aa r X i v : . [ m a t h . OA ] F e b STABLE ELEMENTS AND PROPERTY (S)
JOAN BOSA
Abstract.
We study the relation (and differences) between stability and Property (S)in the simple and stably finite framework. This leads us to characterize stable elementsin terms of its support, and study these concepts from different sides : hereditarysubalgebras, projections in the multiplier algebra and order properties in the Cuntzsemigroup. We use these approaches to show both that cancellation at infinity onthe Cuntz semigroup just holds when its Cuntz equivalence is given by isomorphismat the level of Hilbert right-modules, and that different notions as Regularity, ω -comparison, Corona Factorization Property, property R, etc.. are equivalent undermild assumptions. introduction Characterizing and understanding the class of separable simple C*-algebras is animportant problem since the origins of operator algebra theory. Concerning structuralquestions, one of the most natural (and old) problems wonders how to determine stabilityfor separable and simple C*-algebras. Recall that a separable C*-algebra A is said tobe stable if A ∼ = A ⊗ K , where K is the C*-algebra of compact operators on a separable,infinite dimensional Hilbert space. Furthermore, we say that a positive element a ∈ A is stable if its associated hereditary subalgebra aAa is stable.Related answers of the above fundamental question were given by Brown, Cuntz,Hjelmborg, Rørdam, Winter and many others ([10, 7, 13, 14, 21, 24]). Indeed, in [14]the authors shown that an AF -algebra is stable if and only if it admits no boundedtraces. Moreover, they wondered if a C*-algebra A is stable if and only if satisfies whatis known as property (S) ( A admits no bounded 2-quasitrace and no quotient of A isunital). Rørdam answers negatively the last question in [24], where he built the firstexample of a C*-algebra B such that M ( B ) is stable, but B is not stable. Hence,stability is not an stable property.In the literature, there exist several properties that relate property (S) and stability.We say that a C*-algebra A is regular (asymptotically regular) if any full subalgebra D of A ⊗ K satisfying property (S) is itself stable (there exists n ≥ M n ( D )is stable). The notion of regularity was coined by Rørdam in [25] to study stable C*-algebras, and it is equivalent to both a pure algebraic condition known as ω -comparisonfor the Cuntz semigroup of A (see [21]), and a more analytical property concerningprojections of M ( A ⊗ K ). Indeed, it is shown in Proposition 3.11 that a multiplierprojection P belongs to a non-regular ideal of M ( A ⊗ K ) if and only if the hereditaryC*-subalgebra P ( A ⊗ K ) P satisfies property (S). Date : February 19, 2021.The author was partially supported both by the DGI-MINECO and European Regional DevelopmentFund through the grant MTM2017-83487-P and by the Generalitat de Catalunya through the grant 2017-SGR-1725. Also, this research was carried out thanks to the Beatriu de Pin´os postdoctoral programmeof the Government of Catalonia’s Secretariat for Universities and Research of the Ministry of Economyand Knowledge.
Another important property for C*-algebras linked to stability is the Corona Factor-ization Property (CFP). This was introduced by Elliott-Kucerovsky in [12] in order tostudy absorbing extensions, and it is one of the natural candidates to determine ”nice”C*-algebras. The CFP was used by Zhang (and later by Ortega-Perera-Rørdam in [21])to get a dichotomy result (simple C*-algebras with real rank zero and CFP are eitherstably finite or purely infinite). There exist several equivalent definitions of CFP (seefor instances [5, 19, 20, 28]). We take as the definition the following property, whichis shown to be equivalent to the CFP in [17, Theorem 4.2] and [21, Theorem 5.13]. Aseparable C*-algebra A satisfies the CFP if given a full, hereditary subalgebra D of A ⊗ K such that M n ( D ) is stable for some n ≥
1, then D itself is stable.Focusing on simple, stably finite and separable C*-algebras, we study all the abovementioned properties looking at the Cuntz semigroup invariant associated to C*-algebras.In this setting, the Cuntz semigroup has a unique properly infinite element, which weusually denote by ∞ (see [5] for further details in this direction). The ∞ element, whichcoincides with the largest element of the Cuntz semigroup, has different properly in-finite representatives in ( A ⊗ K ) + (a natural representative of ∞ is a strictly positiveelement of A ⊗ K ); however, one wonders whether all of the other representatives of ∞ are also stable elements. A deep study of (properly) infinite elements was run byKirchberg-Rørdam, and they explained the main differences between these elements in[15]. Building up from there, in Lemma 2.3 of the current paper, stable elements arecharacterized showing that this notion is equivalent to ask that any small portion atthe beginning of its spectrum is properly infinite. Using this description, we show inTheorem 2.6 that asking for all properly infinite elements to be stable is equivalent to analgebraic cancellation property of the Cuntz semigroup known as cancellation of smallelements at infinite. In particular, Theorem 2.6 shows that weak cancellation propertyfor the Cuntz semigroup just holds when the Cuntz equivalence of positive elements isinduced by isomorphism of the associated Hilbert right-modules. Furthermore, we useLemma 2.3 and Theorem 2.6 to study ( ω, n )-decomposable elements as defined in [28].It is important to point out that the relation between these elements and stability wasalready stated in [28, Proposition 9.7]; however, the proof of the implication (i) ⇒ (ii)shown in [28] is not correct. We are able to clarify this relationship in Lemma 2.12.As a result of a more analytical approach, it was shown by Brown in [7] that anelement a in A ⊗ K is stable if and only if the multiplier projection associated to itshereditary subalgebra is Murray-von Neumann equivalent to the unit of M ( A ⊗ K ).Combining this result with our study of property (S) displayed in Proposition 3.11,we show in Theorem 3.12 that asymptotic regularity is equivalent to property R (if aprojection is contained in a proper ideal of M ( A ⊗ K ), it is also contained in a regularideal). As application of Theorem 3.12, we show in Corollary 3.13 that both property R and asymptotic regularity imply dichotomy between stably finite and purely infiniteC*-algebras in the simple setting.We briefly outline the contents of this paper. In the first section we recall notation andsome background needed to understand the sequel. In order to ease reading this section,it is divided in two parts: we first provide the necessary knowledge about Cu-semigroups,and we subsequently recall the comparison properties such as ω -comparison and CFPused in the current paper. Section 2 is dedicated to stable elements. In particular, westate the characterization of stable elements explained along the introduction and someof its implications. We finish this section studying the notion of ( ω, n )-decomposableelements previously introduced in [28]. In section 3 we develop all the machinery needed TABLE ELEMENTS AND PROPERTY (S) 3 to get Theorem 3.12. This part was initiated some time ago as a collaboration with M.Christensen, and some results in it already appeared in his PhD-thesis [9]; though, theyhave never been published in an article before.1.
