The product systems

Let Lbe an étale groupoid. Let S⊂Bis(L) be an inverse semigroup. Letu∈S be a bisection (see Definition 3.5). Then the mapα(u) :r(u)→s(u), s(g)7→r(g) is a homeomorphism. And the map

α:S7→I(L⁰), u7→α(u), (3.46) is a semigroup homomorphism. See [16, Proposition 5.3]. This gives an action ofS onL⁰. Furthermore, the action ofS onL⁰ by partial homeomorphisms induces an action ofS on C₀(L⁰) by partial isomorphisms, that is, an action by isomorphisms between closed ideals in C₀(L⁰). For simplicity, we also denote the induced action by α. In what follows, we assume that S is wide inverse semigroup. That is, the inclusion mapS →Bis(L) is awiderepresentation, see [28, Definition 2.18], and S

u∈Su = L¹. Equivalently, the transformation groupoid SnL⁰ is isomorphic to the groupoid L. For instance, we may take S = Bis(L) (see [9, Proposition 5.1]). For a wide inverse semigroup S⊂Bis(G), the groupoid C^∗-algebra C^∗(L) is naturally isomorphic to the crossed productSnC₀(L⁰) (see [16, Proposition 9.7]).

Definition 3.32. LetSbe a unital inverse semigroup. LetDbe a C^∗-algebra and F a HilbertD-module. A representation ofS onF is a semigroup homomorphism fromSto the inverse semigroupI(F) of isomorphisms between sub-Hilbert modules of F. Let α:S →I(F) be a representation. Foru∈S, letF_u be the domain of the isomorphism α(u).

Definition 3.33. Let X be a locally compact Hausdorff space. And letα:S → I(X) be an action of S on X by partial homeomorphisms. Let E_u denote the domain of the homeomorphism α(u). We also denote by α the action of S by partial isomorphisms on C₀(X). A covariant representation of (S, X) is a pair (β,F), where F is a correspondence from C₀(X) to a C^∗-algebraDand β is a representation ofS onF such that

(1) the domain ofβ(u) isF_u= C₀(E_u^∗)F and its codomain is C₀(E_u)F;

3.3 Actions of Ore monoids by proper correspondences

(2) β(u)ϕ_C0(X)(a)β(u)^∗ =ϕ_C0(X)(α(u)(a)) for allu∈S and all a∈C₀(Eu^∗).

Covariant representations of inverse semigroup actions have been introduced in [43, Definition 4.5], see also [16, Definition 8.1].

Proposition 3.34. Let L be an étale groupoid and let S ⊂ Bis(L) be a wide inverse semigroup. LetD be a C^∗-algebra. Then there is a bijection between the set of correspondences F: C^∗(L) → D and the set of covariant representations of (S,L⁰) on Hilbert D-modules.

Proof. LetF: C^∗(L)→Dbe a correspondence. The correspondence L:L⁰ → L is proper. Hence, byProposition 2.38, the C^∗-correspondence C^∗(L) : C₀(L⁰)→ C^∗(L) is a proper.

We denote B_u := C₀(u^∗)F and

β(u) : B_u^∗ →B_u, f·ξ 7→α(u)(f)·ξ.

See(3.46)for the definition ofα. Then we have

hβ(u)(f₁·ξ1), β(u)(f2·ξ2)i=hα(u)(f₁)·ξ1, α(u)(f2)·ξ2i

=hξ₁,(α(u)(f1))^∗α(u)(f2)·ξ2i

=hξ₁, α(u^∗u)(f₁^∗f₂)·ξ₂i

=hξ₁, f₁^∗f₂·ξ₂i=hf₁·ξ₁, f₂·ξ₂i.

It follows thatβ(u) is an isometry. It is surjective since α(u) is an isomorphism.

Henceβ(u) : Bu → Bu^∗ is an isomorphism of Hilbert D-modules. We will show that (β,F) is a covariant representation. Condition(1)is satisfied by construction.

Next we check condition(2). Letu∈S and leta∈C₀(L⁰). Then for allf·ξ∈B_u^∗ we have

β(u)aβ(u)^∗(f·ξ) =α(u)(aα(u^∗)(f))·ξ =α(u)(a)f ·ξ.

