Almost finiteness, comparison, and tracial Z -stability
aa r X i v : . [ m a t h . OA ] M a y ALMOST FINITENESS, COMPARISON, AND TRACIAL Z -STABILITY HUNG-CHANG LIAO AND AARON TIKUISIS
Abstract.
Inspired by Kerr’s work on topological dynamics, wedefine tracial Z -stability for sub- C ∗ -algebras. We prove that fora countable discrete amenable group G acting freely and mini-mally on a compact metrizable space X , tracial Z -stability for thesub- C ∗ -algebra ( C ( X ) ⊆ C ( X ) ⋊ G ) implies that the action hasdynamical comparison. Consequently, tracial Z -stability is equiv-alent to almost finiteness of the action, provided that the actionhas the small boundary property. Introduction
Operator algebras and dynamical systems had long enjoyed a richand sometimes surprising relationship. Murray and von Neumann’suniqueness theorem on hyperfinite II factors ([19]) is paralleled byDye’s uniqueness of hyperfinite ergodic measure-preserving equivalencerelations [4]. Connes’ celebrated “injectivity implies hyperfiniteness”theorem ([2]) also has a direct analogue — the Connes–Feldman–Weisstheorem — which asserts that amenable nonsingular measured equiva-lence relations are hyperfinite ([3]). The connection between these twofields goes far beyond mere analogies. There is a natural constructionof a von Neumann algebra from a measure-preserving equivalence rela-tion, and amenable (respectively, hyperfinite) equivalence relations areexactly those whose associated von Neumann algebras are amenable(respectively, hyperfinite). In fact, Feldman and Moore showed in [5]that there is a full-fledged correspondence between Cartan subalgebrasof von Neumann algebras and (twisted) measured equivalence relations.The same types of analogies and connections are emerging between C ∗ -algebras and topological dynamics. Here the interest largely sur-rounds properties in the Toms–Winter conjecture — a C ∗ -algebraicconjecture designed to create a robust notion of regularity, primarilyfor the purpose of identifying classifiable C ∗ -algebras. More precisely,
MSC 2010: 54H20, 46L35.
Keywords: Topological dynamics; Cartan subalgebras of C ∗ -algebras; Z -stability;comparison; almost finiteness . Amenability is also of interest, but developments on both the C ∗ - and dynam-ical sides (e.g., [27, 6]) show that amenability alone has weaknesses in terms ofclassifying C ∗ -algebras. Hence the question here is: under the base assumption ofamenability, what additional conditions are needed to ensure regularity? the Toms–Winter conjecture predicts that for simple unital separablenon-elementary nuclear C ∗ -algebras, the following are equivalent:(C1) finite nuclear dimension;(C2) Z -stability;(C3) strict comparison.The implications (C1) ⇒ (C2) and (C2) ⇒ (C3) were established infull generality in [28] and [23], respectively. (C2) ⇒ (C1) was recentlysettled in [1], building on the groundbreaking work of Matui and Sato([18]). The only implication which is not completely settled is (C3) ⇒ (C2). The most general result at the moment was obtained in [1](based on another fundamental work of Matui and Sato [17]): strictcomparison implies Z -stability provided the the C ∗ -algebra has “uni-form property Γ”.On the dynamical side, Kerr defined for actions of countable discreteamenable groups on compact metrizable spaces the following properties([10]), which are analogous to the three conditions in the Toms–Winterconjecture:(D1) finite tower dimension;(D2) almost finiteness; (D3) dynamical comparison. Kerr proved that (among other things) for free minimal actions(1) almost finiteness implies dynamical comparison;(2) if the action is almost finite, then the crossed product is (tra-cially) Z -stable.In [11], Kerr and Szab´o showed a relationship between almost finite-ness and dynamical comparison which is reminiscent of the relationshipbetween Z -stability and strict comparison above. More precisely, theyshowed that dynamical comparison implies almost finiteness providedthat the action has the small boundary property in the sense of [14].Inspired by the apparent analogy between almost finiteness and Z -stability, we sought to establish a formal link here, provided the Car-tan structure is taken into account on the C ∗ -algebraic side. We tooka McDuff-type characterization of Z -stability for nuclear C ∗ -algebrasdue to Hirshberg and Orovitz (called tracial Z -stability [8], and theircharacterization makes key use of ideas of Matui and Sato [17]), andstrengthened it in ways naturally related to the Cartan structure, toproduce a property we call tracial Z -stability for a sub- C ∗ -algebra (Def-inition 3.2). Our definition is, largely, inspired by a close analysis ofKerr’s proof that minimal free actions of amenable groups which arealmost finite give rise to Z -stable crossed products. Our main result This partially generalizes Matui’s almost finiteness for topological groupoidswith totally disconnected unit spaces. Comparison for topological dynamics first appeared in talks of Wilhelm Winter.
LMOST FINITENESS, COMPARISON, AND TRACIAL Z -STABILITY 3 shows that this definition is precisely how the Cartan structure en-codes almost finiteness, at least in the presence of the small boundaryproperty. Theorem A (Corollary 3.7) . Let G be a countable discrete amenablegroup, let X be a compact metrizable space, and let α : G y X be afree minimal action. Consider the following conditions:(i) α is almost finite;(ii) ( C ( X ) ⊆ C ( X ) ⋊ α G ) is tracially Z -stable;(iii) α has dynamical comparison.Then ( i ) ⇒ ( ii ) ⇒ ( iii ) , and if α has the small boundary property thenall three conditions are equivalent. The key novelty in our definition lies in the use of “one-sided nor-malizers”. The study of (two-sided) normalizers has a long history inoperator algebras, and they play a fundamental role in the connectionbetween dynamics and operator algebras. Indeed, Feldman and Mooreproved in [5] that one can reconstruct an equivalence relation from itsassociated Cartan subalgebra and normalizers (similar results in the C ∗ -setting were obtained in [13] and [21]). In this paper we show thatone-sided normalizers also carry a significant amount of dynamical in-formation. In particular, the dynamical subequivalence that appears inthe definition of dynamical comparison can be completely characterizedusing one-sided normalizers (Proposition 2.9).The paper is organized as follows. In Section 1, we establish ournotations and record several facts about sub- C ∗ -algebras and normal-izers. In Section 2 we study one-sided normalizers and prove a C ∗ -characterization of the dynamical subequivalence defined in [10] usingthese normalizers. In Section 3 we prove Theorem A (as Corollary3.7).
Acknowledgements.
This research is supported by an NSERC Dis-covery Grant. H.L. is also supported by the Fields Institute.1.
Preliminaries
For a C ∗ -algebra A we write A + for the set of positive elements in A , A for the set of elements of norm at most 1, and A for the intersection(the set of positive contractions). We write T ( A ) for the set of tracialstates on A . For two elements a, b in A and η > a ≈ η b if k a − b k < η . Definition 1.1. A sub- C ∗ -algebra ( D ⊆ A ) refers to a C ∗ -algebra A together with a C ∗ -subalgebra D . We say a sub- C ∗ -algebra ( D ⊆ A ) is nondegenerate if D contains an approximate unit for A . As mentioned in [10], this subequivalence relation first appeared in talks ofWinter.
HUNG-CHANG LIAO AND AARON TIKUISIS
Definition 1.2.
Let ( D ⊆ A ) be a sub- C ∗ -algebra. An element a ∈ A issaid to normalize D if a ∗ Da + aDa ∗ ⊆ D (we also say a is a normalizer of D in A ). The set of normalizers of D in A is denoted by N A ( D ) . It follows directly from the definition that the set of normalizers isclosed under multiplication, involution, and norm-limits.Although we won’t explicitly need the definition, we recall the notionof a ( C ∗ -algebra) Cartan subalgebra, as this is main context to keepin mind when we work with sub- C ∗ -algebras. A Cartan subalgebra is anondegenerate sub- C ∗ -algebra ( D ⊆ A ) where D is a maximal abeliansubalgebra, such that there exists a faithful conditonal expectation E : A → D , and such that N A ( D ) generates A as a C ∗ -algebra.It is useful to know that, in many cases, if a is a normalizer of D then a ∗ a, aa ∗ belong to D . This is not true in general (for example,take D = { } ), but does under the assumption of nondegeneracy, asthe following shows. Lemma 1.3. [21, Lemma 4.6]
Let ( D ⊆ A ) be a nondegenerate sub- C ∗ -subalgebra and a ∈ N A ( D ) . Then a ∗ a and aa ∗ belong to D .Proof. Let ( u λ ) λ be an approximate unit in D for A . Then by definition a ∗ u λ a is in D for every λ . Since N A ( D ) is closed under norm-limits,we see that a ∗ a ∈ D . The same argument shows that aa ∗ also belongsto D . (cid:3) Although a sum of normalizers is not necessarily a normalizer, it willbe if the subalgebra is abelian and a certain orthogonality condition issatisfied.