Notation and Preliminaries
Cuntz semigroup.
In this part we recall the main facts about the Cuntz semi-group that will be used along the sequel. This semigroup has been deeply studied in thelast years due to its relation to the classification programme of separable simple nuclearC*-algebras. We encourage the reader to look at the overview article [1] for furtherdetails on this semigroup. Let’s start by its definition.
Definition 1.1.
Let A be an stable C ∗ -algebra, i.e. A ∼ = A ⊗ K (where K denotes theC*-algebra of compact operators in an infinite dimensional Hilbert space), and let a , b ∈ A + . We say that a is Cuntz subequivalent to b , in symbols a - b , provided there isa sequence ( x n ) in A such that x n bx ∗ n converges to a in norm, i.e. k a − x n bx ∗ n k → .We say that a and b are Cuntz equivalent if a - b and b - a , and in this case we write a ∼ b . Considering this equivalence relation in A ⊗ K , one obtains the abelian semigroup Cu( A ) := ( A ⊗ K ) + / ∼ . We denote the equivalence classes by h a i .This is a partially ordered abelian semigroup where the operation and order are givenby h a i + h b i = h (cid:18) a b (cid:19) i = h a ⊕ b i , h a i ≤ h b i if a - b. In particular, the semigroup
Cu( A ) is referred to as the Cuntz semigroup . As explained in [6, 11, 22], among others, the Cuntz semigroup of a separable C*-algebra A can be described via different frameworks, i.e. positive elements, hereditarysubalgebras, open projections in A ∗∗ and projections in the multiplier algebra M ( A ⊗K ).We will use all these settings along the sequel, so let us shortly recall the main facts andnotation.Any positive element in a C*-algebra A naturally defines: an hereditary subalgebra A a := aAa , an open projection p a := (strong) lim a /n in A ∗∗ , and a Hilbert A -module E a := aA . One can define the Cuntz semigroup of a C*-algebra A looking at each ofthese different pictures (see [22] for further details). In the Hilbert A -modules picture,for two Hilbert A -modules E, F , we write E ⋐ F if there exists x ∈ K ( F ), the compactoperators of L ( F ), such that xe = e for all e ∈ E . We use it to define the Cuntzsubequivalence between E and F , written as E - Cu F , as if for every Hilbert A -submodule E ′ ⋐ E there exists F ′ ⋐ F with E ′ ∼ = F ′ (isometric isomorphism). It isimportant to notice that, given positive element a, b ∈ A + , it follows that E a ∼ = E b if and only A a ∼ = A b . However, the equivalence relation induced by isomorphism onHilbert A -modules is stronger than the Cuntz equivalence just defined (see [8] for aconcrete counterexample). It is known that under the extra assumption of stable rankone, the Cuntz relation is equivalent to the equivalence relation induced by isomorphismon Hilbert A -modules.Moving to the algebraic framework, the Cuntz semigroup of a C*-algebra alwaysbelongs to an algebraic category of ordered complete semigroups called Cu. The objectsof this category are called Cu-semigroups, and we usually denote them by S and T (see[2] for further details).Fixing S a Cu-semigroup, let us remind some of its main properties. Every increasingsequence has a supremum in S , and S has an auxiliary relation usually denoted by ≪ , JOAN BOSA and called way-below. In particular, one writes x ≪ y if, whenever { x n } is an increasingsequence satisfying y ≤ sup x n , then x ≤ x n for some n . An element x in S is called compact , if x ≪ x . We write y < s x if there exists k ∈ N such that ( k + 1) y ≤ kx , and y ∝ x if there exists n ∈ N such that y ≤ nx . An element x in S is called full if forany y ′ , y ∈ S with y ′ ≪ y , one has y ′ ∝ x , denoted by y ¯ ∝ x . A sequence { x n } in S issaid to be full if it is increasing and for any y ′ , y ∈ S with y ′ ≪ y , one has y ′ ∝ x n forsome (hence all sufficiently large) n. Notice that if x ∈ S is an order unit, it is also afull element, but the reverse is not true.Using these notions, we say that S is simple if x ¯ ∝ y for all nonzero x, y ∈ S . Inother words, S is simple if every nonzero element is full.Focusing on the behaviour of elements in S , let us recall that an element a ∈ S is finite if for every element b ∈ S such that a + b ≤ a , one has b = 0. An element is infinite if it is not finite. An infinite element a ∈ S is properly infinite if 2 a ≤ a . Wesay that S is stably finite if an element a ∈ S is finite whenever there exists ˜ a ∈ S with a ≪ ˜ a . In particular, if S contains a largest element (always happens when it iscountably based or simple), usually denoted by ∞ , then the condition of being stablyfinite is equivalent to a ≪ ∞ ([2, Paragraph 5.2.2.]). Furthermore, if S is simple, then S is purely infinite if S = { , ∞} , i.e. S only contains the zero and largest elements.As shown in [2, Proposition 5.2.10], the ∞ element in a simple Cu-semigroup S is notcompact if and only if S is stably finite. We will denote the elements way-below ∞ by S ≪∞ := { s ∈ S | s ≪ ∞} . In fact, to ease notation, we will often denote it without ∞ , i.e. S ≪ .We finish this part about Cu-semigroups describing their functionals. By a functionalon a Cu-semigroup S we mean a map λ : S → [0 , ∞ ] that preserves addition, order,the zero element, and suprema of increasing sequences. Note that if S is a simple Cu-semigroup, then every functional is faithful (i.e. for nonzero λ ∈ F ( S ) , λ ( x ) = 0 if x =0). We denote the set of functionals on S by F ( S ). The differences and relations betweenfunctionals and states that one can associate to a Cu-semigroup are deeply studied in[5]. We also encourage the reader to look at [27] to know more about functionals in thisframework.1.2. Comparison properties.
Let us now recall the comparison conditions that formthe heart of this article. The majority of the results stated here were proved in [5], andwe encourage the reader to check it for further details.We start this part studying ω -comparison property for Cu-semigroups. This propertyhas been studied in several articles such as [5, 19, 20, 21], and it is implied for themajority of known regularity conditions of separable simple stably finite C*-algebras.Indeed, Ng coined this property as regularity, and it was in [3, Corollary 4.2.5] where theauthors shown that regularity is equivalent to an algebraic condition on Cu( A ) called ω -comparison. The equivalence described in Definition 1.3 is shown in [5]. Definition 1.2.
