ON A CONTINUOUS TRACE C∗-GROUPOID ALGEBRA

(1)

C

^∗

-GROUPOID ALGEBRA

DANIEL TUDOR

This paper analyzes the conditions for aC^∗-groupoid algebra to be a continuous traceC^∗-algebra. Firstly, conditions for the special case of a transitive groupoid are given. Secondly, the case of a certain groupoid is analyzed.

AMS 2010 Subject Classification: 46L05,46L06, 47L65, 22A22.

Key words: C^∗-algebras, crossed product algebras, topological groupoids.

1. INTRODUCTION

According to Dixmier [1, Definition II.9], aC^∗-algebraA is a continuous trace C^∗-algebra ifJ(A) =A, whereJ(A) is the closure of the following ideal M. The elements xfromA⁺with the property that the mapπ →Tr(π(x)) is continuous and finite on the spectrum Abof A compose the positive part of a two-sided, self-adjoint ideal denoted by M. For a group C^∗-algebra, Raeburn and Rosenberg established in [8, Theorem 1.1 (3)] the conditions to be a continuous traceC^∗-algebra. For the particular case of locally compact groupoid, the locally compact transformation group, D. Williams proved in [12, Theo- rem 2.7] that the C^∗-algebra associated to a locally compact transformation group (G, X) has continuous trace if theC^∗-algebra associated to the stability groups has continuous trace, the stability groups vary continuously and every compact subset of X is G-wandering. In this theorem, Williams shows, in fact, the transfer of the property of continuous trace of a C^∗-algebra from a smaller C^∗-algebra, C^∗-algebra associated to the stability groups, to entire C^∗-algebra associated to the transformation group. In this paper, something similar is proved in Proposition 4 and Theorem 4. In Proposition 4 we analyze the case of a transitive groupoid and we manage to transfer the property of the continuous trace, without any other additional conditions, from C^∗-groupoid algebra to theC^∗-algebra associated to one of the stability groups (in this case all the stability groups are isomorphic between them) and conversely. For this proposition we offer two kind of proofs, one based on a classical result of a

MATH. REPORTS14(64),3 (2012), 307–315

(2)

tensor product of continuous trace C^∗-algebras and the other using the property of topological equivalence groupoids to have the associated C^∗-algebras Morita equivalent. The second proof of Proposition 4 suggested the idea of the proof of Theorem 4. In this theorem, we transfer the property of continuous trace from C^∗-algebra of the subgroupoid of stability groups to C^∗-groupoid algebra, using the same technique like in the second proof of Proposition 4.

2. PRELIMINARIES

This part of the paper contains some basic properties of the topological groupoids and the known results about them and the continuous trace C^∗- algebras which we will use later.

Definition 1. LetGbe a set and let G⁽²⁾ be a subset ofG×G. Suppose there is a map (x, y) → xy from G⁽²⁾ to G and an involution x → x⁻¹ such that the following conditions hold:

a) If (x, y) ∈G⁽²⁾, (y, z) ∈ G⁽²⁾ then (xy, z) ∈ G⁽²⁾, (x, yz) ∈ G⁽²⁾ and (xy)z=x(yz).

b) (x⁻¹)⁻¹=x,∀x∈G.

c) (x⁻¹, x) ∈ G⁽²⁾, ∀x ∈ G and if (x, y) ∈ G⁽²⁾ then x⁻¹(xy) = y and (xy)y⁻¹=x.

Then G with this structure will be called a groupoid. The set G⁽²⁾ is called the set of composable pairs andx⁻¹ is called theinverse of x.

The mapsr andsonGdefined by the formulae r(x) =xx⁻¹ ands(x) = x⁻¹x, are called the range and the source maps and the common image of r and sis called theunit space of Gand denoted by G⁽⁰⁾.

Remark 1. On a groupoidGthe following properties hold (see [4, Propo- sition 1.5, p. 2]):

1) (x, y)∈G⁽²⁾⇔s(x) =r(y);

2)r(xy) =r(x),s(xy) =s(y),∀x, y∈G⁽²⁾; 3)x∈G⁽⁰⁾⇔x=x⁻¹=x² =r(x) =s(x);

4)r(x⁻¹) =s(x),s(x⁻¹) =r(x), ∀x∈G;

5)∀(x, y)∈G⁽²⁾⇒(y⁻¹, x⁻¹)∈G⁽²⁾, (xy)⁻¹ =y⁻¹x⁻¹.

