group. Among the many equivalent definitions ofamenability for G , we select the three following ones :

Existence of a left invariant mean : i.e. a state m on `∞(G ) suchthat g .m = m, where g .m(f ) = m(f (g ·)).

Weak containment property : i.e. C ∗(G ) = C ∗r (G ).

Nuclearity of C ∗r (G ).

For the semigroups we consider and for groupoids, these notions are alsowell defined and interesting.

Problem : Are they equivalent ?

An exampleFor P = Pn (the sub-semigroup of G = Fn generated by the freegenerators), one has the exact sequence

0→ K(`2(Pn))→ C ∗r (Pn)→ O(n)→ 0

Therefore C ∗r (Pn) is nuclear. Moreover Pn has the weak containmentproperty.

For n = 1 this is the famous Coburn’s result (1967) about the uniquenessof the C ∗-algebra generated by a non-unitary isometry.

For n ≥ 2 this was observed by Nica (1992) as a consequence of theuniqueness of the Cuntz algebra On. The weak containment propertymeans that C ∗r (Pn) is the unique C ∗-algebra generated by n isometriesv1, · · · , vn such that

∑nk=1 vkv

∗k � 1.

However, Pn is not left amenable.

Recall that when S is an inverse semigroup then

C ∗(S) = C ∗(GS) and C ∗r (S) = C ∗r (GS),

and that in case where P is a sub-semigroup of a group, we have

C ∗(P) = C ∗(S(P)) = C ∗(GS(P))−� C ∗r (GS(P)) = C ∗r (S(P))h−� C ∗r (P).

Since groupoids have a representation theory similar to that of groups, it iseasier to understand the notion of amenability for groupoids than forsemigroups. The strategy is then to use the above observations to getinformations for semigroups.

The groupoid approach in the study of semigroup C ∗-algebras wasinitiated by Muhly-Renault, carried on by Nica, Renault and many others,recently Xin Li, Sundar,...

The USUAL DEFINITION of AMENABILITY for GROUPOIDS

There are many equivalent definitions, as in the group case. We onlymention one of them.

An etale groupoid G is amenable if there exists a net (mi ), where eachmi is a family (mx

i )x∈G(0) of probability measures mxi on Gx = r−1(x), such

that

(i) for every i and every continuous function with compact support on Gthe function x 7→ mx

i (f ) is continuous ;

(ii) limi

∥∥∥γ ·ms(γ)i −m

r(γ)i

∥∥∥1

= 0 uniformly on compact subsets of G.

When G is a group G , we recover a familiar definition of amenability :there exists a net (fi ) of non negative elements of `1(G ) with ‖fi‖1 = 1such that limi ‖γ · fi − fi‖1 = 0 pointwise.

IS AMENABILITY EQUIVALENT TO WEAK CONTAINMENT forGROUPOIDS ?

Let G be an etale groupoid and let us consider the following conditions :

(1) G is amenable

(2) C ∗r (G) is nuclear

(3) C ∗(G) = C ∗r (G) (weak containment property)

Then we have (1)⇔ (2)⇒ (3)

J. Renault and I, we left unsolved the problem of whether theequality C ∗(G) = C ∗r (G) implies the amenability of G.

The answer was proved to be positive

when G is transitive (i.e. for every x , y ∈ G(0) there exists γ ∈ G withs(γ) = y and r(γ) = x) (Buneci 2001) ;

when G is the transformation groupoid G n X for an exact group Gacting by homeomorphisms on a compact space X (Matsumura2014).

The case of bundles of groups

Let G be an etale groupoid which is a bundle of groups over X = G(0).

This means that r−1(x) = s−1(x) is a group that we denote by G(x). Weset Ux = X \ {x} and denote by G(Ux) the subgroupoid of those γ ∈ Gsuch that r(γ) ∈ Ux .

Let qx : C ∗(G)→ C ∗(G(x)) and πx : C ∗r (G)→ C ∗r (G(x)) be thesurjections extending the restriction map Cc(G)→ Cc(G(x)). Then

C ∗(G) is an upper semicontinuous (u.s.c.) field of C ∗-algebras withfibres C ∗(G(x)) (i.e. x 7→ ‖qx(a)‖C∗(G(x)) is u.s.c for a ∈ C ∗(G)).

C ∗r (G) is a lower semicontinuous (l.s.c.) field of C ∗-algebras withfibres C ∗r (G(x)) (i.e. x 7→ ‖πx(a)‖C∗r (G(x)) is l.s.c for a ∈ C ∗r (G)).

The following diagram is commutative

0 // C ∗(G(Ux))

��

// C ∗(G)

p

��

qx // C ∗(G(x)) //

px��

0

0 // C ∗r (G(Ux)) // C ∗r (G)πx // C ∗r (G(x)) // 0

where the first line is exact.