A C*-algebra A is said to be regular if every full hereditary subalgebraof A , with no nonzero unital quotients and no nonzero bounded 2-quasitraces, is stable. Definition 1.3.
Let S be a simple Cu -semigroup. Then, we say S satisfies ω -comparisonif any of the following equivalent conditions (and then all) holds:(1) Whenever ( y n ) is a sequence of nonzero elements in S ≪∞ such that y n < s y n +1 for all n , then P ∞ n =1 y n = ∞ (in S ). TABLE ELEMENTS AND PROPERTY (S) 5 (2) Whenever ( y n ) is a sequence of nonzero elements in S ≪∞ such that λ ( P ∞ n =1 y n ) = ∞ for all functionals λ ∈ F ( S ) , then P ∞ n =1 y n = ∞ . We continue this part with the Corona Factorization Property (CFP for short). Asstated in the introduction this was introduced by Elliott-Kucerovsky in order to studywhen extensions of C*-algebras are absorbing ([12]). This property has many equivalentdefinitions depending on the setting one works (see [19] for further study). We recallbelow the characterization of this property found in [19, Theorem 4.2]. This determinesstructural results about stable C*-algebras.
Definition 1.4. ([19, Theorem 4.2])
Let A be a separable stable C*-algebra. It satisfiesthe Corona Factorization Property if given a full, hereditary subalgebra D of A such that M n ( D ) is stable for some integer n ≥ , then D itself is stable. Moving to the simple Cu-semigroup setting, this property was characterized in [20]as a certain comparison property associated to the Cuntz semigroup. Indeed, a σ -unitalC*-algebra A has the corona factorization property if and only if its Cuntz semigroupCu( A ) has the CFP as defined below ([21]). Let’s us recall a couple of equivalentdefinitions of CFP in this setting. Definition 1.5. ([5, 28])
Let S be a simple Cu -semigroup. Then it satisfies CFP if anyof the following equivalent properties (and then all) hold: • Given any full sequence ( x n ) n in S , any sequence ( y n ) n in S , an element x ′ in S such that x ′ ≪ x , and a positive integer m satisfying x n ≤ my n for all n , thenthere exists a positive integer k such that x ′ ≤ y + . . . + y k . • Given a sequence ( y n ) n in S ≪∞ such that m · P ∞ n = k y n = ∞ for some m and all k ∈ N , then P ∞ n =1 y n = ∞ . The last condition we want to recall is property (QQ). This was originally introducedin [21], and it is deeply related to both conditions just defined (see [5] for further details).As stated before, the behaviour of the largest element in a simple Cu-semigroup iscomplicated to characterize, and the following property just concerns about this element.
Definition 1.6. ([21]) A Cu -semigroup S satisfies the property (QQ) if every elementin S , for which a multiple is properly infinite, is itself properly infinite. For simple Cu-semigroups there is a unique properly infinite element, usually denotedby ∞ . Hence, the above means that if mx = ∞ for some m ∈ N , then x = ∞ .2. stable elements The notion of stable element can be found in [15], where the authors study finite,infinite and properly infinite positive elements in a C*-algebra. They show in [15,Proposition 3.7] that any stable element is properly infinite, but few is known about theconverse. We start recalling the definition of stable elements in A . Definition 2.1.
A positive element a in a C*-algebra A is called stable if aAa is astable C*-algebra. Along this section we deeply study the above notion, and we relate it to the next:
Definition 2.2.
Let A be a separable simple C*-algebra. We say that A is Cu -stablewhether for any representative a ∈ ( A ⊗ K ) + of ∞ ∈ Cu( A ) , i.e. h a i = ∞ in Cu( A ) ,one has that a ( A ⊗ K ) a is a stable C*-algebra. JOAN BOSA
Note that the above definition mainly concerns about the C*-algebra rather its Cuntzsemigroup. In Theorem 2.6 we show that this property can be characterized as a can-cellation property for the Cuntz semigroup. Furthermore, A being Cu-stable impliesthat A is stably finite (Theorem 2.6). Hence, we will often assume this fact wheneverCu-stability is asked. It is worth to point out that, equivalently, the above definitionasks that any properly infinite element is stable.In order to study stable elements, we will use the following two continuous f δ , g δ : R + → R + functions. We thank J. Gabe for suggesting this direction. f δ ( x ) = t, for t ∈ [0 , δ/ δ − t, for t ∈ [ δ/ , δ ]0 for t ≥ δ g δ ( x ) = , for t ∈ [0 , δ/ − √ δt − − , for t ∈ [ δ/ , δ ]1 for t ≥ δ Note that f δ ( t ) = (1 − g δ ( t )) t (1 − g δ ( t )) and that f δ ( t ) ⊥ g δ ( t ) for all δ >
0. Moreover, { g δ ( a ) } δ> defines an approximate unit on A a = aAa , for any positive element a ∈ A + . Lemma 2.3.
Let a ∈ A + be a positive element. The following are equivalent: (i) a is stable (ii) f δ ( a ) is properly infinite and full in aAa for every δ > f δ ( a ) Cuntz dominates every element in aAa for every δ > .Proof. (i) ⇒ (ii) Let δ > f δ ( a ) = (1 − g δ ( a )) a (1 − g δ ( a )) is stable(and thus properly infinite) by [14, Corollary 4.3] (notice separability is not needed inthe proof, only σ -unitality). Suppose for contradiction that f δ ( a ) is not full, and let I be the two-sided closed ideal it generates. Then the spectrum of a + I is contained in[ δ, k a k ], so aAa/I is unital. This contradicts stability of a , indeed every quotient of anstable C*-algebra is also stable, and thus f δ ( a ) must be full for all δ > ⇒ (iii) This follows from [15, Proposition 3.5].(iii) ⇒ (i) Let x ∈ aAa and ε >
0. Since { g δ ( a ) } is an approximate identity for aAa ,pick δ > k g δ ( a ) xg δ ( a ) − x k < ε/
2, and define x := g δ ( a ) xg δ ( a ). Byassumptions, f δ ( a ) Cuntz-dominates x ; hence, there exists y such that y ∗ y = ( x − ε/ + and yy ∗ ∈ f δ ( a ) Af δ ( a ) . Recall that by construction, one has f δ ( a ) ⊥ g δ ( a ); therefore, yy ∗ ⊥ y ∗ y with k y ∗ y − x k < ε . By the criteria described in [14, Theorem 2.2], one obtains that aAa is stableas desired. (cid:3) After settling the background of stable elements, let us show the relation between thecomparison properties exposed in Section 1 and stable elements. Let us start by thefollowing result.
