A local characterization for the Cuntz semigroup of AI-algebras
aa r X i v : . [ m a t h . OA ] F e b A LOCAL CHARACTERIZATION FOR THE CUNTZ SEMIGROUP OFAI-ALGEBRAS
EDUARD VILALTA
Abstract.
We give a local characterization for the Cuntz semigroup of AI-algebras building uponShen’s characterization of dimension groups. Using this result, we provide an abstract characterizationfor the Cuntz semigroup of AI-algebras. Introduction
The Cuntz semigroup of a C ∗ -algebra A , denoted by Cu( A ), is a powerful invariant introducedby Cuntz in [Cun78] that generalizes the construction of the Murray-von Neumann semigroup ofprojections. In fact, for the class of simple, nuclear Z -stable C ∗ -algebras, the Cuntz semigroup functoris, suitably interpreted, equivalent to the Elliott invariant (see [ADPS14]).There are a number of regularity properties formulated for Cuntz semigroups that appear in theclassification program of simple C ∗ -algebras. Among them, almost unperforation stands out, as itappeared in Toms’ example [Tom08] to distinguish two simple C ∗ -algebras that otherwise agreed ontheir Elliott invariant and many reasonable extensions of it. It also features prominently in the Toms-Winter conjecture (see [Win12], [Rør04], [CET + C ∗ -algebras of stable rank one have been uncovered,which lead to the solution of three open problems for this class [APRT21].In [CEI08], some of the properties of Cu( A ) were abstracted into a category of semigroups, termedCu, in order to reflect its continuous nature; see Paragraph 2.1 below. One of the relevant notionswhile studying the objects in Cu, often called Cu-semigroups, is that of compact containment, insymbols ≪ , which is the analogue of the compact containment relation in the lattice of open setsof a compact topological space (see, for example, [GHK +
03, Proposition I-1.22.1]). Projections in a C ∗ -algebra are the natural examples (and in relevant cases the only examples) of compact elements,that is, elements that are ≪ -below themselves. For a C ∗ -algebra A of stable rank one, Cu( A ) furthersatisfies a weaker form of cancellation, as proved in [RW10, Theorem 4.3]; see Paragraph 2.3.The Cuntz semigroup of a C ∗ -algebra is not usually algebraically ordered (it is, for example, inthe finite-dimensional situation). However, an appropriate substitute for this property was proved in[RW10], and has been termed since axiom (O5) (see also [Rob13, Proposition 5.1.1]).The Cuntz semigroup has also been succesfully used in the classification of certain non-simple C ∗ -algebras. Namely, Robert used it in [Rob12] to classify, up to approximate unitary equivalence, *-homomorphisms from a limit of one-dimensional NCCW complexes to a C ∗ -algebra of stable rank one.He subsequently obtained a classification of 1-dimensional NCCW complexes with trivial K -groupusing their Cuntz semigroup.In [CE08], Ciuperca and Elliott established a one-to-one correspondence between the so-calledThomsen semigroup and the Cuntz semigroup, which as a consequence yielded a classification of allseparable AI-algebras by means of their Cuntz semigroup; see also [Tho92].We focus in this paper on (separable) AI-algebras, and more concretely on the range problem for theCuntz semigroup for this class, that is, to determine a natural set of properties that a Cu-semigroup S must satisfy in order to be isomorphic to Cu( A ) for such a C ∗ -algebra A . In order to ease the Date : March 1, 2021.2010
Mathematics Subject Classification.
Primary 46L05, 46L85, 06F05. Secondary 06A06, 46L80, 19K99.
Key words and phrases. C ∗ -algebras, Cuntz semigroups, AI-algebras.The author was partially supported by MINECO (grant No. PRE2018-083419 and No. MTM2017-83487-P), and bythe Comissionat per Universitats i Recerca de la Generalitat de Catalunya (grant No. 2017SGR01725). notation, we will say that a C ∗ -algebra is an AI-algebra if it is *-isomorphic to an inductive limit ofthe form lim n C [0 , ⊗ F n with F n finite dimensional for every n .Our line of attack consists of adapting the strategy used to obtain a local characterization ofdimension groups due to Shen [She79, Theorem 3.1], which was then utilized in the Effros-Handelman-Shen theorem [EHS80, Theorem 2.2]. More explicitly, recall that as a combination of these two resultsone obtains that a countable unperforated ordered (abelian) group G is order isomorphic to the ordered K -group of an AF-algebra if and only if, for every order homomorphism ϕ : Z r → G and element α ∈ ker( ϕ ), there exist order homomorphisms θ, φ such that the diagram Z r ϕ / / θ (cid:15) (cid:15) G Z s φ > > ⑥⑥⑥⑥⑥⑥⑥⑥ commutes and α ∈ ker( θ ).Following the structure of the proof of the abovementioned result (but with additional care), wegive a local characterization for the Cuntz semigroup of AI-algebras; see Theorem C below. We brieflydiscuss some of the details:There are two key ingredients that lead to the proof of our main result. The first one is a suitableanalogue for Cu-semigroups of the well known fact that every element of a group is the image of 1 ∈ Z through a group homomorphism. Theorem A (cf 3.1) . Let S be a Cu-semigroup satisfying (O5) and weak cancellation, and letLsc([0 , , N ) be the Cu-semigroup of lower-semicontinuous functions [0 , → N . Then, given an ele-ment s and a compact element p in S such that s ≤ p , there exists a Cu-morphism Lsc([0 , , N ) → S mapping χ (0 , to s and 1 to p .Given a Cu-semigroup S , in Section 4 we study Cauchy sequences ( ϕ i ) i of Cu-morphisms, where ϕ i : Lsc([0 , , N ) → S for each i , with respect to the distance introduced in [CE08] (see also [CES11]and [RS10]). We show that, under certain conditions, such sequences have a unique limit. Theorem B (cf 4.12) . Let S be a weakly cancellative Cu-semigroup satisfying (O5), and considera sequence of Cu-morphisms ϕ i : Lsc([0 , , N ) → S such that d ( ϕ i , ϕ i +1 ) < ǫ i with ( ǫ i ) i strictlydecreasing and P ∞ i =1 ǫ i < ∞ . Also, assume that ϕ i (1) = ϕ i +1 (1) for each i .Then, there exists a unique Cu-morphism ϕ : Lsc([0 , , N ) → S with d ( ϕ, ϕ i ) → ϕ (1) = ϕ i (1)for every i .Since countably generated Cu-semigroups are generally far bigger than countably generated groups(the former being usually uncountable and the latter being always countable), one cannot hope to ob-tain an exact analogue of Shen’s theorem [She79, Theorem 3.1] with a commutative diagram. Instead,what one does get is an approximate version of it. Theorem C (4.15, 4.16) . Let S be a countably based Cu-semigroup satisfying weak cancellation and(O5) where every compactly bounded element is bounded by a compact. Then, S is Cu-isomorphicto the Cuntz semigroup of an AI-algebra if and only if for every Cu-morphism ϕ : Lsc([0 , , N ) r → S ,every ǫ > x, x ′ , y in Lsc([0 , , N ) r such that x ≪ x ′ with ϕ ( x ′ ) ≪ ϕ ( y ), there existCu-morphisms θ, φ such that the diagramLsc([0 , , N ) r ϕ / / θ (cid:15) (cid:15) S Lsc([0 , , N ) s φ ttttttttttt satisfies:(i) d ( φθ, ϕ ) < ǫ .(ii) θ ( x ) ≪ θ ( y ).(iii) ϕ (1 j ) = φθ (1 j ) for every 1 ≤ j ≤ r . LOCAL CHARACTERIZATION FOR THE CUNTZ SEMIGROUP OF AI-ALGEBRAS 3
Finally, in Section 5 we introduce property I (see Definition 5.30) and provide, using a discreteversion of Theorem C (see Theorem 4.17), an abstract characterization for the Cuntz semigroup ofAI-algebras.
