Ideal structure and pure infiniteness of inverse semigroup crossed products

Let $A\subseteq B$ be a $C^*$-inclusion. We give efficient conditions under which $A$ separates ideals in $B$, and $B$ is purely infinite if every positive element in $A$ is properly infinite in $B$. We specialise to the case when $B$ is a crossed product for an inverse semigroup action by Hilbert bimodules or a section $C^*$-algebra of a Fell bundle over an \'etale, possibly non-Hausdorff, groupoid. Then our theory works provided $B$ is the recently introduced essential crossed product and the action is essentially exact and residually aperiodic or residually topologically free. These last notions are developed in the article.


Introduction
Many authors have given sufficient criteria for crossed products by discrete group actions or for C˚-algebras associated to étale locally compact groupoids to be purely infinite (see, for instance, [2, 6, 19, 22, 34-36, 38, 41, 43]). These articles mostly deal with the case when the bigger C˚-algebra B is simple or when the C˚-subalgebra A Ď B on which the action takes place is commutative and has totally disconnected spectrum. In addition, étale groupoids are required to be Hausdorff. These pure infiniteness criteria also imply that A separates ideals in B. Then the ideal lattice of B is isomorphic to the lattice of invariant ideals in A. Here we formulate sufficient conditions for A to separate ideals in B and for B to be purely infinite, which allow A to be noncommutative and which impose no Hausdorffness restrictions. In this generality, it is natural to study actions of inverse semigroups by Hilbert bimodules (see [11]) or, equivalently, section algebras of Fell bundles over inverse semigroups. This contains Fell bundles over discrete groups and over étale groupoids -possibly non-Hausdorff -as special cases. Another special case are Exel's noncommutative Cartan C˚-inclusions (see [15,31]), which generalise Renault's (commutative) Cartan subalgebras.
This article is based on our recent papers, [28][29][30]32], where two key concepts are developed. The first one is the essential crossed product introduced in [30], which is a variation on the reduced crossed product that "always" has the expected ideal structure -even for general actions of inverse semigroups and for actions of non-Hausdorff groupoids. The second concept is aperiodicity. It is a strong regularity property, abstracted from the work of Kishimoto, Olesen-Pedersen, and others, defined for a general C˚-inclusion A Ď B in [30]. As shown in [32], aperiodicity implies that there is a unique pseudo-expectation -a unique generalised conditional expectation for A Ď B taking values in Hamana's injective hull of A. If in addition this pseudo-expectation is almost faithful, then A supports B in the sense that every element in B`zt0u is supported by an element in A`zt0u in the Cuntz preorder of [12]. This, in turn, implies that A detects ideals in B, that is, J X A ‰ 0 for any ideal 0 ‰ J Ÿ B. All these properties are closely related. In fact, when B is an essential crossed product and A is separable or of Type I, then the following conditions are equivalent: aperiodicity, unique pseudo-expectation, supporting positive elements in all intermediate C˚-algebras, detection of ideals in all intermediate C˚-algebras, and topological freeness of the dual groupoid (see [32]).
In the present paper, we study "residual" versions of these conditions, that is, when they hold for quotient inclusions A{I Ď B{BIB for all ideals I Ÿ A that are restricted from B. The relationship between ideals in B and restricted ideals in A is thoroughly studied in [29]. For a crossed product inclusion, the restricted ideals in A are exactly those that are invariant under the action that produces B from A. The residual version of detection of ideals is separation of ideals. We say that A separates ideals in B if I X A " J X A for ideals I, J Ÿ B implies I " J. This identifies the ideal lattice of B with the lattice of restricted ideals in A. Under some extra assumptions, we may also identify the primitive ideal space of B with the quasi-orbit space of the induced action on the primitive ideal space of A (see [29]). A residual version of supporting is closely related to the formally stronger condition called filling, which was used in [22,23] to prove strong pure infiniteness. For a large class of C˚-inclusion A Ď B, we show that A residually supports B if and only if A fills B. Namely, this holds for the symmetric inclusions defined in [29], which include all sorts of C˚-inclusions coming from crossed products. For general residually supporting C˚-inclusions A Ď B and a family F Ď A`of elements in A that are properly infinite in B, we give sufficient conditions for B to be purely infinite (see Theorem 2.37 below).
To ensure that A`residually supports B we assume that the inclusion A Ď B is residually aperiodic, in the sense that for any restricted ideal I Ÿ A, the inclusion A{I Ñ B{BIB is aperiodic. We also need to assume that the pseudo-expectations for these quotient inclusions are almost faithful. For actions by discrete groups this "residual faithfulness of conditional expectations" is also called exactness (see [1,34,44]) and for groupoids inner exactness (see [3,6]). We generalise this concept to inverse semigroup actions by Hilbert bimodules and Fell bundles over étale groupoids. In particular, we prove that the full crossed product is an exact functor and the reduced crossed product is an injective functor, but only when restricted to special homomorphisms between actions (Propositions 4.2 and 4.15). The essential crossed product is not functorial. Therefore, exactness for these crossed products, which we call essential exactness, is more subtle.
As an appetizer, we formulate here a theorem that summarises and illustrates some of our results. An action of a unital inverse semigroup S by Hilbert bimodules on a C˚-algebra A is a semigroup E " pE t q tPS , where each fibre E t is a Hilbert A-bimodule and the semigroup product is compatible with the internal tensor product (see Definition 3.1 below). This induces an S-action by partial homeomorphisms on the spectrum p A and the primitive ideal spaceǍ. The corresponding transformation groupoids p A¸S andǍ¸S are called dual groupoids of E. If the unit spaces in these groupoids are closed, we call E a closed action. Then the essential crossed product A¸e ss S coincides with the reduced crossed product A¸r S. The following theorem combines Theorems 5.8, 5.9 and 5.15 and Corollary 5.22: Theorem 1. Let E " pE t q tPS be an inverse semigroup action by Hilbert bimodules on a C˚-algebra A. Assume that E is essentially exact (or exact when the action is closed) and residually aperiodic (this holds when p A¸S is residually topologically free). Then (1) A fills the essential crossed product A¸e ss S. So A¸e ss S is strongly purely infinite if and only if every pair of elements in A`satisfies the matrix factorisation property of [23]; (2) the ideal lattice of A¸e ss S is isomorphic to the lattice I E pAq of E-invariant ideals in A.
IfǍ is second countable, the primitive ideal space of A¸e ss S is homeomorphic to the quasi-orbit spaceǍ{" of the dual groupoidǍ¸S; (3) if F Ď A`residually supports A and consists of residually E-infinite elements (see Definition 5.18), and I E pAq is finite or the projections in F separate the ideals in I E pAq, then A¸e ss S is purely infinite and has the ideal property.
The above theorem directly applies to (twisted) crossed products by discrete groups. As we explain in Section 5.2, it covers all the pure infiniteness results in [19,22,34,35,38,43]. Theorem 1 has an analogue for Fell bundles over étale groupoids (Corollary 5.10). In particular, it can be used to generalise the results in [6,36,41] to twisted (not necessarily Hausdorff) étale groupoids. The twisted version covers all Cartan inclusions of Renault [42] (see Corollary 5.29 and Remark 5.30). In fact, Theorem 1 may be applied to all regular residually aperiodic C˚-inclusions A Ď B with a residually faithful conditional expectation (see Proposition 5.12). This includes a large class of noncommutative Cartan inclusions in the sense of Exel [15]. See [31,Theorem 4.3] for a number of equivalent characterisations of such inclusions.
(1) If p is a primitive ideal in B, then rppq P I B pAq is prime, and this defines a continuous map r :B Ñ Prime B pAq.