Corollary 2.4.
Let A be a simple separable stably finite C*-algebra such that Cu( A ) satisfies property (QQ) . Then, A is Cu -stable.Proof. Let a ∈ A ⊗ K be positive element such that h a i = ∞ in Cu( A ); namely, a is aproperly infinite element in A ⊗ K . Since A is stably finite, it follows that ∞ 6≪ ∞ by[2, Proposition 5.2.10]; therefore, f δ ( a ) = 0 for all δ > a - f δ ( a ) + g δ ( a ) - f δ ( a ) ⊕ g δ ( a ) for all δ >
0. We can use simplicity of Cu( A ) to find n ∈ N such that h g δ ( a ) i ≤ ( n − h f δ ( a ) i ; TABLE ELEMENTS AND PROPERTY (S) 7 then, h a i ≤ n h f δ ( a ) i . Using property (QQ), one has that h f δ ( a ) i is equal to ∞ in Cu( A ),and so f δ ( a ) is properly infinite for all δ . Therefore, a is stable by Lemma 2.3. (cid:3) Notice that Cu-stability of A is a natural property to ask for any stably finite C*-algebra. Indeed, looking at Cu( A ) from the Hilbert A -modules picture (see [11] forfurther details about this approach), Cu-stability of A asks for all Hilbert A-modulesassociated to different representatives of ∞ ∈ Cu( A ) to be isomorphic to ℓ ( A ⊗ K ).It is well-known that the Cuntz equivalence between Hilbert A -modules does not im-ply isomorphism between them (see [8] for a concrete counterexample); however, thatproperty holds under stable rank one assumption. Another extra property arising fromstable rank one assumption is that Cu( A ) becomes weak cancellative (as shown in [26,Proposition 4.2]). Next lemma uses the approximation displayed by Rørdam-Winter toshow that Cu-stability of A also gives us cancellation for big elements on Cu( A ). Lemma 2.5.
Let A be a separable stably finite C*-algebra and a, b, p ∈ ( A ⊗ K ) + with p a projection. Assume further that b ⊕ p is an stable element in A ⊗ K . If a ⊕ p - b ⊕ p, then a - b .Proof. Let us assume, without loss of generality, that a, b and p are orthogonal elementsin A ⊗ K . By [29, Corollary 2.56], the Cuntz comparison for stable C*-algebras isunitarily implemented. Namely, there exists a unitary in the unitarization of A ⊗ K suchthat u (( a − ε ) + + p ) u ∗ ∈ ( b + p )( A ⊗ K )( b + p ) =: B ⊗ K . Notice the latter equality holds due to the fact that ( b + p ) is an stable element byassumptions.Now, upu ∗ and p are Murray-von Neumann equivalent in B ⊗K ; therefore they are alsoPeligrad-Szid´o equivalent (see [22]). Using [6, Corollary 1.11], we extend the isometrydefining the Peligrad-Szid´o equivalence between p and upu ∗ to a unitary v in M ( B ⊗ K )satisfying that vpv ∗ = upu ∗ in B ⊗ K .Now, we have that v ∗ u ( a − ε ) + u ∗ v ∈ B ⊗ K , v ∗ u ( a − ε ) + u ∗ v ⊥ , v ∗ upu ∗ v = p, which provides that v ∗ u ( a − ε ) + u ∗ v belongs to b ( A ⊗ K ) b . This shows that ( a − ε ) + - b ;and as ε > a - b . (cid:3) A property of cancellation for big elements in Cu-semigroups was already introducedin [5]. Indeed, in the simple case, we say that a Cu-semigroup S satisfies Cancellation ofSmall elements at Infinity (CSE ∞ for short) whether x + y = ∞ with x ≪ ∞ , impliesthat y = ∞ . Next result shows the equivalence between the two notions under study:Cu-stability of A and CSE ∞ . In particular, it characterizes cancellation of big elementsin Cu( A ). In order to show that, we need to ensure the existence of a projection in A ⊗ K . That is the reason why we assume the algebra to be unital. Theorem 2.6.
Let A be a unital separable simple C*-algebra. Then A is Cu -stable ifand only if Cu( A ) satisfies CSE ∞ . In particular, A is Cu -stable implies that it is stablyfinite.Proof. For the ”if” direction, let us consider a properly infinite element, denoted by a , such that f δ ( a ) = 0 for all δ >
0. By construction it follows that ∞ = h a i ≤ JOAN BOSA h f δ ( a ) i ⊕ h g δ ( a ) i , with h g δ ( a ) i ≪ h a i . By (CSE ∞ ), one has that h f δ ( a ) i is properlyinfinite for all δ >
0; hence, Lemma 2.3 implies that a is stable as desired.For the converse direction, let us first check that Cu-stability of A implies it is stablyfinite. Then, we will show the desired implication in this case.If A was neither stably finite neither purely infinite, for any x ∈ Cu( A ) there exists n x ∈ N such that n x x = ∞ (see [5] for further details). Hence, considering the Cuntzclass of the unit on A , we have that n h A i = ∞ for some n ∈ N . By Cu-stability of A , one has that 1 M n ⊗ A is a stable element. Hence, by Lemma 2.3, f δ (1 M n ⊗ A ) =1 M n ⊗ f δ (1 A ) is properly infinite for all δ >
0. Since 1 A is a projection, there exists δ ′ > f δ ′ (1 A ) = 0, what provides a contradiction since we requested f δ (1 A ) = 0 forall δ > A is stably finite, let us show that Cu-stability of A implies (CSE ∞ ).To this end, let x + y = ∞ with x ≪ ∞ . As before, assuming A is unital, we may furtherassume that x ≤ n h A i for some n ; hence, x may be considered a compact element inCu( A ). Let a projection p be one of its representatives ([8]). This, indeed, can be donedue x ≪ ∞ and Cu( A ) is simple.Since any representative of the above Cuntz class is stable, denoting x = h p i and y = h b i , one has that p ⊕ b is an stable element in A ⊗ K . Applying functional calculusto p ⊕ b , one has that f δ ( p ⊕ b ) = f δ ( p ) ⊕ f δ ( b ) for all δ >
0. Hence, by Lemma 2.3 f δ ( p ) ⊕ f δ ( b ) is properly infinite for all δ >
0. Since p is a projection, there exists δ > f δ ( p ) = 0; therefore, f δ ′ ( p ) ⊕ f δ ′ ( b ) = f δ ′ ( b ) is properly infinite for all δ ′ < δ .That implies that b is an stable element by Lemma 2.3. (cid:3) As stated in [5], it is natural to wonder the following:
Question 2.7.
