The local bisection hypothesis for twisted groupoid C*-algebras

In this note, we present criteria that are equivalent to a locally compact Hausdorff groupoid $G$ being effective. One of these conditions is that $G$ satisfies the"C*-algebraic local bisection hypothesis"; that is, that every normaliser in the reduced twisted groupoid C*-algebra is supported on an open bisection. The semigroup of normalisers plays a fundamental role in our proof, as does the semigroup of normalisers in cyclic group C*-algebras.


Introduction
The connection between twisted groupoid C*-algebras and diagonal and Cartan pairs has become a cornerstone of C*-algebraic theory.Given a C*-algebra A and a commutative subalgebra B, Kumjian and Renault show in [9] and [15], respectively, that: • (A, B) is a diagonal pair if and only if there is a twisted groupoid Σ → G with G principal such that A ∼ = C * r (G; Σ) and B ∼ = C 0 (G (0) ); and • (A, B) is a Cartan pair if and only if there is a twisted groupoid Σ → G with G effective such that A ∼ = C * r (G; Σ) and B ∼ = C 0 (G (0) ).Twisted Steinberg algebras were introduced in [3] as a purely algebraic analogue to twisted groupoid C*-algebras, and algebraic versions of Kumjian and Renault's findings were established in [2].The authors (which include most of the authors of the current note) define algebraic diagonal and algebraic Cartan pairs (A 0 , B 0 ) in [2], and they prove that A 0 is isomorphic to the twisted Steinberg algebra A(G; Σ) of a "discrete" twist Σ over an effective ample Hausdorff groupoid G (which is principal if B 0 is an algebraic diagonal subalgebra of A 0 ).An algebraic Cartan subalgebra B 0 of A 0 is a maximal commutative subalgebra (just like a C*-algebraic Cartan subalgebra).
The authors of [2] also define algebraic quasi-Cartan pairs, in which the maximality requirement is relaxed to requiring that the conditional expectation E : A 0 → B 0 be implemented by idempotents, in the sense that for each algebraic normaliser n ∈ A 0 , there exists an idempotent p ∈ B 0 such that E(n) = pn = np.This corresponds to a relaxation of the effectiveness assumption as follows: a discrete twist (Σ, i, q) over an ample Hausdorff groupoid G satisfies the local bisection hypothesis [2, Definition 4.1] if, for every normaliser n ∈ A(G; Σ) of A(G (0) ), q(supp • (n)) is an open bisection of G. Several examples are provided to demonstrate that there are indeed discrete twists over ample Hausdorff groupoids that satisfy the local bisection hypothesis but for which the groupoid is not effective (see [2,Section 9]).However, every discrete twist over an effective ample groupoid does satisfy the local bisection hypothesis (see [2,Lemma 4.7(b)]).The authors then show that: • (A 0 , B 0 ) is an algebraic quasi-Cartan pair if and only if there is a discrete twisted groupoid Σ → G that satisfies the local bisection hypothesis such that A 0 ∼ = A(G; Σ), and B 0 ∼ = A(G (0) ).Thus, a natural question arises: what does the C*-algebraic local bisection hypothesis tell us about a twisted groupoid?Is this a more general property than effectiveness?It turns out that the answer is no.We show that, unlike in the algebraic setting, if we insist that each normaliser in the reduced twisted groupoid C*-algebra C * r (G; Σ) is supported on the preimage under the quotient map of an open bisection of G (when viewed as a function on Σ via the map j; see, for example, [5, Proposition 2.8]), then G is effective.That is, we prove the converse to Renault's [15,Proposition 4.8(ii)].With this characterisation, we see that being quasi-Cartan but not Cartan is a purely algebraic phenomenon.
Outline.We begin by discussing the theory of topological groupoids, twisted groupoid C*-algebras, and Cartan pairs.We then state our main result in Theorem 3.1, which is a list of conditions that are equivalent to a groupoid being effective.We also present the special case of the groupoid being ample, in which effectiveness is equivalent to the conditional expectation onto the Cartan subalgebra being "implemented by projections" (see Corollary 3.2).(This is similar to the condition (AQP) in [2] of the algebraic conditional expectation being implemented by idempotents.)We then show in Section 4 that the discrete group of integers, which is an ample groupoid, satisfies the algebraic local bisection hypothesis but does not satisfy the C*-algebraic version.We use this, along with an analysis of normalisers in finite cyclic group C*-algebras, to complete the proof of Theorem 3.1 in Section 5 (see Proposition 5.1).