(2) If p is a primitive ideal in A, then there is a largest restricted ideal in A that is contained
in p, which we denote by πppq. This element of I B pAq is prime, and the resulting map π :Ǎ Ñ Prime B pAq is continuous. Define an equivalence relation onǍ by p " q if and only if πppq " πpqq. So π descends to a continuous map r π :Ǎ{" Ñ Prime B pAq. Remark 2.8. If B is the full or reduced crossed product -or an exotic crossed product, for an action of a discrete group G on a C˚-algebra A, thenǍ{" coincides with the usual quasi-orbit space of the dual action of G onǍ. In [29], the quasi-orbit space is also described in several other cases.
Next we turn to regular inclusions. To link them to crossed products for inverse semigroup actions, we describe them through gradings by inverse semigroups. Definition 2.9 ([29, Definition 6.15]). Let S be an inverse semigroup with unit 1 P S. An S-graded C˚-algebra is a C˚-algebra B with a family of closed linear subspaces pB t q tPS such that Bg " B g˚, (1) A is a regular subalgebra of B; (2) A is the unit fibre for some S-grading on B; (3) A is the unit fibre for some saturated S-grading on B. If the inclusion is regular, then it is symmetric, and the set SpA, Bq of all closed linear A-subbimodules M Ď B that consist entirely of normalisers is an inverse semigroup with the operations M¨N :" spantmn : m P M, n P N u and M˚:" tm˚: m P M u, and it gives a saturated grading on B.
(1) If pB t q tPS is an S-grading of B with A as unit fibre, then pqpB t qq tPS is an S-grading on B{J with unit fibre A{I. And qpB t q -B t {B t I as Banach spaces -and even as Hilbert A{I-bimodules -for all t P S.
Proof. The canonical map from A{I to B{J is injective because J X A " I. Thus we may view A{I as a C˚-subalgebra of B{J. We prove (1). It is easy to see that pqpB t qq tPS is an S-grading on B{J. Each B t is naturally a right Hilbert A-module with inner product xa | by :" a˚b P A for a, b P B t and the right multiplication in B. The proof of the Rieffel correspondence between ideals in A and KpB t q shows that B t I " tb P B t : xb | by P Iu. The quotient Banach space B t {B t I is a right Hilbert A{I-module with the induced multiplication and the inner product xa`B t I | b`B t Iy :" xa | by`I " qpa˚bq P A{I for a, b P B t . We claim that the norm defined by this inner product is equal to the quotient norm on B t {B t I. To show this, let pu n q nPN be an approximate unit for I. Then This finishes the proof of (1). Assertion (2) follows from (1)  Let E : B ÑÃ Ě A be a generalised expectation. It is called faithful if Epb˚bq " 0 for some b P B implies b " 0, almost faithful if Eppbcq˚bcq " 0 for all c P B and some b P B implies b " 0, and symmetric if Epb˚bq " 0 for some b P B implies Epbb˚q " 0. The largest two-sided ideal in B contained in ker E is equal to N E :" tb P B : Eppbcq˚bcq " 0 for all c P Bu " tb P B : Epxbyq " 0 for all x, y P Bu (see [30,Proposition 3.6]), and N E " 0 if and only if E is almost faithful.
Since E| A " Id A and E| N E " 0, it follows that A X N E " 0. Hence the composite map A Ñ B Ñ B{N E is injective and we may identify A with its image in B r :" B{N E . The map E descends to a generalised expectation E r : B r ÑÃ Ě A that we call the reduced generalised expectation associated to E (see [30,Definition 3.5]). The reduced generalised expectation E r is always almost faithful. It is faithful if and only if E is symmetric (see [30,Corollary 3.8]).
We will mainly work with pseudo-expectations below. The injectivity of IpAq implies that any C˚-inclusion has a pseudo-expectation. The following lemma links detection of ideals to almost faithfulness of pseudo-expectations: The following are equivalent: (2). That (2)  In general, it is not practical to check that all pseudo-expectations B Ñ IpAq are faithful or almost faithful. And the residual version of this statement looks even more hopeless. There are, however, inclusions with a unique pseudo-expectation. This is the case for aperiodic inclusions by [32,Theorem 3.6]. When the inclusion is even "residually" aperiodic, then the inclusions A{I ãÑ B{BIB have a unique pseudo-expectation for all I P I B pAq. And if we know pseudoexpectations E I : B{BIB Ñ IpA{Iq for all I P I B pAq, then it becomes possible to check whether they are all (almost) faithful (see Theorem 2.34 below).
The residual version of Lemma 2.16 discussed above uses pseudo-expectations for the inclusion A{I ãÑ B{BIB for I P I B pAq.
The following examples show that these need not be closely related to pseudo-expectations for the original inclusion A Ď B. In fact, for inclusions that are not symmetric, even a genuine conditional expectation E : B Ñ A need not "induce" a conditional expectation E : B{BIB Ñ A{I.
Example 2.18. Let B " BpHq be the algebra of bounded operators on a separable Hilbert space H, and let A " KpHq`1 be the minimal unitisation of the compacts. The inclusion A Ď B is symmetric, I B pAq " t0, KpHq, Au, and IpAq " BpHq " B. We may take the identity map as the pseudo-expectation E : B Ñ B for A Ď B. Let I :" KpHq. Then, on the one hand, E descends to the identity map E I : B{I Ñ B{I on the Calkin algebra. On the other hand, pseudo-expectations for C -A{I Ď B{I are just states on the Calkin algebra. It seems that there is no universal way how to produce a state from the identity map. Let J " M 2 pCq ' 0. Then I :" J X A " C¨pP 00 , 0q is a restricted ideal in A -C 3 . We have J " BIB and EpJq " tλ 1 pP 00 , 0q`λ 2 pP 11 , P 11 q : λ 1 , λ 2 P Cu Ę I. Proof. Let J P I A pBq and put I " J X A P I B pAq. Since A Ď B is symmetric we have J " IBI. Thus EpJq " EpIBIq Ď IEpBqI " I because E is A-bilinear. Since I Ď EpJq always holds, this is equivalent to I " EpJq. Then E I is well defined. □ Many C˚-algebras, including section algebras for Fell bundles over Hausdorff étale groupoids, are naturally equipped with a conditional expectation, which is residually symmetric in the sense described in the following proposition. Then the residual faithfulness of E is called exactness of the corresponding action [1,3,6,34,44].  We equip B`with the Cuntz preorder À introduced in [12]: for a, b P B`, we write a À b and say that a supports b if, for every ε ą 0, there is x P B with ∥a´x˚bx∥ ă ε. We call a, b P B`Cuntz equivalent and write a « b if a À b and b À a. We call a, b P B`Murray-von Neumann equivalent and write a " b if there is z P B with a " z˚z and b " zz˚. Both " and « are equivalence relations, and a " b implies a « b. In the converse direction, only a weaker result is true (see [21, Lemma 2.3(iv)]). Namely, for ε ą 0 and a P B`zt0u, let pa´εq`P B be the positive part of a´ε¨1 P MpBq. Let bBb be the hereditary subalgebra generated by b, that is, the closure of tbxb : x P Au. Then a À b if and only if every ε-cut-down of a is Murray-von Neumann equivalent to an element in bBb, that is a À b ðñ @ εą0 D zPB pa´εq`" z˚z and zz˚P bBb.
In particular, a P bBb implies a À b, aBa " bBb implies a « b, and a À b implies a P BbB. A C˚-algebra B is purely infinite [20] if it admits no characters and a, b P B`zt0u satisfy a ĺ b if and only if a P BbB.
(2) Let F residually support B. If J P IpBq, then J X F residually supports J. In addition, F separates ideals in B.
Proof. We first prove (1). Let J P IpBq and b P J`zt0u. Then x˚bx P J for each x P B. If a À b for some a P B`zt0u, then a " lim xk bx k for some sequence px k q kPN in B. Therefore, any a P F`zt0u with a ĺ b and b P J`zt0u belongs to J. So if F supports B and J ‰ t0u, then J X F ‰ t0u.
Next we prove (2). Let J P IpBq and I P IpJq. Let q : B Ñ B{I be the quotient map. For any b P qpJq`zt0u, there is a P F with 0 ‰ qpaq À b. The proof above shows that qpaq P qpJq. Since qpaq ‰ 0, we have a P JzI. Thus J X F residually supports J. If I, J P IpBq and I ‰ J, then I X J is a proper ideal of I or J. Assume, say, that I X J Ĺ J. As we have seen above, there is a P pJ X FqzI. Hence I X F ‰ J X F. □ (1) F supports B; (2) for each b P B`zt0u, there is x P B with x˚bx P Fzt0u; (3) for each D P HpBq, there is z P B with zz˚P D and z˚z P Fzt0u.
(2)ñ(3): Let b P D`zt0u and choose x P B with x˚bx P Fzt0u as in (2). Then z :" ? b¨x satisfies zz˚P D and z˚z P Fzt0u, as required in (3). In this definition, we may also require z P B with zz˚P D and z˚z P FzJ because z P J if and only if z˚z P J. For J " 0, this is the condition in Lemma 2.26.(3). This suggests that filling and residually supporting families are closely related. We are going to prove some results to this effect.  Hence I A pBq " IpBq. So J " BIB with I :" A X J. Let q : B Ñ B{J be the quotient map. There is d P D`zJ. Let b :" qpdq P pB{Jq`zt0u. Lemma 2.26 gives x P B{J with a :" x˚bx P pA{Iq`zt0u.
There are c P A`with qpcq " a and w P B with qpwq " x. Then qpcq " x˚bx " qpw˚dwq. So c " w˚dw`v for some v P J. Let ε :" ∥a∥{2. By assumption, an approximate unit in I is also one for J. So there is f P Iẁ ith ∥f ∥ ď 1 and ∥v´f v∥ ă ε. Let 1 denote the formal unit in the unitisation of B and let g :" 1´f P MpAq`. Then ∥g∥ ď 1 and ∥gw˚dwg´gcg∥ " ∥gvg∥ ď ∥v´f v∥ ă ε.
Hence z˚z R J, which is equivalent to zz˚R J. □ Example 2.30. Let B " C 0 pΩq be commutative and let F Ď B`. The following conditions are equivalent: (1) F fills B; (2) F residually supports B; (3) the open supports of elements of F form a basis of the topology of Ω.
Proposition 2.28 implies (1)ñ(2), and (3)ñ(1) is straightforward. We show (2)ñ (3). Fix an open subset U Ď Ω and a point x 0 P U . There is a function b P C 0 pΩq with bpx 0 q " 1 and b| ΩzU " 0. Let J be the ideal in B consisting of functions vanishing on ΩzU Y tx 0 u. There is a P FzJ with a`J À b`J. Then V :" tx P Ω : apxq ą 0u is an open subset of U that contains x 0 . This implies (3).
Example 2.30 suggests to view filling families and residually supporting subsets as noncommutative analogues of bases for topologies.
2.5. Residual aperiodicity and criteria for pure infiniteness. We introduce the residual version of aperiodicity and use it to characterise when a C˚-inclusion A Ď B separates ideals. We also formulate some criteria for B to be purely infinite.