Lemma 1.4.
Let ( D ⊆ A ) be a sub- C ∗ -algebra with D abelian, and x , ..., x n be normalizers of D in A . Set z = P ni =1 x i .(i) If x ∗ i x j = 0 whenever i = j , then z ∗ Dz ⊆ D .(ii) If x i x ∗ j = 0 whenever i = j , then zDz ∗ ⊆ D .As a consequence, if x ∗ i x j = 0 = x i x ∗ j whenever i = j , then z belongsto N A ( D ) .Proof. We only prove (i) as the other assertion is completely analogous.It suffices to show that if i = j then x ∗ i dx j = 0 for every d ∈ D . Forthis observe that k x ∗ i dx j k = k x ∗ j d ∗ x i x ∗ i dx j k = k x ∗ j d ∗ x i x ∗ i dx j x ∗ j d ∗ x i x ∗ i dx j k≤ k x j k k x ∗ j d ∗ x i x ∗ i dd ∗ x i x ∗ i dx j k . Note that x i x ∗ i dd ∗ x i x ∗ i = x i ( x ∗ i dd ∗ x i ) x ∗ i ∈ x i Dx ∗ i ⊆ D. Since D is abelian, x ∗ j d ∗ ( x i x ∗ i dd ∗ x i x ∗ i ) dx j = x ∗ j d ∗ d ( x i x ∗ i dd ∗ x i x ∗ i ) x j = 0 . It follows that x ∗ i dx j = 0 and the proof is complete. (cid:3) LMOST FINITENESS, COMPARISON, AND TRACIAL Z -STABILITY 5 Throughout the paper we write M n for the algebra of all complex-valued n -by- n matrices, and D n for the subalgebra of M n that consistsof all diagonal matrices. Example . [13, Example 2] An element a ∈ M n belongs to N M n ( D n )if and only if a has at most one nonzero entry in each row and eachcolumn.In this paper, normalizer-preserving maps between sub- C ∗ -subalgebraswill play an essential role. Lemma 1.6.
Let ( D A ⊆ A ) and ( D B ⊆ B ) be two sub- C ∗ -algebrasand φ : A → B be a positive linear map. Suppose ( D B ⊆ B ) isnondegenerate.(i) If φ ( N A ( D A )) ⊆ N B ( D B ) , then φ ( D A ) ⊆ D B .(ii) Suppose in addition that ( D A ⊆ A ) is nondegenerate and φ isa ∗ -homomorphism. If φ ( N A ( D A )) = N B ( D B ) , then φ ( D A ) = D B .Proof. (i) Let d be a positive element in D A . Then φ ( d ) belongsto N B ( D B ) because D A is contained in N A ( D A ). Since φ ispositive, φ ( d ) = φ ( d ) ∗ φ ( d ) and the later belongs to D B byLemma 1.3. It follows that φ ( d ) ∈ D B and from linearity weconclude that φ ( D A ) ⊆ D B .(ii) By (i) we only need to show that D B ⊆ φ ( D A ), so let e be apositive element in D B . Since D B ⊆ N B ( D B ), by assumptionwe can find an element a ∈ N A ( D A ) such that φ ( a ) = e . Usingthe fact that φ is a ∗ -homomorphism, we see that φ ( | a | ) = φ (( a ∗ a ) ) = ( φ ( a ) ∗ φ ( a )) = ( e ) = e. As ( D A ⊆ A ) is nondegenerate, | a | belongs to D A by Lemma1.3 and the proof is complete. (cid:3) Remark . In the previous lemma nondegeneracy is necessary in bothassertions. For example, if we take D B = { } then any map from A into B is normalizer-preserving (because N B ( D B ) = B ) but in mostcases it does not map D A into D B . For (ii) we can take D A = { } and D B = B . Then any surjective ∗ -homomorphism from A onto B maps N A ( D A ) (which is all of A ) onto N B ( D B ) (which is all of B ), but theimage of D A is { } .Recall that a c.p.c. map φ : A → B be two C ∗ -algebras is orderzero if φ ( a ) φ ( b ) = 0 whenever a, b are positive elements in A satisfying ab = 0. Using the structure theorem ([29, Theorem 3.3]) we see thatorder zero maps in fact preserve arbitrary orthogonality: φ ( a ) φ ( b ) = 0if ab = 0. The next observation follows from Example 1.5 and Lemma1.4. HUNG-CHANG LIAO AND AARON TIKUISIS
Lemma 1.8.
Let ( D ⊆ A ) be a nondegenerate sub- C ∗ -algebra, φ : M n → A be a c.p.c. order zero map. Then φ ( N M n ( D n )) ⊆ N A ( D ) ifand only if φ ( e ij ) ∈ N A ( D ) for every matrix unit e ij . C ∗ -Algebraic characterization of the DynamicalSubequivalence Given a group acting on a compact Hausdorff space, one obtains apreorder on the open sets of the space which encodes certain informa-tion about the dynamics.
Definition 2.1. [10, Definition 3.1]
Let G be a countable discrete group,let X be a compact Hausdorff space, and let α : G y X be an actionby homeomorphisms. For a closed set F and an open set V in X , wewrite F ≺ V if there are open sets U , ..., U n and s , ..., s n ∈ G suchthat F ⊆ S ni =1 U i and that s i U i are pairwise disjoint subsets of V .For open sets O, V in X , we write O ≺ V if F ≺ V for every closedsubset F of O . This preorder was first defined in talks of Wilhelm Winter, and fea-tures prominently in [10, 11].In [15], this definition was extended to tuples of open sets, which werecall below. The main motivation was to study a generalized versionof the classical type semigroup, and use it to characterize dynamicalcomparison.For a continuous function f ∈ C ( X ) on a locally compact Hausdorffspace X , we denote(2.1) supp ◦ ( f ) := f − ( C \{ } )(the open support of f ). Definition 2.2 ([15, Definitions 1.4 and 2.1]) . Let G be a count-able discrete group, let X be a compact Hausdorff space, and let α : G y X be an action by homeomorphisms. For a tuple of compactsets F , . . . , F n ⊆ X and a tuple of open sets V , . . . , V m ⊆ X , write ( F , . . . , F n ) ≺ ( V , . . . , V m ) if there are open sets U i,j ⊆ X , group el-ements s i,j ∈ G , and indices k i,j ∈ { , . . . , m } for i = 1 , . . . , n and j = 1 , . . . , J i such that:(i) For each i , (2.2) F i ⊆ U i, ∪ · · · ∪ U i,J i , (ii) (2.3) n a i =1 J i a j =1 s i,j U i,j × { k i,j } ⊆ m a l =1 V l × { l } . For a = diag( a , . . . , a n ) ∈ ( D n ⊗ C ( X )) + and b = diag( b , . . . , b m ) ∈ ( D m ⊗ C ( X )) + , we write a b if ( F , . . . , F n ) ≺ (supp ◦ ( b ) , . . . , supp ◦ ( b m )) for every tuple of compact sets F , . . . , F n with F i ⊆ supp ◦ ( a i ) for all i . LMOST FINITENESS, COMPARISON, AND TRACIAL Z -STABILITY 7 It was noted in [15] that the preorder is closely related to theCuntz subequivalence. Indeed, [15, Proposition 2.3] shows that for a = diag( a , ..., a n ) ∈ ( D n ⊗ C ( X )) + and b = diag( b , . . . , b m ) ∈ ( D m ⊗ C ( X )) + , if a b then a is Cuntz subequivalent to b in C ( X ) ⋊ α G .The main result of this section shows that this preorder is completelycharacterized by a refined Cuntz subequivalence that makes use of “one-sided normalizers”. Definition 2.3.