We will denote G^x = r⁻¹(x), G_x = s⁻¹(x), and if E ⊆ G⁽⁰⁾, G^E will denote r⁻¹(E) and GE will denote s⁻¹(E).

Definition 2. LetGbe a groupoid andX a set.Gacts (to the left) onX if there exists a surjection p:X →G⁽⁰⁾and a map (g, x)→gxfrom the space G∗X:={(g, x)|s(g) =p(x)} toX such that the following properties hold:

a)p(gx) =r(g), (g, x)∈G∗X;

(3)

b) if (g1, x)∈G∗X and (g2, g1)∈G⁽²⁾ then (g2g1, x),(g2, g1x)∈G∗X and g₂(g₁x) = (g₂g₁)x;

c)p(x)x=x,∀x∈X.

Note 1. If the groupoid G acts (to the left) on X, the set G∗X will become a groupoid, if we consider (G∗X)⁽²⁾:={((g₁, x1),(g2, x2))|x1=g2x2} with the operations (g1, g2x2)(g2, x2) = (g1g2, x2); (g, x)⁻¹ = (g⁻¹, gx). The unit space of G∗X will be identified with X through the map x→(r(x), x).

The right action ofGon X is defined similarly to the left action.

Definition 3. LetGbe a groupoid andXa set. Gacts(to the right) onX if there exists a surjectionσ:X →G⁽⁰⁾and a map (x, g)→xgfrom the space X∗G:={(x, g)|r(g) =σ(x)} to X such that the following properties hold:

a)σ(xg) =s(g),(x, g)∈X∗G;

b) if (x, g₁)∈X∗G and (g₁, g₂)∈G⁽²⁾ then (x, g₁g₂),(xg₁, g₂)∈X∗G and (xg1)g2 =x(g1g2);

c)xσ(x) =x,∀x∈X.

Definition 4. Let G be a groupoid and let φ:G→ G⁽⁰⁾×G⁽⁰⁾,φ(g) = (r(g), s(g)).

1) The map φdefines an equivalence relation on G⁽⁰⁾. Two elements u and v from G⁽⁰⁾ are equivalent if there exists x inG such that φ(x) = (u, v) (i.e., r(x) =u,s(x) =v).

2) For u ∈ G⁽⁰⁾, its equivalence class given by map φ from 1) is called the orbit of u. G⁽⁰⁾/Gwill denote the factor space of G⁽⁰⁾ determined by the equivalence relation of orbits.

3) A groupoidGis called principal if the mapφis one to one.

4) A groupoidGis called transitive if the map φis surjective.

Note 2. If G is a groupoid and E a non empty subset of G⁽⁰⁾, the set G/E will denoter⁻¹(E)∩s⁻¹(E).G/E will become a groupoid with the unit space the set E, if we consider (G/E)⁽²⁾ = G⁽²⁾ ∩(G/E ×G/E) and the algebrical operations the restriction of G operations. If E contains only one elementu,u∈G⁽⁰⁾,G/{u}will be a group, calledthe stability groupof uand G/{u}={g∈G|r(g) =s(g) =u}.

Definition 5. A subset H of a groupoid G is called subgroupoid of G if H containsG⁽⁰⁾,H⁽²⁾= (H×H)∩G⁽²⁾ and H is closed under Goperations.

Definition 6. If G is a groupoid, Γ = {x ∈ G | r(x) = s(x)} will be a subgroupoid ofG calledstability groups bundle orstabilizer subgroupoid.

Note 3. Evidently, Γ is the disjoint union of the stability groupsG/{u}, u∈G⁽⁰⁾.

(4)

Definition 7. LetG be a groupoid which acts on the right on the setX.

The action ofGonXis calledfree if from equationxg=xresultsg=σ(x) = r(g) (org∈G⁽⁰⁾). Similarly, the notion offree left action can be defined.