Definition 2.31 ([30, Definition 5.14]). A C˚-inclusion A Ď B is aperiodic if the Banach
A-bimodule B{A is aperiodic, that is, if for every x P B, D P HpAq and ε ą 0, there are a P D`and y P A with ∥axa´y∥ ă ε and ∥a∥ " 1. (1) the unique pseudo-expectation E I is almost faithful for all I P I B pAq; If A Ď B is symmetric,Ǎ is second countable, and the above equivalent conditions hold, theň B -Ǎ{" via the quasi-orbit map.
Proof. It follows from [32, Theorem 3.6] that the expectation E I is unique and that (1) implies (3). Lemma 2.23 shows that (3) implies (2). Next we prove that (2) (3) is equivalent to (4). The claims in the last sentence follow from Theorem 2.6. □ We end this section with pure infiniteness criteria that use filling and residually supporting families. Infinite and properly infinite elements in B`are defined in [20]. We recall their equivalent descriptions in [34, Lemma 2.1]. We also recall the notion of a separated pair of elements in Bf rom [28, Definition 5.1] and relate it to the matrix diagonalisation property. We write a « ε b if ∥a´b∥ ď ε for a, b P B.
y˚y « ε a and x˚y « ε 0. We say that a pair of elements a, b P B`has the matrix diagonalisation property in B, if for each x P B with`a xx b˘P M 2 pBq`and each ε ą 0 there are d 1 P B and d 2 P B such that We say that a subset F Ď B`is invariant under ε-cut-downs if pa´εq`P F for all a P F and arbitrarily small ε ą 0. Proof. Assume that A separates ideals in B. Suppose first that B is purely infinite. By Proposition 2.21, for any b P B`zt0u, Epbq is in the ideal in B generated by b. Then Epbq À b because b is properly infinite (see [20,Theorem 4.16]). Now suppose that every a P A`zt0u is properly If A separates ideals in B and B is purely infinite, the same holds for all the quotient inclusions A{I Ď B{J, I " J X A, J P IpBq (see [20,Proposition 4.3]). By Proposition 2.21, the conditional expectation E I : B{J Ñ A{I is faithful. Hence the first part of the assertion shows that b P pB{Jq`zt0u implies 0 ‰ E I pbq À b. Thus A`residually supports B. Then A`fills B by Proposition 2.29. □

Inverse semigroup actions and their crossed products
In this section, we briefly recall inverse semigroup actions by Hilbert bimodules and their crossed products, referring to [10,30] for more details.

Inverse semigroup actions by Hilbert bimodules.
Throughout this paper, S is an inverse semigroup with unit 1 P S.

Definition 3.1 ([11]). An action of S on a C˚-algebra A (by Hilbert bimodules) consists of Hilbert
A-bimodules E t for t P S and Hilbert bimodule isomorphisms µ t,u : Ý Ñ E tu for t, u P S, such that (A1) for all t, u, v P S, the following diagram commutes (associativity): Ý Ñ E t for t P S are the maps defined by µ 1,t pabξq " a¨ξ and µ t,1 pξ b aq " ξ¨a for a P A, ξ P E t . Any S-action by Hilbert bimodules comes with canonical involutions J t : Et Ñ E t˚a nd inclusion maps j u,t : E t Ñ E u for t ď u that satisfy the conditions required for a saturated Fell bundle in [15] (see [11,Theorem 4.8]). Thus S-actions by Hilbert bimodules are equivalent to saturated Fell bundles over S. A nonsaturated Fell bundle over S is turned into a saturated Fell bundle over another inverse semigroup in [9], such that the full and reduced section C˚-algebras stay the same. Therefore, we usually restrict attention to saturated Fell bundles, which we may replace by inverse semigroup actions as in Definition 3.1. Definition 3.1 contains (twisted) actions by partial automorphisms.
Example 3.2 (Twisted actions of inverse semigroups, see [7, Definition 4.1]). A twisted action of an inverse semigroup S by partial automorphisms on a C˚-algebra A consists of partial automorphisms α t : D t˚Ñ D t of A for t P S -that is, D t is an ideal in A and α t is a˚-isomorphism -and unitary multipliers ωpt, uq P UMpD tu q for t, u P S, such that D 1 " A and the following conditions hold for r, t, u P S and e, f P EpSq :" ts P S : s 2 " su: (1) α r˝αt " Ad ωpr,tq α rt ; (2) α r`a ωpt, uq˘ωpr, tuq " α r paqωpr, tqωprt, uq for a P D r˚X D tu ; (3) ωpe, f q " 1 ef and ωpr, r˚rq " ωprr˚, rq " 1 r , where 1 r is the unit of MpD r q; (4) ωpt˚, eqωpt˚e, tqa " ωpt˚, tqa for all a P D t˚et . Let ppα t q tPS , ωpt, uqq t,uPS be a twisted action as above. For t P S, let E t be the Hilbert A-bimodule associated to the partial homeomorphism α t ; this is D t as a Banach space, and we denote elements by bδ t to highlight the t P S in which we view b P D t Ď A as an element; the Hilbert A-bimodule structure is defined by [7,Theorem 4.12]. And pE t , µ t,u q t,uPS is an action of S on A by Hilbert bimodules. Fell bundles over S that come from twisted partial actions are characterised in [7,Corollary 4.16], where they are called "regular." Example 3.3 (Inverse semigroup gradings). An S-grading pB t q tPS of a C˚-algebra B as in Definition 2.9 gives a Fell bundle over S, using the multiplication and involution in B. This bundle is saturated if and only the grading is saturated. Then it is an action of S by Hilbert bimodules on A :" B 1 Ď B. Thus inverse semigroup actions by Hilbert bimodules are inevitable in the study of regular inclusions (see also Proposition 2.11). Any inverse semigroup action E "`pE t q tPS , pµ t,u q t,uPS˘o n a C˚-algebra A comes from an S-graded C˚-algebra. Namely, embed the spaces E t for t P S into, say, the full crossed product. They form an S-grading of the full crossed product.
Fell bundles over étale groupoids may be described through S-actions.
Example 3.4 (Fell bundles over groupoids). Let G be an étale groupoid with locally compact and Hausdorff unit space X. So the range and source maps r, s : G Ñ X are local homeomorphisms. A Fell bundle over G is defined, for instance, in [8,Section 2]. It is an upper semicontinous bundle A " pA γ q γPG of complex Banach spaces equipped with a continuous involution˚: A Ñ A and a continuous multiplication¨: tpa, bq P AˆA : a P A γ1 , b P A γ2 , pγ 1 , γ 2 q P G p2q u Ñ A, which satisfy some natural properties. The set of (open) bisections BispGq :" tU Ď G : U is open and s| U , r| U are injectiveu is a unital inverse semigroup with U¨V :" tγ¨η : γ P U, η P V u for U, V P BispGq. Namely, X P BispGq is the unit element and U˚:" tγ´1 : γ P U u for U P BispGq. Let A U for U P BispGq be the space of continuous sections of pA γ q γPG vanishing outside U . The spaces A rpU q and A spU q are closed two-sided ideals in A " A X , and A U becomes a Hilbert A rpU q -A spU q -bimodule with the bimodule structure pa¨ξ¨bqpγq :" aprpγqqξpγqbpspγqq and the right and left inner products This data defines a Fell bundle over BispGq, which is saturated if A is (see [8,30] for details). If A is not saturated, this may be naturally turned into a saturated Fell bundle over another inverse semigroup in a number of ways. For instance, for any inverse subsemigroup S Ď BispGq we may letS be the family of all Hilbert subbimodules of A U for U P S. Equivalently, elements ofS are of the form A U¨I for U P S and I Ÿ A. ThenS, with operations defined as above, forms an inverse semigroup that acts by Hilbert bimodules on A (see [30,Lemma 7.3]).
Let A be a C˚-algebra with an action E of a unital inverse semigroup S. Let p A and q A " PrimpAq be the space of irreducible representations and the primitive ideal space of A, respectively. The action of S on A induces actions p E " p p E t q tPS and q E " p q E t q tPS of S by partial homeomorphisms on p A and q A, respectively (see [11,Lemma 6.12], [30, Section 2.3]). The homeomorphisms are given by Rieffel's correspondence and induction of representations, respectively. Any action by partial homeomorphisms has a transformation groupoid, which is étale (see [14,Section 4]  Example 3.6 (Dual groupoids to Fell bundles). Let A be a Fell bundle over an étale groupoid G with locally compact Hausdorff object space X. Then G acts naturally both on the primitive ideal spaceǍ and the spectrum p A of the C˚-algebra A :" C 0 pX, Aq. More specifically, every irreducible representation of A factors through the evaluation map A Ñ A x for some x P X, and this defines a continuous map ψ : p A Ñ X, which is the anchor map of the G-action on p A given by the partial [18,Section 2]). The corresponding transformation groupoid is p A¸G :" tprπs, γq P p AˆG : ψprπsq " spγqu.
Two elements prρs, ηq and prπs, γq are composable if and only if rρs " ψ γ prπsq, and then their composite is prπs, ηγq. The inverse is prπs, γq´1 " pψ γ prπsq, γ´1q. The maps ψ : p A Ñ X and ψ γ : z A spγq Ñ z A rpγq for γ P G, factor through to a G-action onǍ, which defines a transformation groupoidǍ¸G (see [18]). These actions give rise to transformation groupoidsǍ¸G and p A¸G.
Let S Ď BispGq be a unital, inverse subsemigroup of bisections of G which is wide in the sense that Ť S " G and U X V is a union of bisections in S for all U, V P S. Then turning this into the inverse semigroup actionS on A described in Example 3.4, we have natural isomorphisms of groupoids (see [30,Remark 7.4 We callǍ¸G and p A¸G dual groupoids for A.