Does any separable simple and stably finite C*-algebra A satisfy that itis Cu -stable? Remark 2.8.
Using the equivalence of both properties described in Theorem 2.6, oneobserves that Question 2.7 wonders about the uniqueness of orthogonal complement onHilbert A -modules. Indeed, looking at Cu( A ) from the Hilbert A -modules picture de-scribed in [11] , Cu -stability of A says that whether x + y = ∞ in Cu( A ) with x ≪ ∞ ,then y = ∞ . Namely, if x = h a i and y = h b i , for a, b ∈ ( A ⊗ K ) + , stability gives us: E a ⊕ E b ∼ ℓ ( A ⊗ K ) ⇒ E a ⊕ E b ∼ = ℓ ( A ⊗ K ) ⇐⇒ E b ∼ = ℓ ( A ⊗ K ) . The above fact holds in the stable rank one setting. Indeed, in this framework the Cuntzequivalence is described by isomorphism of the associated Hilbert A -modules as shown in [22] , and we have a weak cancellation by [26] . Notice that Cu -stability of A asks for a”weak” version of stable rank one since we just need the above properties on the largestelement of Cu( A ) . We express it in the next corollary. Corollary 2.9.
Let A be a unital separable simple stably finite C*-algebra, and a ∈ ( A ⊗ K ) + . Then, the following are equivalent: • orthogonal complement of E a on ℓ ( A ⊗ K ) is unique (up to isomorphism). • Cuntz equivalence at ∞ ∈
Cu( A ) is induced by isomorphism of Hilbert ( A ⊗ K ) -modules. • A is Cu -stable. If Question 2.7 had an affirmative answer, it would answer [15, Question 3.4] in thesetting under study, as we next state.
TABLE ELEMENTS AND PROPERTY (S) 9
Corollary 2.10.
Let A be a unital separable simple Cu -stable C*-algebra. Then, M ( aAa ) is properly infinite for any properly infinite element a in A .Proof. By assumptions any properly infinite element a ∈ A is automatically stable;namely aAa is stable. The multiplier algebra of a stable C*-algebra contains the boundedoperators on an infinite-dimensional Hilbert space as a unital sub-C*-algebra; therefore,it is properly infinite (cid:3) We finish this section relating properly infiniteness with the notion of ( ω, n )-decompo-sable as defined in [28]. Let us first recall this notion.
Definition 2.11.
Let A be a C*-algebra, n ≥ be an integer, and u be an element in Cu( A ) . We say that u is ( w, n ) -decomposable if there exists x , x , . . . , different thanzero, in Cu( A ) such that P ∞ i =1 x i ≤ u and u ≤ nx i for all i . In [28, Lemma 9.2], the authors give several equivalent conditions to determinewhether an element is ( ω, n )-decomposable. These different notions are used in [28,Proposition 9.7] to provide a result relating stable C*-algebras and ( ω, n )-decomposability.However, the proof of the implication (i) ⇒ (ii) in [28, Proposition 9.7] is wrong. Indeed,they claim to find infinite mutually orthogonal positive elements in a non-stable C*-algebra, and this can not be done in general. Note that if [28, Proposition 9.7] wastrue, it would imply that properly infinite elements are always stable (answering [15,Question 3.4] in the general setting). Assuming Cu-stability for A , one get the followingcharacterization of ( ω, n )-decomposable elements. Lemma 2.12.
Let A be a simple separable Cu -stable C*-algebra, and x = h a i ∈ Cu( A ) .Then,(1) If x is ( ω, n ) -decomposable, then nx is properly infinite.(2) If nx is properly infinite for some n ∈ N , then x is ( ω, n ) -decomposable.Proof. The first part is trivial from the definition of ( ω, n )-decomposable. Indeed, byDefinition 2.11 one has that n · P ∞ i =1 x i ≤ n · x and x ≤ n · x i ; hence, ∞ · x ≤ n · ∞ X i =1 x i = ∞ X i =1 n · x i ≤ n · x, showing that nx is properly infinite.For the second statement, let x ∈ Cu( A ) such that nx is properly infinite for some n ∈ N , and consider the representative of x given in the statement, i.e. a positive element a ∈ A ⊗ K such that h a i = x . By Cu-stability of A , it follows that a ⊗ n is an stableelement; therefore, M n (( A ⊗ K ) a ) is stable. Use [20, Lemma 5.3] to find a sequence { a k } of pairwise elements in ( A ⊗ K ) a such that h ( a − /k ) + i ≤ n h ( a − /k ) + i ≤ n h a k i forall k in Cu(( A ⊗ K ) a ). Then, considering u = h a i , x k = h a k i and y k = h ( a − /k ) + i , itfollows from [28, Lemma 9.2(iii)] that h a i is ( ω, n )-decomposable as desired. (cid:3) Corollary 2.13.
Let A be a simple separable Cu -stable C*-algebra. Then, A satisfiesthe Corona Factorization property if and only if Cu( A ) satisfies the property (QQ) .Proof. The ”only if” part is already proven in [5], so let us show the converse. Let x ∈ Cu( A ) such that nx = ∞ for some n . Then, one has that x is ( ω, n )-decomposable byLemma 2.12. Using the characterization of Corona Factorization Property described in[28, Proposition 9.3], one has that any ( ω, n )-decomposable element is properly infinite;therefore, the desired result follows. (cid:3) Focusing on the real rank zero setting, recall that if A is simple and neither stably finitenor purely infinite case, then we have that any element in Cu( A ) is ( ω, n )-decomposablefor some n by [20, Proposition 4.2]. Using this fact, following the lines of the aboveproof, we obtain the well-known dichotomy between stably finite and purely infiniteunder the assumptions of simplicity, real rank zero and Corona Factorization property(Zhang’s Theorem).3. Projections on M ( A ⊗ K ) and Asymptotic ω -comparison In this section we seek to study asymptotic regularity for separable simple and stablyfinite C*-algebras, as defined by Ng in [19]. The final goal is to show the equivalencebetween this property and property R (Theorem 3.12). This section has been build onfrom a collaboration effort started some years ago with M. Christensen. In particular,some parts of this section already appeared in M. Christensen’s PhD-thesis.As happened with both ω -comparison and the Corona Factorization property in [21],we start rewriting asymptotic regularity condition in terms of an order property asso-ciated to the Cuntz semigroup. The equivalence between these properties is exposedin Lemma 3.6. After rephrasing this condition, we immerse into the M ( A ⊗ K ) frame-work to show the desired equivalence between property R and asymptotic regularity(Theorem 3.12).Let’s start recalling that a C*-algebra D is said to have property (S) if it has no unitalquotients and admits no bounded 2-quasitraces. This property tries to characterizestable C*-algebras as explained in the introduction, and it is the milestone of the currentsection. An equivalent definition of property (S) is given in [21, Proposition 4.5]. Thisstates that D satisfies property (S) if and only if for all a ∈ F ( D ) + , there exists b ∈ D + such that ab = 0 and h a i < s h b i in Cu( D ), where F ( D ) := { a ∈ A + | ae = e for some e ∈ A + } . We provide an equivalent condition in Proposition 3.11 using the projections of M ( A ⊗ K ). With this in mind, we recall the next definition. Definition 3.1 (Asymptotically regular) . Given a separable C*-algebra A , we say that A is asymptotically regular if, for any full hereditary C*-subalgebra D of A ⊗ K withproperty (S) , there exists an integer n ∈ N such that M n ( D ) is stable. In order to determine the above property, we will relate it to the following conditionin the Cuntz semigroup of A . Due to its relationship with ω -comparison, it is naturalto call it asymptotic ω -comparison. Definition 3.2.