Theorem D (5.35) . Let S be a Cu-semigroup. Then, S is Cu-isomorphic to the Cuntz semigroupof an AI-algebra if and only if S is countably based, compactly bounded and it satisfies (O5), (O6),weak cancellation, and property I. Acknowledgments.
This paper constitutes a part of the Ph.D. dissertation of the author. He isgrateful to his advisor Francesc Perera for his guidance and many fruitful discussions on the subject.2.
Preliminaries
Let S be a positively ordered monoid and let x, y be elements in S . Recall that we write x ≪ y , andsay that x is compactly contained in (or way-below ) y , if for every increasing sequence ( z n ) n whosesupremum exists and is such that y ≤ sup z n , we have x ≤ z n for some n ∈ N . As defined in [CEI08], a Cu -semigroup S is a positively ordered monoid that satisfies the followingproperties:(O1) Every increasing sequence in S has a supremum.(O2) Every element in S can be written as the supremum of an ≪ -increasing sequence.(O3) For every x ′ ≪ x and y ′ ≪ y , we have x ′ + y ′ ≪ x + y .(O4) For every pair of increasing sequences ( x n ) n and ( y n ) n we have sup n x n + sup n y n = sup n ( x n + y n ) n .The category Cu is defined as the subcategory of positively ordered monoids whose objects areCu-semigroups and whose morphisms are positively ordered monoid morphisms preserving the way-below relation and suprema of increasing sequences. The morphisms in Cu are usually referred to asCu -morphisms . It was proven in [CEI08] that there exists a functor between the category of C ∗ -algebras and Cu.We briefly recall the construction here:Given two positive elements a, b in a C ∗ -algebra A , we say that a is Cuntz subequivalent to b ,denoted by a - b , if there exists a sequence ( r n ) n in A such that a = lim r n br ∗ n . The elements a, b aresaid to be Cuntz equivalent if a - b and b - a .The Cuntz semigroup of A , denoted by Cu( A ), is defined as the quotient ( A ⊗ K ) + / ∼ . Writting theclass of a positive element a by [ a ], the Cuntz semigroup of A becomes a Cu-semigroup when endowedwith the order induced by - and the addition induced by [ a ] + [ b ] = (cid:2)(cid:0) a b (cid:1)(cid:3) .Given a *-homomorphism ϕ : A → B , let ϕ also denote the amplification ϕ : A ⊗ K → B ⊗ K . Then, ϕ induces a Cu-morphism between Cu( A ) and Cu( B ), denoted by Cu( ϕ ), by sending an element[ a ] ∈ Cu( A ) to [ ϕ ( a )].By [APT18, Corollary 3.2.9], the category Cu has inductive limits and the functor Cu : C ∗ → Cu iscontinuous.
The Cuntz semigroup of a C ∗ -algebra always satisfies the following additional properties (see[APT18, Proposition 4.6] and [Rob13, Proposition 5.1.1] respectively):(O5) Given x + y ≤ z , x ′ ≪ x and y ′ ≪ y , there exists c such that x ′ + c ≤ z ≤ x + c and y ′ ≪ c .(O6) Given x ′ ≪ x ≤ y + z there exist elements v ≤ x, y and w ≤ x, z such that x ′ ≤ v + w .A Cu-semigroup is said to be countably based if it contains a countable sup-dense subset, that is tosay a countable subset such that each element in the semigroup can be written as the supremum ofan increasing sequence of elements in the subset. Every separable C ∗ -algebra has a countably basedCuntz semigroup; see, for example, [APS11].Also recall that a Cu-semigroup is weakly cancellative if x ≪ y whenever x + z ≪ y + z . Itwas proven in [RW10, Theorem 4.3] that stable rank one C ∗ -algebras have weakly cancellative Cuntzsemigroups. We say that a C ∗ -algebra is an AI-algebra if it is *-isomorphic to an inductive limit of the formlim n C [0 , ⊗ F n with F n finite dimensional for every n ∈ N . EDUARD VILALTA
AI-algebras were classified in [CE08] using the Cuntz semigroup. In fact, following the proof of[RLL00, Proposition 7.2.8] and using the result from [CE08] one can also prove the next theorem, whererecall that Cu( C [0 , , , N ), the Cu-semigroup of lower-semicontinuousfunctions [0 , → N (see, for example, [APS11, Theorem 3.4]). Theorem 2.5.
The Cuntz semigroup of an AI -algebra is Cu -isomorphic to the inductive limit of asystem of the form (Lsc([0 , , N ) k i , ϕ i ) . Conversely, for every inductive system (Lsc([0 , , N ) k i , ϕ i ) there exists an AI -algebra such that its Cuntz semigroup is Cu -isomorphic to the limit of the system. It is readily checked that finite sums of elements of the form χ ( t, , χ ( s,t ) , χ [0 ,t ) , and 1are a basis for Lsc([0 , , N ). Definition 2.6.
An element in Lsc([0 , , N ) will be called a basic indicator function if it is of theform χ ( t, , χ ( s,t ) , χ [0 ,t ) or 1 for some s, t .Also, we will say that an element f ∈ Lsc([0 , , N ) is basic if it is the finite sum of basic indicatorfunctions.Given any M ∈ N , we say that an element in Lsc([0 , , N ) M is a basic indicator function (resp. basic element ) if each of its components is a basic indicator function (resp. basic element). Remark 2.7.
Let F be the free abelian semigroup generated by the basic indicator functions (assymbols) on Lsc([0 , , N ). Given two elements g, h ∈ F , we write g ∼ h if and only if there exist f ∈ F and two open intervals U, V such that g = f + χ U + χ V and h = f + χ U ∪ V + χ U ∩ V . We write f ∼ g if f = g , f ∼ g or if g ∼ f .Let ≃ be the transitive relation induced by ∼ . Then, the quotient F/ ≃ is isomorphic to the monoidof basic elements in Lsc([0 , , N ), since for any f, g ∈ Lsc([0 , , N ) we have f + g = f ∨ g + f ∧ g .Indeed, this is related to distributive lattice ordered semigroups, as defined in [Vil21, Definition 4.1],and is generally true in Lsc( X, N ). Notation 2.8.