Crossed products.
Fix an action E "`pE t q tPS , pµ t,u q t,uPS˘o f S on A. For any t P S, let rpE t q and spE t q be the ideals in A generated by the left and right inner products of vectors in E t , respectively.
be the closed ideal generated by spE v q for v ď t, u. It is contained in spE t q X spE u q, and the inclusion may be strict. There is a unique Hilbert bimodule isomorphism The˚-algebra A¸a lg S of the action E is defined as the quotient vector space of À tPS E t by the linear span of ϑ u,t pξqδ u´ξ δ t for all t, u P S and ξ P E t¨It,u . The algebraic structure on A¸a lg S is given by the multiplication maps µ t,u and the involutions J t . Definition 3.9. The (full) crossed product A¸S of the action E is the maximal C˚-completion of the˚-algebra A¸a lg S described above.
Any representation π of E in B induces a˚-homomorphism π¸S : A¸S Ñ B. Conversely, every˚-homomorphism A¸S Ñ B is equal to π¸S for a unique representation π (compare [10, Proposition 2.9]). This universal property determines A¸S uniquely up to isomorphism.
The C˚-algebra A¸S is canonically isomorphic to the full section C˚-algebra of the Fell bundle over S corresponding to E. The reduced section C˚-algebra of a Fell bundle over S was first defined using inducing pure states, see [15]. An equivalent definition appears in [10], where this is called the reduced crossed product A¸r S of the action E (the reduced C˚-algebra obtained in [5], using a regular representation, is in general different). The main ingredient in the construction in [10] is the weak conditional expectation for A¸S Ñ A 2 described in [10, Lemma 4.5] through the formula where ξ P E t , t P S, pu i q is an approximate unit for I 1,t and s-lim denotes the limit in the strict topology on MpI 1,t q Ď A 2 . This weak expectation is symmetric by [30,Theorem 3.22]. So the induced weak expectation E r on A¸r S is faithful.
Remark 3.14. The canonical maps from A¸a lg S to A¸S and to A¸r S are injective by [10,Proposition 4.3]. In particular, both A¸S and A¸r S are naturally S-graded with the same Fell bundle pE t q tPS over S.  .7) is complemented in spE t q for each t P S.
Example 3.17 (C˚-algebras of Fell bundles over groupoids). Retain the notation from Example 3.4. The˚-algebra associated to the Fell bundle A over the étale groupoid G is denoted by SpG, Aq.
It is the linear span of compactly supported continuous sections A U " C c pU, Aq for all bisections U P BispGq with a convolution and involution given by 3.3. Essential crossed products. The local multiplier algebra M loc pAq of A is the inductive limit of the multiplier algebras MpJq, where J runs through the directed set of essential ideals in A (see [4]). A key idea in [30] is a natural generalised expectation EL : A¸S Ñ M loc pAq with values in M loc pAq. It is defined as follows: for each ξ P E t , t P S, the element ELpξq P MpI 1,t ' I K 1,t q Ď M loc pAq is given by ELpξqpu`vq :" ϑ 1,t pξuq P A for u P I 1,t , v P I K 1,t . This generalised expectation is symmetric by [30,Theorem 4.11]. Hence EL factors through a faithful pseudo-expectation on the quotient There is a canonical embedding ι : M loc pAq ãÑ IpAq compatible with the inclusions A Ď M loc pAq and A Ď IpAq (see [16,Theorem 1]). Thus the canonical M loc -expectation EL may be viewed as a pseudo-expectation. A C˚-algebra B with˚-epimorphisms A¸S ↠ B ↠ A¸e ss S that compose to the canonical quotient map A¸S Ñ A¸e ss S is called an exotic crossed product (see [30]). The following proposition characterises when the reduced and essential crossed products coincide: Example 3.21 (Essential twisted groupoid C˚-algebras). We define the essential groupoid C˚-algebra Ce ss pG, Σq of a twisted groupoid pG, Σq as Ce ss pG, Lq for the corresponding Fell line bundle L.
Denoting by X the unit space of G, [30,Proposition 7.18] implies that the following are equivalent: (1) Cr pG, Σq " Ce ss pG, Σq; (2) tx P X : E r pf qpxq ‰ 0u is not meagre for every f P Cr pG, Σq`zt0u; (3) if f P Cr pG, Σq`zt0u, then tx P X : ∥E r pf qpxq∥ ą εu has nonempty interior for some ε ą 0. Here E r : Cr pG, Σq Ñ BpXq is the canonical generalised expectation that restricts sections of the corresponding Fell line bundle to X.
Example 3.22 (Twisted crossed products by partial automorphisms). Let pα, ωq be a twisted action of an inverse semigroup S by partial automorphisms on a C˚-algebra A as in Example 3.2. By [7, Definition 6.2], a covariant representation of pα, ωq on a Hilbert space H is a pair pρ, vq consisting of a˚-homomorphism ρ : A Ñ BpHq and a family v " pv t q tPS of partial isometries in BpHq such that for all b P D t , t, u P S. Here ρ is the extension of ρ to the enveloping von Neumann algebra of A, so that ρpωpt, uqq and ρp1 e q make sense. By definition, the full crossed product for pα, ωq is the universal C˚-algebra for covariant representations, and by [7, Theorem 6.3] it is naturally isomorphic to the full crossed product A¸S for the associated inverse semigroup action E by Hilbert bimodules (see Example 3.2). The reduced crossed product for pα, ωq may be identified with the reduced crossed product A¸r S by [7, Definition 6.6]. We define the essential crossed product for pα, ωq as A¸e ss S. By [7, Theorem 7.2], for any twisted groupoid pG, Σq the bisections S of G that trivialise the twist Σ give rise to a twisted inverse semigroup action pα, ωq by partial automorphisms of A :" C 0 pG 0 q such that Cr pG, Σq -A¸r S. By definition, this descends to an isomorphism Ce ss pG, Σq -A¸e ss S.
The reduced section C˚-algebra Cr pG, Aq is usually defined through the regular representation, which is the direct sum of representations λ x : C˚pG, Aq Ñ B`ℓ 2 pG x , Aq˘.
Here ℓ 2 pG x , Aq is the Hilbert A x -module completion of À spγq"x A γ with the obvious right multiplication and the standard inner product xf | gy :" ř spγq"x f pγq˚gpγq. For f P SpH, Aq and g P À spγq"x A γ , define λ x pf qpgqpγq :" ř rpηq"rpγq f pηqgpη´1γq. The kernel of À xPX λ x is N E . So the reduced C˚-algebra Cr pH, Aq is isomorphic to the completion of SpH, Aq in the reduced norm ∥f ∥ r :" sup xPX ∥λ x pf q∥. We now describe essential algebras in a similar fashion: Definition 3.23 ([30, Definition 7.14]). Call x P X dangerous if there is a net pγ n q in G that converges towards two different points γ ‰ γ 1 in G with spγq " spγ 1 q " x.

Proposition 3.24. Let A be a continuous Fell bundle over an étale groupoid G with locally compact and Hausdorff unit space X. Assume G is covered by countably many bisections and let D Ď X be the set of dangerous points. Then
That is, Ce ss pG, Aq is isomorphic to the Hausdorff completion of SpG, Aq in the seminorm ∥f ∥ ess :" sup xPXzD ∥λ x pf q∥.
Proof. For each x P X, γ P G x " s´1pxq and f P C˚pG, Aq, Thus ker λ x " tf P C˚pG, Aq : Epf˚f qprpG x qq " 0u. We claim that The set of dangerous points is G-invariant. Indeed, if η P s´1pxq and there is a net pγ n q that converges towards two different γ ‰ γ 1 P H with spγq " spγ 1 q " x, then the net pγ n η´1q converges to γη´1 ‰ γ 1 η´1 P H with spγη´1q " spγ 1 η´1q " rpηq. Hence x P D implies that rpG x q Ď D. Then ker à xPXzD λ x " č xPXzD ker λ x " tf P C˚pG, Aq : Epf˚f qpxq " 0 for all x P XzDu.
The set on the right hand side is equal to N EL by [30,Proposition 7.18]. □

Exactness of inverse semigroup actions
4.1. Functoriality. For group actions, the full and reduced crossed products are functors, and the reduced one preserves injective homomorphisms. We extend this to inverse semigroup actions by Hilbert bimodules.
Definition 4.1. Let E "`pE t q tPS , pµ t,u q t,uPS˘a nd F "`pF t q tPS , pν t,u q t,uPS˘b e two actions of S by Hilbert bimodules on A and B, respectively. A homomorphism ψ from E to F is a family of linear maps ψ t : E t Ñ F t for t P S such that for all t, u P S, ξ P E t , η P E u we have ψ tu pµ t,u pξ b ηqq " ν t,u pψ t pξq b ψ u pηqq, xψ t pξq | ψ t pηqy " ψ 1 pxξ | ηyq, xxψ t pξq | ψ t pηqyy " ψ 1 pxxξ | ηyyq.
The maps ψ t are always contractive. We call ψ injective if ψ 1 is injective; then the maps ψ t are isometric for all t P S. We call ψ an isomorphism if all ψ t are isomorphisms.
We use the superscripts E and F to distinguish between objects defined for the actions E and F.