Assume that p is injective. By diagram chasing, we see that px isinjective if and only if the bottom line is exact. But px injective isequivalent to the amenability of the group G(x).

Since the groupoid G is amenable if and only if every fibre G(x) isamenable we see that

weak containment property ⇒ amenability if and only if for every x ∈ Xthe above bottom line is exact.

Problem : Is it always exact ?

Example : Higson-Lafforgue-Skandalis (HLS) groupoidsLet Γ be a finitely generated residually finite group and letΓ ⊂ N0 ⊃ N1 · · · ⊃ Nk ⊃ · · · be a decreasing sequence of finite indexnormal subgroups with ∩kNk = {e}. We set Γk = Γ/Nk and Γ∞ = Γ anddenote by ρk : Γ→ Γk the quotient map. Let N = N ∪ {∞} be theAlexandroff compactification of N. Let G be the quotient of N× Γ by theequivalence relation

(k , s) ∼ (l , t) if k = l and ρk(s) = ρk(t).

Equipped with the quotient topology, G is an etale Hausdorff groupoid, abundle of groups, whose fibre at k is G(k) = Γk .

A B

For f ∈ Cc(G), recall that πk(f ) acts on `2(Γk) and we have, for x ∈ Γk ,(πk(f )ξ

)(x) =

∑y∈Γk

f (xy−1)ξ(y) =((f |Γk

) ∗ ξ)(x).

πk(C ∗r (G)) = C ∗r (Γk) = λk(C ∗r (Γ)).

Higson-Lafforgue-Skandalis groupoids

C ∗r (G) is a lower semicontinuous field of C ∗-algebras over N with fibreC ∗r (Γk) at k ∈ N.

Let π = ⊕k∈Nλk . We set C ∗π(Γ) = π(C ∗(Γ)). There is an isometricembedding θ : C ∗π(Γ) ↪→ C ∗r (G) such that for f ∈ CΓ,

θ(f ) =∑t∈Γ

f (t)1St ∈ Cc(G)

where St = {(k , ρk(t)) : k ∈ N}.

We have πk ◦ θ = λk , where λk is the quasi-regular representation of Γ in`2(Γk). Since k 7→ ‖πk(a)‖C∗r (Γk ) is l.s.c., we have for a ∈ C ∗r (G),

‖λ∞(f )‖ = ‖π∞ ◦ θ(f )‖ ≤ supk∈N‖πk ◦ θ(f )‖ = ‖θ(f )‖C∗r (G) = ‖π(f )‖.

In particular, the regular representation, λ = λ∞ of Γ factors through π.

Higson-Lafforgue-Skandalis have proved that whenever the trivialrepresentation of Γ is isolated in the support of π (e.g. if Γ has Kazhdanproperty T) then

0 −→ C ∗r (G(N)) −→ C ∗r (G)π∞−→ C ∗r (G(∞)) = C ∗r (Γ) −→ 0

is not exact in the middle.

Indeed, there exists a projection e ∈ C ∗π(Γ) such that whenever ω is arepresentation of Γ that factors through π, then ω(e) is the projection onthe subspace of Γ-invariant vectors. Therefore π∞(θ(e)) = λ(e) = 0 andπk(θ(e)) = λk(e) 6= 0 if k ∈ N.

Thus θ(e) is in the kernel of π∞ but not in the image ofC ∗r (G(N)) =

⊕k∈N πk(C ∗r (G)) =

⊕k∈N C

∗r (Γk) =

⊕k∈N λk

(C ∗r (Γ)

).

In this example, G has not the weak containment property.

The restriction map f ∈ Cc(G) 7→ f|G(∞)from Cc(G) onto C[Γ] induces an

isomorphism from C ∗r (G)/C ∗r (G(N)) onto C ∗π(Γ) := π(C ∗(Γ)).

So we have the following commutative diagram

0 // C ∗(G(N)) // C ∗(G)

p

��

// C ∗(Γ) //

pπ��

0

0 // C ∗r (G(N)) // C ∗r (G) // C ∗π(Γ)

��

// 0

0 // C ∗r (G(N)) // C ∗r (G) // C ∗r (Γ) // 0

where the two first lines are exact.In particular we see that p is injective (i.e. G has the weak containmentproperty) iff pπ is injective, that is

∀a ∈ C ∗(Γ), ‖a‖C∗(Γ) = supk‖λk(a)‖.

G has the weak containment property iff

∀a ∈ C ∗(Γ), ‖a‖C∗(Γ) = supk‖λk(a)‖ (1)

G is amenable iff the group Γ is amenable

Willett has recently (2015) provided an example of a non amenablegroup Γ (namely Γ = F2) and a sequence (Nk)k≥0 of subgroups forwhich (1) holds.