Let ( D ⊆ A ) be a sub- C ∗ -algebra. An element a ∈ A is an r -normalizer of D if a ∗ Da ⊆ D . It is an s -normalizer of D if aDa ∗ ⊆ D .The set of r -normalizers of D in A is denoted RN A ( D ) , and the setof s -normalizers of D in A is denoted SN A ( D ) . The names “ r -normalizer” and “ s -normalizer” are motivated by aconnection to r - and s -sections for groupoids, established in Proposition2.6 below. The following facts are evident: • The product of two r -normalizers is an r -normalizer, and like-wise for s -normalizers; • RN A ( D ) = SN A ( D ) ∗ ; • an element is a normalizer if and only if it is both an r - andan s -normalizer.Here is a useful descriptions of r -normalizers in matrix amplifica-tions. Lemma 2.4.
Let ( D ⊆ A ) be a sub- C ∗ -algebra and let (2.4) x = x · · · x n ... ... x n · · · x nn ∈ M n ⊗ A. Then x ∈ RN M n ⊗ A ( D n ⊗ D ) if and only if(i) x ij ∈ RN A ( D ) for all i, j , and(ii) for all i, j, k with i = j and all a ∈ D , x ∗ ki ax kj = 0 .Proof. For b = diag( b , . . . , b n ) ∈ D n ⊗ D , we compute that the ( i, j )-entry of x ∗ bx is(2.5) n X k =1 x ∗ ki b k v kj . Suppose that x ∈ RN M n ⊗ A ( D n ⊗ D ), so this must always be in D ,and it must moreover be 0 whenever i = j . By setting b k := a and b l := 0 for l = k , we thus get that x ∗ ki ax kj ∈ D , and is moreover 0 if i = j . This shows both (i) and (ii).Conversely, suppose that (i) and (ii) hold. By (ii), it follows thatthe ( i, j )-entry of x ∗ bx is 0 whenever i = j , i.e., x ∗ bx is a diagonalmatrix. Moreover by (i), it follows that x ∗ ki b k x ki ∈ D for all i, k , andthus x ∗ bx ∈ D n ⊗ D . This shows that x ∈ RN M n ⊗ A ( D n ⊗ D ). (cid:3) HUNG-CHANG LIAO AND AARON TIKUISIS
These one-sided normalizers are best understood in the context ofgroupoids. For a locally compact Hausdorff ´etale groupoid G we write G (0) for its unit space, r and s for the range and source map, respec-tively. If x is a point in G (0) then we write G x := { γ ∈ G : s ( γ ) = x } and G x := { γ ∈ G : r ( γ ) = x } . We refer the readers to [25] for more on´etale groupoids and their C ∗ -algebras. Definition 2.5 ([21, Section 3]) . Let G be a locally compact Hausdorff´etale groupoid. A subset A of G is an r -section if r | A : A → G (0) isinjective; it is an s -section if s | A : A → G (0) is injective. We note for context that the more familiar notion of a bisection is asubset A ⊆ G which is both an r -section and an s -section.It is known that when the groupoid is topologically principal, nor-malizers are exactly functions that are supported in bisections ([21,Proposition 4.8]; in the case of principal ´etale groupoids, see [13, Propo-sition 1.6]), a fact classically proven using the polar decomposition ofa normalizer. We generalize this fact, at least in the principal case, to r -normalizers and r -sections; however, we give a completely differentargument, since the partial isometry in the polar decomposition of an r -normalizer no longer leads to an r -section. Proposition 2.6.
Let G be a locally compact Hausdorff principal ´etalegroupoid, and let a ∈ C ∗ r ( G ) . Then a ∈ RN C ∗ r ( G ) ( C ( G (0) )) if and onlyif supp ◦ ( a ) is an r -section. Likewise, a ∈ SN C ∗ r ( G ) ( C ( G (0) )) if andonly if supp ◦ ( a ) is an s -section.Proof. Using the adjoint, the second statement is equivalent to the first,which is the one we’ll prove. Set A := supp ◦ ( a ).For f ∈ C ( G (0) ) and γ ∈ G , we compute( a ∗ f a )( γ ) = X s ( α )= r ( γ ) a ∗ ( α − ) f ( r ( α )) a ( αγ )= X s ( α )= r ( γ ) a ( α ) f ( r ( α )) a ( αγ ) , (2.6)and note that the summand can only be nonzero when both α and αγ are in A .Thus, if A is an r -section, then nonzero summands can only arise if γ is a unit, so that a ∗ f a ∈ C ( G (0) ).For the other direction, suppose for a contradiction that there existdistinct elements γ , γ ∈ A such that r ( γ ) = r ( γ ); then we set(2.7) γ := γ − γ ∈ G\G (0) . By [20, Proposition II.4.1 (i)], the sums P s ( α )= r ( γ ) | a ( α ) | and P s ( α )= r ( γ ) | a ( αγ ) | converge; thus by the Cauchy–Schwarz inequality, so does(2.8) X s ( α )= r ( γ ) | a ( α ) a ( αγ ) | . LMOST FINITENESS, COMPARISON, AND TRACIAL Z -STABILITY 9 Therefore we may find a finite set F of G r ( γ ) such that(2.9) X α ∈ G r ( γ ) \ F | a ( α ) a ( αγ ) | < | a ( γ ) a ( γ ) | . Choose a function f ∈ C ( G (0) ) of norm 1 such that f ( r ( γ )) = 1 and f ( r ( α )) = 0 for α ∈ F \{ γ } . (Since G is principal, r ( γ ) = r ( α ) forany α ∈ F \{ γ } , so this is possible.) Then | ( a ∗ f a )( γ ) | (2.6) = (cid:12)(cid:12) X s ( α )= r ( γ ) a ( α ) f ( r ( α )) a ( αγ ) (cid:12)(cid:12) (2.9) > (cid:12)(cid:12) X α ∈ F a ( α ) f ( r ( α )) a ( αγ ) (cid:12)(cid:12) − | a ( γ ) a ( γ ) | = | a ( γ ) f ( r ( γ )) a ( γ γ ) | − | a ( γ ) a ( γ ) | = 0 . (2.10)Since γ
6∈ G (0) , this implies that a ∗ f a C ( G (0) ), which contradicts thehypothesis that a ∈ RN C ∗ r ( G ) ( C ( G (0) )). (cid:3) Specializing the above to the case of interest in this paper – that G is a transformation groupoid G × X – yields the next corollary. In thefollowing, for an element a of the crossed product C ( X ) ⋊ α G , we write(2.11) a = X g ∈ G a g u g to mean that E ( au ∗ g ) = a g (an element of C ( X )) for all g ∈ G , where E : C ( X ) ⋊ α G → C ( X ) is the canonical conditional expectation. Wedo not mean that the sum converges in any sense. Corollary 2.7.
Let G be a countable discrete group, let X be a compactHausdorff space, let α : G y X be a free action, and let (2.12) a = X g ∈ G a g u g ∈ C ( X ) ⋊ α G. Then a ∈ RN C ( X ) ⋊ α G ( C ( X )) if and only if the collection (2.13) { supp ◦ ( a g ) : g ∈ G } is pairwise disjoint.Proof. The crossed product C ( X ) ⋊ α G is the groupoid C ∗ -algebra ofthe transformation groupoid G = G × X (see [25, Example 2.1.15] forexample), and upon making this identification, one can easily computesupp ◦ ( a ) = [ g ∈ G { g } × g − . supp ◦ ( a g ) . Moreover, since r ( g, g − x ) = x for ( g, x ) ∈ G , this is an r -section if andonly if { supp ◦ ( a g ) : g ∈ G } is pairwise disjoint. (cid:3) We also use the above to give an interpretation of the conditions inLemma 2.4 in the case of a free group action.
Corollary 2.8.