Definition 8. Let G and H be two groupoids. G and H are (algebri- cally) equivalent if there exists a space denoted by X such that the following statements hold:

1)Gacts freely to the left on X, and H acts freely to the right on X;

2) the actions ofGandH commute (i.e., (g·x)·h=g·(x·h) forgfrom G,x from X and h fromH);

3) the mapp:X →G⁽⁰⁾induces a bijection betweenG/X andH⁽⁰⁾, i.e., p(x) = p(y) if and only if there is an η ∈ H such thatxη = y, and the map σ :X →H⁽⁰⁾ induces a bijection betweenX/H and G⁽⁰⁾, i.e.,σ(x) =σ(y) if and only if there is an ω ∈Gsuch that ωx=y.

Remark 2. The notations ofpand σ from Definition 7 and 8 is in accor- dance with the notations from Definition 2 and Definition 3.

Definition 9. SupposeGis a groupoid with a topology and giveG⁽²⁾the relative product topology coming from G×G. Then Gis called a topological groupoid in case the map (x, y)→ xy from G⁽²⁾ to G, and the map x→ x⁻¹ are continuous.

Remark 3. In this paper, we assume that our groupoids are locally compact, Hausdorff and second countable.

Note 4. If G is a topological groupoid, then the range and the source maps are continuous. It follows that G⁽²⁾ is closed in G×G, the unit space G⁽⁰⁾ is closed in Gand every stability group is a closed group ofG.

Definition 10. A (left) Haar system on a topological groupoid is a family {λ^u}_u∈G(0) of non-negative measures onG such that:

(i) supp(λ^u) =G^u, u∈G⁽⁰⁾;

(ii) forf ∈Cc(G), the function u→R

fdλ^u on G⁽⁰⁾ is inCc(G⁽⁰⁾);

(iii) for x∈G,R

f(xy) dλ^s(x)(y) =R

f(y) dλ^r(x)(y).

Remark 4. Similarly, we can define a right Haar system. In this paper, unless otherwise stated or implied by context, by a Haar system we mean a left Haar system.

Proposition 1[4, Proposition 1.24, p. 10]. If a topological groupoid G has a Haar system then the range and the source maps are open.

Definition 11. If Γ is the stability groups bundle of a topological groupoid G, then the stability groups vary continuously if the map u → G/{u} is

(5)

continuous from G⁽⁰⁾ with the topology induced by Gand the set of stability groups with Fell topology (for the Fell topology, see [2]).

Proposition 2 [4, Remark 1.44, p. 15]. The stability groups bundle Γ has its own Haar system if and only if the stability groups vary continuously.

Definition 12. A groupoidGactsproperly (to the right) on a topological space X if the map (x, g)→(x, xg) is proper.

Definition 13. (see [5, Definition 5.32, p. 151]) Let G and H two topological groupoids.GandH are (topologically)equivalent if there exists a topological, local compact space denoted by X such that the following statements hold:

1) G acts freely and properly to the left on X, and H acts freely and properly to the right on X;

2) the actions ofGand H commute;

3) the map p:X → G⁽⁰⁾ induces a homeomorphism between G/X and H⁽⁰⁾, i.e.,p(x) =p(y) if and only if there is an uniqueη∈H such thatxη =y, and the mapσ:X→H⁽⁰⁾ induces a homeomorphism betweenX/H andG⁽⁰⁾, i.e., σ(x) =σ(y) if and only if there is an unique ω∈Gsuch thatωx=y;

4) the mapsp:X→G⁽⁰⁾ and σ:X→G⁽⁰⁾ are open.

Proposition 3(see [6, Examples 5.33 (3), p. 155]). If Gis a topological, transitive groupoid then for every u ∈ G⁽⁰⁾, G and G/{u} are topologically equivalent.