Proposition 4.2.
Let ψ be a homomorphism from an action E " pE t q tPS on A to an action F " pF t q tPS on B. It induces a˚-homomorphism ψ¸S : A¸S Ñ B¸S where pψ¸Sqpξq " ψ t pξq for ξ P E t , t P S. In particular, ψ respects the involution and inclusions maps on E and F, and ψ¸S restricts to a˚-homomorphism ψ¸a lg S : A¸a lg S Ñ B¸a lg S. Moreover the following conditions are equivalent: (1) ψ¸S descends to a˚-homomorphism ψ¸r S : A¸r S Ñ B¸r S that respects the canonical weak expectations, that is the following diagram comutes where Λ E (resp. Λ F ) is the regular representation and E E (resp. E F ) is the canonical weak conditional expectation associated to the action E (resp. F). (2) ψ 2 1 prI E 1,t sq " ψ 2 1 prspE t qsqrI F 1,t s, for all t P S, where rI E 1,t s, rspE t qs P A 2 and rI F 1,t s P B 2 are the support projections of the ideals I E 1,t , spE t q Ď A and I F 1,t Ď B, respectively. If the above equivalent conditions hold and ψ is injective, then so are ψ¸r S, ψ¸a lg S, and ψ 2 1 .
Proof. The Hilbert bimodules F t for t P S embed into B¸S. Hence we may treat the maps ψ t : E t Ñ F t as taking values in B¸S. Then ψ is a representation of E in B¸S. It integrates to a˚-homomorphism ψ¸S : A¸S Ñ B¸S by [10,Proposition 2.9]. This restricts to å -homomorphism ψ¸a lg S : A¸a lg S Ñ B¸a lg S and therefore ψ respects the induced involutions and inclusions maps on E and F. Our first goal is to show that (1) is equivalent to It is clear that (1) implies (4.3). Conversely, assume (4.3). For any a P pA¸Sq`, we get Hence pψ¸Sqpker Λ E q Ď kerpΛ F q. So ψ¸S descends to a˚-homomorphism ψ¸r S as in (1). If, in addition, ψ is injective, then so is ψ 2 1 , and then the only one-sided implication in (4.4) may be reversed. Thus pψ¸Sqpker Λ E q " kerpΛ F q, so that ψ¸r S is injective. Since the canonical map A¸a lg S Ñ A¸r S is still injective, it also follows that ψ¸a lg S is injective. So we get all assertions in (1).
Next, we prove that (2) is equivalent to (4.3). By passing to biduals, we get a weakly continuous -homomorphism pψ¸Sq 2 : pA¸Sq 2 Ñ pB¸Sq 2 . The C˚-algebra A 2 and the Hilbert bimodules E 2 t for t P S embed naturally into pA¸Sq 2 . Similarly, F 2 t for t P S are embedded into pB¸Sq 2 .
So condition (2) only asserts the inverse inequality. This condition is always satisfied when S " G is a group, as then rI E 1,t s " 0 for t ‰ 1 (and rI E 1,1 s " 1). For general actions, (2) holds whenever ψ 1 pI E 1,t q " ψ 1 pspE t qqI F 1,t for all t P S. The latter equality is automatic for inclusion and quotient homomorphisms in Proposition 4.15 below.

Condition (2) in Proposition 4.2 may fail (and perhaps for some purposes one might want to include it in the definition of a homomorphism). We thank Alcides Buss, Diego Martínez and Jonathan Taylor for point this to us.
Example 4.7. Consider the actions whose crossed products are described in [10,Proposition 8.5]. Namely, let S " t´1, 0, 1u be the inverse semigroup with the usual number multiplication. Take any C˚-algebra A and any ideal I in A different from A. Let E 1 " E´1 :" A and E 0 " I be trivial Hilbert bimodules over A, and let µ t,u pa b bq " a¨b for t, u P S be just the multiplication in A. Then A¸E S " A¸E r S " A¸E alg S -A ' A{I (see [10, (8.6)]). We let F be the similar action with I replaced by A. Then A¸F S " A¸F r S " A¸F alg S -A. The inclusion maps yield a homomorphism ψ from E to F where ψ¸S " ψ¸a lg S : A ' A{I Ñ A is given by pa ' b`Iq Þ Ñ a.
This homomorphism is not injective, although ψ is. So conditions in Proposition 4.2 are not satisfied. Indeed, note that ψ¸r S " ψ¸S exists in this example, but it does not intertwine the canonical weak expectations, which are given by E E pa ' pb`Iqq " a`b 2`a´b 2 rIs and E F paq " a, for a, b P A, where rIs P A 2 is the support projection of I.

Restrictions of actions.
We fix an action E "`pE t q tPS , pµ t,u q t,uPS˘o f a unital inverse semigroup S on a C˚-algebra A by Hilbert bimodules. Definition 4.8. Let I E pAq :" tI P IpAq : IE t " E t I for every t P Su be the set of E-invariant ideals in A. Lemma 4.9. Let I P IpAq and let B be any S-graded C˚-algebra with grading pE t q tPS . The following are equivalent: (1) I is E-invariant, that is I P I E pAq; (2) I is restricted, that is, Proof. Proposition 2.13 implies that (1) and (2) are equivalent. It is easy to see that (3) and (4) are equivalent. If t P S, [26, page 645] or the proof of [1, Proposition 3.10]). That is, (1) and (3) Ý Ñ E tu I for t, u P S, forms an inverse semigroup action by Hilbert bimodules on the C˚-algebra I. For t P S, the quotient Banach space E t {E t I is a Hilbert A{I-bimodule in a natural way because E t I " IE t . If t, u P S, then the isomorphism There are natural isomorphisms of Hilbert bimodules Remark 4.11. Let I P I E pAq. The inclusions E t I Ď E t for t P S yield an injective homomorphism from E| I to E. The quotient maps E t Ñ E t {E t I for t P S yield a homomorphism from E to E| A{I . Remark 4.12. Let B be an S-graded C˚-algebra with grading pB t q tPS and let I P I B pAq. The induced ideal BIB carries the S-grading BIB X B t by Proposition 2.13. The quotient B{BIB is S-graded by the images of B t by Lemma 2.14.  Proof. Note that ιpI EI 1,t q " I E 1,t¨I " ιpspE t Iqq¨I E 1,t and κpI E 1,t q " I E{EI 1,t , for all t P S. Hence ι and κ satisfy the equivalent conditions in Proposition 4.2 by the last part of Remark 4.6. Thus, by Proposition 4.2, not only the˚-homomorphisms ι¸S, κ¸S but also ι¸r S, κ¸r S exist, and ι¸r S is injective. The maps κ¸S and κ¸r S are surjective because their images contain the dense˚-subalgebra A{I¸a lg S.
We prove that ι¸S is injective. Let π be a faithful, nondegenerate representation of I¸S on a Hilbert space H. Since I is nondegenerate in I¸S, the representation π| I is also nondegenerate. Therefore, for each ξ t P E t , the formula r πpξ t qπpaqh :" πpξ t aqh for a P I and h P H defines a bounded operator on H. Alternatively, r πpξ t q could be defined using an approximate identity pµ λ q for I, as the limit of the strongly convergent net πpξ t µ λ q. A standard proof shows that pr π t q tPS is a representation r π of E. The integrated representation r π¸S : A¸S Ñ BpHq satisfies pr π¸Sq˝pι¸Sq " π. Hence ι¸S is injective. The composite maps pκ¸Sq˝pι¸Sq and pκ¸r Sq˝pι¸r Sq vanish. Hence the range of ι¸S is contained in kerpκ¸Sq and the range of ι¸r S is contained in kerpκ¸r Sq. Conversely, we claim that the range of ι¸S contains kerpκ¸Sq. We identify I¸S with its image in A¸S. Let q : A¸S Ñ pA¸Sq { pI¸Sq be the quotient map. For t P S, the restriction of q to E t I vanishes. Hence q induces maps ψ t : E t {E t I Ñ pA¸Sq{pI¸Sq. They form a representation of E A{I . It integrates to a homomorphism ψ¸S : pA{Iq¸S Ñ pA¸Sq{pI¸Sq.

