The set of isomorphism classes of certainequivariant continuous bundles over the space of units of a groupoid Gcan be made into a group. This group is denoted by Br(G) and is called the Brauer group of G. Elements of the Brauer group of Gare equivalence classes of equivariant continuous trace C∗-algebrasC0(G(0),A) with spectrumG(0).
In [21], the authors define the Brauer group for locally compact groupoids and prove two isomorphism theorems for it. The first isomorphism, which is our point of attention, says that an equivalence X from H to G produces an isomorphism from Br(G) to Br(H). We show that a Hilsum–Skandalis morphism fromH to Ggives a homomorphism from Br(G) to Br(H).
Important convention: In this section we shall work with locally compact, Hausdorff, second countable groupoids whichneed not have Haar systems. Only the topological properties of the groupoids will be used. We shall work with Hilsum–Skandalis morphism in this section and not the topological correspondences we have beed discussing so far. Since the right anchor map in a Hilsum-Skandalis morphism is not required to be open, there usually cannot be any continuous invariant family of measures with full support. So Hilsum-Skandalis morphisms are quite different from the topological correspondences introduced here. The following results show, therefore, that Brauer groups and groupoid C*-algebras are functorial for very different kinds of morphisms of groupoids.
3.4.1 The Brauer group
Definition 3.4.1 (Hilsum–Skandalis morphism). A Hilsum–Skandalis morphismfrom a groupoid H to a groupoidG is anH-G-bispace X such that
i) the action ofG is free and proper;
ii) the left momentum map induces a bijection fromX/Gto H(0).
This is a bibundle functor in the notation of Meyer and Zhu [27]. The terminology ‘Hilsum–
Skandalis morphism’ is originally from geometry and we continue using it.
If the action of H is proper and sX is open, then an Hilsum–Skandalis morphism is a corre-spondence in the sense of Macho Stadler and O’uchi as in Example 3.1.8.
Definition 3.4.2 (Upper semicontinuous Banach bundle). An upper semicontinuous Banach bundle over a topological space X is a topological space A together with a continuous open surjection πX:A → X and complex Banach space structures on each fibre Ax := p−1(x) satisfying the following axioms:
i) ifA ∗ A:={(a, b)∈ A × A:p(a) =p(b)}, then (a, b)7→a+b is continuous from A ∗ A toA;
ii) for eachλ∈C, the map A → Asendinga7→λais continuous;
iii) if {ai} is a net in A, with p(ai)→x and ||a|| →0, thenai →0∈ Ax; iv) the map a7→ ||a|| is upper semicontinuous from A to R+.
We abbreviate the phrase “upper semicontinuous” as u. s. c.. We callA thetotal space of the u. s. c. Banach bundle, X the base space and πX the bundle projection of the bundle. In the fourth condition, if the norm function is continuous instead of being u. s. c., then the bundle is called a continuous Banach bundle. In this section, we shall be working with continuous Banach bundles only.
Definition 3.4.3. A C∗-bundle over X is a Banach bundle πX:A →X such that each fibre is a C∗-algebra satisfying, in addition to all axioms in Definition 3.4.2, the following axioms.
v) The map(a, b)7→abis continuous fromA ∗ A → A.
vi) The map a7→a∗ is continuous from A → A.
An elementary C∗-algebra is a C∗-algebra which is isomorphic to the compact operators on a Hilbert space.
Definition 3.4.4 (Elementary C∗-bundle). A C∗-bundle is called elementary if every fibre is an elementary C∗-algebra.
Definition 3.4.5(Fell’s condition). An elementary C∗-bundle over a spaceX satisfiesFell’s condition if each x∈X has a neighbourhood U such that there is a section f for which f(y) is a rank-one projection for each y∈U.
Proposition 10.5.8 of [12] says that an elementary C∗-bundle satisfies Fell’s condition if and only if its algebra of sections vanishing at infinity is a continuous trace C∗-algebra.
Definition 3.4.6 (Right action of a groupoid on a continuous Banach bundle). Let πX:E →X be a Banach bundle, letG be a groupoid acting on X on the right. AG-action on E is a G-action on X by isometric isomorphisms αx,γ:Exγ → Ex for each γ ∈G such that
i) αx,s(x)=idEx for each s(x)∈G(0);
ii) if γ and γ0 are composable then αγγ0 =αγ◦αγ0; iii) α makes E into a continuous left G-space.
There is a similar definition for a left G-spaceX. WhenG acts on a Banach bundleE, then we say that E is a G-bundle.
Definition 3.4.7 (C∗-G-bundle). A G-C∗-bundle is a pair (A, α) where πG(0): A → G(0) is a G-bundle andα is an action of Gon A by *-isomorphisms.
Definition 3.4.8. For a groupoidG, let Br(G) denote the collection of continuous C∗-G-bundles (A, α), where A is an elementary C∗-bundle with separable fibres and which satisfies Fell’s
conditions.