Let S be a simple Cu -semigroup. We say that S satisfies asymptotic ω -comparison if the following holds: • whenever y , y , . . . is a sequence of non-zero elements in S ≪ , such that y i < s y i +1 for all i ≥ , there exists n ∈ N such that n P ∞ i = m y i = ∞ for all m ≥ . In a similar fashion that it is shown an equivalent definition of ω -comparison via theuse of functionals of Cu( A ) in [5], we are able to conclude the following proposition. Weomit the proof, and recommend to look at [5] for further details. Proposition 3.3.
Let S be a simple Cu -semigroup. Then the following are equivalent:(1) S has asymptotic ω -comparison.(2) Whenever y , y , . . . is a sequence of non-zero elements in S ≪ satisfying thecondition that λ ( P ∞ i =1 y i ) = ∞ for all non-zero functionals λ on S , there exists n ∈ N such that n P ∞ i = m y i = ∞ , for all m ∈ N . TABLE ELEMENTS AND PROPERTY (S) 11
Let us start with an easy lemma, which rewrites the first part of Proposition 1.5 (iv).
Lemma 3.4.
Let S be a Cu-semigroup and y , y , . . . be any sequence of elements in S .Then the next holds:(1) If there is an n ∈ N such that n P ∞ i = m y i = ∞ for all m ∈ N , then, for every j ≥ , it holds that P jk =1 y k ≤ n P ∞ i = j +1 y i . (2) If there is an n ∈ N such that n P ∞ i =1 y i = ∞ and n P jk =1 y k ≤ n P ∞ l = j +1 y l forall j ≥ , then n P ∞ l = m y l = ∞ for all m ≥ .Proof. The first statement is obvious, so let’s show the second. To this end, let y , y , . . . a sequence in S and n ∈ N as described in the statement. Then, for an arbitrary m ∈ N it follows 2 n ∞ X l = m y l = n ∞ X l = m y l + n ∞ X l = m y l ≥ n ∞ X i =1 y i = ∞ , getting the desired inequality. (cid:3) In order to conclude the desired equivalence we need the following lemmas. The firstis a mild elaboration of [21, Lemma 4.3] and needs the next definition.Given a C*-algebra D , and a strictly positive contraction c ∈ D + , let L c ( D ) := { a ∈ D + | g ε ( c ) a = a for some ε > } , where g ε is the continuous function defined before Lemma 2.3.Note that, if a, b ∈ L c ( D ), then a + b ∈ L c ( D ) and g δ ( c ) dg δ ( c ) ∈ L c ( D ), for every δ > d ∈ D + . Lemma 3.5.
Let D be a σ -unital C*-algebra with property (S) , and let c ∈ D be astrictly positive contraction. Then, for every a ∈ L c ( D ) , there exists b ∈ D + such that ab = 0 , h a i < s h b i , and b ∈ L c ( D ) .Proof. Choose ε ′ > g ε ′ ( c ) a = a and denote by e := g ε ′ ( c ). Note that a - ( e − / + . Since e ∈ L c ( D ), and D has property (S), there exists b ∈ D + suchthat eb = 0 and h e i < s h b i by [21, Proposition 4.5]. Moreover, there exists δ > h ( e − / + i < s h ( b − δ ) + i .Since { g /m ( c ) } is an approximate unit for D , we may choose m ∈ N such that ε :=1 /m < ε ′ / k b − g /m ( c ) b g /m ( c ) k < δ . Moreover, the element g /m ( c ) b g /m ( c )belongs to L c ( D ) and it is orthogonal to a . Hence, by [15, Lemma 2.5] one has h a i ≤ h ( e − / + i < s h ( b − δ ) + i ≤ h g /m ( c ) b g /m ( c ) i , as desired. (cid:3) We are now in the position of showing the equivalence between both notions.
Lemma 3.6.