Preliminaries
In this section, we recall some terminology relating to groupoids, twists, twisted groupoid C*-algebras, and Cartan pairs.For groupoids, twists, and twisted groupoid C*algebras, we primarily follow the conventions and notation of [16].

Groupoid terminology.
A groupoid G is a small category in which every morphism γ ∈ G has a unique inverse γ −1 ∈ G.We write G (0) for the set of units (or objects) in G, and we write G (2) for the collection of composable pairs in G × G.We read composition of elements from right to left, so the range and source maps r, s : G → G (0) are given by r(γ) = γγ −1 and s(γ) = γ −1 γ, respectively.For each x, y ∈ G (0) , we define and G x y := G x ∩ G y .We call a groupoid G a topological groupoid if it is endowed with a topology with respect to which composition and inversion are continuous, and the unit space G (0) is Hausdorff.In what follows, we will restrict our attention to locally compact Hausdorff groupoids; however, we do not require our groupoids to be second-countable.We say that a topological groupoid G is étale if the range map r : G → G (0) is a local homeomorphism (or, equivalently, if the source map s : Note that if A and B are open bisections of G, then so are A −1 and AB, and in this case, AA −1 = r(A) and A −1 A = s(A).
The isotropy of a groupoid G is the subgroupoid Iso(G) := We denote the topological interior of the isotropy of G by Iso(G) • , and we observe that if G is a locally compact Hausdorff étale groupoid, then so is Iso(G) where G (0) × T is a trivial group bundle with fibres T, and Σ is a Hausdorff groupoid with unit space Σ (0) = i(G (0) × {1}), such that • i and q are continuous groupoid homomorphisms that restrict to homeomorphisms of unit spaces; • the sequence is exact, in the sense that i is injective, q is surjective, and in the sense that i(r(σ), z) σ = σ i(s(σ), z) for all σ ∈ Σ and z ∈ T; and • Σ is a locally trivial T-bundle over G, in the sense that for every α ∈ G, there is an open bisection U α of G such that α ∈ U α , and there is a continuous map There are variations on how twisted groupoids are defined in the literature, but these definitions are generally equivalent; see, for example, [1,Remark 2.6].Twists over principal groupoids are of particular importance because they characterise C*-diagonals; see Kumjian's definition [9] and subsequent papers [11,8,7].Twists over effective groupoids are also of significant interest because they characterise Cartan pairs; see Renault's work [15, Theorems 5.2 and 5.9].