Given ǫ > ≤ a < b ≤
1, we define the ǫ -retraction of the intervals ( a, b ), ( a, , b ) as ( a + ǫ, b − ǫ ) , ( a + ǫ, , and [0 , b − ǫ )respectively.Given an interval U , we denote by R ǫ ( U ) its ǫ -retraction. Also, given a finite disjoint union ofintervals, we define its ǫ -retraction to be the finite disjoint union of the ǫ -retractions of the intervals.Given a basic indicator function χ U , we define R ǫ ( χ U ) = χ R ǫ ( U ) . Whenever we do not need tospecify ǫ , we will simply write R ( χ U ). Also, if ǫ > R ǫ ( U ) = ∅ , it will be understoodthat R ǫ ( χ U ) = 0. 3. Lifting morphisms
Given a group G and an element g ∈ G , there always exists a group morphism Z → G mapping1 to g . Even though this is trivial, it is a key feature in both Shen’s and Effros-Handelman-Shen’stheorems (see [She79] and [EHS80] respectively).In this section, we study an analogue of such a property in the category Cu. Namely, given aCu-semigroup S and an element s ∈ S , we wish to prove that, under the right assumptions, thereexists a Cu-morphism Lsc([0 , , N ) → S mapping χ (0 , to s . In fact, we will prove more: Theorem 3.1.
Let S be a weakly cancellative Cu -semigroup satisfying (O5). Then, given a finite ≪ -increasing sequence s ≪ · · · ≪ s n and a compact element p such that s n ≤ p , there exists a Cu -morphism φ : Lsc([0 , , N ) → S such that φ ( χ ( n − k/n, ) = s k and φ (1) = p . Recall that in a locally small category an object G is called a generator if for every pair of morphisms g, f : X → Y there exists a morphism h : G → X such that g ◦ h = f ◦ h .The proof of Theorem 3.1 will rely on the fact that the sub-Cu-semigroup of Lsc([0 , , N ) definedas G = { f ∈ Lsc([0 , , N ) | f (0) = 0 , f increasing } is a generator for the category Cu (see [Sch18,Section 5.2]). In particular, [APT20, Proposition 2.10] and [Sch18, Lemma 5.16] imply that, given LOCAL CHARACTERIZATION FOR THE CUNTZ SEMIGROUP OF AI-ALGEBRAS 5 any Cu-semigroup S and any finite ≪ -increasing sequence s ≪ · · · ≪ s n , there always exists aCu-morphism G → S mapping χ ( n − k/n, to s k for every k .Thus, the theorem will follow if we can prove that certain Cu-morphisms G → S can be lifted toCu-morphisms Lsc([0 , , N ) → S . This is done in Proposition 3.4.Some remarks are in order: Remark 3.2. (1) A Cu-morphism from G to a Cu-semigroup S may not always be liftable.For example, set S = Lsc([0 , , N ). Take ∞ ∈ Lsc([0 , , N ) and let α : G → S be a Cu-morphism with α ( χ (0 , ) = ∞ . Recall that such a map must exist because G is a generator.If such a map could be lifted, one would have ∞ = α ( χ [0 , ) ≪ α (1) ≤ ∞ , which would imply that ∞ is compact, a contradiction.(2) Even if a map can be lifted, the lift may not necessarily be unique. As an example, for every n ∈ N define the Cu-morphism i n : Lsc([0 , , N ) → Lsc([0 , , N ) as i n ( χ [0 ,b ) ) = ( n −
1) + χ [0 ,b ) , i n ( χ ( a, ) = χ ( a, , i n ( χ ( a,b ) ) = χ ( a,b ) , and i n (1) = n .Then, i n | G = i n +1 | G for every n , so each i n is a lift of the inclusion G →
Lsc([0 , , N ).Given a Cu-semigroup S satisfying weak cancellation, we know that for any compact element p andany pair of elements a, b ∈ S such that p + a ≤ p + b , we have a ≤ b . Indeed, taking a ′ ≪ a , one has p + a ′ ≪ p + b . Applying weak cancellation, we obtain a ′ ≪ b for any a ′ ≪ a and, consequently, a ≤ b .In particular, this implies that if p + a = p + b , we must have a = b . This fact will be used repeatedlyin the proof of the following lemma.We will also consider the ordered set { χ ( t, , χ [0 ,t ) , } t ∈ [0 , with the order inherited from Lsc([0 , , N ).Note that in this set suprema of increasing sequences always exist.Indeed, every increasing sequence ( s n ) n is of one of the following forms s n = 1 for every n, s n = χ ( a n , with a n +1 ≤ a n , or s n = χ [0 ,b n ) with b n ≤ b n +1 , with supremum 1, χ (inf( b n ) , and χ [0 , sup( a n )) respectively. Note that all three elements belong to ourset. Lemma 3.3.
Let S be a Cu -semigroup satisfying (O5) and weak cancellation, and let φ : { χ ( t, , χ [0 ,t ) , } t ∈ [0 , → S be an order and suprema preserving map such that for every t ≤ s < t ′ we have φ ( χ [0 ,t ) ) + φ ( χ ( s, ) ≤ φ (1) ≪ φ (1) ≤ φ ( χ [0 ,t ′ ) ) + φ ( χ ( s, ) . Then, there is a unique Cu -morphism from Lsc([0 , , N ) to S lifting φ . Conversely, for any Cu -morphism φ from Lsc([0 , , N ) to S , the previous condition holds.Proof. Necessity is clear, so we only need to prove the other implication. That is to say, we need todefine a Cu-morphism Lsc([0 , , N ) → S , which we will also call φ , extending the inital assignments.First, note that φ ( χ ( t, ) , φ ( χ [0 ,t ) ) ≤ φ (1) for every t . Also, given any t ′ > t and s ′ < s , we can applyweak cancellation to the inequalities φ ( χ [0 ,t ) ) + φ ( χ ( t, ) ≤ φ (1) ≪ φ (1) ≤ φ ( χ [0 ,t ′ ) ) + φ ( χ ( t, ) ,φ ( χ [0 ,s ) ) + φ ( χ ( s, ) ≤ φ (1) ≪ φ (1) ≤ φ ( χ [0 ,s ) ) + φ ( χ ( s ′ , ) , to obtain φ ( χ [0 ,t ) ) ≪ φ ( χ [0 ,t ′ ) ) and φ ( χ ( s, ) ≪ φ ( χ ( s ′ , ).Let s < t . Then, since φ (1) ≤ φ ( χ ( s, ) + φ ( χ [0 ,t ) ), we know by (O5) and using that φ (1) is compactthat there exists an element x ∈ S such that φ (1) + x = φ ( χ ( s, ) + φ ( χ [0 ,t ) ) . Note that this element is unique by the remark preceding the lemma. Thus, we can define φ ( χ ( s,t ) ) = x . EDUARD VILALTA
Now let s < t in [0 ,
1] and let y ≪ φ ( χ ( s,t ) ). Then, we have φ (1) + y ≪ φ (1) + φ ( χ ( s,t ) ) = φ ( χ ( s, ) + φ ( χ [0 ,t ) ) ≤ φ ( χ ( s, ) + φ (1) , φ ( χ [0 ,t ) ) + φ (1) . By weak cancellation, one gets y ≪ φ ( χ ( s, ) , φ ( χ [0 ,t ) ). Taking the supremum on y , one obtains φ ( χ ( s,t ) ) ≤ φ ( χ ( s, ) , φ ( χ [0 ,t ) ).Also, given s ′ < s < t < t ′ , we have, using as proved above that φ ( χ ( s, ) ≪ φ ( χ ( s ′ , ) and φ ( χ [0 ,t ) ) ≪ φ ( χ [0 ,t ′ ) ), φ (1) + φ ( χ ( s,t ) ) = φ ( χ ( s, ) + φ ( χ [0 ,t ) ) ≪ φ ( χ ( s ′ , ) + φ ( χ [0 ,t ′ ) ) = φ (1) + φ ( χ ( s ′ ,t ′ ) ) . Applying weak cancellation once again, we get φ ( χ ( s,t ) ) ≪ φ ( χ ( s ′ ,t ′ ) ).Further, given any increasing sequence t n → t and any decreasing sequence s n → s , we have φ ( χ ( s,t ) ) + φ (1) = φ ( χ [0 ,t ) ) + φ ( χ ( s, ) = sup n ( φ ( χ [0 ,t n ) ) + φ ( χ ( s n , ))= sup n ( φ (1) + φ ( χ ( s n ,t n ) )) = φ (1) + sup n φ ( χ ( s n ,t n ) ) . By the remark preceding the lemma, one gets φ ( χ ( s,t ) ) = sup n φ ( χ ( s n ,t n ) ). Claim 1
Given any two open intervals
U, V , one has φ ( χ U ) + φ ( χ V ) = φ ( χ U ∪ V ) + φ ( χ U ∩ V ) .Proof. Assume that
U, V are of the form U = ( s, t ), V = ( s ′ , t ′ ) with s ≤ s ′ ≤ t ≤ t ′ .Then, one has φ ( χ U ) + φ ( χ V ) + 2 φ (1) = φ ( χ [0 ,t ) ) + φ ( χ ( s, ) + φ ( χ [0 ,t ′ ) ) + φ ( χ ( s ′ , )= φ ( χ ( s,t ′ ) ) + φ ( χ ( s ′ ,t ) ) + 2 φ (1) , which implies φ ( χ U ) + φ ( χ V ) = φ ( χ U ∪ V ) + φ ( χ U ∩ V ), as required.All the other cases are proven analogously. (cid:3) Let B be the set of basic elements in Lsc([0 , , N ). By Remark 2.7 and Claim 1, we can now extend φ to a monoid morphism φ : B → S .Since φ is additive and retractions are applied to each connected component (see Notation 2.8),note that we have sup n φ ( χ R /n ( U ) ) = φ ( χ U ) for every open subset U ⊂ [0 ,
1] that can be written asthe finite disjoint union of intervals.