Definition 4.17.
The action E is exact if the sequence (4.16) is exact for each I P I E pAq.
Example 4.18 (Exactness of twisted groupoids). An action E of an inverse semigroup S on a commutative C˚-algebra A -C 0 pXq corresponds to a twisted étale groupoid pΣ, Gq with unit space X (see [8] 4.4. Exactness for essential crossed products. The essential crossed product is not functorial (see [30,Remark 4.8]). This complicates the definition of "exactness" for essential crossed products.
Only the quotient maps cause extra problems: Lemma 4.20. Let E be an action of S by Hilbert bimodules on a C˚-algebra A and let I P I E pAq. The injective homomorphism ι from E| I into E induces an injective˚-homomorphism ι¸e ss S : I¸e ss S Ñ A¸e ss S. Its image is the ideal in A¸e ss S generated by I.
Proof. If J is an essential ideal in I, then J ' I K is an essential ideal in A. The obvious inclusions MpJq Ñ MpJ ' I K q for the essential ideals J Ď I induce a natural isomorphism from M loc pIq onto an ideal in M loc pAq. Let EL I : I¸S Ñ M loc pIq be the canonical essential expectation. We are going to prove below that the following diagram commutes: (4.21)

I¸S A¸S
M loc pIq M loc pAq ι¸S EL I EL Then N EL X ι¸SpI¸Sq " ι¸SpN EL I q because ι¸SpI¸Sq is an ideal in A¸S. This, in turn, implies that the injective homomorphism ι¸S factors through an injective homomorphism ι¸e ss S : I¸e ss S " I¸S{N EL I Ñ A¸e ss S " A¸S{N EL .
To check (4.21), let t P S. If I t,1 " ř vďt,1 spE v q then I t,1 I " ř vďt,1 spE v Iq and the restriction of the map ϑ t,1 : E t¨It,1 " Ý Ñ I t,1 (defined in (3.8)) to E t I¨I t,1 coincides with the corresponding map defined for the restricted action pE s Iq sPS . Hence for each ξ P E t I the element ELpξq P MpI 1,t ' I K 1,t q Ď MpI 1,t I ' I K 1,t I ' I 1,t I K ' I K 1,t I K q acts on u P I 1,t I in the same way as EL I pξq: ELpξqu " ϑ 1,t pξuq " EL I pξqu.
Since ELpξqpI K 1,t I ' I 1,t I K ' I K 1,t I K q " 0, the embedding M loc pIq Ď M loc pAq maps EL I pξq to ELpξq. This proves (4.21). □ ). Let S :" G Y t0u be the inverse semigroup obtained by adjoining a zero element to an amenable discrete group G. Let G act on A " Cr0, 1s by E g " Cr0, 1s for g P G and E 0 " C 0 p0, 1s, equipped with the usual involution and multiplication maps. Then A¸e ss S " A. Every ideal I in A is E-invariant and I¸e ss S " I, and so I¸e ss S Ď A¸e ss S. However, if I :" C 0 p0, 1s, then A{I¸e ss S " C¸G " C˚pGq and the quotient homomorphism κ from E onto E| A{I does not induce a map from A to C˚pGq.
Definition 4.23. We call the action E essentially exact if for each I P I E pAq there is å -homomorphism κ¸e ss S : A¸e ss S Ñ A{I¸e ss S whose restriction to each fibre E t is the quotient map onto E t {E t I, t P S, and the kernel of κ¸e ss S is ι¸e ss SpI¸e ss Sq. Remark 4.24. When the action E is closed, then the reduced and essential crossed products coincide for all restrictions of E (see Proposition 4.14). Thus essential exactness is the same as exactness for closed actions of inverse semigroups and for Fell bundles over Hausdorff groupoids.
Example 4.25 (Essentially exact Fell bundles). Consistently with Example 4.19, we will call a Fell bundle A " pA γ q γPG over an étale groupoid essentially exact if the corresponding action E is essentially exact. More specifically, by Lemma 4.20, for any G-invariant ideal I in A, the inclusion C c pA| I q Ď C c pAq extends to an injective˚-homomorphism Ce ss pA| I q ↣ Ce ss pAq. So A is essentially exact if and only if, for every G-invariant ideal I in A, restriction of sections gives a well-defined˚-homomorphism Ce ss pAq ↠ Ce ss pA| A{I q and the following sequence is exact: Ce ss pA| I q ↣ Ce ss pAq ↠ Ce ss pA| A{I q.
If the S-action on A is residually aperiodic, then A separates ideals in A¸e ss S if and only if the S-action on A is essentially exact (see Theorem 5.9 below). In Example 4.22, however, A separates ideals in A¸e ss S although the S-action on A is not essentially exact. The following proposition shows that the S-action on A must be essentially exact if A separates ideals in A¸r S. (1) A separates ideals in the reduced crossed product A¸r S; (2) the action is exact and for each I P I E pAq, A{I detects ideals in A{I¸r S. If these equivalent conditions hold, then A{I¸r S " A{I¸e ss S for every I P I E pAq and the action is essentially exact.
Proof. Lemma 2.4 and Proposition 2.13 show that A separates ideals in A¸r S if and only if A{I detects ideals in A¸r S{ι r¸S pI¸r Sq for all I P I E pAq. Then the action is exact because the kernel of κ¸r S has to be ι r¸S pI¸r Sq, and then A¸r S{ι r¸S pI¸r Sq -A{I¸S for all I P I E pAq. Thus (1) and (2) are equivalent. Condition (2) implies A{I¸r S " A{I¸e ss S because otherwise A{I¸e ss S would be a quotient of A{I¸r S by a nonzero ideal not detected by A{I. □ We illustrate by an example what can go wrong with the exactness of essential crossed products. Our example is closely related to the Reeb foliation or, more precisely, to its restriction to a transversal.
Example 4.27. Let ϑ : R Ñ R be a homeomorphism with ϑptq " t for t ď 0 and ϑptq ą t for all t ą 0. Let G be the germ groupoid of the transformation groupoid Rˆϑ Z. We claim that Cr pGq -Ce ss pGq. To see this, we use that the restrictions of G to r0, 8q and p´8, 0q are Hausdorff. Indeed, G| p0,8q is the Hausdorff transformation groupoid p0, 8q¸ϑ Z because Z acts freely (and properly) on p0, 8q. Similarly, G| r0,8q is the Hausdorff transformation groupoid r0, 8q¸ϑ Z; the action of Z on r0, 8q fixes 0, but the germ of any n P Z at 0 is nontrivial because ϑ n acts nontrivially on p0, 8q. Therefore, the support of any nonzero element of Cr pGq must intersect r0, 8q and p´8, 0q in relatively open subsets. Hence the support cannot be meagre, and this proves our claim. This equality of reduced and essential groupoid C˚-algebras for G is not inherited by the restriction to the closed invariant subset p´8, 0s. Indeed, the restriction of G to p´8, 0s is the non-Hausdorff group bundle with trivial fibre over p´8, 0q and the fibre Z at 0 (see [30,Example 4.7] and Example 4.22). In this case, the full and reduced groupoid C˚-algebras are obtained by gluing together C 0 pp´8, 0qq and the fibre C˚pZq at 0. However, Ce ss pG| p´8,0s q " C 0 pp´8, 0sq, so Cr pG| p´8,0s q fl Ce ss pG| p´8,0s q.
Since the essential and reduced crossed products coincide for G and G| p0,8q , but not for G| p´8,0s , the following sequence of essential crossed products exists, but fails to be exact: 0 Ñ Ce ss pG| p0,8q q Ñ Ce ss pGq Ñ Ce ss pG| p´8,0s q Ñ 0. This is one way how essential crossed products may fail to be exact. The restriction G| t0u is simply the group Z. So Cr pG| t0u q " Ce ss pG| t0u q -C˚pZq. The restriction˚-homomorphism C˚pG| p´8,0s q Ñ C˚pG| t0u q does not descend to the essential crossed products. That is, there is no canonical map from Ce ss pG| p´8,0s q to Ce ss pG| t0u q. This is the second way how essential crossed products may fail to be exact.

Amenability vs exactness.
Definition 4.28. Let E be an S-action by Hilbert bimodules on a C˚-algebra A. We call the action E amenable if the regular representation Λ : A¸S Ñ A¸r S is an isomorphism.   where the top horizontal sequence is exact, and Λ I , Λ A , Λ A{I denote the respective regular representations. If Λ A{I is injective, then kerpκ¸r Sq " Λ A pkerpκ¸Sqq " Λ A pι¸SpI¸Sqq " ι¸r SpI¸r Sq.
Hence I¸r S ↣ A¸r S ↠ A{I¸r S is exact. In general, if this sequence is exact, the Snake Lemma from homological algebra yields a short exact sequence Hence kerpΛ A q " 0 if and only if kerpΛ I q " 0 and kerpΛ A{I q " 0. □ Remark 4.31. It is unclear whether the quotient action E| A{I for I P I E pAq is amenable if E is (the proof of [34, Lemma 3.9] is incorrect). If so, then by Lemma 4.30, the amenable action is also exact. Let S " G be a group. If E satisfies the approximation property in [13,Definition 4.4], then also every restriction E| A{I satisfies the approximation property. Hence the approximation property of a Fell bundle over a group implies both amenability and exactness. In particular, if there is an amenable group action E such that E| A{I is not amenable for some I P I E pAq, then the approximation property is strictly stronger then amenability (see also [13,Page 169]