Let A be a simple and separable C*-algebra. Then the following areequivalent:(1) Cu( A ) has asymptotic ω -comparison.(2) A is asymptotically regularProof. Assume Cu( A ) has asymptotic ω -comparison, and let D ⊆ A ⊗ K be a non-zerohereditary sub-C*-algebra, with property (S). Let c be a strictly positive element in D .Then, for every m ≥
1, the element c ⊗ m ∈ D ⊗ M m ∼ = M m ( D ) is strictly positive.Hence, proving the stability of M m ( D ), it suffices to prove the existence of n ∈ N such that, for all ε >
0, there exists b ∈ ( D ⊗ M m ) + satisfying that ( c ⊗ m − ε ) + ⊥ b , and( c ⊗ m − ε ) + - b by [14, Theorem 2.1].Let ( ε n ) n ≥ be a decreasing sequence of positive real numbers such that ε →
0. Weprove, by induction, that there exists a sequence ( b n ) of pairwise orthogonal, positiveelements in D , such that h b i i < s h b i +1 i , b i ∈ L c ( D ), h ( c − ε i ) + i < s h b i i , and ( c − ε i ) + ⊥ b i for all i ≥
1. The induction starts from Lemma 3.5. Now, let b , . . . , b n satisfying thedesired properties. By the properties of L c ( D ), we have that( c − ε n +1 ) + + b + . . . + b n ∈ L c ( D ) , whence, applying Lemma 3.5, it follows that there exists b n +1 satisfying the desiredproperties.For i ≥
1, let y i := h b i i ∈ Cu( A ). Now, by asymptotic ω -comparison it follows thatthere exists n ∈ N such that n P ∞ i = m y i = ∞ for all m ∈ N . Let ε > m ∈ N such that ε m < ε . Then, there exists N ∈ N such that h ( c ⊗ n − ε m ) + i ≤ n N X i = m h b i i = h N X i = m b i ⊗ n i . Since ( c ⊗ n − ε ) ≤ ( c ⊗ n − ε m ) ⊥ P Ni = m b i ⊗ n , the desired result follows.For the converse, let y , y , . . . a sequence of non-zero elements in Cu( A ) such that y i < s y i +1 for all i ≥
1. Choose pairwise orthogonal positive elements b i ∈ A ⊗ K suchthat k b i k ≤ − i and y i = h b i i . Set b := P ∞ i =1 b i and let D ⊆ A ⊗ K denote the hereditarysub-C*-algebra generated by b . We show that D has property (S). It is easy to seethat λ ( h b i ) = ∞ for all functionals λ on Cu( A ); hence, D does not admit any bounded2-quasitrace. Similarly, assuming for a contradiction that D is unital, it follows that b isinvertible, and therefore P mi =1 b i is invertible for some m ∈ N . Therefore, b k = 0 for all k > m since these elements are orthogonal to an invertible element. This implies that b i = 0 for all i due h b i i < s h b j i whenever i < j , a contradiction.Therefore, D has property (S) and D ⊗ M n is stable for some n ∈ N by asymptoticregularity of A . Now, by Lemma 3.4 one needs to show j X k =1 h ( b k ⊗ n − ε ) + i ≤ ∞ X i = j +1 h b i ⊗ n i for every j ≥ ε > ε >
0, and denoting d i := b i ⊗ n , for all i ≥ e jm := m X i = jg /m ( d i ) and e m := e m for every j ≥ m ≥ . By functional calculus, it follows that ( e m ) m ≥ is an approximate unit for D ⊗ M n .Moreover, e jm - P mi = j d i for all j, m ∈ N .Let us now use that D ⊗ M n is stable. Then, there exists a ∈ D ⊗ M n such that P jk =1 d k ⊥ a and P jk =1 d k - a . We may therefore choose δ > j X k =1 ( d k − ε ) + - ( a − δ ) + . TABLE ELEMENTS AND PROPERTY (S) 13
By construction, we can choose m ∈ N such that k a − e m ae m k < δ ; therefore, the aboveorthogonality implies e m a / = ( m X i =1 g /m ( d i )) a / = ( m X i = j +1 g /m ( d i )) a / = e j +1 m a / . In particular, e m ae m ≤ k a k ( e j +1 m ) ;hence, j X k =1 ( d k − ε ) + - ( a − δ )+ - e m ae m - ( e j +1 m ) - m X i = j +1 d i , as desired. (cid:3) The above equivalence shows that asymptotically regular implies dichotomy in oursetting.
Proposition 3.7.
Let A be a simple and separable C*-algebra. Then it is either stablyfinite or purely infinite if it is asymptotically regular.Proof. Assuming A is neither stably finite nor purely infinite, then every z ∈ Cu( A )satisfies nz = ∞ and there exists x ∈ Cu( A ) such that x ≪ ∞ and x = ∞ . UsingGlimm’s halving property (i.e. for each y ∈ Cu( A ) there exists z ∈ Cu( A ) such that2 z ≤ y ) we may find a sequence x := x, x , x , . . . ∈ Cu( A ) such that 2 i +1 x i ≤ x forall i ≥
1. Moreover, by induction it follows that P ∞ i = j x i ≤ x j for all j ≥ j P ∞ i = j x i ≤ j +1 x j ≤ x a finite element. Sinceevery element in Cu( A ) is eventually infinite, it follows that x i < s x i +1 for all i , butthere exists no n ∈ N such that n P ∞ i = m x i = ∞ for all m ∈ N . Namely, A cannot beasymptotically regular. (cid:3) We finish this section showing the equivalence between asymptotic regularity andproperty R , as defined in [19]. Let us first introduce some notation and background inorder to understand better the result.As explained in section 1, the Cuntz semigroup of a separable C*-algebra A canbe described via different frameworks. A concrete characterization between hereditarysubalgebras of A ⊗ K and projections in M ( A ⊗ K ) was explicitly written by Kucerovskyin [16, Lemma 10]: Lemma 3.8. ([16])
Let A ⊗ K be a separable C*-algebra. Then, for every hereditarysubalgebra B ⊂ A ⊗ K , there exists a multiplier projection P ∈ M ( A ⊗ K ) such that P ( A ⊗ K ) P ∼ = B . This approach is very useful to determine when a full hereditary subalgebra of A ⊗ K is stable. Indeed, we have the next important result shown by Brown in [7, Theorem4.23]. Theorem 3.9. ([7])
Let A ⊗ K be a separable C*-algebra, and P a multiplier projectionin M ( A ⊗ K ) . Then P ( A ⊗ K ) P is a stable, full, hereditary subalgebra of A ⊗ K if andonly if P is Murray-von Neumann equivalent to the unit of M ( A ⊗ K ) . This analytical side provides us a new definition of property (S), as stated below. Westart recalling some definitions.
Definition 3.10.
Let A be a unital C*-algebra. We say that A ⊗ K has property R, ifwhenever p is a projection contained inside a proper ideal of M ( A ⊗ K ) , then p is alsocontained inside a regular ideal. Recall that any unital trace on A extends canonically to a trace on the positive cone of M ( A ⊗K ). We use this extension to say that a proper ideal J of M ( A ⊗K ) is regular if J is contained in an ideal of the form J τ (norm-closure of { b ∈ M ( A ⊗ K ) | τ ( b ∗ b ) < ∞} ),for some unital τ on A . Otherwise, we say that J is non-regular. Proposition 3.11.