Claim 2
Given a finite strictly increasing sequence s < · · · < s n in [0 , , we have φ ( χ [0 ,s )) + φ ( χ ( s ,s ) ) + · · · + φ ( χ ( s n − ,s n ) ) + φ ( χ ( s n , ) ≤ φ (1) ≤ φ ( χ [0 ,s )) + φ ( χ ( s ,s ) ) + · · · + φ ( χ ( s n − ,s n ) ) + φ ( χ ( s n − , ) In particular, given
U, V open subsets that can be written as the finite disjoint union of open inter-vals, we have φ ( χ U ) ≤ φ ( χ V ) whenever U ⊂ V and φ ( χ U ) ≪ φ ( χ V ) whenever U ⋐ V .Proof. Using the equality φ ( χ ( s,t ) ) + φ (1) = φ ( χ [0 ,t ) ) + φ ( χ ( s, ) at the first step, and the inequality φ ( χ [0 ,s ) ) + φ ( χ ( s, ) ≤ φ (1) at the second step, we get φ ( χ [0 ,s )) + φ ( χ ( s ,s ) ) + · · · + φ ( χ ( s n − ,s n ) ) + φ ( χ ( s n , ) + nφ (1)= n X i =0 φ ( χ [0 ,s i ) ) + φ ( χ ( s i , ) ≤ ( n + 1) φ (1) . Similarly, but now using φ ( χ [0 ,t ) ) + φ ( χ ( s, ) ≥ φ (1) for every s < t at the second step, we have φ ( χ [0 ,s )) + φ ( χ ( s ,s ) ) + · · · + φ ( χ ( s n − ,s n ) ) + φ ( χ ( s n − , ) + ( n − φ (1)= n X i =1 φ ( χ [0 ,s i +1 ) ) + φ ( χ ( s i − , ) ≥ nφ (1) . Since φ (1) is compact, we can apply weak cancellation to cancel nφ (1) in the first inequality and( n − φ (1) in the second. This proves the first part of the claim. LOCAL CHARACTERIZATION FOR THE CUNTZ SEMIGROUP OF AI-ALGEBRAS 7
Given a subset Y ⊂ [0 , Y ) denote its interior. Then, given U ⋐ V as in the statement,and using the result above in both inequalities, one has φ ( χ U ) + φ ( χ Int([0 , − U ) ) ≤ φ (1) ≤ φ ( χ V ) + φ ( χ Int([0 , − U ) ) . Applying weak cancellation once again, we have φ ( χ U ) ≪ φ ( χ V ).If U ⊂ V , take n > R /n ( U ) ⋐ U ⊂ V . Thus, we have φ ( χ R /n ( U ) ) ≪ φ ( χ V ).Taking the supremum on n one obtains the required result. (cid:3) Now let f, g ∈ B , and note that f ≤ g (resp. f ≪ g ) if and only if { f ≥ k } ⊂ { g ≥ k } for every k ≤ sup( g ) (resp. { f ≥ k } ⋐ { g ≥ k } ). Since φ is additive, it follows from Claim 2 that φ preservesboth the order and the ≪ -relation.Take f ∈ B , which can be written as f = P i χ U i with U i finite disjoint unions of intervals, and let f n be P i χ R /n ( U i ) , which is still an element in B with f n ≪ f n +1 for all n . Clearly, since φ is additiveand sup n f n = f , we have that φ ( f n ) is an ≪ -increasing sequence with supremum φ ( f ).Thus, given an increasing sequence ( f ′ m ) m in B with supremum f , we have that for every n thereexists some m with f n ≤ f ′ m ≤ f. Since φ preserves the order, one gets φ ( f n ) ≤ φ ( f ′ m ) ≤ φ ( f ). Taking suprema, it follows thatsup m φ ( f ′ m ) = φ ( f ) whenever f ′ m is an increasing sequence with supremum f .Finally, since every element in Lsc([0 , , N ) can be written as the supremum of an ≪ -increasingsequence of elements in B , we can define φ : Lsc([0 , , N ) → S as φ ( g ) = sup n φ ( g n ) for g n ≪ -increasingsequence in B with supremum g .It is now easy to check that φ is a Cu-morphism (see, for example, the proof of [Vil21, Theo-rem 4.40]). Note that it is a lift for our previously defined map φ : B → S because we already knowthat sup m φ ( f ′ m ) = φ ( f ) whenever f ′ m is an increasing sequence with supremum f . (cid:3) Recall that G is the sub-Cu-semigroup of Lsc([0 , , N ) defined as G = { f ∈ Lsc([0 , , N ) | f (0) = 0 , f increasing } . Proposition 3.4.