Let G be a countable discrete group, let X be a compactHausdorff space, let α : G y X be a free action, and let (2.14) x = x · · · x n ... ... x n · · · x nn ∈ M n ⊗ ( C ( X ) ⋊ α G ) , where for each i, j = 1 , . . . , n , (2.15) x ij = X g ∈ G x i,j,g u g . Then x ∈ RN M n ⊗ ( C ( X ) ⋊ α G ) ( D n ⊗ C ( X )) if and only if, for every i =1 , . . . , n , the collection (2.16) { supp ◦ ( x i,j,g ) : j = 1 , . . . , n, g ∈ G } is pairwise disjoint.Proof. Consider the action ( π × α ) : Z /n × G y { , . . . , n } × X where π is the canonical action of the cyclic group Z /n on { , . . . , n } ; thisproduct action is free since both π and α are. The sub- C ∗ -algebra ( D n ⊗ C ( X ) ⊆ M n ⊗ ( C ( X ) ⋊ α G )) identifies canonically with ( C ( { , . . . , n }× X ) ⊆ C ( { , . . . , n } × X ) ⋊ π × α ( Z /n × G )), and this identification maps x to(2.17) y := X g ∈ G n X i,j =1 ( χ { i } ⊗ x i,j,g ) u ( i − j,g ) ∈ C ( { , . . . , n } × X ) ⋊ π × α ( Z /n × G ) . Thus, x ∈ RN M n ⊗ ( C ( X ) ⋊ α G ) ( D n ⊗ C ( X )) if and only if(2.18) y ∈ RN C ( { ,...,n }× X ) ⋊ π × α ( Z /n × G ) ( C ( { , . . . , n } × X )) . By Corollary 2.7, this is equivalent to the collection(2.19) { supp ◦ ( χ { i } ⊗ x i,j,g ) : i, j = 1 , . . . , n, g ∈ G } of subsets of { , . . . , n } × G being pairwise disjoint. Since supp ◦ ( χ { i } ⊗ x i,j,g ) = { i } × supp ◦ ( x i,j,g ), this is the same as requiring that(2.20) { supp ◦ ( x i,j,g ) : j = 1 , . . . , n, g ∈ G } be pairwise disjoint, for each i . (cid:3) Here is our algebraic characterization of the preorder from Def-inition 2.2. Note that although is defined for diagonal matrices in C ( X ) of different sizes, we may always pad one of them with zeroes toarrange that they have the same size. Proposition 2.9.
Let G be a countable discrete group, let X be acompact Hausdorff space, and let α : G y X be a free action. Let a, b ∈ ( D n ⊗ C ( X )) + . The following are equivalent: LMOST FINITENESS, COMPARISON, AND TRACIAL Z -STABILITY 11 (i) a b ;(ii) there exists a sequence ( t k ) ∞ k =1 in RN M n ⊗ ( C ( X ) ⋊ α G ) ( D n ⊗ C ( X )) such that (2.21) lim k →∞ k t ∗ k bt k − a k = 0; (iii) for every ǫ > there exists δ > and t ∈ RN M n ⊗ ( C ( X ) ⋊ α G ) ( D n ⊗ C ( X )) such that (2.22) t ∗ ( b − δ ) + t = ( a − ǫ ) + . Proof.
Let us write a = diag( a , . . . , a n ) and b = diag( b , . . . , b n ).(i) ⇒ (iii): This is a variant on the proofs of [10, Lemma 12.3] and[15, Proposition 2.3]. In both of those proofs, it is shown (roughly)that a b implies that a is Cuntz subequivalent to b in C ( X ) ⋊ α G ; the main novelty here is to verify that the Cuntz subequivalence canbe witnessed using r -normalizers.Let ǫ > F i := supp ◦ (( a i − ǫ ) + ) for i = 1 , . . . , n , sothat F i is a compact set contained in supp ◦ ( a i ). Then apply Definition2.2 to obtain open sets U i,j ⊆ X , group elements s i,j ∈ G , and indices k i,j ∈ { , . . . , m } for i = 1 , . . . , n and j = 1 , . . . , J i such that F i ⊆ U i, ∪ · · · ∪ U i,J i for i = 1 , . . . , n , and(2.23) n a i =1 J i a j =1 s i,j U i,j × { k i,j } ⊆ m a l =1 supp ◦ ( b l ) × { l } . Next find open sets V i,j such that V i,j ⊆ U i,j and(2.24) F i ⊆ V i, ∪ · · · ∪ V i,J i , i = 1 , . . . , n. It follows that(2.25) n a i =1 J i a j =1 s i,j V i,j × { k i,j } ⊆ m a l =1 supp ◦ ( b l ) × { l } , so by compactness of the left-hand side, there exists δ > n a i =1 J i a j =1 s i,j V i,j × { k i,j } ⊆ m a l =1 supp ◦ (( b l − δ ) + ) × { l } . By (2.24) and the definition of F i , we may choose a continuous function h i,j ∈ C ( V i,j ) + for each i = 1 , . . . , n and j = 1 , . . . , J i such that(2.27) J i X j =1 h i,j = ( a i − ǫ ) + , i = 1 , . . . , n. In [10, Lemma 12.3], the hypothesis is formally stronger than just a b . Using functional calculus, let ˆ b l ∈ C ∗ ( b l ) be a function such that(2.28) ˆ b l ( x ) ( b l − δ ) + ( x ) = 1 , x ∈ supp ◦ (( b l − δ ) + ) . Now define(2.29) t = t · · · t n ... ... t n · · · t nn ∈ M n ⊗ ( C ( X ) ⋊ α G )by(2.30) t li := X j : k i,j = l ˆ b l u s i,j h i,j = X j : k i,j = l ˆ b l ( h i,j ◦ α − s i,j ) u s i,j , i, l = 1 , . . . , n. By (2.23) and since h i,j ∈ C ( V i,j ), for each l the collection(2.31) { supp ◦ ( h i,j ◦ α − s i,j ) : k i,j = l } = { α s i,j (supp ◦ ( h i,j )) : k ( i, j ) = l } is pairwise disjoint. Therefore by Corollary 2.8, t ∈ RN M n ⊗ ( C ( X ) ⋊ α G ) ( D n ⊗ C ( X )).In particular, t ∗ ( b − δ ) + t is a diagonal matrix. Moreover, for i =1 , . . . , n , we compute the ( i, i )-entry of t ∗ ( b − δ ) + t to be n X k =1 t ∗ ki ( b k − δ ) + t ki = X j,j ′ : k i,j = k i,j ′ h i,j u ∗ s i,j ˆ b k i,j ( b k i,j − δ ) + u s i,j ′ h i,j ′ (2.28) = X j,j ′ : k i,j = k i,j ′ u ∗ s i,j ( h i,j ◦ α − s i,j )( h i ′ ,j ′ ◦ α − s i,j ′ ) u s i,j ′ . (2.32)By pairwise disjointness of the collection (2.31), we have that ( h i,j ◦ α − s i,j )( h i,j ′ ◦ α s i,j ′ ) = 0 whenever k i,j = k i,j ′ and j = j ′ ; thus the abovesimplifies to(2.33) J i X j =1 u ∗ s i,j ( h i,j ◦ α − s i,j ) u s i,j (2.27) = J i X j =1 h i,j = ( a i − ǫ ) + , as required.