Furthermore, the mapβ:S →I(F) is a semigroup homomorphism since α:S → I(C₀(L⁰)) is a semigroup homomorphism.

To prove the converse, let (β,F) be a covariant representation of the pair (S,L⁰).

By [16, Theorem 9.8], C^∗(L) ∼= C₀(L⁰)oS. We define an action of C^∗(L) on F as follows. Letf ∈S(L). Then there is a natural number n∈N such thatf = Pn

k=1fk◦s|_uk for bisections u1, . . . , un ∈ S and fk ∈C_c(u^∗_kuk) for k= 1, . . . , n.

We set

ϕ(f)(ξ) =f·ξ :=

n

X

k=1

β(u_k)ϕ_C0(L⁰₎(f_k)(ξ). (3.47)

Equation (3.47) gives a well defined map^L_u∈SC_c(u)→B(F). For the mapϕto extend to a map S(L)→B(F), it has to vanish on the kernel of the natural map E: ^L_u∈SC_c(u) → S(L). By [10, Proposition B.2], the kernel of E is given by the closed linear span of the set of elements of the form f δ_u−f δ_v forf ∈C_c(u), u, v∈S and u⊂v, where δu is the characteristic function on u. Let f δu−f δv

be such an element. Thenf =g◦s|_u withg∈C_c(u^∗u). We have β(u) =β(v)|_Fu sinceu⊂v and (β,F) is a representation. Hence

(f δu−f δv)·ξ=β(u)ϕ_C0(L⁰)(g)(ξ)−β(v)ϕ_C0(L⁰)(g)(ξ) = 0

for all ξ∈ F. It follows thatϕ(f δu−f δv) = 0. Sinceϕis continuous, it vanishes on the kernel ofE and gives a well-defined mapS(L)→B(F). This map extends to a non-degenerate ^∗-homomorphism ϕ_C^∗(L): C^∗(L) →B(F). The map ϕ_C^∗(L)

restricts to (β,F) by construction. This finishes the proof.

LetX be a locally compact Hausdorff space. LetL be a groupoid acting onX.

LetS ⊂Bis(L) be such that SnL⁰∼=L. Then the action of LonX is equivalent to an action ofS on X by partial homeomorphisms, see [9, Theorem 3.7]. The action of S is given as follows. Let e∈ S be an idempotent. Let X_e := r⁻¹(e).

Now for eachu∈S, define α_u:X_u^∗_u→X_uu^∗ by

αu(x) =g·x, (3.48)

where g∈u is the unique element withr(x) =s(g).

Lemma 3.35. Let X be a locally compact Hausdorff space. Let L be a groupoid acting on X. Let S ⊂Bis(L) be a wide inverse semigroup. Let D be a C^∗-algebra.

There is a bijection between the set of correspondences F: C^∗(LnX)→D and the set of non-degenerate covariant representations of (S, X).

Proof. The inverse semigroupS is wide in the groupoidLnX. Hence the claim follows fromProposition 3.34.

Theorem 3.36. The product systems (C^∗( ˜X_p),µ˜p,q) and (C^∗(X_p), µp,q) have the same colimits in Corr and in Corr_prop. The groupoid H is a groupoid model for the Cuntz–Pimsner algebra of the product system (C^∗(X_p), µ_p,q).

Proof. Let D be a C^∗-algebra and let (F,V˜p) be a transformation from the dia-gram (C^∗( ˜X_p),µ˜_p,q) to const_D. Here F: C^∗( ˜G)→C^∗(D) is a correspondence and V˜p: C^∗(X_p)⊗_C∗( ˜G)F → F is an isomorphism for all p∈P; the isomorphisms ˜Vp

satisfy condition (2.32).

Now the natural (left) action of G on ˜G commutes with the right action of ˜G on itself by multiplication. Hence ˜G:G →G˜is a correspondence (an actor in fact). It

3.3 Actions of Ore monoids by proper correspondences

is proper sincer∗: ˜G/G˜=H⁰ → G⁰ is proper byLemma 3.25; and hence it induces a non-degenerate^∗-homomorphismι: C^∗(G)→C^∗( ˜G).