Let S be a Cu -semigroup satisfying (O5) and weak cancellation, and let α : G → S be a Cu -morphism such that α ( χ (0 , ) ≤ p with p a compact element in S .Then, there exists a unique Cu -morphism φ : Lsc([0 , , N ) → S extending α such that φ (1) = p .Proof. We will show that there exists a map φ : { χ ( t, , χ [0 ,t ) , } t ∈ [0 , → S satisfying the conditions in Lemma 3.3. The result will then follow from the application of this lemma.First, set φ (1) = p and φ ( χ ( s, ) = α ( χ ( s, ) for every s .Take t ∈ (0 ,
1] and consider a strictly increasing sequence s n → t . Since φ ( χ ( s n +1 , ) ≪ φ ( χ ( s n , ) ≪ φ (1) for every n , we know by (O5) that there exist elements x n such that φ ( χ ( s n +1 , ) + x n ≤ φ (1) ≤ φ ( χ ( s n , ) + x n for every n .Using φ (1) ≪ φ (1) and weak cancellation, one gets from φ ( χ ( s n +1 , ) + x n ≤ φ (1) ≤ φ ( χ ( s n +1 , ) + x n +1 that x n ≪ x n +1 . Let x = sup x n .Now take t ′ ≥ t , and consider an stricly increasing sequence s ′ n → t ′ . Using the same argument, weobtain an associated ≪ -increasing sequence ( y n ) n .Further, note that for every n there must exist some m ≥ n such that s ′ m ≥ s n +1 . Using this at thethird step we obtain φ ( χ ( s n +1 , ) + x n ≤ φ (1) ≤ φ ( χ ( s ′ m , ) + y m ≤ φ ( χ ( s n +1 , ) + y m . Applying weak cancellation and taking suprema over m , one obtains x ≤ sup m y m . EDUARD VILALTA
This argument shows that we can define φ ( χ [0 ,t ) ) as x (by taking t ′ = t ), and that φ ( χ [0 ,t ) ) ≤ φ ( χ [0 ,t ′ ) )whenever t ′ ≥ t . Thus, φ is order preserving.With the previous notation, note that, for any t ′ < t , we get φ ( χ ( t, ) + x n ≤ φ ( χ ( s n +1 , ) + x n ≤ φ (1) ≤ φ ( χ ( s n , ) + x n ≤ φ ( χ ( t ′ , ) + x n , for some large enough n such that s n > t ′ .This shows that φ ( χ ( t, ) + φ ( χ [0 ,t ) ) ≤ φ (1) ≤ φ ( χ ( t ′ , ) + φ ( χ [0 ,t ) ) whenever t ′ < t .Now take an increasing sequence t m → t . In order to show that sup m φ ( χ [0 ,t m ) ) = φ ( χ [0 ,t ) ), we onlyneed to prove sup m φ ( χ [0 ,t m ) ) ≥ φ ( χ [0 ,t ) ), as we already know that φ is order preserving.Using the same notation as above, let s n → t be a strictly increasing sequence and let x n ∈ S betheir associated elements. Since s n is strictly increasing, we must have that for every n there exists m ≥ n such that s n +1 < t m . Using that φ (1) ≤ φ ( χ ( t ′ , ) + φ ( χ [0 ,t ) ) whenever t ′ < t in the secondinequality, we get φ ( χ ( s n +1 , ) + x n ≤ φ (1) ≤ φ ( χ ( s n +1 , ) + φ ( χ [0 ,t m ) ) . By weak cancellation and taking suprema, we have sup m φ ( χ [0 ,t m ) ) ≥ sup n x n = φ ( χ [0 ,t ) ). Thisshows that φ also preserves suprema and, consequently, that Lemma 3.3 can be applied.We now get a (unique) Cu-morphism φ : Lsc([0 , , N ) → S extending our map φ . Since, by con-struction, φ ( χ ( s, ) = α ( χ ( s, ) for every s and the submonoid of G generated by { χ ( s, } s is sup-densein G , it follows that φ extends α as desired. (cid:3) Cauchy sequences and a local characterization
We now turn our attention to the proof of the main result of this paper, Theorem 4.15, and itsdiscrete counterpart, Theorem 4.17. Since countably based Cu-semigroups are usually far bigger thancountably generated groups (for instance, a countably based Cu-semigroup is not generally countable),one cannot hope to get an exact analogue of [She79, Theorem 3.1]. Instead, one has to make do withan approximate version of it.This is why we define a distance on the morphisms from Lsc([0 , , N ) to a Cu-semigroup S .4.1. Distance between maps.
Let ϕ , ϕ : Lsc([0 , , N ) → S be Cu-morphisms with ϕ (1) = ϕ (1).We define the distance between ϕ and ϕ as d ( ϕ , ϕ ) := inf { ǫ ∈ [0 , | ϕ ( χ ( t + ǫ, ) ≤ ϕ ( χ ( t, ) , ϕ ( χ ( t + ǫ, ) ≤ ϕ ( χ ( t, ) ∀ t ∈ [0 , } . Note that the distance between ϕ , ϕ is the distance between ϕ | Lsc((0 , , N ) , ϕ | Lsc((0 , , N ) consideredin [CE08] and [CES11] (see also [RS10]). Remark 4.1.
Let s, t ∈ [0 ,
1] with s − t > d ( ϕ , ϕ ). Then, we have ϕ ( χ ( s, ) ≪ ϕ ( χ ( t, ) , and ϕ ( χ ( s, ) ≪ ϕ ( χ ( t, ) . Indeed, given η > s − t > η > d ( ϕ , ϕ ), we have ϕ ( χ ( s, ) ≪ ϕ ( χ ( t + η, ) ≤ ϕ ( χ ( t, ) , and ϕ ( χ ( s, ) ≪ ϕ ( χ ( t + η, ) ≤ ϕ ( χ ( t, ) . This remark will be used throughout the section.Under the hypothesis of weak cancellation, Ciuperca and Elliott proved in [CE08, Theorem 4.1]that their distance is a metric. To see that our distance is also a metric, we recall the following result:
Lemma 4.2.
Let S be a Cu -semigroup satisfying weak cancellation and (O5). If two Cu -morphisms ϕ , ϕ from Lsc([0 , , N ) to S agree on Lsc((0 , , N ) and , they are the same.Proof. If ϕ , ϕ agree on Lsc((0 , , N ), then they also agree on the sub-Cu-semigroup G consisting ofincreasing lower-semicontinuous functions.Thus, since ϕ (1) = ϕ (1), we know from Proposition 3.4 that there exists a unique lift for ϕ | G sending 1 to ϕ (1), which, by uniqueness, must be the same for ϕ | G that sends 1 to ϕ (1).By Proposition 3.4, this shows ϕ = ϕ . (cid:3) Following [CES11], one can generalize the previous notion of distance to pairs of morphisms fromfinite direct sums of Lsc([0 , , N ) to S . LOCAL CHARACTERIZATION FOR THE CUNTZ SEMIGROUP OF AI-ALGEBRAS 9
Definition 4.3.
Let S be a Cu-semigroup. Let L = ⊕ ri =1 L i with L i = Lsc([0 , , N ) for each i , andconsider a pair of morphisms ϕ , ϕ : L → S with ϕ (1 i ) = ϕ (1 i ) for each i ≤ r .We define the distance between ϕ and ϕ as d ( ϕ , ϕ ) := sup ≤ i ≤ r d ( ϕ | L i , ϕ | L i ) . Remark 4.4.
Note that d ( · , · ) is clearly a metric as well. Lemma 4.5.