Ideals and pure infiniteness for inverse semigroup crossed products
In this section we present efficient criteria for separation of ideals and pure infiniteness in essential crossed products. 5.1. Residual aperiodicity and ideal structure. Let E " pE t q tPS be an action of a unital inverse semigroup S by Hilbert bimodules on a C˚-algebra A. . Let G be an étale groupoid and X Ď G its unit space. The isotropy group of a point x P X is Gpxq :" s´1pxq X r´1pxq Ď G. We call G topologically free if, for every open U Ď GzX, the set tx P X : Gpxq X U ‰ Hu has empty interior.
Effective groupoids are topologically free. The converse implication holds if X is closed in G, but not in general.
Aperiodicity is related in [28,30,32] to topological freeness and several other conditions. We now introduce residual versions of aperiodicity and topological freeness.
Definition 5.5. The action E is residually aperiodic if, for each I P I E pAq, the restricted action E| A{I is aperiodic. Definition 5.6. An étale groupoid G with unit space X Ď G is residually topologically free if, for each nonempty closed G-invariant subset Y Ď X, the restricted groupoid r´1pY q " s´1pY q is topologically free.
The following lemma may help to show that a transformation groupoid is (residually) topologically free. (  (2) assume that G˙Y is topologically free and let U Ď pG˙XqzX. Then f˚pU q is open and contained in pG˙Y qzY . Let I U :" tx P X : Gpxq X U ‰ Hu and I f˚pU q :" ty P Y : Gpyq X f˚pU q ‰ Hu. We claim that f pI U q " I f˚pU q . To see this, let x P I U . Then there is an arrow px, g, xq P U ; here g P G is such that ϱpxq " spgq and g¨x " x. Since f is G-equivariant, then pf pxq, g, f pxqq P f˚pU q. This witnesses that f pxq P I f˚pU q . Since G˙Y is topologically free, I f˚pU q has empty interior.
Since f is open, the preimage of I f˚pU q in X has empty interior as well. It follows that I U has empty interior. This witnesses that G˙X is topologically free and proves (2). Finally, for any G-invariant set D Ď X the set f pDq is G-invariant. If we assume f pDq is closed in Y and G˙Y is residually topologically free, then G˙f pDq is topologically free. Since the restriction of f˚to G˙D is a continuous open map onto G˙f pDq, (2) implies that G˙D is topologically free. This proves (3). □ Theorem 5.8. Let E be an action of a unital inverse semigroup S on a C˚-algebra A by Hilbert bimodules. If A is separable or of Type I, then the following are equivalent: (1) the dual groupoid p A¸S is residually topologically free; (2) the action E is residually aperiodic; (3) for any I P I E pAq, the full crossed product for the restricted action E| A{I has a unique pseudo-expectation namely, the canonical M loc -expectation; (4) for any I P I E pAq, pA{Iq`supports C for each intermediate C˚-subalgebra A{I Ď C Ď A{I¸e ss S for the restricted action E| A{I ; (5) for any I P I E pAq, A{I detects ideals in each intermediate C˚-subalgebra A{I Ď C Ď A{I¸e ss S for the restricted action E| A{I .

Proof. A subset of p
A is closed and invariant if and only if it is of the form X " y A{I for an E-invariant ideal I. The dual groupoid of the induced action on A{I is the restriction X¸S of the dual groupoid to X. Hence the implications (1)ñ(2)ñ(3), (4) follow from [32,Corollary 4.8 and Theorem 3.6], applied to the quotients A{I and intermediate C˚-algebras. Lemma 2.23 shows that (4) implies (5), and [40,Theorem 3.5] shows that (3) implies (5). If A is separable or of Type I, so are its quotients A{I, and then [32,Proposition 6.1] shows that (5) implies (1). □ The following result justifies introducing the notion of essential exactness -it is an instance of condition (1) in Theorem 2.34. Theorem 5.9. Let B be an S-graded C˚-algebra with a grading E " pE t q tPS that forms a residually aperiodic action of S on A :" E 1 (this holds if the dual groupoid p A¸S is residually topologically free). The following are equivalent: (1) B -A¸e ss S and E is essentially exact; If the above equivalent conditions hold and the primitive ideal spaceǍ is second countable, then the quasi-orbit map induces a homeomorphismB -Ǎ{", whereǍ{" is the quasi-orbit space of the dual groupoidǍ¸S, that is, p 1 , p 2 PǍ satisfy p 1 " p 2 if and only if pǍ¸Sq¨p 1 " pǍ¸Sq¨p 2 .
Proof. The C˚-inclusion A Ď B is symmetric and residually aperiodic by Propositions 2.13 and 5.2. Hence by Theorem 2.34 conditions (2)-(4) are equivalent to the condition that for each I P I B pAq " I E pAq the unique pseudo-expectation E I : B{BIB Ñ A{I is almost faithful. Let us assume this. Then there is a commutative diagram where Ψ is the homomorphism that exists by universality of A{I¸S because B{BIB is graded by E| A{I by Lemma 2.14, EL I is the canonical essential expectation for A{I¸S, and M loc pA{Iq ãÑ IpA{Iq is the canonical embedding. The diagram commutes because the inclusion A{I Ď A{I¸S is aperiodic and hence there is a unique pseudo-expectation by [32,Theorem 3.6]. As a consequence, ker Ψ " Ψ´1p0q " Ψ´1pN E I q " N EL and thus Ψ factors through an isomorphism A{I¸e ss S -B{BIB. Since this holds for every I P I E pAq we get that B -A¸e ss S and that E is essentially exact (A¸e ss S{I¸e ss S -B{BIB -pA{Iq¸e ss S for I P I E pAq). Theorem 2.34 implies easily that (1) implies (2). This finishes the proof that the four conditions are equivalent. The remaining claims follow mostly from the last part of Theorem 2.34. That the quasi-orbit space has the asserted form follows from [29, Theorem 6.22]. □ Corollary 5.10. Let A " pA γ q γPG be a Fell bundle over an étale groupoid G with locally compact, Hausdorff unit space X, and put A " C 0 pA| X q. Define the dual groupoid p A¸G as in Example 3.6.