2.3.
Twisted groupoid C*-algebras.Given a locally compact Hausdorff space X and a continuous function f ∈ C(X), we define supp • (f ) := f −1 (C\{0}) and supp(f ) := supp • (f ).We say that f ∈ C(X) is compactly supported if supp(f ) is compact, and we write C c (X) for the collection of compactly supported functions in C(X).We say that f ∈ C(X) vanishes at infinity if, for every > 0, the set {x ∈ X : |f (x)| ≥ } is compact.We write C 0 (X) for the collection of functions in C(X) that vanish at infinity, and we note that C 0 (X) is the C*-completion of C c (X) with respect to the uniform norm • ∞ .
Let G be a locally compact Hausdorff étale groupoid.Then C c (G) is a *-algebra under the operations [17,Section 4] and [6, Section 3]) is dense in both C c (G) and C * r (G) with respect to the reduced norm.For f ∈ C c (G) with supp(f ) ⊆ G (0) , we have f r = f ∞ , and so we identify the completion of C c (G (0) ) ⊆ C c (G) with respect to the reduced norm with C 0 (G (0) ).
The construction of twisted groupoid C*-algebras is similar.Let Σ be a twist over ) is a twist over G (0) , and the completion of C c (G (0) ; q −1 (G (0) )) ⊆ C c (G; Σ) with respect to the reduced norm is isomorphic to C 0 (G (0) ).We write C 0 (G (0) ) ⊆ C * r (G; Σ) with this identification in mind.Note that in some literature (such as [5]) the reduced C*-algebra of a twisted groupoid Σ → G is denoted by C * r (Σ; G) and is defined using continuous sections of a complex line bundle instead (see [15]).
Given a twisted groupoid Σ → G, we write C 0 (G; Σ) for the collection of T-contravariant functions in C 0 (Σ).As used in [15] and as detailed in [5], we can view elements of C * r (G; Σ) as T-contravariant functions in C 0 (G; Σ) via a norm-decreasing injective linear map j : C * r (G; Σ) → C 0 (G; Σ) that preserves convolution and involution and satisfies j(f ) = f for all f ∈ C c (G; Σ).Given a ∈ C * r (G; Σ), we define S a := q(supp • (j(a))).This is consistent with the definition of supp (a) in [15, Section 4]; see also [ Most definitions of a Cartan pair that appear in the literature include the additional assumption that B contains an approximate identity for A. However, Pitts shows in [12,Theorem 2.6] that this condition follows automatically from the other three.It follows that for all n ∈ N (B), we have nn * , n * n ∈ B.
C*-algebras containing Cartan subalgebras are precisely the twisted C*-algebras of effective Hausdorff étale groupoids (see Theorem 2.2 below).This was proven in the separable setting by Renault [15] and was later extended to the non-separable setting by Raad [13].
Theorem 2.2 ([15, Theorems 5.2 and 5.9] and [13, Theorem 1.2]).Let Σ be a twist over an effective locally compact Hausdorff étale groupoid G. Then the map that restricts functions in C c (G; Σ) from Σ to q −1 (G (0) ) extends to a faithful conditional expectation ) is a Cartan pair, then there exists a twist Σ over an effective locally compact Hausdorff étale groupoid G such that there is an isomorphism from A to C * r (G; Σ) that maps B onto C 0 (G (0) ).

Effective groupoids
In [15, Proposition 4.8(i)] Renault shows that if (Σ, i, q) is a twist over a secondcountable locally compact Hausdorff étale groupoid G, then for any element a ∈ C * r (G; Σ) such that S a = q(supp • (j(a))) is an open bisection of G, we have that a is a normaliser of C 0 (G (0) ), regardless of whether C 0 (G (0) ) is a Cartan subalgebra of C * r (G; Σ).Raad [13] also makes this observation in the non-second-countable setting.
In this paper, we say that a twist Σ over a Hausdorff étale groupoid G satisfies the C*-algebraic local bisection hypothesis if, for every normaliser n ∈ C * r (G; Σ) of C 0 (G (0) ), S n is an open bisection of G. Renault shows in [15, Proposition 4.8(ii)] that the C*-algebraic local bisection hypothesis holds for twists over effective groupoids.The purpose of our paper is to show that the C*-algebraic local bisection hypothesis is in fact equivalent to the groupoid being effective.
The remainder of the paper is devoted to establishing the following theorem.
Theorem 3.1.Let G be a locally compact Hausdorff étale groupoid.The following are equivalent.
We conclude this section by showing that if the groupoid G is ample, then for any twist Σ over G, the conditional expectation E : ) is implemented by projections, in the sense that the functions f k ∈ C 0 (G (0) ) appearing in condition (4) of Theorem 3.1 can be chosen to be projections.Corollary 3.2.Let G be an ample locally compact Hausdorff groupoid.Then G is effective if and only if for any twist (Σ, i, q) over G and any normaliser n ∈ C * r (G; Σ) of C 0 (G (0) ), there exists a sequence of projections Proof.Suppose that G is effective.Let (Σ, i, q) be a twist over G, and fix a normaliser n ∈ C * r (G; Σ) of C 0 (G (0) ).By the implication (1) =⇒ (3) of Theorem 3.1, we know that S n = q(supp • (j(n))) is an open bisection of G. Following the proof of the implication (3) =⇒ (4) in Theorem 3.1, for each k ∈ N\{0}, let and since W k is compact and G is ample, we can find a finite cover of W k consisting of compact open subsets ) such that supp(p k ) ⊆ S E(n) and p k (x) = 1 for all x ∈ W k .Thus the remainder of the proof follows exactly as in the proof of the implication (3) =⇒ (4) in Theorem 3.1.Finally, since ample groupoids are étale, the converse follows trivially from the implication (4) =⇒ (1) of Theorem 3.1, which itself follows from Proposition 5.1.