Let ϕ , ϕ : Lsc([0 , , N ) → S with S a weakly cancellative Cu -semigroup. If d ( ϕ , ϕ ) ≤ ǫ with ϕ (1) = ϕ (1) , then for every basic indicator function f ∈ Lsc([0 , , N ) , as defined in Definition 2.6,we have ϕ ( R ǫ ( f )) ≤ ϕ ( f ) , and ϕ ( R ǫ ( f )) ≤ ϕ ( f ) , where R ǫ denotes the ǫ -retraction of f ; see Notation 2.8.Proof. We will only prove one of the inequalities, since the other one is proved analogously.If f = 1 or f = χ ( s, for some s ∈ [0 , f = χ ( s,t ) for some s < t , let η > R ǫ + η ( f ) + χ ( t − ǫ − η, = χ ( s + η + ǫ,t − η − ǫ ) + χ ( t − η − ǫ, ≪ χ ( s + ǫ, = R ǫ ( χ ( s, ) . Therefore, using that ϕ , ϕ are Cu-morphisms at the first and third steps and the assumption that d ( ϕ , ϕ ) ≤ ǫ at the second and fourth steps, we get ϕ ( R ǫ + η ( f )) + ϕ ( χ ( t − ǫ − η, ) ≪ ϕ ( R ǫ ( χ ( s, ) ≤ ϕ ( χ ( s, ) ≤ ϕ ( χ ( s,t ) ) + ϕ ( χ ( t − η, ) ≤ ϕ ( f ) + ϕ ( χ ( t − ǫ − η, ) . Applying weak cancellation we get ϕ ( R ǫ + η ( f )) ≪ ϕ ( f ). Since sup η → R ǫ + η ( f ) = R ǫ ( f ), it followsthat ϕ ( R ǫ ( f )) ≤ ϕ ( f ) as required.Finally, if f = χ [0 ,t ) for some t ∈ [0 , η > ϕ (1) = ϕ (1), ϕ ( R ǫ + η ( f )) + ϕ ( χ ( t − η, ) ≤ ϕ ( R ǫ + η ( f )) + ϕ ( χ ( t − ǫ − η, ) ≪ ϕ (1)= ϕ (1) ≤ ϕ ( χ [0 ,t ) ) + ϕ ( χ ( t − ǫ, ) = ϕ ( f ) + ϕ ( χ ( t − ǫ, )As before, we get ϕ ( R ǫ + η ( f )) ≪ ϕ ( f ) for every η > ϕ ( R ǫ ( f )) ≤ ϕ ( f ). (cid:3) Cauchy sequences and their limits.
In this section we will prove that suitable Cauchy se-quences of Cu-morphisms ϕ i : Lsc([0 , , N ) → S have a limit. From this point onwards, and until theend of the section, S will denote a Cu-semigroup satisfying (O5) and weak cancellation.The following lemmas will be of importance: Lemma 4.6.
Suppose that f ′ ≪ f in Lsc([0 , , N ) and let φ : Lsc([0 , , N ) → S be a Cu -morphism.Then, there exists ǫ = ǫ ( f, f ′ ) > and f ′′ ∈ Lsc([0 , , N ) such that(i) f ′ ≪ f ′′ ≪ f .(ii) For every Cu -morphism ϕ : Lsc([0 , , N ) → S with φ (1) = ϕ (1) and d ( φ, ϕ ) < ǫ , we have φ ( f ′ ) ≪ ϕ ( f ′′ ) ≪ φ ( f ) .Proof. First, let us prove the result when f ′ , f are basic indicator functions. If f ′ = f = 1, the resultfollows trivially.If f ′ = χ ( t, and f = χ ( s, with t > s , let ǫ > s < s + 2 ǫ < t . Then, if d ( φ, ϕ ) < ǫ ,one has, using Remark 4.1 at the last step, φ ( χ ( t, ) ≪ φ ( χ ( s +2 ǫ, ) ≤ ϕ ( χ ( s + ǫ, ) ≪ φ ( χ ( s, ) , so we can set f ′′ = χ ( s + ǫ, .If f = χ [0 ,t ) and f ′ = χ [0 ,s ) with t > s , we can take ǫ as before to get, if d ( φ, ϕ ) < ǫ and usingLemma 4.5 at the first step, that φ ( χ [0 ,s ) ) ≪ ϕ ( χ [0 ,s + ǫ ) ) ≪ φ ( χ [0 ,s +2 ǫ ) ) ≪ φ ( χ [0 ,t ) )whenever d ( φ, ϕ ) < ǫ . This shows that we can set f ′′ = χ [0 ,s + ǫ ) . Finally, given f = χ ( s,t ) and f ′ = χ ( s ′ ,t ′ ) with s < s ′ < t ′ < t , let ǫ > s < s ′ − ǫ and t ′ < t ′ + 2 ǫ < t . Then, if d ( φ, ϕ ) < ǫ , we have φ (1) + φ ( χ ( s ′ ,t ′ ) ) = φ ( χ ( s ′ , ) + φ ( χ [0 ,t ′ ) ) ≪ ϕ ( χ ( s ′ − ǫ, ) + ϕ ( χ [0 ,t ′ + ǫ ) )= ϕ (1) + ϕ ( χ ( s ′ − ǫ,t ′ + ǫ ) ) = ϕ ( χ ( s ′ − ǫ, ) + ϕ ( χ [0 ,t ′ + ǫ ) ) ≪ φ ( χ ( s ′ − ǫ, ) + φ ( χ [0 ,t ′ +2 ǫ ) ) ≪ φ ( χ ( s, ) + φ ( χ [0 ,t ) ) = φ (1) + φ ( χ ( s,t ) ) . In particular, one gets φ (1) + φ ( f ′ ) ≪ ϕ (1) + ϕ ( χ ( s ′ − ǫ,t ′ + ǫ ) ) ≪ φ (1) + φ ( f ) . Applying weak cancellation, we have φ ( f ′ ) ≪ ϕ ( χ ( s ′ − ǫ,t ′ + ǫ ) ) ≪ φ ( f )and in this case we set f ′′ = χ ( s ′ − ǫ,t ′ + ǫ ) .Now, given any pair f ′ ≪ f , let g, g ′ ∈ Lsc([0 , , N ) be such that f ′ ≪ g ′ ≪ g ≪ f with g ′ = P ni =1 h ′ i , g = P ni =1 h i and h ′ i ≪ h i basic indicators for each i . This can be done because the finitesums of basic indicators are a basis for Lsc([0 , , N ).Since h ′ i ≪ h i , it follows from our previous case that there exist ǫ i = ǫ i ( h i , h ′ i ) and h ′′ i with h ′ i ≪ h ′′ i ≪ h i and such that, whenever d ( ϕ, φ ) < ǫ i and ϕ (1) = φ (1), we have φ ( h ′ i ) ≪ ϕ ( h ′′ i ) ≪ φ ( h i ) . Let ǫ = ǫ ( f, f ′ ) = min( ǫ i ) and set f ′′ = P i h ′′ i .Now let ϕ be a Cu-morphism such that d ( ϕ, φ ) < ǫ and φ (1) = ϕ (1). Since d ( ϕ, φ ) < ǫ ≤ ǫ i forevery i , we have φ ( h ′ i ) ≪ ϕ ( h ′′ i ) ≪ φ ( h i )for every i .Thus, one gets φ ( f ′ ) ≪ φ ( g ′ ) = X i φ ( h ′ i ) ≪ X i ϕ ( h ′′ i ) = ϕ ( f ′′ ) ≪ X i φ ( h i ) = φ ( g ) ≪ φ ( f )as required. (cid:3) Corollary 4.7.
