aa r X i v : . [ m a t h . OA ] J a n A GENERALIZATION OF RENAULT’S THEOREM FORCARTAN SUBALGEBRAS
ALI RAAD
Abstract.
We prove a generalized version of Renault’s theorem for Cartansubalgebras. We show that the original assumptions of second countabilityand separability are not needed. This weakens the assumption of topologicalprincipality of the underlying groupoid to effectiveness. Introduction and statement of Results
Jean Renault, in [7], manages to find a correspondence between Cartan subalgebrasof separable C ∗ -algebras, and ´etale twisted groupoids. Specifically, we have: Theorem 1.1 (Renault’s Theorem, 5.2 and 5.9 of [7]) . Let ( G, Σ) be a twisted ´etaleHausdorff locally compact second countable topologically principal groupoid. Then C ( G ) is a Cartan subalgebra of C ∗ r ( G, Σ) .Conversely, if B is a Cartan subalgebra of a separable C ∗ -algebra A , then thereexists a twisted ´etale Hausdorff locally compact second countable topologically prin-cipal groupoid ( G, Σ) and an isomorphism which carries A onto C ∗ r ( G, Σ) and B onto C ( G ) . In this paper we generalize this theorem. Specifically, for the first statement of The-orem 1.1, we remove the second countability assumption and also weaken topolog-ical principality to merely assuming effectiveness of the groupoid. For the conversestatement we remove separability, but pay the price by obtaining a groupoid thatis not necessarily second countable and thus not necessarily topologically principal.We obtain:
Theorem 1.2.
Let ( G, Σ) be a twisted ´etale Hausdorff locally compact effectivegroupoid. Then C ( G ) is a Cartan subalgebra of C ∗ r ( G, Σ) .Conversely, if B is a Cartan subalgebra of a C ∗ -algebra A , then there exists a twisted´etale Hausdorff locally compact effective groupoid ( G, Σ) and an isomorphism whichcarries A onto C ∗ r ( G, Σ) and B onto C ( G ) . Recall the definition of Cartan subalgebras:
Definition 1.3. A C ∗ -subalgebra B of a C ∗ -algebra A is a Cartan subalgebra if • B contains an approximate unit for A , • B is a masa (maximal Abelian subalgebra) in A , • B is regular in A (the normalizer set of B , N ( B ) = { a ∈ A : aBa ∗ ⊂ B and a ∗ Ba ⊂ B } , generates A as a C ∗ -algebra), and finally, • there exists a faithful conditional expectation P : A ։ B . Section 2 of this paper is devoted to summarising Renault’s proofs from [7]. Herewe will highlight the main ideas in the construction, especially those ideas that wewill eventually generalize.Section 3 provides the argument for Theorem 1.2. Here we develop the techniquesrequired to generalize the relevant sections of Renault’s proof, and highlight wherethey are used.The first novelty in our approach is that we make use of an Urysohn type lemmafor locally compact Hausdorff spaces that are not necessarily second countable, inorder to obtain certain separation functions which are used in Renault’s proofs.Of course, if one assumes second countability, then paracompactness follows andhence so does normality, which yields the usual version of Urysohn’s lemma, andobtaining certain separation functions becomes trivial.A second novelty is that we use these weakened properties, together with Dini’stheorem, to get an approximate identity in C ( G ) for C ∗ r ( G, Σ), without an as-sumption of second countability on G . Of course with second countability thisis a trivial task as the space becomes σ -compact and hence can be exhausted bycompact subsets.By removing the assumptions of second countability and separability one obtainsgroupoids that are not necessarily topologically principal, but rather effective.One of the advantages of Theorem 1.2 is that it may be applied to Cartan sub-algebras of C ∗ -algebras that are not necessarily separable. An important class ofnon-separable C ∗ -algebras are the uniform Roe algebras, which are of interest asthey build a link to coarse geometry (see Section 1 in [4]). These have Cartan sub-algebras (see Section 6 in [3]) which fall outside that which Renault’s theorem cancapture. In addition, the authors of [3] obtain a distinguished Cartan subalgebra byusing a slight modification of Renault’s theorem, where second countability of thegroupoid is weakened to σ -compactness. Of course with the generalized Theorem1.2 this is not necessary. Acknowledgements:
This research was conducted as part of my PhD project inQueen Mary University of London and the University of Glasgow. This project hasreceived funding from the European Research Council (ERC) under the EuropeanUnion’s Horizon 2020 research and innovation programme (grant agreement No.817597). I would like to thank Prof. Xin Li for his supervision in this project.2.