We define

ν_p: ˜X_p→ X_p◦GG,˜ (x_p, ξ)7→[x_p, ξ].

The mapνp is injective andG-equivariant. Letx= [xp,(g, ξ)]∈ X_p◦_GG, then˜ x= [xpg, ξ] =νp(xpg, ξ). Henceνp is surjective. Let (g, η)∈G˜and (xp, ξ)∈X˜_p satisfy gη=ξ. Then

νp(xp, ξ)(g, η) = [xp, ξ](g, η) = [xpg, η] = [xpg, g⁻¹ξ].

Therefore, ν_p is ˜G-equivariant. It is also open and continuous. Hence it is an isomorphism. Thus its image under the functorGr→Corris an isomorphism

C^∗( ˜X_p)∼= C^∗(X_p)⊗_C^∗_(G)C^∗( ˜G).

As a result, we obtain isomorphisms

C^∗(X_p)⊗_C^∗_(G)F −→^∼= C^∗(X_p)⊗_C^∗_(G)C^∗( ˜G)⊗_C∗( ˜G)F −→^∼= C^∗( ˜X_p)⊗_C∗( ˜G)F −^∼=→ F.

(3.49) Let V_p: C^∗(X_p)⊗_C^∗_(G)F → F for p ∈ P be the product of the isomorphisms in(3.49). Since ˜Vpsatisfies condition(2.27), the isomorphismsVp also satisfy(2.27).

This implies that (F, V_p) is a transformation from (C^∗(X_p), µp,q) to the constant diagram const_D.

Conversely, let (F, V_p) be a transformation from (C^∗(X_p), µp,q) to the constant diagram const_D. We want to construct a transformation from (C^∗( ˜X_p),µ˜p,q) to const_D. The spaceX_pis a correspondence from the spaceX_p/GtoG, where the left anchor mapr =p_p:X_p → X_p/G is the quotient map. The image ofX_p:X_p/G → G under the functorGr→Corris a C^∗-correspondence C^∗(X_p) : C₀(X_p/G)→C^∗(G).

That is, the C^∗-algebra C₀(X/G) acts on the Hilbert C^∗(G)-module C^∗(X_p) by pointwise multiplication. The Hilbert D-modules C^∗(X_p) ⊗_C^∗_(G)F and F are isomorphic via the map Vp. Hence we may view F as a C^∗-correspondence from C₀(X_p/G) toD. The left action of C₀(X_p/G) onF is given byϕ_C0(Xp/G)(a) :=

V_paV_p⁻¹. Furthermore, for allp, q∈P, the map r_p,q:X_pq/G → X_p/G is proper by Lemma 3.25. Hence it induces a map r_p,q^∗ : C₀(X_p/G) → C₀(X_pq/G). We claim that

ϕ_C0(X_pq/G)r_p,q^∗ =ϕ_C0(X_p/G). (3.50) First, we have

Vpq=Vp(id_C^∗_(Xp)⊗_C^∗_(G)Vq)(u⁻¹_p,q⊗_C^∗_(Xp)id_F).

Hence

V_pq⁻¹ = (up,q⊗_C^∗_(G)id_F)(id_C^∗_(Xp)⊗_C^∗_(G)V_q⁻¹)V_p⁻¹. We also have

(u⁻¹_p,q⊗_C^∗_(G)id_F)(r^∗_p,q(a))(up,q⊗_C^∗_(Xp)id_F) =a⊗_C^∗_(G)id_C^∗_(Xq)⊗_C^∗_(G)id_F. So

ϕ_C0(Xpq/G)r_p,q^∗ (a) =V_pq(a⊗_C^∗_(G)id_F)V_pq⁻¹

=Vp(id_C^∗_(Xp)⊗_C^∗_(G)Vq)(a⊗_C^∗_(G)id_C^∗_(Xq)⊗_C^∗_(G)id_F)·

(id_C^∗_(Xp)⊗_C^∗_(G)V_q⁻¹)V_p⁻¹

=Vp(a⊗_C^∗_(G)id_F)V_p⁻¹=ϕ_C0(Xp/G)(a).