(iii) ⇒ (i): As in Definition 2.2, let F i be a compact subset ofsupp ◦ ( a i ) for i = 1 , . . . , n . By compactness, there exists ǫ > F i ⊆ supp ◦ (( a i − ǫ ) + ). By (iii), let t ∈ RN M n ⊗ ( C ( X ) ⋊ α G ) ( D n ⊗ C ( X ))and δ > t ∗ ( b − δ ) + t = ( a − ǫ ) + . Write(2.34) t = t · · · t n ... ... t n · · · t nn and for each i, j , write(2.35) t ij = X g ∈ G t i,j,g u g LMOST FINITENESS, COMPARISON, AND TRACIAL Z -STABILITY 13 where t i,j,g ∈ C ( X ) for each i, j, g . By Corollary 2.8, for each i (2.36) { supp ◦ ( t i,j,g ) : j = 1 , . . . , n, g ∈ G } is pairwise disjoint . We now compute that ( a i − ǫ ) + , which is the ( i, i )-entry of t ∗ ( b − δ ) + t ,is equal to n X k =1 t ∗ ki ( b k − δ ) + t ki = n X k =1 X g,h ∈ G u ∗ g t ∗ k,i,g ( b k − δ ) + t k,i,h u h (2.36) = n X k =1 X s ∈ G u ∗ s | t k,i,s | b k u s = n X k =1 X s ∈ G ( | t k,i,s | ( b k − δ ) + ) ◦ α s . (2.37)Since F i ⊆ supp ◦ (( a i − ǫ ) + ), it follows that(2.38) F i ⊆ n [ k =1 [ s ∈ G supp ◦ (( | t k,i,s | b k ) ◦ α s ) . By compactness, we may choose k i, , . . . , k i,J i ∈ { , . . . , n } and s i, , . . . , s i,J i ∈ G such that, upon setting(2.39) U i,j := supp ◦ (( | t k i,j ,i,s i,j | b k i,j ) ◦ α s i,j ) = α − s i,j (supp ◦ ( t k i,j ,i,s i,j ) ∩ supp ◦ ( b k i,j )) , we have(2.40) F i ⊆ J i [ j =1 U i,j . Also, the collection of sets of the form(2.41) { k i,j } × α s i,j ( U i,j ) = { k i,j } × (supp ◦ ( t k i,j ,i,s i,j ) ∩ supp ◦ ( b k i,j ))(where i ranges over { , . . . , n } and j ranges over { , . . . , J i } ) is con-tained in the collection of sets of the form(2.42) { k } × (supp ◦ ( t k,i,s ) ∩ supp ◦ ( b k )) . Each of these is evidently contained in ` k { k } × supp ◦ ( b k ), and by(2.36), they are pairwise disjoint. Therefore, we have(2.43) a i,j { k i,j } × α s i,j ( U i,j ) ⊆ a k { k } × supp ◦ ( b k ) , as required.(iii) ⇒ (ii) is immediate.(ii) ⇒ (iii): Assume (ii) holds and let ǫ > k we have k t ∗ k bt k − a k < ǫ/
2, and thus there exists δ > k t ∗ k ( b − δ ) + t k − a k < ǫ . Since t k ∈ RN M n ⊗ ( C ( X ) ⋊ α G ) ( D n ⊗ C ( X )), itfollows that t ∗ k ( b − δ ) + t k ∈ D n ⊗ C ( X ), so applying [22, Proposition 2.2] to this algebra (and using that it is commutative), we see that thereexists s ∈ D N ⊗ C ( X ) such that(2.44) s ∗ t ∗ k ( b − δ ) + t k s = ( a − ǫ ) + . Thus (iii) holds with t := t k s . (cid:3) Remark . As noted in the above proof, the argument for (i) ⇒ (iii)is a variant on the proof of [10, Lemma 12.3]. We note for use laterthat, in fact, the element v constructed in the proof of [10, Lemma12.3] is an r -normalizer, for example by writing it as(2.45) v = n X i =1 (( f h i ) / ◦ α − s i ) u s i (where we use α : G y X to denote the action) and then using Corol-lary 2.7.3. Almost finiteness, dynamical comparison, and tracial Z -stability In this section we define tracial Z -stability for sub- C ∗ -algebras andprove Theorem A (as Corollary 3.7). We first recall the definition ofdynamical comparison. Definition 3.1. [10, Definition 3.2]
Let G be a countable discrete group,let X be a compact metrizable space, and let α : G y X be an action byhomeomorphisms. We say α has dynamical comparison if O ≺ V forall open sets O, V ⊆ X satisfying µ ( O ) < µ ( V ) for every G -invariantBorel probability measure µ . Given τ ∈ T ( C ( X ) ⋊ α G ), and a ∈ C ( X ) + , define(3.1) d τ ( a ) := lim n →∞ τ ( a /n ) , i.e., the value of the measure associated to τ evaluated on supp ◦ ( a ).When the action α is free , the G -invariant Borel probability measureson X correspond exactly to tracial states on C ( X ) ⋊ α G (see, for ex-ample, [7, Theorem 11.1.22]). Therefore in this case dynamical com-parison can be reformulated as follows: the action α has dynamicalcomparison if and only if for any a, b ∈ C ( X ) + , if d τ ( a ) < d τ ( b ) for all τ ∈ T ( C ( X ) ⋊ α G ) then a b (as in Definition 2.2).In C ∗ -algebra theory, Z -stability is the property of tensorially ab-sorbing a certain canonical C ∗ -algebra, called the Jiang–Su algebra Z .This C ∗ -algebra was defined in [9], but in practice it is often a McDuff-type characterization of Z -stability that is used (see [28, Proposition2.14], a combination of results by Rørdam–Winter [24], Kirchberg [12],and Toms–Winter [26]). Building on ideas of Matui and Sato ([17]),Hirshberg and Orovitz defined “tracial Z -stability”, an a priori weak-ening of this McDuff-type condition, and proved that it is equivalent to LMOST FINITENESS, COMPARISON, AND TRACIAL Z -STABILITY 15 Z -stability for simple separable unital nuclear C ∗ -algebras (see [8, Def-inition 2.1, Proposition 2.2, and Theorem 4.1]) (nuclearity is the keyhypothesis that enables this equivalence). The following is a version oftracial Z -stability for a sub- C ∗ -algebra, where a key role is played by(one-sided) normalizers of the smaller algebra; tracial Z -stability forthe algebra A is precisely the case D = A . Definition 3.2.
Let ( D ⊆ A ) be a sub- C ∗ -algebra with A unital, andsuch that A ∈ D . For a, b ∈ D + , write a - ( D ⊆ A ) b if there is asequence ( t k ) ∞ k =1 in RN A ( D ) such that lim k →∞ k t ∗ k bt k − a k = 0 .We say that ( D ⊆ A ) is tracially Z -stable if for every n ∈ N , everytolerance ǫ > , every finite set F ⊂ A , and every h ∈ D + \{ } , thereexists a c.p.c. order zero map φ : M n → A such that:(i) φ ( N M n ( D n )) ⊆ N A ( D ) ,(ii) A − φ ( n ) - ( D ⊆ A ) h , and(iii) k [ a, φ ( x )] k < ǫ for all a ∈ F and every contraction x ∈ M n . We will make use of the generalized type semigroup defined in [15].The main result of [15] shows that dynamical comparison is equivalentto almost unperforation of this semigroup.
Definition 3.3. [15, the paragraphs after Lemma 2.2]
Let G be acountable discrete group, let X be a compact Hausdorff space, and let α : G y X be a free action by homeomorphisms. Define a ≈ b to meanthat both a b and b a (as in Definition 2.2), and set (3.2) W ( X, G ) := ∞ [ n =1 ( D n ⊗ C ( X )) + / ≈ . For a ∈ ( D n ⊗ C ( X )) + , we use [ a ] to denote its equivalence class in W ( X, G ) . The preorder induces an order ≤ on W ( X, G ) , and thereis a well-defined addition operation on W ( X, G ) given by (3.3) [ a ] + [ b ] = [ a ⊕ b ] .W ( X, G ) is then a partially ordered abelian semigroup. Theorem 3.4.