Summary of Renault’s Proof
This section serves as a summary of the constructions in [7], which yield Theorem1.1. We assume throughout that the reader is familiar with the basic notions inthe theory of ´etale groupoids. Information on this may be found, amongst otherplaces, in Chapter 3 of [5], Chapters 1.1 and 1.2 of [6], and/or Chapters 2 and 3 of[9].
GENERALIZATION OF RENAULT’S THEOREM FOR CARTAN SUBALGEBRAS 3
The first statement in Theorem 1.1 is that C ( G ) is a Cartan subalgebra of C ∗ r ( G, Σ) for a twisted ´etale Hausdorff locally compact second countable topologi-cally principal groupoid. In order to explain how this arises, we start by defining thenotions of topological principality, the twist Σ, and how one obtains a C ∗ -algebra C ∗ r ( G, Σ) from such groupoids. Thereafter we show that C ( G ) is a Cartan sub-algebra of C ∗ r ( G, Σ), which involves showing that it satisfies the requirements inDefinition 1.3.
Definition 2.1.
An ´etale groupoid G is topologically principal if the set of pointsin G with trivial isotropy is dense in G . We now summarize pages 39-41 of [7] and pages 975-976 of [2], which defines thetwist and shows how to get a C ∗ -algebra. Let Σ be a topological groupoid that isalso a principal T -space. Let G := Σ / T , which is made into a topological groupoidin the natural way.This gives rise to a T -bundle: T → Σ → G. We say Σ is a twist over G . In the language of exact sequences this is equivalentto a central extension: T × G ֒ → Σ ։ G. It is convenient to consider another T -bundle: T → C × Σ → ( C × Σ) / T . The T -action is given by t ( z, γ ) = ( tz, tγ ), and of course the projection onto thebase space is the canonical projection onto orbit classes [ z, γ ]. Set L := ( C × Σ) / T and form the complex line bundle: C → L → G. The projection map onto the base space is given by [ z, γ ] → ˙ γ , where ˙ is the canoni-cal projection Σ → G . Continuous sections of this line bundle have a representationvia equivariant continuous maps Σ → C (maps satisfying f ( tγ ) = tf ( γ )).Let G be a locally compact Hausdorff groupoid with Haar system { λ x : x ∈ G } ,and let Σ be a twist over G . We denote this pairing of a groupoid and its twistby ( G, Σ). Consider the space of compactly supported continuous sections G → L ,which we denote by C C ( G, Σ). Define a multiplication and involution on C C ( G, Σ)which turns it into a *-algebra, as follows: for f, g ∈ C C ( G, Σ), let(1) f g ( σ ) = Z G f ( στ − ) g ( τ )d λ s ( σ ) ( ˙ τ ) , f ∗ ( σ ) = f ( σ − ) . For each x ∈ G , let H x = L ( G x , L x , λ x ), where L x := p − ( G x ) where p is theprojection L → G . Then define π x : C C ( G, Σ) → B ( H x ) by π x ( f ) ζ ( σ ) = Z G f ( στ − ) ζ ( τ )d λ x ( ˙ τ ) , where ζ ∈ H x . ALI RAAD
Finally, define the norm k f k := sup x ∈ G k π x ( f ) k B ( H x ) , and complete C C ( G, Σ) withrespect to this norm, obtaining the reduced twisted groupoid C ∗ -algebra C ∗ r ( G, Σ).It can be shown that for every f ∈ C C ( G, Σ), we have k π x ( f ) k ≤ k f k I :=max sup y ∈ G R G | f | d λ y , sup y ∈ G R G | f ∗ | d λ y ! , which can in turn be shown to be a normon C C ( G, Σ). This whole construction does not assume second countability ortopological principality of G . For thorough details, consult Chapter 2, Section 1,of [6].Let supp ′ ( f ) = { γ ∈ G : f ( γ ) = 0 } . Renault shows in Proposition 4.1 and itsconsequences in [7] that we obtain the following properties: Proposition 2.2.