Assume that it is residually topologically free (this holds, for instance, if G is residually topologically free and the base map for the C˚-bundle A is open and closed). Assume also one of the following
(1) B :" Ce ss pAq and A is essentially exact; (2) B :" Cr pAq, A is exact, and the unit space inǍ¸G is closed (the latter is automatic if G is Hausdorff ); (3) B :" C˚pAq, A is separable, and G is amenable and Hausdorff.
Then A separates ideals in B and, even more, A`fills B. The lattice IpBq is naturally isomorphic to the lattice of G-invariant ideals in A. If, in addition,Ǎ is second countable, thenB -Ǎ{", whereǍ{" is the quasi-orbit space of the dual groupoidǍ¸G, that is, p 1 , p 2 PǍ satisfy p 1 " p 2 if and only if pǍ¸Gq¨p 1 " pǍ¸Gq¨p 2 .
Proof. The claims in brackets follow from Lemma 5.7. The claim in case (1) follows from Theorem 5.9 (see also Example 4.25). Case (2) follows from (1) and Remarks 3.16 and 4.24. Example 4.32 explains why (3) is a special case of (2). □ Theorem 5.9 allows us to describe the ideal structure of B in terms of A under the following assumptions: E is an essentially exact, residually aperiodic action and B " A¸e ss S.
We are going to study whether B is purely infinite using the same assumption. Before we do this, we simplify (5.11) in the presence of a conditional expectation. In order to use these results, we need conditions that suffice for a, b P A`to have the matrix diagonalisation property in B or for a P A`zt0u to be properly infinite in B. Checking the matrix diagonalisation property is usually difficult. Nevertheless, the following lemma may be useful (see [22,28]): Lemma 5.17. Let a, b P A`zt0u. Suppose that for each c P E t , t P S, and each ε ą 0 there are n, m P N and a i P aE si , s i P S, for i " 1, . . . , n and b j P bE tj , t j P S, for j " 1, . . . , m such that and ř n,m i"1,j"1 ai cb j « ε 0. Then a, b P A`zt0u have the matrix diagonalisation property in B. Proof. Let C :" Ť tPS E t and S :" Ť tPS E t . We claim that a, b P A`zt0u have the matrix diagonalisation property with respect to C and S as introduced in [23, Definition 4.6]. Indeed, let x P E t be such that`a xx b˘P M 2 pBq`and let ε ą 0. Let a i P aE si and b j P bE tj satisfy the conditions described in the assertion with c :" a 1 {2 xb 1 {2 . We may write a i " a 1 {2 x i and b j " b 1 {2 y j for some x i , y j . Let d 1 :" ř n i"1 x i and d 2 :" ř n j"1 y j . The assumed estimates imply that d1 ad 1 This proves our claim. Clearly, S is a multiplicative subsemigroup of B, S˚SS Ď S, ASA Ď S, and the closed linear span of C is B. Let us compare the definitions of infinite and properly infinite elements in Definition 2.35 to the definitions of E-infinite and properly E-infinite elements in Definition 5.18. There are two differences. First, we now choose the elements x, y P aB in the subalgebra A¸a lg S, so that we may write them as a finite sum ř a i with a i P aE ti . Secondly, we estimate each product ai a j for i ‰ j separately. The first change does not achieve much because A¸a lg S is dense in B and we only aim for approximate equalities anyway. The second change simplify the estimates a lot because ∥ai a j ∥ is computed in the Hilbert A-bimodule E ti tj , whereas the norm estimates in Definition 2.35 involve the C˚-norm of B. For an E-paradoxical element, we even assume the products ai a j for i ‰ j to vanish exactly. This is once again much easier to check. Paradoxical elements are also important because they are related to paradoxical decompositions, which were studied already by Banach and Tarski. In the setting of purely infinite crossed products, their importance was highlighted by Rørdam and Sierakowski [43]. The implications among our infiniteness conditions hinted at above are summarised in the following proposition: Proposition 5.20. Assume that A separates ideals in B. Consider the following conditions a P A`zt0u may satisfy: (1) a is properly infinite in B; (2) for each ε ą 0 there are n, m P N, t i P S, and a i P aE ti for 1 ď i ď n`m, such that ai a j « ε 0; (3) a is residually E-infinite; (4) a is properly E-infinite; (5) a is E-paradoxical. Then (1)ô(2)ð(3)ð(4)ð(5).
Proof. The implications (5)ñ(4)ñ(3) are straightforward. By [20,Proposition 3.14], a is properly infinite if and only if it is residually infinite. Since A separates ideals in B, any ideal in B comes from an invariant ideal in A, as in the definition that a is residually E-infinite. Together with Lemma 5.19, this shows that (3) implies (1).
According to Definition 2.35, a P A`zt0u is properly infinite in B if and only if, for all ε ą 0, there are x, y P a¨B with x˚x « ε a, y˚y « ε b and x˚y « ε 0. Without loss of generality, we may pick x, y P a¨p ř tPS E t q because ř tPS E t is dense in B. So x " ř n j"1 a j and y " ř n`m j"n`1 a j for some n, m P N, t i P S, and a i P aE ti for 1 ď i ď n`m. The relations x˚x « ε a, y˚y « ε b and x˚y « ε 0 translate to those described in (2). This proves that (1) and (2) are equivalent. □ It is unclear whether the implications in Proposition 5.20 may be reversed.
Remark 5.21. The example of graph C˚-algebras shows that it may be much easier to check that an element is residually E-infinite than that it is properly E-infinite (see also [34,Remark 7.10] We may simplify our conditions further if A is commutative. Then A¸e ss S -Ce ss pG, Σq for a twisted étale groupoid G with object spaceÂ. The twist Σ is always locally trivial. Therefore, the bisections that trivialise the twist Σ form a wide inverse subsemigroup S 1 among all bisections of G (see [7,Theorem 7.2]). Then Ce ss pG, Σq -A¸e ss S 1 . The action of S 1 on A is equivalent to a twisted action as in [7,Definition 4.1], that is, each E t for t P S comes from an isomorphism between two ideals in A. We assume this because it allows us to identify elements of E t with there is 0 ‰ a 2 ď a that is E-infinite; namely, choose U " supp a and then a 2 with supp a 2 " V and a 2 ď a for V as above.
Condition (3) in Lemma 5.23 could be relaxed so that it still implies E-infiniteness, by using compact subsets of V . We formulate the relevant condition implying E-proper infiniteness: Lemma 5.26. Retain the assumptions of Lemma 5.23. In particular, let a P A`and V :" tx P p A : apxq ‰ 0u. If for each compact subset K Ď V there are n, m P N, t 1 , . . . , t n`m P S, and open subsets V 1 , . . . , V n`m Ď V such that pt i¨Vi q X pt j¨Vj q " H if 1 ď i ă j ď n`m, K Ď Ť n i"1 V i and K Ď Ť n`m i"n`1 V i , then a is E-properly infinite. Proof. Fix ε ą 0. Let K :" tx P p A : apxq ě εu. Choose n, m, t i , V i as in the assumption of the lemma. Let w 1 , . . . , w n P A and w n`1 , . . . , w n`m P A be partitions of unity subordinate to the open coverings K Ď Ť n i"1 V i and K Ď Ť n`m i"n`1 V i , respectively. Let a i :" pa´εq 1{2¨w1{2 i for i " 1, . . . , n`m. As in the proof of the implication (3)ñ(4) in Lemma 5.23 one sees that treating a i as an element of E ti , the elements a i satisfy the relations in Definition 5.18. (3). □ Now we assume, in addition, that the spectrum p A is totally disconnected. This implies that the compact open bisections form a basis for the topology and that A is spanned by projections. We are going to see that a projection is E-paradoxical if and only if its support is p2, 1q-paradoxical as defined in [6]. Such open subsets give purely infinite elements in the type semigroup considered in [6,36,41]. spU i q, rpU i q X rpU j q " H for i ‰ j.
Proposition 5.28. Let E be an action of an inverse semigroup S by Hilbert bimodules on a commutative C˚-algebra A with totally disconected spectrum p A; equivalently, the dual groupoid G :" p A¸S is ample. A projection a P A`is E-paradoxical if and only if its support V :" tx P p A : apxq ‰ 0u is p2, 1q-paradoxical.
Proof. Suppose first that a P A`zt0u is E-paradoxical. That is, there are n, m P N, t 1 , . . . , t n`m P S, and a i P aE ti such that a " ř n i"1 ai a i " ř n`m i"n`1 ai a i and ai a j " 0 for i ‰ j. Let 1 ď i ď n`m. Recall that we may treat E ti as spaces of sections A Ui of a line bundle over G " p A¸S that are supported on open bisections U i P BispGq. Thus U i :" tγ P G : ∥a i pγq∥ ą 0u is an open bisection of G contained in U i . Since a i P aA Ui , we have rpU i q " tx P p A : pa i ai qpxq ą 0u Ď V . And ai a j " 0 implies that rpU i q X rpU j q " H for all i ‰ j. Since tx P X : pai a i qpxq ą 0u " spU i q, the equalities a " ř n i"1 ai a i and a " ř n`m i"n`1 ai a i imply V " Ů n i"1 spU i q and V " Ů n`m i"n`1 spU i q. Hence the family U i P BispGq for 1 ď i ď n`m has all the desired properties, except that U i need not be compact. However, since G is ample, every U i is a union of some compact open bisections. Since V is compact and V " Ů n i"1 spU i q " Ů n`m i"n`1 spU i q, we may, in fact, replace each U i for 1 ď i ď n`m by a finite union of compact open bisections. This gives a compact open bisection.
Conversely, let U i Ď G for 1 ď i ď n`m be a family of bisection as in Definition 5.27. Let S 1 Ď BispGq be the family of open compact bisections that trivialise the twist, that is, the restrictions of the associated line bundle over G to sets in S 1 are trivial. Note that S 1 forms an inverse semigroup and a basis for the topology of G; this holds for the family of all open bisections that trivialise the twist, by the proof of [7,Theorem 7.2], and for the family of all compact open bisections because G is ample. Since V " Ů n i"1 spU i q " Ů n`m j"n`1 spU j q is compact, for each i " 1, . . . , n we may find a finite family of sets pU i,j q ni j"1 Ď S 1 such that Ť ni j"1 U i,j Ď U i and V " Ť n i"1 Ť ni j"1 spU i,j q. Since the bisections pU i,j q ni j"1 Ď U i are closed and open, we may arrange that the sets pspU i,j qq ni j"1 are pairwise disjoint. Then the sets pU i,j q n,ni i"1,j"1 are pairwise disjoint, and since Ť ni j"1 U i,j Ď U i , for i " 1, . . . , n, also prpU i,j qq n,ni i"1,j"1 are pairwise disjoint. We put a i,j :" 1 Ui,j , for i " 1, . . . , n, j " 1, . . . , n i . By the choice of bisections in S 1 , we may treat a i,j as an element of the space C c pU ij q of sections of the line bundle over G. By the construction of the Fell bundle over p A¸S, by passing if necessary to smaller sets, we may assume that each space C c pU i,j q is contained in E tij for some t ij P S. Hence a i,j P E tij for all i, j. Using the Fell bundle structure, we get n,ni ÿ i"1,j"1 ai ,j¨ai,j " n,ni ÿ i"1,j"1 1 spUi,j q " 1 V " a.
Similarly, we get ř n`m,ni i"n`1,j"1 ai ,j¨a i,j " a and ai ,j a i 1 ,j 1 " 0 for all pi, jq ‰ pi 1 , j 1 q. Hence a is E-paradoxical. □ Corollary 5.29. Let pG, Σq be an essentially exact twisted groupoid where G is ample and residually topologically free with locally compact Hausdorff X :" G 0 . If every compact open subset of X is p2, 1q-paradoxical, then the essential C˚-algebra Ce ss pG, Σq is purely infinite (and has the ideal property).
Proof. View Ce ss pG, Σq as the essential crossed product by an inverse semigroup action E on C 0 pXq as in Examples 3.21 and 3.22. The assertion follows from Proposition 5.28 and Corollary 5.22. □ Remark 5.30. When G is Hausdorff, then Ce ss pG, Σq " Cr pG, Σq and pG, Σq is essentially exact if and only if it is inner exact. Thus Corollary 5.29 generalises the pure infiniteness criteria in [6,41], where the authors considered Hausdorff ample groupoids without a twist. They proved, in addition, that if the type semigroup associated to G is almost unperforated, then the implication in Corollary 5.29 may be reversed. We will generalise this and some other results of Ma [36] to étale twisted groupoids in the forthcoming paper [33].