Let A be a unital separable simple exact C*-algebra, and P a mul-tiplier projection in M ( A ⊗ K ) defining a full hereditary C*-subalgebra P ( A ⊗ K ) P of A ⊗ K . Then, P ( A ⊗ K ) P satisfies property (S) if and only if P belongs to a non-regularideal of M ( A ⊗ K ) .Proof. Let P be a multiplier projection in M ( A ⊗ K ) as in the statement. Then, assumethat the hereditary C*-subalgebra P ( A ⊗ K ) P satisfies property (S). The definition ofproperty (S) asks for the lack of bounded 2-quasitraces; hence, P cannot be containedin any regular ideal of A ⊗ K . Namely, P belongs to a non-regular ideal.For the converse, let us assume that P belongs to a non-regular ideal of M ( A ⊗ K ).Then, the hereditary C*-subalgebra in A ⊗ K defined via P , i.e. P ( A ⊗ K ) P ⊆ A ⊗ K ,has no nonzero bounded 2-quasitraces and no quotient is unital due simplicity of A .Therefore, P ( A ⊗ K ) P satisfies property (S) by definition. (cid:3) Using Brown’s result described in Theorem 3.9, we use the above Proposition tosee when property (S) implies stability (up to some matrix extension). Indeed, thishappens when there is no proper non-regular ideals in M ( A ⊗ K ). In other words,when property R holds. Next result shows the equivalence between this property andasymptotic regularity. Theorem 3.12.
Let A be a unital separable simple exact C*-algebra. Then A ⊗ K isasymptotically regular if and only if A ⊗ K satisfies property R .Proof. Let us first show the ”only-if” direction. Let P be a multiplier projection in M ( A ⊗ K ) such that P is contained in a proper ideal of M ( A ⊗ K ). If P ∈ A ⊗ K , thenautomatically P is contained in a regular ideal of M ( A ⊗ K ); hence, we assume that P is not in A ⊗ K .In this case, suppose that P is not contained in any regular ideal of M ( A ⊗ K ). Thisimplies that P ( A ⊗ K ) P must be a full hereditary subalgebra of A ⊗ K , with no unitalquotient and no nonzero bounded 2 quasitraces, i.e., it satisfies property (S). By theassumption of asymptotic regularity, there exists n ∈ N such that M n ( P ( A ⊗ K ) P ) isstable; namely, the sum of n -copies of P , i.e. ⊕ n P , is Murray-von Neumann equivalentto the unit of M ( A ⊗ K ) (see Theorem 3.9). This contradicts our assumption that P is contained in a proper ideal of M ( A ⊗ K ); therefore, P belongs to a regular ideal of M ( A ⊗ K ), and property R holds.For the converse, let D be a nonzero hereditary subalgebra of A ⊗K satisfying property(S). Use Lemma 3.8 to find the multiplier projection P in M ( A ⊗K ) such that P ( A ⊗K ) P is isomorphic to D . By the lack of nonzero bounded 2-quasitraces given by property(S), the projection P cannot be contained in a regular ideal of M ( A ⊗ K ). Therefore, P is a norm-full element in M ( A ⊗ K ) due to property R .Since P is norm-full element, there exists n ∈ N such that ⊕ n P is Murray-von Neu-mann equivalent to the unit of M ( A ⊗ K ). Therefore, M n ( P ( A ⊗ K ) P ) ∼ = M n ( D ) isstable by Theorem 3.9 as desired. (cid:3) A simple combination of Proposition 3.7 and Theorem 3.12 provides the following:
Corollary 3.13.
Let A be a unital separable simple exact C*-algebra such that A ⊗ K satisfies property R . Then A is either stably finite or purely infinite. TABLE ELEMENTS AND PROPERTY (S) 15
Remark 3.14.
One may wonder whether the converse of last corollary (or equivalentlyProposition 3.7) is true. The answer to that is negative, i.e. there exist a separablesimple stably finite C*-algebra that does not satisfy property R . The counterexamplesatisfying that condition appeared in [23] . In there it is constructed a stably finite pro-jection Q in the multiplier algebra of a separable stable simple C*-algebra A , whichsatisfies that Q = P ∞ j =1 p j , where p j are pairwise orthogonal projections in A satisfying λ ( p i ) = λ ( p j ) < ∞ for all i, j and all functionals λ .Stably finiteness is not specified in [23] , but it follows from the definition of projec-tion Q . Indeed, if for the contrary ∞ ∈ Cu( A ) was a compact element, then somemultiple of h P ∞ i =1 p j i would be equal than ∞ . Using compacity of ∞ , that implies that n P mi =1 p i = ∞ for some n, m ∈ N . Namely, λ ( p i ) = ∞ for all functionals, contradictingthe finiteness of them.This example shows that not all stably finite exact C*-algebras admits a bounded trace. An equivalent formulation of Corona Factorization Property for separable simple C*-algebras is that any norm-full multiplier projection P ∈ M ( A ⊗ K ) is Murray-vonNeumann equivalent to 1 M ( A ⊗K ) (see [17] for further details). Roughly speaking, CFPallows us to pass from n -size of stability, to stability itself. Hence, Theorem 3.12 inducesthe following: Corollary 3.15.
Let A be a unital separable simple exact C*-algebra. Then, the follow-ing are equivalent:(1) A is regular(2) A is asymptotically regular and satisfies CFP (3) A satisfies property R and CFPFocusing on the real rank zero setting, Ng also described asymptotic regularity in [19,Proposition 4.9]. In particular, he showed the following, which we recall (without proof)by completeness.
Proposition 3.16.
Let A be separable simple exact real rank zero C*-algebra. If thereexists a norm-full multiplier projection P ∈ M ( A ⊗ K ) not Murray-von Neumann equiv-alent to M ( A ⊗K ) , then there exists a full hereditary subalgebra D ⊆ A ⊗ K satisfyingproperty (S) such that M n ( D ) is not stable for any n ∈ N . Above proposition easily provides the argument that, in the real rank zero framework,asymptotic regularity (or property R) implies Corona Factorization Property. Indeed,if there was a norm-full projection different than the unit in M ( A ⊗ K ), then asymp-totic regularity would not hold. Therefore, by Corollary 3.15 it follows that Regularity,Asymptotic regularity and Property R are equivalent in this setting (as shown in [19,Theorem 4.14]).By Proposition 3.11, one has the latter condition exposed in Proposition 3.16 is equiv-alent to the existence of a projection P in a proper non-regular ideal in M ( A ⊗ K ). Inthis context, to build an example satisfying CFP and not being regular, one needs tobuild an algebra A such that both M ( A ⊗ K ) has a projection in a proper non-regularideal and all norm-full multiplier projections are Murray-von Neumann equivalent to1 M ( A ⊗K ) . We recommend to look at [18] to learn more about non-regular ideals in themultiplier algebra of an stable C*-algebra. Acknowledgments . The author would like to thank J. Gabe for suggesting Lemma2.3 and M. Christensen for preliminary discussions about asymptotic regularity. More-over, I would also like to thank P. Ara for their comments on an earlier version of thispaper.
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Dpt. de Matem`atiques, Fac. C, Universitat Aut`onoma de Barcelona, 08193 Bellaterra,Barcelona (Spain)
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