Let G be a countable discrete infinite amenable group,let X be a compact metrizable space, and let α : G y X be a freeminimal action. If ( C ( X ) ⊆ C ( X ) ⋊ α G ) is tracially Z -stable then α has dynamical comparison.Proof. Let f, g ∈ C ( X ) + , and assume that d τ ( f ) < d τ ( g ) for all τ ∈ T ( C ( X ) ⋊ α G ). We need to show that f g (as in Definition 2.2).Since the action is free and minimal and G is infinite, X has noisolated points. Choose any point x ∈ X such that g ( x ) = 0, let h ∈ C ( X ) + be a positive contraction which vanishes at x and is nonzeroeverywhere else, and consider g ′ := hg . Since the action is minimal, By (i) and Lemma 1.6 φ ( n ) ∈ D , so this makes sense. µ ( { x } ) = 0 for every G -invariant measure on X . Since the G -invariantprobability measures on X correspond to traces on C ( X ) ⋊ α G , itfollows that d τ ( g ) = d τ ( g ′ ) for all τ ∈ T ( C ( X ) ⋊ α G ). Also, 0 is not anisolated point of the spectrum of g ′ . Thus by replacing g with g ′ , wemay assume that 0 is not an isolated point in the spectrum of g .Now by [15, Theorem 3.9], and again since the G -invariant probabil-ity measures on X correspond to traces on C ( X ) ⋊ α G , the condition d τ ( f ) < d τ ( g ) for all τ ∈ T ( C ( X ) ⋊ α G ) is equivalent to ( n +1)[ f ] ≤ n [ g ]in W ( X, G ), for some n ∈ N .Since n [ f ] ≤ ( n + 1)[ f ] ≤ n [ g ], this means that M n ⊗ f M n ⊗ g .Let ǫ >
0; we will show that there exists t ∈ RN C ( X ) ⋊ α G ( C ( X )) suchthat k t ∗ gt − f k < ǫ , which suffices to show f g by Proposition 2.9.By that same proposition, we have that there exists δ > v = v · · · v n ... ... v n · · · v nn ∈ RN M n ⊗ C ( X ) ⋊ α G ( D n ⊗ C ( X ))such that v ∗ ( M n ⊗ ( g − δ ) + ) v = M n ⊗ ( f − ǫ ) + . By Lemma 2.4, wehave v ij ∈ RN C ( X ) ⋊ ¯ α G ( C ( X )) , i, j = 1 , . . . , n, and(3.5) v ∗ ki av kj = 0 , i = j, a ∈ C ( X ) . (3.6)By looking at the entries of v ∗ ( M n ⊗ ( g − δ ) + ) v = M n ⊗ ( f − ǫ ) + , weobtain for all j (3.7) n X i =1 v ∗ ij ( g − δ ) + v ij = ( f − ǫ + . Since 0 is not an isolated point of the spectrum of g , we may usefunctional calculus to find a nonzero element d ∈ C ∗ ( g ) + ⊆ C ( X ) + ,along with orthogonal elements ˆ d, ˆ g ∈ C ∗ ( g ) + ⊆ C ( X ) + such that(3.8) g ˆ d = d and g ˆ g = ( g − δ ) + . Set(3.9) η := ǫ n + 3and using tracial Z -stability of ( C ( X ) ⊆ C ( X ) ⋊ α G ), let φ : M n → C ( X ) ⋊ α G be an order zero map such that:(i) φ ( N M n ( D n )) ⊆ N C ( X ) ⋊ α G ( C ( X )),(ii) A − φ ( n ) d , and(iii) k [ a, φ ( x )] k < η for all a ∈ { v ij : i, j = 1 , . . . , n } ∪ { ( g − δ ) + } and every contraction x ∈ M n . Note that - ( C ( X ) ⊆ C ( X ) ⋊ α G ) in Definition 3.2 is the same as from Definition2.2, by Proposition 2.9. LMOST FINITENESS, COMPARISON, AND TRACIAL Z -STABILITY 17 Define(3.10) r := n X i,j =1 φ ( e ii ) v ij φ ( e ij ) . Note that by (i) and Lemma 1.6, φ ( e ii ) ∈ C ( X ). Using this, for a ∈ C ( X ), we have r ∗ ar = n X i,j,k,l =1 φ ( e ji ) v ∗ ij φ ( e ii ) aφ ( e kk ) v kl φ ( e kl )= n X i,j,k,l =1 φ ( e ji ) v ∗ ij φ ( e ii ) φ ( e kk ) av kl φ ( e kl )= n X i,j,l =1 φ ( e ji ) v ∗ ij φ ( e ii ) av il φ ( e il ) (3.6) = n X i,j =1 φ ( e ji ) v ∗ ij φ ( e ii ) av ij φ ( e ij ) , (3.11)and by (3.5), and (i), this is in C ( X ). This shows that r ∈ RN C ( X ) ⋊ α G ( C ( X )).Next, in the case that a = ( g − δ ) + , we get r ∗ ( g − δ ) + r = n X i,j =1 φ ( e ji ) v ∗ ij φ ( e ii )( g − δ ) + v ij φ ( e ij ) ≈ n η n X i,j =1 φ ( e ji ) φ ( e ii ) φ ( e ij ) v ∗ ij ( g − δ ) + v ij = n X i,j =1 φ ( e jj ) v ∗ ij ( g − δ ) + v ij (3.7) = n X j =1 φ ( e jj ) ( f − ǫ + ≈ ǫ/ φ ( M n ) f. (3.12)Next, we note that C ( X ) ⋊ α G − φ ( M n ) and C ( X ) ⋊ α G − φ ( M n ) areCuntz equivalent (in C ∗ ( C ( X ) ⋊ α G , φ ( M n ))), and so combining thiswith (ii), we obtain some s ∈ RN C ( X ) ⋊ α G ( C ( X )) such that(3.13) s ∗ ds ≈ η ( C ( X ) ⋊ α G − φ ( M n ) ) f. Define t := ˆ gr + ˆ ds (using ˆ g, ˆ d defined just above (3.8)). Since ˆ g, ˆ d are orthogonal and in C ( X ), it follows that (ˆ gr ) ∗ a ( ˆ ds ) = 0 for all a ∈ C ( X ), and thus t ∈ RN C ( X ) ⋊ α G ( C ( X )) by Lemma 1.4. Moreover, t ∗ gt = r ∗ ˆ gg ˆ gr + s ∗ ˆ dg ˆ ds (3.8) = r ∗ ( g − δ ) + r + s ∗ ds ≈ n η + ǫ + η φ ( M n ) f + (1 − φ ( M n ) ) f = f. (3.14)Since 4 n η + η < ǫ , we are done. (cid:3) We now establish Theorem A. The following definition of a “castle”is borrowed from David Kerr, except that (for later use) we allow thecastle to possibly have infinitely many towers.
Definition 3.5 (cf. [10, Definitions 4.1 and 5.7]) . Let G be a countablediscrete group, let X be a compact Hausdorff space, and let α : G y X be a free action. A castle is a collection { ( V i , S i ) } i ∈ I where each V i is asubset of X and each S i is a finite subset of G , such that the collection (3.15) { sV i : s ∈ S i , i ∈ I } is pairwise disjoint. Each ( V i , S i ) is called a tower , the sets S i arecalled shapes , and the sets sV i (where s ∈ S i ) are called levels of thecastle. We recall the definition of almost finiteness for a group acting byhomeomorphisms. Kerr’s definition partially generalizes an earlier con-cept for locally compact ´etale groupoids with compact totally discon-nected unit spaces, which Matui defined and used to prove strong re-sults about the associated topological full group (see [16], particularlyDefinition 6.2).