Let f, g, h ∈ Lsc([0 , , N ) and let φ : Lsc([0 , , N ) → S be such that f ≪ h and φ ( h ) ≪ φ ( g ) . Then, there exists ǫ = ǫ ( f, g, h ) > such that, for every ϕ : Lsc([0 , , N ) → S with φ (1) = ϕ (1) and d ( φ, ϕ ) < ǫ , we have ϕ ( f ) ≪ ϕ ( g ) .In fact, there exists f ′′ such that f ≪ f ′′ and ϕ ( f ′′ ) ≪ ϕ ( g ) .Proof. Let ǫ ( h, f ) > f ′′ ∈ S be the number and element given by Lemma 4.6 applied to f ≪ h .Also, for every g ′ ≪ g , let ǫ ( g, g ′ ) > g ′′ be a choice of number and element given by Lemma 4.6applied to g ′ ≪ g .Set ǫ = ǫ ( f, g, h ) = min( ǫ ( h, f ) , sup { ǫ ( g, g ′ ) | φ ( h ) ≪ φ ( g ′ ) } ).Take a Cu-morphism ϕ : Lsc([0 , , N ) → S with φ (1) = ϕ (1) and d ( φ, ϕ ) < ǫ . Thus, there exists anelement g ′ ≪ g with φ ( h ) ≪ φ ( g ′ ) such that d ( φ, ϕ ) < ǫ ( g, g ′ ) , ǫ ( h, f ).Using Lemma 4.6 at the first and third step, we have ϕ ( f ′′ ) ≪ φ ( h ) ≪ φ ( g ′ ) ≪ ϕ ( g ′′ ) ≪ ϕ ( g )as desired.Note, in particular, that ϕ ( f ) ≪ ϕ ( g ) because f ≪ f ′′ . (cid:3) Lemma 4.8.
Let n ∈ N and ǫ ∈ (0 , . Also, let t i = i/n be a partition of [0 , and consider two Cu -morphisms ϕ , ϕ : Lsc([0 , , N ) → S such that ϕ (1) = ϕ (1) , ϕ ( χ ( t i + ǫ, ) ≤ ϕ ( χ ( t i , ) , and ϕ ( χ ( t i + ǫ, ) ≤ ϕ ( χ ( t i , ) for every ≤ i ≤ n .Then, d ( ϕ , ϕ ) ≤ ǫ + 1 /n . LOCAL CHARACTERIZATION FOR THE CUNTZ SEMIGROUP OF AI-ALGEBRAS 11
Proof.
Let t ∈ [0 ,
1] and take i minimal such that t ≤ t i . Thus, t i ≤ t + 1 /n .We have, using our assumptions and that ϕ i are Cu-morphisms, ϕ ( χ ( t +1 /n + ǫ, ) ≤ ϕ ( χ ( t i + ǫ, ) ≤ ϕ ( χ ( t i , ) ≤ ϕ ( χ ( t, ) . By an analogous argument, one sees that ϕ ( χ ( t +1 /n + ǫ, ) ≤ ϕ ( χ ( t, ) as required. (cid:3) Proposition 4.9.
Let ǫ > and let φ : Lsc([0 , , N ) → Lsc([0 , , N ) be a Cu -morphism. There exists ǫ ′ > such that, for any pair of Cu -morphisms ϕ , ϕ : Lsc([0 , , N ) → S with ϕ (1) = ϕ (1) and atdistance at most ǫ ′ , we have d ( ϕ φ, ϕ φ ) < ǫ. Proof.
We begin the proof with the following claim:
Claim.
For every t ∈ [0 , there exists ǫ ( t ) > such that, for every pair of Cu -morphisms ϕ , ϕ : Lsc([0 , , N ) → S with ϕ (1) = ϕ (1) and at distance at most ǫ ( t ) , we have ϕ ( φ ( χ ( t + ǫ/ , )) ≤ ϕ ( φ ( χ ( t, )) , and ϕ ( φ ( χ ( t + ǫ/ , )) ≤ ϕ ( φ ( χ ( t, )) . Proof.
Let t ∈ [0 ,
1] and consider the basic indicators χ ( t, and χ ( t + ǫ/ , of Lsc([0 , , N ). Since χ ( t, , χ ( t + ǫ/ , ≪
1, it follows that φ ( χ ( t, ) , φ ( χ ( t + ǫ/ , ) ≤ m for some m ∈ N .Thus, since we also know that φ ( χ ( t + ǫ/ , ) ≪ φ ( χ ( t, ), there exist functions f i , g i ∈ Lsc([0 , , N )with f i ≪ g i ≪ φ ( χ ( t + ǫ/ , ) = P i ≤ m f i and φ ( χ ( t, ) = P i ≤ m g i (see, for example, [Vil21,Lemma 4.19]).Also, since f i ≪ g i ≪ i , there exist basic indicators a i,j such that f i ≪ a i, + · · · + a i,k ( i ) ≪ g i for every i .Therefore, for every i ≤ m there exists δ i such that f i ≪ R δ i ( a i, ) + · · · + R δ i ( a i,k ( i ) ) ≪ a i, + · · · + a i,k ( i ) ≪ g i . Set ǫ ( t ) = min( δ i ). By Lemma 4.5, for every pair of Cu-morphisms ϕ , ϕ with ϕ (1) = ϕ (1) andat distance at most ǫ ( t ), we have ϕ ( f i ) ≤ ϕ ( R δ i ( a i, )) + · · · + ϕ ( R δ i ( a i,k ( i ) )) ≤ ϕ ( a i, ) + · · · + ϕ ( a i,k ( i ) ) ≪ ϕ ( g i ) . Adding these inequalities, one gets ϕ ( φ ( χ ( t + ǫ/ , )) ≤ ϕ ( φ ( χ ( t, ))and, by an analogous argument, we also have ϕ ( φ ( χ ( t + ǫ/ , )) ≤ ϕ ( φ ( χ ( t, )) as required. (cid:3) Let n ∈ N be such that 1 /n ≤ ǫ/
2, and consider the partition t i = i/n of [0 , i there exists ǫ ( t i ) such that ϕ ( φ ( χ ( t i + ǫ/ , )) ≤ ϕ ( φ ( χ ( t i , )) , ϕ ( φ ( χ ( t i + ǫ/ , )) ≤ ϕ ( φ ( χ ( t i , ))whenever d ( ϕ φ, ϕ φ ) < ǫ ( t i ).Define ǫ ′ = min( ǫ ( t i )). Then, by Lemma 4.8, we get d ( ϕ φ, ϕ φ ) ≤ ǫ/ /n ≤ ǫ as desired. (cid:3) Corollary 4.10.