Let ( G, Σ) be a twisted ´etale locally compact Hausdorff groupoid.Then: • For all f ∈ C C ( G, Σ) we have | f ( σ ) | ≤ k f k for every σ ∈ Σ , and R G | f | d λ x ≤k f k for every x ∈ G . • The elements of C ∗ r ( G, Σ) can be represented as continuous sections of theline bundle L . • The multiplication and involution equations in (1) are valid for elements of C ∗ r ( G, Σ) . • We have the identification C ( G ) = { f ∈ C ∗ r ( G, Σ) : supp ′ ( f ) ⊂ G } . Renault proves that starting with a Hausdorff ´etale locally compact second count-able topologically principal twisted groupoid ( G, Σ), one can obtain that C ( G ) isa Cartan subalgebra of C ∗ r ( G, Σ) (we will say that ( C ∗ r ( G, Σ) , C ( G )) is a Cartanpair).The first thing to check is that the subalgebra contains an approximate unit forthe C ∗ -algebra. Since the groupoid is locally compact and second countable it is σ -compact so admits an exhaustion by compact sets, and finding an approximateunit becomes trivial.Theorem 4.2 in [7] shows that an element of C ∗ r ( G, Σ) commutes with C ( G ) ifand only if its open support is contained in G ′ which then yields that C ( G ) is amasa in C ∗ r ( G, Σ) (since G is topologically principal).Proposition 4.3 in [7] asserts the existence of a unique faithful conditional expec-tation P : C ∗ r ( G, Σ) → C ( G ) defined by restriction. That this is a faithfulconditional expectation can be checked directly by definitions, but uniqueness isobtained through topological principality and second countability of the groupoid.Indeed, Renault shows that any other conditional expectation Q would have toagree with P on C C ( G, Σ), by dividing the argument into two cases. First, byconsidering elements in C C ( G, Σ) whose compact support is contained in an openbisection that does not meet G , and second by considering an arbitrary element f in C C ( G, Σ) but reducing to the first case by covering the support of of f by thosebisections that do not meet G , and one that does, and then using a partition ofunity subordinate to such a finite cover. GENERALIZATION OF RENAULT’S THEOREM FOR CARTAN SUBALGEBRAS 5
In this argument, Renault makes crucial use of Urysohn’s lemma which meansthat one can find elements in C C ( G ) that separate closed subsets from disjointpoints. Of course with the space assumed second countable and locally compact,it is regular hence paracompact hence normal, and so Urysohn’s lemma applies.Finally, one needs the regularity of C ( G ) in C ∗ r ( G, Σ). Renault shows this inProposition 4.8 and Corollary 4.9 in [7]. He proves that the elements of the nor-malizer set N ( C ( G )) are exactly those elements of C ∗ r ( G, Σ) whose open supportis a bisection. Then since an element in C C ( G, Σ) can be written as a finite sum ofelements each of whose open support is an open bisection (this is because G is ´etaleand so has a basis of open bisections, and so one can use a partition of unity withrespect to a finite cover), and each summand is a normalizer, the result follows.This proves how one goes from a Hausdorff ´etale locally compact second countabletopologically principal twisted groupoid ( G, Σ) to a Cartan pair( C ( G ) , C ∗ r ( G, Σ)), which is the first statement in Theorem 1.1.For the reverse statement, Renault starts with an arbitrary Cartan pair (
A, B ),and constructs a Hausdorff ´etale locally compact second countable topologicallyprincipal twisted groupoid ( G ( B ) , Σ( B )). The construction is given in [7].Start by letting X = Spec( B ), and define D ( B ) = { ( x, n, y ) ∈ X × N ( B ) × X : α n ( y ) = x } . Here, α n : { x ∈ X : n ∗ n ( x ) > } → { x ∈ X : nn ∗ ( x ) > } is theunique homeomorphism satisfying n ∗ bn ( x ) = b ( α n ( x )) n ∗ n ( x )for all b ∈ B , x ∈ { x ∈ X : n ∗ n ( x ) > } (see Proposition 4.7 in [7]). Let Σ( B ) = D/ ∼ where ( x, n, y ) ∼ ( x ′ , n ′ , y ′ ) if and only if y = y ′ and there exists a b, b ′ ∈ B with b ( y ) , b ′ ( y ) > nb = n ′ b ′ . Σ( B ) is the given the groupoid of germs structure(see Section 3 in [7]). G ( B ) is defined as the image of the map taking [ x, n, y ] inΣ( B ) to [ x, α n , y ]. The map is well-defined and G ( B ) also inherits the groupoid ofgerms structure.We may identify the set B := { [ x, b, x ] : b ∈ B, b ( x ) = 0 } with T × X via the map[ x, b, x ] → (cid:16) b ( x ) | b ( x ) | , x (cid:17) . Proposition 4.14 in [7] gives an algebraic extension B → Σ( B ) → G ( B ) . It makes use of the fact that given a point and an open neighbourhood around itin X , we may find a compactly supported continuous function on X with compactsupport inside U .The topology is recovered in Lemma 4.16 in [7], so that Σ( B ) becomes a locallytrivial topological twist over G ( B ).Let L ( B ) be the complex line bundle arising, as we described in this section, as C → L ( B ) → G ( B ). The aim is to construct continuous sections of this linebundle, which is equivalent to having equivariant continuous maps Σ( B ) → C .This is done in Lemma 5.3 in [7], where one defines, for a ∈ A and ( x, n, y ) ∈ D ,ˆ a ( x, n, y ) = P ( n ∗ a )( y ) √ n ∗ n ( y ) ( P is the conditional expectation associated to the Cartanpair), showing that this is independent of choice of representative for a class inΣ( B ), and so this map can be defined on the quotient, and is continuous and ALI RAAD equivariant. The lemma also shows that the map ˆ which sends a to ˆ a is injectiveand linear. Injectivity makes use of the fact that elements in B can separate closedsets from points, which is automatic when A is separable as it implies that X is second countable and locally compact Hausdorff and one may apply Urysohn’slemma.Proposition 5.7 in [7] argues that because the elements { ˆ a : a ∈ A } separate thepoints of G ( B ), this groupoid is Hausdorff. Being a groupoid of germs it is also´etale. Again this makes use of Urysohn’s lemma.Lemma 5.8 in [7] proves that the map ˆ is a *-algebra isomorphism from A C to C C ( G ( B ) , Σ( B )), where A C is the linear span of elements of N ( B ) whose imageunder ˆ has compact support. Theorem 5.9 in [7] then extends this to showingthat the map is an isometry with respect to the C ∗ -algebra norms, sending A onto C ∗ r ( G ( B ) , Σ( B )) and B onto C ( G ( B ) ). Separability of A implies secondcountability of the groupoid. Topological principality is concluded from Proposition3.6 in [7].Hence the construction completes the second statement in Theorem 1.1. Startingfrom a Cartan pair ( A, B ) with A separable we obtain a locally compact Hausdorff´etale topologically principal second countable twisted groupoid ( G ( B ) , Σ( B )).The two procedures, going from twisted groupoids to Cartan pairs and from Cartanpairs to twisted groupoids are inverse to each other. Indeed, Proposition 4.15 in [7]tells us that if ( G, Σ) is a locally compact Hausdorff ´etale topologically principalsecond countable twisted groupoid then if we let A = C ∗ r ( G, Σ) and B = C ( G )then we have an isomorphism of extensions: Diagram A B Σ( B ) G ( B ) T × G Σ G We already have noted the isomorphism of the left vertical arrow. The isomorphismof the right vertical arrow is given by Proposition 4.13 in [7], and the middle verticalarrow is discussed in Proposition 4.15 in [7]. In Section 3 we will give alternateproofs of these statements that do not rely on any second countability or topologicalprincipality assumptions.Diagram A tells us that the process( G, Σ) ( A = C ∗ r ( G, Σ) , B = C ( G )) ( G ( B ) , Σ( B ))is the identity (up to isomorphism). Theorem 5.9 in [7] tells us that for a Cartanpair ( A, B ), the process(
A, B ) ( G ( B ) , Σ( B )) ( C ∗ r ( G ( B ) , Σ( B )) , C ( G ( B ) ))is the identity (up to isomorphism).Note that the automorphism group of the twisted groupoids can thus be identifiedwith the automorphism group of the associated Cartan pair. GENERALIZATION OF RENAULT’S THEOREM FOR CARTAN SUBALGEBRAS 7 Generalizing Renault’s Theorem
This section proves Theorem 1.2. We will first prove the first statement in thetheorem by focusing on the areas in Renault’s construction that make use of thesecond countability of the groupoid or the topological principality.We will need the following definition:
Definition 3.1.