Note 5. In this paper, we will consider the same construction of the groupoidC^∗-algebra as in [5, Chapter 2,§3, p. 45–51]. If a topological groupoid G has the Haar system of measures {λ^u}_u∈G(0), we may define an involutive structure on the space C_c(G) of continuous, with compact support functions on G by the formulae

f∗g= Z

f(x)g(x⁻¹y)dλ^r(y)(x) = Z

f(yx)g(x⁻¹)dλ^s(y)(x), and

f^∗(x) =f(x⁻¹),

f, g ∈ C_c(G). Moreover, for f ∈ C_c(G), the quantity kfk = sup{kπ(f)k | π a representation of Cc(G)} is finite and defines a C^∗-norm on Cc(G). We will denote by C^∗(G, λ) the completion ofC_c(G) in k · k, hence C^∗(G, λ) is a C^∗-algebra called the C^∗-algebra associated to the topological groupoidG.

Theorem 1 (see [6, Corollary 5.38, p. 162]). If G and H are two topological groupoids with Haar systems λ:= {λ^u}_u∈G(0) and β :={β^u}_u∈H(0) respectively, and G and H are topologically equivalent, then C^∗(G, λ) is Morita equivalent with C^∗(H, β).

(6)

Theorem 2 [12, Theorem 2.14]. If A and B are Morita equivalent C^∗- algebras, then A is a continuous trace C^∗-algebra if and only if B is a continuous trace C^∗-algebra.

Theorem 3 [6, Theorem 3.1]. Let G be a second countable, transitive groupoid with Haar system {λ^u}_u∈G(0) and let u ∈ G⁽⁰⁾. Then, there is a positive measure µon G⁽⁰⁾ such thatC^∗(G, λ) is isomorphic to C^∗(G/{u})⊗ K(L²(G⁽⁰⁾, µ)), whereK(L²(G⁽⁰⁾, µ))denotes the algebra of compact operators on L²(G⁽⁰⁾, µ)).

3. THE MAIN RESULTS

The first case analyzed here is the C^∗-algebra associated to a locally compact, transitive, second countable, Hausdorff groupoid G, with a Haar system of measures{λ^u}_u∈G(0), whereG⁽⁰⁾ is the unit space ofG. The following proposition will show that if G is a groupoid with the mentioned above properties, theC^∗-algebra associated toGwill be with continuous trace if and only if the C^∗-algebra associated to a stability group is a continuous trace C^∗-algebra.

Proposition 4. Let G be a local compact, transitive, second countable, Hausdorff groupoid G, with a Haar system of measures {λ^u}_u∈G(0) and let u ∈G⁽⁰⁾. The C^∗-groupoid algebra C^∗(G, λ) is a continuous trace C^∗-algebra if and only if the group algebra associated to G/{u}, C^∗(G/{u}), is with continuous trace.

Proof. We propose two kind of proofs for this proposition.

1) According to Theorem 3, C^∗(G, λ) is isomorphic, because of the hypothesis of transitivity of a groupoid G, to the tensor product C^∗-algebra, C^∗(G/{u})⊗K(L²(G⁽⁰⁾, µ)). But theC^∗-algebra of the compact operators on a Hilbert space is a continuous trace C^∗-algebra and using Tomyiama’s theorem [10, Theorem 2(a)], C^∗(G, λ) is with continuous trace if and only if C^∗(G/{u}) is.

2) The second proof uses Proposition 3, and from it follows that in this case, G and G/{u} are topologically equivalent. From this, by Theorem 1, C^∗(G, λ) andC^∗(G/{u}) are Morita equivalent, therefore, according to Theo- rem 2, the property of continuous trace can be transferred from C^∗(G, λ) to C^∗(G/{u}) and conversely.

Because in the case of a transitive groupoid the property of aC^∗-groupoid algebra to be with continuous trace can be transferred from the C^∗-groupoid algebra to the group algebra associated to the only one stability group (in this case all the stability groups are isomorphic), in the general case of a certain

(7)

groupoid, the following proposition establishes the conditions to transfer the properties of continuous trace from C^∗-algebra associated to the subgroupoid of stability groups of a groupoid Gto the entireC^∗-groupoid algebra.

Theorem 4. Let G be a local compact, second countable, Hausdorff groupoid,G, with a Haar system of measures{λ^u}_u∈G(0) andΓthe subgroupoid of stability groups. C^∗(G, λ) will be a continuous trace C^∗-algebra if the following conditions are verified:

1)the stability groups vary continuously;

2) for every pair (g₁, g₂) ∈ G×G such that (g₁, g₂⁻¹) ∈ G⁽²⁾ we have g1g₂⁻¹∈Γ;

3)C^∗(Γ) is a continuous trace C^∗-algebra.