Definition 3.6 ([10, Definition 8.2]) . Let G be a countable discretegroup, let X be a compact metrizable space, and let α : G y X be afree action. The action is almost finite if for every finite subset K ⊂ G ,and every δ > , there exists:(i) a castle { ( V i , S i ) } i ∈ I such that I is finite, each level is openwith diameter at most δ , and each shape is ( K, δ ) -invariant(i.e., | gS i △ S i | / | S i | < δ for all g ∈ K and all i ∈ I ), and(ii) a set S ′ i ⊆ S i for each i ∈ I such that | S ′ i | < δ | S i | and (3.16) X \ a i ∈ I S i V i ≺ a i ∈ I S ′ i V i , using ≺ from Definition 2.2. The following corollary shows how tracial Z -stability (for sub- C ∗ -algebras) fits into different regularity-type dynamical properties. Corollary 3.7. (Theorem A) Let G be a countable discrete amenablegroup, let X be a compact metrizable space, and let α : G y X be afree minimal action. Consider the following conditions:(i) α is almost finite;(ii) ( C ( X ) ⊆ C ( X ) ⋊ α G ) is tracially Z -stable;(iii) α has dynamical comparison.Then ( i ) ⇒ ( ii ) ⇒ ( iii ) , and if α has the small boundary property thenall three conditions are equivalent. LMOST FINITENESS, COMPARISON, AND TRACIAL Z -STABILITY 19 As mentioned in the introduction, our definition of tracial Z -stabilityfor a sub- C ∗ -algebra is largely inspired by Kerr’s proof that almostfiniteness implies Z -stability of C ( X ) ⋊ α G ([10, Theorem 12.4]), andindeed his proof shows the implication (i) ⇒ (ii) (we flesh out thedetails below). For the rest, one need only combine Theorem 3.4 withKerr and Szab´o’s proof that dynamical comparison combined with thesmall boundary property implies almost finiteness ([11, Theorem 6.1]).To verify (i) ⇒ (ii), we take a closer look at the normalizer-preservingcondition in the definition of Z -stability for a sub- C ∗ -algebra. We shallconsider a general construction of a c.p.c. map into a crossed product,given the following data. Let G be a countable discrete group, let X be a compact metrizable space, and let α : G y X be an action. Let T be a countable set, and for each t ∈ T , let f t ∈ C ( X ) + and let S t = { s t, , . . . , s t,n } be a subset of G of size n . Suppose that:(i) lim t →∞ k f t k = 0, and(ii) ((supp ◦ ( f t ) , S t ) ∞ t =1 is a castle.Also for each t ∈ T and i = 1 , . . . , n , let θ t,i : supp ◦ ( f t ) → T be acontinuous function. Define φ : M n → C ( X ) ⋊ α G by(3.17) φ ( e ij ) := X t ∈ T u s t,i θ t,i ¯ θ t,j f t u ∗ s t,j and extending linearly. We call such a map a castle order zero map . Itis not hard to check the following facts.(i) The sum defining φ ( e ij ) converges in norm.(ii) φ is c.p.c. order zero and normalizer-preserving, i.e., φ ( N M n ( D n )) ⊆N C ( X ) ⋊ α G ( C ( X )) (by Lemma 1.8 it is enough to check that φ ( e ij ) is a normalizer for all i, j ). Remark . We check that the map ϕ defined in Equation (18) in theproof of [10, Theorem 12.4] is a castle order zero map (again with afinite sum). To see this, using the notation of that proof, set I := { ( k, l, m, q, c, t ) : k = 1 , . . . , K, l = 1 , . . . , L, (3.18) m = 1 , . . . , M, c ∈ C (1) k,l,m , q = 1 , . . . , Q, t ∈ B k,l,c,q } , and for ( k, l, m, q, c, t ) ∈ I , set S ( k,l,m,q,c,t ) := { t Λ k, ( c ) , . . . , t Λ k,n ( c ) } ,f ( k,l,m,q,c,t ) := qQ h k , and θ t,i ≡ . (3.19) We note that for each k , the sets S ( k,l,m,q,c,t ) are pairwise disjoint andcontained in S k (where S k is defined in [10]) Since h k ∈ C ( U k ) andthe sets sU k for k = 1 , . . . , K are pairwise disjoint, it follows that(3.20) (supp ◦ ( f ( k,l,m,q,c,t ) ) , S ( k,l,m,q,c,t ) ) ( k,l,m,q,c,t ) ∈ I is a castle, and so defines a castle order zero map.To see that the map it defines is ϕ , using that Λ k,i,j = Λ k,i ◦ Λ − k,j andthat Λ k,j : C (1) k,l,m → C ( j ) k,l,m is a bijection, we see that h k,l,c,i,j = Q X q =1 X t ∈ B k,l,c,q qQ u t Λ k,i,j ( c ) c − t − ( h k ◦ α tc )= Q X q =1 X t ∈ B k,l,c,q u t Λ k,i ( c ) f ( k,l,m,q,c,t ) u ∗ t Λ k,j ( c ) . (3.21)Thus(3.22) ϕ ( e ij ) = X ( k,l,m,q,c,t ) ∈ I u t Λ k,i ( c ) f ( k,l,m,q,c,t ) u ∗ t Λ k,j ( c ) . Proof of Corollary 3.7.
As explained earlier, (ii) ⇒ (iii) is Theorem 3.4.When α has the small boundary property,(iii) ⇒ (i) is [11, Theorem6.1](i) ⇒ (ii): The proof of [10, Theorem 12.4] essentially shows thisimplication, although since tracial Z -stability for a sub- C ∗ -algebra isnot defined there, it is not explicitly stated in this way. Let us explaincarefully how to obtain (ii) from the proof of [10, Theorem 12.4].The proof begins with a ∈ C ( X ) + nonzero, a finite set Γ of the unitball of C ( X ), a finite symmetric set F of G , and a tolerance ǫ >
0. Itproduces a c.p.c. order zero map φ : M n → C ( X ) ⋊ α G such that: • C ( X ) ⋊ α G − φ ( n ) is Cuntz subequivalent to a , and • k [ x, φ ( b )] k < ǫ for all x ∈ Γ ∪ { u g : g ∈ F } .As argued in the proof of [10, Theorem 12.4], since the crossed productis generated by C ( X ) and the canonical unitaries, given any finitesubset F ′ of C ( X ) ⋊ α G , by choosing Γ, F , and ǫ appropriately, the Suppose that x ∈ S ( k,l,m,q,c,t ) ∩ S ( k,l ′ ,m ′ ,q ′ ,c ′ ,t ′ ) . Then x = t Λ k,i ( c ) = t ′ Λ k,i ′ ( c ′ )for some i, i ′ ∈ { , . . . , n } . We have t ∈ B k,l,c,q ⊆ T ′ k,l,c ⊆ T k,l,c and t ′ ∈ B k ′ ,l ′ ,c ′ ,q ′ ⊆ T ′ k ′ ,l ′ ,c ′ ⊆ T k ′ ,l ′ ,c ′ , while Λ k,i ( c ) ∈ C ( i ) k,l,m ⊆ C k,l and Λ k ′ ,i ′ ( c ′ ) ∈ C ( i ′ ) k ′ ,l ′ ,m ′ ⊆ C k ′ ,l ′ .Since the collection of sets T k,l,c γ for l = 1 , . . . , L and γ ∈ C k,l are disjoint, itfollows that l = l ′ , t = t ′ , and Λ k,i ( c ) = Λ k,i ′ ( c ′ ). Since C ( i ) k,l,m for i = 1 , . . . , n and m = 1 , . . . , M are pairwise disjoint, it then follows that i = i ′ and m = m ′ .Injectivity of Λ k,i then implies that c = c ′ . Finally, the sets B k,l,c,q for q = 1 , . . . , Q are pairwise disjoint, so q = q ′ . Continuing from the previous footnote, if x = t Λ k,i ( c ) for some i ∈ { , . . . , n } then x ∈ T k,l,c C k,l,m ⊆ S k (by the use of [10, Theorem 12.2] right after [10, Eq.(14)]). LMOST FINITENESS, COMPARISON, AND TRACIAL Z -STABILITY 21 second condition will imply that k [ x, φ ( b )] k < ǫ for all x ∈ F ′ . ByRemark 3.8, the order zero map φ is in fact a “castle order zero map”and therefore φ ( N M n ( D n )) ⊆ N C ( X ) ⋊ α G ( C ( X )). Finally, we note thatthe first of the above conditions is obtained in the proof of [10, Theorem12.4] by invoking [10, Lemma 12.3], and so by Remark 2.10, we get thestronger conclusion that C ( X ) − φ ( n ) - ( C ( X ) ⊆ C ( X ) ⋊ α G ) a . Hence, weget precisely our definition of ( C ( X ) ⊆ C ( X ) ⋊ α G ) being tracially Z -stable, as required. (cid:3) We close by noting that normalizer-preserving c.p.c. order zero mapsfrom M n into the crossed product C ( X ) ⋊ α G are closely related tocastles. In fact, every such a map must be a castle order zero map. Proposition 3.9.