This sequence is such that for every finite index normal subgroup N of F2

there exists k such that Nk < N.

The conclusion follows from a result of Lubotsky-Shalom (2004) statingthat

‖a‖C∗(F2) = sup ‖ρ(a)‖

where ρ ranges over the irreducible representations that factors through afinite quotient of Γ. But if ρ factors through Γ/N we have‖ρ(a)‖ ≤ ‖λk(a)‖.

WEAK CONTAINMENT ⇒ AMENABILITY : WHEN ?

Let us have a look to the case of a group G . We have C ∗(G ) = C ∗r (G )• if and only if the trivial representation ι of G is weakly contained in theregular representation λ,

• if and only if there is a net ϕi of coefficients of λ converging to 1uniformly on compact subsets of G . Each ϕi is of the formϕi (t) = 〈ξi , λ(t)ξi 〉, where ξi ∈ L2(G ) may be taken of norm 1. Then (ξi )is a net of almost invariant vectors in L2(G ). Then if mi is the probabilitymeasure of density |ξi |2 with respect to the Haar measure, we havelimi ‖t ·mi −mi‖1 = 0 uniformly on compact subsets of G .

It is important in this context that representations of C ∗(G ) correspondcanonically to unitary representations of G .

For groupoids we have the same kind of correspondence.

Unitary representation of a groupoid 2

Let G be a groupoid and X = G(0). We say that a measure µ on X isquasi-invariant if the measure ν defined by∫

Gf dν =

∫X

(∑γ∈Gx

f (γ))dµ(x)

is equivalent to its image ν−1 by the inverse map γ 7→ γ−1.

A unitary representation of G consists of (H = (Hx)x∈X ,U, µ) where• µ is a quasi-invariant measure on X ;• H is a Borel field of Hilbert spaces on X ;• U is a measurable family of isometries U(γ) : Hs(γ) → Hr(γ) withU(γ1γ2) = U(γ1)U(γ2), U(x) = IdHx if x ∈ X .

It can be integrated to give a representation πU of C ∗(G) in the Hilbertspace

∫ ⊕Hx dµ(x). A very useful result of J. Renault states that everyrepresentation of C ∗(G) is of this form.

2. For simplicity we assume that the groupoid is etale.

MEASURED GROUPOIDS

When G is equipped with a quasi-invariant measure µ we say that (G, µ) isa measured groupoid.

Its left regular representation is ((`2(Gx)x∈X , L, µ) where

L(γ) : `2(Gs(γ))→ `2(Gr(γ))

is the left translation by γ. Its integrated form πL acts on L2(G, ν) and weset C ∗r (G, µ) = πL(C ∗(G)).

We denote by C ∗(G, µ) the C ∗-algebra obtained from Cc(G) by separationand completion with respect to the norm supπ ‖π(f )‖ where π ranges overall representations of Cc(G) that disintegrate over µ.

A measured groupoid (G, µ) is amenable if there exists a net (mi ),where each mi is a family (mx

i )x∈G(0) of probability measures mxi on

Gx = r−1(x), such that

(i) for every i and every bounded Borel function f on G the functionx 7→ mx

i (f ) is µ-mesurable ;

(ii) limi

∥∥∥γ ·ms(γ)i −m

r(γ)i

∥∥∥1

= 0 in the weak*-topology of L∞(G, ν).

(G, µ) is amenable iff C ∗(G, µ) = C ∗r (G, µ) (J. Renault, 1997).

G is amenable iff (G, µ) is amenable for every quasi-invariant measureµ (C. AD, J. Renault 2000)

A subset F of X = G(0) is said to be invariant if whenever r(γ) ∈ F iffs(γ) ∈ F . The support of a quasi-invariant measure µ is an invariantclosed subset of X .

We say that a groupoid G is inner exact if for every closed invariantsubset F of X = G(0), the following sequence is exact :

0 // C ∗r (G(U)) // C ∗r (G) // C ∗r (G(F )) // 0

where U = X \ F , G(U) = {γ ∈ G : r(γ) ∈ U}, and similarly for G(F ).

Examples of inner exact groupoids :

Minimal groupoids, i.e such that the only invariant closed subsets are∅ and X . In particular, groups are inner exact.

Transformation groupoids G n X where G is an exact group.

Theorem : Let G be an inner exact etale groupoid. The followingconditions are equivalent :

(i) G is amenable ;

(ii) G has the weak containment property ;

(iii) C ∗r (G) is nuclear.

Proof of (ii) ⇒ (i) : Let µ be a quasi-invariant measure, F its supportand set U = X \ F . The following diagram is commutative

0 // C ∗(G(U))

��

// C ∗(G)

p

��

// C ∗(G(F )) //

pF��

0

0 // C ∗r (G(U)) // C ∗r (G) // C ∗r (G(F )) // 0

where the two lines are exact and q is injective. Then an elementarydiagram chasing shows that pF is injective.