Proof. The ideea of this proof is analogous to the second version of the proof, in the case of a transitive groupoid. If G and Γ will be topologically equivalent and with the first condition that stability groups vary continuously (which means that, from Proposition 2, Γ will have his own Haar system of measures),C^∗(G, λ) will be Morita equivalent withC^∗(Γ). The third condition from hypothesis which makes C^∗(Γ) a continuous trace C^∗-algebra and the Morita equivalence between C^∗(G, λ) and C^∗(Γ) makeC^∗(G, λ) a continuous traceC^∗-algebra. It remains to show thatGand Γ are topologically equivalent.

This equivalence will be constructed via G, which means a construction of a left action of Γ onGand a right action ofGon Gsuch that these actions will generate an equivalence, by Definition 13, between Gand Γ.

For the construction of the left action of Γ onG, let the surjectionp:G→ Γ⁽⁰⁾=G⁽⁰⁾,p(g) =r(g), forg∈Gand Γ∗G={(γ, g)|p(g) =s_Γ(γ) =r(g)}, wheres_Γdenoted the restriction ofsat subgroupoidG. The space Γ∗Gis the space of the pairs (γ, g) with the property that r(γ) = s(γ) = r(g), equality between s(γ) and r(g) makes the elementsγ ∈Γ andg∈Gcomposable, and we can define the action of Γ on G as the classical left multiplication of a subgroupoid on the entire groupoid: γ·g=γg.

For the construction of the right action of G on G, the space G∗G= G⁽²⁾ = {(g₁, g₂) | s(g₁) = r(g₂)} will be considered, with σ : G → G⁽⁰⁾, σ(g) =s(g). The action ofGonGis defined, for the elements fromG∗G, by the ordinary composition fromG, i.e., for (g₁, g₂)∈G⁽²⁾we haveg₁·g₂ =g₁g₂. The following sentences establishes that the actions defined above fulfill the conditions from the definition of topological equivalence:

1) Both actions defined above are classical examples of free and proper actions (see [4, Example 1.86, p. 32]), taking into account of the fact that they are given as multiplication with elements from Gso they will be continuous.

For example, in the case of the first action, γ·g=g impliesγg=g, therefore γ ∈G⁽⁰⁾ and the action is free. The map (γ, g)→ (γg, g) is the restriction to

(8)

Γ∗Gof the map (g1, g2)→(g1g2, g2) defined onG∗Gand this map is proper because its inverse is continuous, where the inverse map is (u, v)→(uv⁻¹, v).

2) The comutativity of the actions is described by the following relation:

(γ, g₁)∈Γ∗G, (g₁, g₂)∈G∗G⇒(γ·g₁)·g₂ =γ·(g₁·g₂).

But this relation is obviously true, because it represents the property of asso- ciativity of composable elements of G.

3) i) According to 3) from Definition 13, we have to prove that the map p :G → G⁽⁰⁾ has the property p(g₁) = p(g₂) if and only if there exists an unique g ∈ G, g1 = g2 ·g. Because the map p is the range map of the groupoidGand the action is a multiplication of elements fromG, the converse implication will become trivial. If there exists an element g ∈ G such that g1 = g2g, then r(g1) = r(g2g) = r(g2). For the direct implication, we note that s(g₂g₂⁻¹) =s(g₂⁻¹) = r(g₂) = r(g₁), therefore the elements g₂g⁻¹₂ and g₁ are composable and (g1g⁻¹₁ )g1 = (g2g₂⁻¹)g1 if and only ifg1 =g2(g₂⁻¹g1). The fact that g⁻¹₂ andg1 are composable is guaranteed byr(g1) =r(g2) =s(g⁻¹₂ ).