Let G be a countable discrete amenable group, let X be a compact metrizable space, and let α : G y X be a free action. If φ : M n → C ( X ) ⋊ α G is a c.p.c. order zero map such that φ ( N M n ( D n )) ⊆N C ( X ) ⋊ α G ( C ( X )) , then φ is a castle order zero map.Proof. Let(3.23) φ ( e i ) = X g ∈ G h i,g u ∗ g for h i,g ∈ C ( X ). Since φ ( e i ) is a normalizer, by Corollary 2.7, thecollections { supp ◦ ( h i,g ) : g ∈ G } and { supp ◦ ( h i,g ◦ α − g ) : g ∈ G } areboth pairwise disjoint. Since φ is order zero, we have(3.24) φ ( e ) = ( φ ( e i ) φ ( e i ) ∗ ) = X g ∈ G | h i,g | (where the sum is orthogonal and norm-convergent). Since this is truefor all i , it follows that we may find a pairwise disjoint family ( f t ) t ∈ T in C ( X ) + (indexed by some countable set T ) along with a function s : T × { , . . . , n } → G such that(3.25) | h i,g | = X t ∈ T : s ( t,i )= g f t . Since the orthogonal sum P g ∈ G | h i,g | = P t ∈ T f t converges, we musthave k f t k → t → ∞ . For each t ∈ T and i = 1 , . . . , n , we havethat f t = | h i,s ( t,i ) | on supp ◦ ( f t ) (by the orthogonality of the f t ), so wemay define θ t,i : supp ◦ ( f t ) → T by(3.26) θ t,i ( x ) := f t ( x ) h i,s ( t,i ) ( x ) , and this is a continuous function. We obtain(3.27) h i,g = X t ∈ T : s ( t,i )= g ¯ θ t,i f t . Since φ ( e ) ∈ C ( X ) + , we have s ( t,
1) = e and θ t, ≡ t ∈ T .We may therefore rewrite(3.28) φ ( e j ) = X t ∈ T ¯ θ t,j f t u ∗ s ( t,j ) = X t ∈ T u s ( t, θ t,i ¯ θ t,j f t u ∗ s ( t,j ) . Since φ is order zero, we also obtain(3.29) φ ( e ij ) := X t ∈ T u s ( t,i ) θ t,i ¯ θ t,j f t u ∗ s ( t,j ) . It remains only to show that when we set(3.30) S t := { s ( t, , . . . , s ( t, n ) } , we have that ((supp ◦ ( f t ) , S t )) t ∈ T is a castle.For this, first since(3.31) φ ( e i ) = X t ∈ T u s ( t,i ) θ t,i f t = X t ∈ T (( θ t,i f t ) ◦ α − s ( t,i ) ) u s ( t,i ) is a normalizer, it follows from Corollary 2.7 (and the fact that the f t are orthogonal) that(3.32) { supp ◦ ( f t ◦ α − s ( t,i ) ) : t ∈ T } = { α s ( t,i ) (supp ◦ ( f t )) : t ∈ T } is pairwise orthogonal. Moreover, we compute(3.33) φ ( e ii ) = X t ∈ T u s ( t,i ) f t u ∗ s ( t,i ) = X t ∈ T f t ◦ α − s ( t,i ) , so using the orthogonality of the above family,(3.34) supp ◦ ( φ ( e ii )) = a t ∈ T α s ( t,i ) (supp ◦ ( f t )) . Since φ is order zero, we know that φ ( e ii ) and φ ( e jj ) are orthogonal forall i = j . Consequently, we find that the entire family(3.35) { α s ( t,i ) (supp ◦ ( f t )) : t ∈ T, i = 1 , . . . , n } is pairwise disjoint, which means that ((supp ◦ ( f t ) , S t ) t ∈ T is a castle. (cid:3) References [1] Jorge Castillejos, Samuel Evington, Aaron Tikuisis, and Stuart White. Uni-form property Gamma. arXiv:1912.04207.[2] A. Connes. Classification of injective factors. Cases II , II ∞ , III λ , λ = 1. Ann. of Math. (2) , 104(1):73–115, 1976.[3] A. Connes, J. Feldman, and B. Weiss. An amenable equivalence relation isgenerated by a single transformation.
Ergodic Theory Dynamical Systems ,1(4):431–450 (1982), 1981.[4] H. A. Dye. On groups of measure preserving transformations. I.
Amer. J.Math. , 81:119–159, 1959.[5] Jacob Feldman and Calvin C. Moore. Ergodic equivalence relations, cohomol-ogy, and von Neumann algebras. II.
Trans. Amer. Math. Soc. , 234(2):325–359,1977.
LMOST FINITENESS, COMPARISON, AND TRACIAL Z -STABILITY 23 [6] Julien Giol and David Kerr. Subshifts and perforation. J. Reine Angew. Math. ,639:107–119, 2010.[7] Thierry Giordano, David Kerr, N. Christopher Phillips, and Andrew Toms.
Crossed products of C ∗ -algebras, topological dynamics, and classification . Ad-vanced Courses in Mathematics. CRM Barcelona. Birkh¨auser/Springer, Cham,2018. Lecture notes based on the course held at the Centre de RecercaMatem`atica (CRM) Barcelona, June 14–23, 2011, Edited by Francesc Perera.[8] Ilan Hirshberg and Joav Orovitz. Tracially Z -absorbing C ∗ -algebras. J. Funct.Anal. , 265(5):765–785, 2013.[9] Xinhui Jiang and Hongbing Su. On a simple unital projectionless C ∗ -algebra. Amer. J. Math. , 121(2):359–413, 1999.[10] David Kerr. Dimension, comparison, and almost finiteness. To appear in J.Eur. Math. Soc. arXiv:1710.00393.[11] David Kerr and G´abor Szab´o. Almost finiteness and the small boundary prop-erty. To appear in Comm. Math. Phys. arXiv:1807.04326.[12] Eberhard Kirchberg. Central sequences in C ∗ -algebras and strongly purelyinfinite algebras. In Operator Algebras: The Abel Symposium 2004 , volume 1of
Abel Symp. , pages 175–231. Springer, Berlin, 2006.[13] Alexander Kumjian. On C ∗ -diagonals. Canad. J. Math. , 38(4):969–1008, 1986.[14] Elon Lindenstrauss and Benjamin Weiss. Mean topological dimension.
IsraelJ. Math. , 115:1–24, 2000.[15] Xin Ma. A generalized type semigroup and dynamical comparison.
ErgodicTheory and Dynamical Systems , page 118, 2020.[16] Hiroki Matui. Homology and topological full groups of ´etale groupoids on to-tally disconnected spaces.
Proc. Lond. Math. Soc. (3) , 104(1):27–56, 2012.[17] Hiroki Matui and Yasuhiko Sato. Strict comparison and Z -absorption of nu-clear C ∗ -algebras. Acta Math. , 209(1):179–196, 2012.[18] Hiroki Matui and Yasuhiko Sato. Decomposition rank of UHF-absorbing C ∗ -algebras. Duke Math. J. , 163(14):2687–2708, 2014.[19] F. J. Murray and J. von Neumann. On rings of operators. IV.
Ann. of Math.(2) , 44:716–808, 1943.[20] Jean Renault.
A groupoid approach to C ∗ -algebras , volume 793 of Lecture Notesin Mathematics . Springer, Berlin, 1980.[21] Jean Renault. Cartan subalgebras in C ∗ -algebras. Irish Math. Soc. Bull. ,(61):29–63, 2008.[22] Mikael Rørdam. On the structure of simple C ∗ -algebras tensored with a UHF-algebra. II. J. Funct. Anal. , 107(2):255–269, 1992.[23] Mikael Rørdam. The stable and the real rank of Z -absorbing C ∗ -algebras. Internat. J. Math. , 15(10):1065–1084, 2004.[24] Mikael Rørdam and Wilhelm Winter. The Jiang-Su algebra revisited.
J. ReineAngew. Math. , 642:129–155, 2010.[25] Aidan Sims. ´Etale groupoids and their C ∗ -algebras.https://arxiv.org/abs/1710.10897.[26] Andrew S. Toms and Wilhelm Winter. Strongly self-absorbing C ∗ -algebras. Trans. Amer. Math. Soc. , 359(8):3999–4029, 2007.[27] Jesper Villadsen. Simple C ∗ -algebras with perforation. J. Funct. Anal. ,154(1):110–116, 1998.[28] Wilhelm Winter. Nuclear dimension and Z -stability of pure C ∗ -algebras. In-vent. Math. , 187(2):259–342, 2012.[29] Wilhelm Winter and Joachim Zacharias. Completely positive maps of orderzero.
M¨unster J. Math. , 2:311–324, 2009.
Department of Mathematics and Statistics, University of Ottawa,150 Louis-Pasteur Pvt, Ottawa, ON, Canada K1N 6N5
E-mail address : [email protected] Department of Mathematics and Statistics, University of Ottawa,150 Louis-Pasteur Pvt, Ottawa, ON, Canada K1N 6N5
E-mail address ::