An ´etale groupoid G is called effective if G ′ = int( G ) . We will require certain separation properties for locally compact Hausdorff spaces,which are used implicitly throughout [7], and which are trivially respected whenthe topological space is second countable. However, we verify that we have therequired results even for non second countable spaces.
Lemma 3.2.
Let X be a locally compact Hausdorff space. Then (1) Given a compact subset K of X and an open U such that K ⊂ U ⊂ X ,there exists b ∈ C ( X ) with b ≡ on K , and 0 outside U . (2) Given a closed subset C ⊂ X and a point x ∈ X disjoint from C , there is b ∈ C ( X ) where b ( x ) = 1 and b | C ≡ . (3) Given an open set U ⊂ X containing a point x there exists an open set V containing x such that V is a compact subset of U .Proof. Claim (1) is Urysohn’s lemma for locally compact Hausdorff spaces (see[8], 2.12). Since locally compact Hausdorff spaces are regular, we may use (1) todirectly get (2).Let us prove (3). By local compactness, there is an open set O containing x anda compact set K containing O . Let D = K ∩ U C , which is a closed subset of acompact set, hence compact. By regularity, we may find disjoint open sets O D containing D and O x containing x . Let V = O x ∩ O which is open and contains x , and note that V ⊂ V ⊂ O ⊂ K , hence V is compact. Now note also that V ⊂ O x ⊂ O CD ⊂ D C = K C ∪ U . Since V ⊂ K it follows V ⊂ U . (cid:3) We may use the above Lemma to find an approximate unit in C ( G ) for C ∗ r ( G, Σ),without reverting to second countability.
Proposition 3.3.
Let ( G, Σ) be a twisted ´etale locally compact Hausdorff groupoid.Then C ( G ) contains an approximate unit for C ∗ r ( G, Σ) .Proof. Consider the (different) C ∗ -algebra ( C ( G ) , kk ∞ ). We know that this hasan approximate unit ( b i ) i ∈ I . Let a ∈ C C ( G, Σ). Then K := r ( supp ( a )) is compact.From Lemma 3.2 we have that there exists b ∈ C ( G ) with b ≡ K . Then0 ← k b i b − b k ∞ ≥ k ( b i b − b ) | K k ∞ . Hence b i → K .Define f i ( y ) := R G | b i a − a | d λ y . Note that | b i a − a | ( σ ) = | b i ( r ( σ )) a ( σ ) − a ( σ ) | , whichis 0 for σ / ∈ supp ( a ), and converges uniformly to 0 for σ ∈ supp ( a ). Hence, for ALI RAAD fixed y ∈ G , f i ( y ) →
0. Note also that each f i is continuous (by definition of Haarsystem), with support in the compact set s ( supp ( a )). Now because ( b i ) i ∈ I is anapproximate unit for a C ∗ -algebra, we have, for i ≤ j , that | b i a − a | ( σ ) = | b i ( r ( σ )) a ( σ ) − a ( σ ) | = | b i ( r ( σ )) − || a ( σ ) | ≥| b j ( r ( σ )) − || a ( σ ) | = | b j ( r ( σ )) a ( σ ) − a ( σ ) | = | b j a − a | ( σ ) . (2)Hence f i ( y ) ≥ f j ( y ). Hence the f i ’s is a net of monotonically decreasing continuousmaps with compact support, converging pointwise to 0. By the generalized Dini’sTheorem (see [10], Corollary 6) we have that f i → ab i rather than b i a (we just switch the rangemap r to the source map s ), and if we consider ( b i a − a ) ∗ in the integrand (asthe b i ’s are positive). Hence we get that the net ( b i ) i ∈ I is an approximate unit for C C ( G, Σ) with respect to the kk I -norm, and hence with respect to the kk -norm.By density of C C ( G, Σ) inside C ∗ r ( G, Σ), we obtain the result. (cid:3)