Putting g⁻¹₂ g₁ which is, obviously, in G, in place of g, the existence part is proved. For the unicity part, we suppose there exists another h from G, such that g₁ =g₂·h, thereforeg₂h =g₂g (with the multiplication of the groupoid G). But s(g⁻¹₂ ) = r(g₂) = r(g₂g) = r(g₂h) and the element g⁻¹₂ will be composable with g₂h and g₂g, respectively. Moreover, g⁻¹₂ (g₂h) = g₂⁻¹(g₂g), and it follows that h=g, and the unicity of gis proved.

ii) Now, we have to prove that the mapσ :G→ G⁽⁰⁾ has the property σ(g1) =σ(g2) if and only if there exists an uniqueγ ∈Γ,g1 =γ·g2. Just as with the assertion 3), i), because the map σis the source map of the groupoid Gand the action is the multiplication of Gthe converse implication is trivial.

Indeed, if there exists γ ∈ Γ such that g1 = γg2, then s(g1) = s(γg2) = s(g₂). For the direct implication, we notice that the following equalities are equivalent: σ(g₁) =σ(g₂),s(g₁) =s(g₂), g⁻¹₁ g₁ =g₂⁻¹g₂,g₁ = (g₁g₂⁻¹)g₂. The idea to take the element g₁g₂⁻¹ ( these elements are composable) instead ofγ seems to be suggested here. The only problem which remains the condition that γ = g₁g₂⁻¹ has to be an element of Γ, so we need that r(γ) = s(γ).

But r(g1g₂⁻¹) = r(g1), and s(g1g⁻¹₂ ) = s(g⁻¹₂ ) = r(g2) hence, we only have to have r(g₁) =r(g₂) when s(g₁) = s(g₂). In general, this thing is not true, therefore we need the condition that g₁g⁻¹₂ ∈Γ if the elementsg₁ and g⁻¹₂ are composable. But this condition is just the hypothesis (2) from theorem. The unicity part is similar to the unicity part from i).

4) In our construction, the mapspandσhave been taken to be the range and the source maps of the groupoidG. But, because Ghas a Haar system of measures, by Proposition 1, the maps r and sare open, therefore the mapsp

(9)

andσ will be as well. With point 4 proved, it resultsGand Γ are topologically equivalent and our theorem is proved.

REFERENCES

[1] J. Dixmier,Traces sur lesC^∗-algebras. Ann. Inst. Fourier13(1963),1, 219–262.

[2] J.M.G. Fell, A Hausdorff Topology for the Closed Subsets of a Locally Compact Non- Hausdorff Space. Proc. Amer. Math. Soc.13(1962), 472–476.

[3] I. Fulman, P. Muhly and D.P. Williams, Continuous trace groupoid crossed products.

Proc. Amer. Math. Soc.132(1996),3, 3621–3641.

[4] G. Goehle, Groupoid crossed products, Ph.D. Thesis, Dartmouth College, arXiv:0905.4681, (2009).

[5] P. Muhly,Coordinates in operator algebras. American Mathematical Society, 1997.

[6] P. Muhly and J. Renault and D.P. Williams,Equivalence and isomorphism for groupoid C^∗-algebras. J. Operator Theory17(1987), 40–76.

[7] I. Raeburn and D.P. Williams,Morita Equivalence and Continuous Trace C^∗-algebras, Mathematical Surveys and Monographs60(1998).

[8] I. Raeburn and J. Rosenberg, Crossed products of continuous trace C^∗-algebras by smooth actions. Trans. Amer. Math. Soc.305(1988), 1–45.

[9] J. Renault, Representation de produits croises d’alg´ebres de groupoides. J. Operator Theory18(1987), 67–97.

[10] J. Tomyiama, Applications of Fubini type theorem to the tensor products C^∗-algebras.

Tˆohuku Math. J.19(1967),2, 213–226.

[11] D.P. Williams,Crossed products ofC^∗-algebras. Mathematical Surveys and Monographs 134(2007).

[12] D.P. Williams,Transformation groupC^∗-algebras with continuous trace. J. Func. Anal.

41(1981), 40–76.

Received 8 November 2011 Technical University of Civil Engineering Departement of Mathematics and Computer Science

122–124 Bd. Lacul Tei 020396 Bucharest, Romania

danieltudor@gmail.com