Let H and G be groupoids, let X be a Hilsum–Skandalis morphism from H to G and let (p:A → G(0), α) be a G-C∗-bundle. We induce an H-bundle AX as follows: Define the fibre product s∗X(A) :={(x, a)∈X× A:sX(x) =p(a)}. Define an action of G on this fibre product by
(x, a)γ = (xγ, αγ−1(a)).
Then s∗X(A) becomes a principal G-space. Let AX denote the quotient of s∗X(A) by the G-action, and denote the class of (x, a)∈s∗X(A) in the quotient by [x, a]. We show that AX is an H-C∗-bundle.
The C∗-bundle: The assignment [x, a]7→ rX(x) defines a surjection from AX to H(0). If rX is an open map, then this surjection is also open [21, the discussion after Definition 2.14]. The map a7→[x, a] defines an isomorphism from As
X(x) to AXx showing that each fibre of the surjection pX:AX →H(0) is a C∗-algebra.
H-action: For η∈H defineαXη :AXx → AXηx) by
αXη [x, a] := [ηx, a].
Proposition 3.4.9. LetH andGbe groupoids and letX be a Hilsum–Skandalis morphism fromH toG. If(A, α)∈Br(G), then(AX, αX)∈Br(H).
Proof. The proof is the same as the proof of Proposition 2.15 in [21]. To prove continuity of the addition onAX the freeness of theG-action and conditioniii)in the definition of a Hilsum–Skandalis morphism in Definition 3.4.1 are used.
The following is Definition 3.1 in [21]
Definition 3.4.10 (Morita equivalence of G-C∗-bundles). Two G-C∗-bundles (A, α) and (B, β) are Morita equivalent if there is an A-B-imprimitivity bundleπX:X →X with an action V of Gby isomorphisms such that
AhVγ(x), Vγ(y)i=αγ(Ahx , yi), hVγ(x), Vγ(y)iB=βγ(hx , yiB).
In this case, we say that(X, V) implements a Morita equivalence between(A, α)and (B, β)and write(A, α)∼(X,V)(B, β). Morita equivalence is an equivalence relation ofG-C∗-bundles [21, Lemma 3.2].
Definition 3.4.11 (The Brauer group). The set Br(G) of Morita equivalence classes of bundles in Br(G) is called the Brauer group of G.
LetA and B be elementary C∗-bundles over G(0). For every (u, v)∈G(0)×G(0), let T(u,v)0 :=
Au⊗ Bv, whereAu⊗ Bv is the minimal tensor product. Forφ∈Γ0(G(0);A)andψ∈Γ0(G(0);A)the map(u, v)7→ ||φ(u)⊗ψ(v)||is continuous. The set{φ(u)⊗ψ(v) : φ∈Γ0(G(0);A), ψ ∈Γ0(G(0);B)}
is dense in T(u,v) for each (v, u)∈G(0)×G(0). We appeal to Theorem II.13.18 [14], which ensures that there is a Banach bundle T0 over G(0) ×G(0) that has fibres T(u,v)0 so that the functions (u, v) 7→ φ(u)⊗ψ(v) generate Γ0(G(0) ×G(0);T0). We identify G(0) inside G(0)×G(0) via the diagonal embedding u7→(u, u) and denote the restriction of the bundle T0 toG(0) by T. To get a better idea about the fibres of the bundles, we denote the bundle T0 by A ⊗ B and the bundle T by A ⊗G(0) B.
The bundle A ⊗ B and its restrictionA ⊗G(0)B are both elementary C∗-bundles. If Aand B satisfy Fell’s condition, then so do A ⊗ B and A ⊗G(0) B.
Restriction of the actionα⊗β={αu⊗βv}u,v∈G(0) to G(0) gives an action ofG on A ⊗G(0)B.
The continuity of the action is shown in [21, page 18].
Definition 3.4.12(Conjugate Banach bundle). Let(p:A →G(0), α)be aG-C∗-bundle. Theconjugate G-C∗-bundle of(A, α) is given by (¯p: ¯A →G(0),α)¯ where
i) A¯=A as a topological space;
ii) id: A → A¯ is the identity map, p¯: ¯A → G(0) is defined by p(id(a)) =¯ id(p(a)), the fibre A¯id(p(a)) is identified with the conjugate of Ap(a);
iii) α¯γ(id(a)) :=id(αγ(a));
If(A, α)∈Br(G) then ( ¯A,α)¯ ∈Br(G).
LetI denote the trivial line bundle G(0)×Cwith the G-actionI given by (s(γ),z)γ = (r(γ),z).
Theorem 3.4.13. The binary operation
[A, α][B, β] = [A ⊗G(0) B, α⊗G(0)β], (3.4.14) is well defined onBr(G). Furthermore,Br(G)can be made into an abelian group where
a. the addition is defined by(3.4.14);
b. the identity element is the class[I, I];
c. the inverse of [A, α]is given by[ ¯A,α].¯
Proof. Use Proposition 3.6 and Theorem 3.7 of [21].
Theorem 3.4.15. IfX is a Hilsum–Skandalis morphism fromH to Gthen [A, α]7→ [AX, αX]is a homomorphism fromBr(G)toBr(H).
Proof. The proof of Theorem 4.1 in [21] goes through.