The following diagram is commutative and all the maps are surjective :

C ∗(G(F )) // C ∗(G, µ)

��C ∗r (G(F )) C ∗r (G, µ)oo

It follows that C ∗(G, µ) = C ∗r (G, µ) for every quasi-invariant measure µand therefore G is amenable.

Remark : It suffices that G is equivalent to an inner exact groupoid forthe weak containement property to be equivalent to amenability.

BACK TO SEMIGROUPSLet S be an inverse semigroup and consider the following conditions :

(1) Existence of a left invariant mean on `∞(S).(2) Weak containment property .(3) Nuclearity of C ∗r (S).

We have (3) ⇒ (2) but no other implication holds.

Example : Let Γ, (Nk), (ρk) as in the definition of the HLS-groupoid. B

Let S ={ρk(t) : k ∈ N, t ∈ Γ

}. It is an inverse semigroup with respect to

the following operations :

(ρk(t)∗ = ρk(t)−1, ρm(t)ρn(u) = ρn∧m(tu).

Then• GS is the HLS-groupoid.• S is left amenable (since the maximal group homomorphism image isΓ/N0 is an amenable group.• If Γ has the Kazhdan property (T), (2) does not hold. So (1) 6⇒ (2)• In Willett’s example, (2) 6⇒ (3).

If the inverse semigroup S is more similar to a group, in the followingsense : 3

There exists a map ψ : S× to some group G such that

(a) ψ(ab) = ψ(a)ψ(b) if ab 6= 0 ;

(b) ψ−1(e) is the set of non-zero idempotents ;

c) ψ−1(g), if non-empty, has a greatest element for the partial ordera ≤ b if a = ba∗a.

C

Proposition : If these three property holds and if G is exact, the weakcontainment property of S implies that C ∗r (S) is nuclear.

The reason is that in this case the groupoid G is equivalent to atransformation groupoid for an action of G .Many inverse semigroups have these properties (a), (b), (c).

3. We set S× = S \ {0} if S has a zero, and S× = S otherwise.

An example : quasi-lattice ordered groupsLet P be a sub-semigroup of a group G such that P ∩ P−1 = {e}. Wedefine a partial order on G by setting x ≤ y if x−1y ∈ P. We say that(P,G ) is a quasi-lattice ordered groups (q.l.o.g.) if every pair ofelements in G having a common upper bound has a least upper bound.We say that (P,G ) is a lattice ordered group if every pair of elements inG has a least upper bound.

Examples :

(P = N,G = Z) is lattice ordered group ;

(P = Pn,G = Fn)

(P : No N×,G = QoQ×).

q.l.o.g. is stable under products and free products.

Facts : (due to Muhly-Renault 1992 ; Nica 1992,1994 ;Khoshkam-Skandalis 2002 ;Xin Li 2012,2013 ; Norling 2014,2015 ;...)

C ∗(P) is the universal C ∗-algebra generated by isometries vp, p ∈ P suchthat vpvq = vpq, ve = 1,

(vpv∗p )(vqv

∗q = (vp∨qv

∗p∨q) or 0

depending on the existence of the least upper bound p ∨ q.Recall the S(P) is the inverse semigroup generated by the translations byp on P. Here we have C ∗(P) = C ∗(S(P))−� C ∗r (S(P)) = C ∗r (P).ψ = S(P)× → G is well defined by

ψ(L∗p1Lq1 · · · L∗pnLqn) = p−1

1 q1 · · · p−1n qn,

and ψ satisfies conditions (a), (b), (c) C

For instance, g is in the image of ψ iff g ∈ PP−1 and the greatest elementof ψ−1(g) is βg : p ∈ g−1P ∩ P → gP ∩ P.

Proposition : Let (P,G ) be a q.l.o.g. with G exact. Then (P,G ) has theweak containment property if and only if C ∗r (P) is nuclear.

If G is amenable, then C ∗r (P) is nuclear. The converse is not true.

Another example where weak containment implies amenability :

Let Γ be a discrete group. Then C ∗(∂Γo Γ) = C ∗r (∂Γo Γ) iff the groupoid∂Γo Γ is amenable.

Indeed, if C ∗(∂Γo Γ) = C ∗r (∂Γo Γ), using the commutativity of thediagram

0 // C ∗(Γo Γ)

��

// C ∗(βΓo Γ)

��

// C ∗(∂Γo Γ) 0

0 // C ∗r (Γo Γ) // C ∗r (βΓo Γ) // C ∗r (∂Γo Γ) // 0

we see that the bottom line is exact.

Then a result of Roe-Willett (2013) states that this property implies thatthe metric space Γ has Yu’s property A.