In order to get that C ( G ) is a masa in C ∗ r ( G, Σ), it suffices to assume that G iseffective in Theorem 4.2 in [7] rather than topologically principal.As was stated in Section 2, in order to get a unique faithful conditional expectationin Proposition 4.3 in [7], Renault makes use of Urysohn’s lemma. We may nowuse Lemma 3.2 (2) instead and obtain the same result. Additionally, assumingeffectiveness rather than topological principality suffices.To show that the normalizer set is regular, Proposition 4.8 in [7] can easily bereplaced with effectiveness rather than topological principality.Together, this gives the first statement of Theorem 1.2. For the reverse statement,recall from Section 2 that Renault proves that there is an extension B → Σ( B ) → G ( B )by making use of the fact that one can find compactly supported continuous functionwith support inside a given open set. We may now instead use Lemma 3.2 (3) forthis.The rest of the arguments all use separation properties that are found in Lemma3.2 without having to allude to second countability. Of course in Theorem 5.9 in[7] one would not get a second countable groupoid G ( B ) if separability of A isremoved, and hence G ( B ) might not be topologically principal. However, being an´etale groupoid of germs it is automatically effective.Together this yields the second statement of Theorem 1.2. In order to get DiagramA without second countability or topological principality assumptions, Proposition4.13 of [7] must be slightly modified. In particular, we modify the argument thatgives the existence of an n ∈ A for which S = supp ′ ( n ), as in the proof.Indeed if S is an open bisection on which L is trivializable, then there exists anon-vanishing continuous section u : S → L . Since it is non-vanishing, we may justassume | u ( γ ) | = 1 for all γ ∈ S . By Lemma 3.2 there exists an h ∈ C ( G ) withcompact support in s ( S ). Let U = { γ ∈ G : h ( s ( γ )) = 0 } ∩ S . Then U is an openbisection contained in S . Define n : G → L by n ( γ ) = u ( γ ) h ( s ( γ )) if γ ∈ U , and 0otherwise. GENERALIZATION OF RENAULT’S THEOREM FOR CARTAN SUBALGEBRAS 9
Now we may consider the C ∗ -algebra ( C ( s ( S )) , kk ∞ ). We know there exists a net( h α ) in C C ( s ( S )) converging uniformly to h | s ( S ) . By an abuse of notation we shallassume the h α ’s converge uniformly to h on G (we just extend them to the wholespace by declaring them to be 0 outside s ( S )). Now uh α converges uniformly to n on G .Now fix a y ∈ G . Consider the integral R G | uh α − n | d λ y = R S | uh α − n | d λ y . It evaluatesto | h α ( s ( γ y )) − h ( s ( γ y )) | , where γ y is the unique point satisfying s ( γ y ) = y . Thisuniqueness comes from the fact that S is a bisection. Since convergence is uniform,we have that the integral converges uniformly, with respect to y ∈ G , to 0. Hence uh α → n in the kk I -norm and hence n ∈ A . We have supp ′ ( n ) = U and so n ∈ N ( B ).The remaining parts of the proof work by just assuming effectiveness of the groupoid.To get the isomorphism of the middle arrow in Diagram A we offer an alternativeproof to Proposition 4.15 of [7]. We thank Prof. Xin Li for the proof: Proposition 3.4.
Let A = C ∗ r ( G, Σ) and B = C ( G ) , for a twisted ´etale Hausdorfflocally compact effective groupoid ( G, Σ) . Then we have a canonical isomorphismof extensions: B Σ( B ) G ( B ) T × G Σ G Proof.
We have already shown the left and right vertical arrows to be isomorphisms.We just need an isomorphism of the middle vertical arrow. This is achieved asfollows. We define a map Σ( B ) → Σ by [ x, n, y ] → n ( σ ) | n ( σ ) | σ , where σ ∈ Σ chosen sothat ˙ σ ∈ supp ′ ( n ) with s ( ˙ σ ) = y and r ( ˙ σ ) = x . The inverse Σ → Σ( B ) is defined bysending σ → h x, n ( σ ) | n ( σ ) | n, y i where n ∈ N ( B ) chosen so that n ( σ ) = 0, and y = s ( ˙ σ ), x = r ( ˙ σ ). It is a tedious but straightforward task to check that these maps arewell-defined groupoid homomorphisms, and are inverse to each other. (cid:3) Finally, we remark that since the proof of Proposition 5.11 in [7] does not usethe separability of the C ∗ -algebra nor the second countability of the groupoid, itfollows that for any Cartan pair ( A, B ) we have that B satisfies the unique extensionproperty if and only if the groupoid G ( B ) is principal. References [1] Fell, J.M.G.
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