The algebraic vs the C*-algebraic local bisection hypothesis
In this section we present an example that will be used in the proof of Proposition 5.1 to show that if G is not effective, then there is a normaliser n ∈ C * r (G) of C 0 (G (0) ) such that j(n) is not supported on a bisection.Our example also demonstrates how the algebraic and analytic situations differ.We think of condition (5) in Theorem 3.1 as the "untwisted" C*-algebraic local bisection hypothesis for G.Even if G is ample, this is not the same as the "local bisection hypothesis" introduced by Steinberg in [18, Definition 4.9]: if R is a commutative unital ring (for example, the complex numbers endowed with the discrete topology) and A R (G) is the Steinberg R-algebra associated to G, then we say that G satisfies the (algebraic) local bisection hypothesis if every normaliser n ∈ A R (G) of A R (G (0) ) is supported on an open bisection of G.
Example 4.1.The integers satisfy the algebraic local bisection hypothesis but not the C*-algebraic local bisection hypothesis; that is, the integers do not satisfy the condition described in Theorem 3.1 (5).
Then m is continuous and circle-valued since the zeros of z − 2z 2 are z = 0 and z is not a bisection of Z, and hence the C*-algebraic local bisection hypothesis does not hold.

The C*-algebraic local bisection hypothesis implies effectiveness
In this section, we complete the proof of Theorem 3.1 by establishing that (5) implies (1), restated in the following proposition.In our proof of Proposition 5.1, we consider the possible orders of elements in Iso(G) • , the interior of the isotropy of the groupoid G.Most of the work comes in dealing with the existence of torsion.If every element of Iso(G) • has infinite order, an argument like the one in Example 4.1 makes the proof fairly straight forward.We begin with a technical lemma about normalisers in the group algebra of a group with prime order.When k = 0, we have We claim that when k = 0 is fixed, the terms in the sum are a permutation of the pth roots of unity.To see this, note that if k ∈ {1, . . ., p − 1} and k(2 1 − k) ≡ k(2 2 − k) (mod p) for some 1 , 2 ∈ {0, . . ., p − 1}, then we must have

5 , 4 .
Remark 2.4].Note that S a is open in G because j(a) is continuous and q is an open map (see[1, Lemma 2.7(a)]).Similarly, in the untwisted setting, we can view elements of C * r (G) as functions in C 0 (G) via the norm-decreasing injective linear map j : C * r (G) → C 0 (G) extending the identity map on C c (G) (see [14, Proposition II.4.2] for details).In this setting, given a ∈ C * r (G), we define S a := supp • (j(a)), which is again open in G. 2.Cartan pairs.Let A be a C*-algebra and let B be an abelian subalgebra of A. We call an element n ∈ A a normaliser of B if nBn * ∪ n * Bn ⊆ B. We write N (B) for the set of normalisers of B in A. Note that n ∈ A is a normaliser of B if and only if n * is a normaliser of B. There are various equivalent definitions of a Cartan pair in the literature; we use the one from [5, Definition 1.1].Definition 2.1.Let A be a C*-algebra and let B be an abelian subalgebra of A. We call (A, B) a Cartan pair and say that B is a Cartan subalgebra of A if the following conditions are satisfied: (i) B is a maximal abelian subalgebra of A; (ii) span (N (B)) = A; and (iii) there exists a faithful conditional expectation E : A → B.

Proof.
The first claim follows from [2, Corollary 9.3].To prove the second claim, we identify C * r (Z) = C * (Z) with C(T) via the Fourier transform F : C * (Z) → C(T), which sends the generating unitary u = δ 1 of C * (Z) to the identity map on T. The groupoid G = Z has unit space G (0) = {0}, and so F(C 0 (G (0) )) = C1 C(T) .Consider the function m : T → C given by

Proposition 5 . 1 .
Let G be a locally compact Hausdorff étale groupoid.Suppose that for every normaliser n ∈ C * r (G) of C 0 (G (0) ), S n = supp • (j(n)) is an open bisection of G. Then G is effective.
is contained in an open subset U of G such that r| U and s| U are homeomorphisms onto open subsets of G (0) .Every locally compact Hausdorff étale groupoid has a basis of precompact open bisections.We call a topological groupoid ample if it has a basis of compact open bisections.Given subsets A and B of a Hausdorff étale groupoid G, we define