And the claim follows. Equation (3.50)and the universal property of the inductive limit imply that (ϕ_C0(X_p/G))_p∈P induces a map

ϕ: lim

−→C_P

(C₀(X_p/G), r^∗_p,q)→B(F).

This map is non-degenerate since ϕ_C0(Xp/G) is non-degenerate for all p ∈ P. Furthermore,

lim−→

C_P

(C₀(X_p/G), r_p,q^∗ ) = C₀(lim

←−C_P

(X_p/G, r_p,q)) = C₀(H⁰).

Summing up, we obtain a C^∗-correspondence F: C₀(H⁰) → D. And F is a correspondence C^∗(G) → D. We have to show that F extends to a correspon-dence F: C^∗( ˜G) = C^∗(GnH⁰) → D. By Proposition 3.34, the correspondence F: C^∗(G)→ D is equivalent to a non-degenerate covariant representation (α,F) of the pair (S,G⁰). We will show that the pair (α,F), where F is now a cor-respondence from C₀(H⁰) to D, is also a covariant representation. Then using Proposition 3.34again we get a correspondenceF: C^∗( ˜G)→D. First, linearity of the mapsVp for p∈P implies condition(1) in Definition 3.33. Secondly, the action ofG onX_p/G induces an action ofS onX_p/G. The resulting action induces an action ofSon C₀(X_p/G). With a slight abuse of notation, we denote this action also by α. The action is given by

α_u(f_p(x_p)) =f_p[g·x_p] (3.51) for all fp ∈ S(X_p/G), u ∈ S and xp ∈ G, where g ∈ u is the unique ele-ment with s(g) = r(x_p). This action commutes, as expected, with the maps

3.4 Examples

r^∗_p,q: C₀(X_p/G) → C₀(X_pq/G) for all p, q ∈ G and induces the action of S on C₀(H⁰). We only need to check that the condition(2)in Definition 3.33holds for the pairs (S,C₀(X_p/G)) for allp∈P. Now let p∈P. Let u∈S,fp ∈C_c(X_p/G).

Then for allξ_p⊗ξ∈C^∗(X_p)⊗_C^∗_(G)F and all f ∈S(u), we have

β(u)fpβ(u^∗)(Vp(f ξp⊗ξ)) =β(u)Vp(fpα(u^∗)(f)ξp⊗ξ). (3.52) But for allxp ∈ X_p we have

f_pα(u^∗)(f)ξ_p(x_p) =f_p[x_p]f(g⁻¹)ξ_p(g⁻¹x_p),

whereg∈uis the unique element with r(g) =r_p(x_p). It follows that f_pα(u^∗)(f)ξ_p =α(u)(f_p)α(u^∗)(f)ξ_p.

Thus

β(u)fpβ(u^∗)(Vp(f ξp⊗ξ)) =Vp(α(u)(fp)f ξp⊗ξ) =α(u)(fp)Vp(f ξp⊗ξ). (3.53) This implies that the condition (2) is satisfied. Therefore, we get a correspon-dence F: C^∗( ˜G) → D. Furthermore, equality (3.49) holds. Hence we have a transformation (F,V˜_p) from (C^∗( ˜X_p),µ˜_p,q) to const_D. Clearly, the transformation from (C^∗(X_p), µ_p,q) to const_D obtained as before from (F,V˜_p) is again (F, V_p).

Finally, let (F,V˜p) be a transformation from (C^∗( ˜X_p),µ˜p,q) to const_D. Let (β,F) be the representation of (S,G⁰) obtained as before. Let u be a bisection in X_p for p∈P and letf ∈C₀(u^∗u). Then f ◦s|_u∈C₀(u)⊆C₀(X_p/G). So

ϕ(f◦s|_u) =β(u)ϕ(f).

The functions of the form f ◦s|_u span a dense subset in C₀(X_p/G). Hence the action of C₀(X_p/G) is given by the action of C₀(G⁰) on F and the isomorphisms Vp constructed from the action as above. Thus (F,V˜p) is determined uniquely by the transformation from (C^∗(X_p), µ_p,q) to const_D obtained from it.

3.4 Examples

Im Dokument A colimit construction for groupoids (Seite 74-79)