Let L = ⊕ ri =1 Lsc([0 , , N ) and L ′ = ⊕ sj =1 Lsc([0 , , N ) . Then, for any ǫ > and φ : L → L ′ , there exists ǫ ′ > such that, for any pair of morphisms ϕ , ϕ : L ′ → S with d ( ϕ , ϕ ) < ǫ ′ and ϕ (1 j ) = ϕ (1 j ) for every j , we have d ( ϕ φ, ϕ φ ) < ǫ. Now let ϕ i : Lsc([0 , , N ) → S be such that d ( ϕ i , ϕ i +1 ) < ǫ i with ǫ i strictly decreasing and R := P ∞ i =1 ǫ i < ∞ . Also, assume that ϕ i (1) = ϕ i +1 (1) for every i .Define R i := P ik =1 ǫ k and let t ∈ [0 , ϕ ( χ ( t + R, ) ≪ ϕ ( χ ( t + R − R , ) ≪ ϕ ( χ ( t + R − R , ) ≪ · · · ≪ ϕ i +1 ( χ ( t + R − R i , ) ≪ · · · because ( t + R − R i ) − ( t + R − R i +1 ) = ǫ i > d ( ϕ i , ϕ i +1 ).Thus, the sequence ( ϕ i +1 ( χ ( t + R − R i , )) i is ≪ -increasing, and we can consider its supremum. Foreach t , we define ϕ ( χ ( t, ) := sup i ϕ i +1 ( χ ( t + R − R i , ). We will now see that ϕ induces a Cu-morphism, and that such morphism is the limit of our sequence.That is to say, we will see that the Cauchy sequences with summable distances have a limit. Proposition 4.11.
Retain the above assumptions. Then:(i) The sequence ( ϕ i ) i induces a Cu -morphism ϕ : Lsc([0 , , N ) → S .(ii) d ( ϕ, ϕ i ) → as i tends to infinity.Proof. We prove each claim separately:(i) First we will see that the map ϕ : { χ ( t, } t → S preserves order, suprema and the way-below relation.Thus, let χ ( s, ≤ χ ( t, , which happens if and only if s ≥ t . Then, since s + R − R i ≥ t + R − R i foreach i , we have ϕ i +1 ( χ ( s + R − R i , ) ≤ ϕ i +1 ( χ ( t + R − R i , )for every i , and hence ϕ ( χ ( s, ) ≤ ϕ ( χ ( t, ).If χ ( s, ≪ χ ( t, , we know that s − t = d for some d > R − R i →
0, there exists some k ∈ N such that d > R − R k − ). Also, note that forevery i > k , one has d ( ϕ i +1 , ϕ k ) ≤ d ( ϕ i +1 , ϕ i ) + · · · + d ( ϕ k +1 , ϕ k ) < ǫ i + · · · + ǫ k = R i − R k − ≤ R − R k − . Therefore, one has ϕ i +1 ( χ ( s + R − R i , ) = ϕ i +1 ( χ ( t + d + R − R i , ) ≪ ϕ k ( χ ( t + d + R k − − R i , ) ≤ ϕ k ( χ ( t + R − R k − , ) , where in the second step we have used ( t + d + R − R i ) − ( t + d + R k − − R i ) = R − R k − > d ( ϕ i , ϕ k ),and in the third step we have used d ≥ R − R k − ) ≥ ( R − R k − ) + ( R i − R k − ).This shows that ϕ ( χ ( s, ) ≤ ϕ k ( χ ( t + R − R k − , ) ≪ ϕ ( χ ( t, ).Now let ( t n ) n be a decreasing sequence converging to t . Since t ≤ t n for each n , it follows thatsup n ϕ ( χ ( t n , ) ≤ ϕ ( χ ( t, ) . Conversely, let k ∈ N be fixed and take i > k . Recall from the previous argument that we have d ( ϕ i +1 , ϕ k +1 ) < R i − R k . Set ǫ = R i − R k − d ( ϕ i +1 , ϕ k +1 ) > t n − t is positive and tends to zero, there exists n such that ǫ > t n − t . In particular, one gets( t + R − R k ) − ( t n + R − R i ) = ( t − t n ) + R i − R k > − ǫ + R i − R k = d ( ϕ i +1 , ϕ k +1 ) . Using Remark 4.1 we get ϕ k +1 ( χ ( t + R − R k , ) ≪ ϕ i +1 ( χ ( t n + R − R i , ) ≤ ϕ ( χ ( t n , ) ≤ sup n ϕ ( χ ( t n , ) . This shows that ϕ ( χ ( t, ) ≤ sup n ϕ ( χ ( t n , ) and, consequently, sup n ϕ ( χ ( t n , ) = ϕ ( χ ( t, ).Now recall that G is the sub-Cu-semigroup of increasing lower-semicontinuous functions in (0 , ≪ -preserving map { χ ( t, } t → S can be lifted uniquely to a Cu-morphism G → S . Infact, note that in this case the argument is simpler, since G can be seen to be the sup-completion ofthe free abelian semigroup generated by { χ ( t, } t .Thus, let ϕ : G → S be the unique Cu-morphism lifting our map ϕ : { χ ( t, } t → S .Finally, we know by Proposition 3.4 that ϕ has a unique lift ϕ : Lsc([0 , , N ) → S such that ϕ (1) = ϕ i (1) for all i .(ii) Fix ǫ ∈ (0 , m ∈ N such that d ( ϕ, ϕ m ) ≤ ǫ .Since lim R j = R , there exists j ∈ N such that for every j ≥ j we have 0 ≤ R − R j ≤ ǫ/
2. Thus,one gets ϕ j ( χ ( t + ǫ/ , ) ≤ ϕ j ( χ ( t + R − R j , ) ≤ ϕ ( χ ( t, )for every j ≥ j and t ∈ [0 , n ∈ N be such that 1 /n ≤ ǫ/ t i = i/n of [0 , i wehave ϕ ( χ ( t i + ǫ/ , ) ≪ ϕ ( χ ( t i , ) and thus, by the definition of ϕ , there exists j i such that ϕ ( χ ( t i + ǫ/ , ) ≤ ϕ j i ( χ ( t i + R − R ji − , ) . LOCAL CHARACTERIZATION FOR THE CUNTZ SEMIGROUP OF AI-ALGEBRAS 13
Let m = max( j , j , · · · , j n ). Then, ϕ m ( χ ( t + ǫ/ , ) ≤ ϕ ( χ ( t, ) , since m ≥ j , and ϕ ( χ ( t i + ǫ/ , ) ≤ ϕ j i ( χ ( t i + R − R ji − , ) ≤ ϕ m ( χ ( t i + R − R m − , ) ≤ ϕ m ( χ ( t i , ) . Thus, we have by Lemma 4.8 that d ( ϕ, ϕ m ) ≤ ǫ/ /n ≤ ǫ . (cid:3) Using Proposition 4.11, we can prove the following result.
Theorem 4.12.
Let L = ⊕ rj =1 L j with L j = Lsc([0 , , N ) for each j , and let ϕ i : L → S be a sequenceof Cu -morphisms such that d ( ϕ i , ϕ i +1 ) < ǫ i with ( ǫ i ) strictly decreasing and P ∞ i =1 ǫ i < ∞ . Also,assume that ϕ i (1 j ) = ϕ i +1 (1 j ) for each i, j .Then, there exists a unique Cu -morphism ϕ : L → S satisfying d ( ϕ, ϕ i ) → and ϕ (1 j ) = ϕ i (1 j ) forevery j ≤ r .Proof. For each fixed j ≤ r , apply the previous proposition to the sequence ( ϕ i τ j ) i , where τ j : L j → L is the canonical inclusion. This produces a Cu-morphism ϕ ( j ) such that d ( ϕ i τ j , ϕ ( j ) ) → j .The Cu-morphism ϕ := ϕ (1) ⊕ · · · ⊕ ϕ ( r ) : L → S satisfies d ( ϕ, ϕ i ) → φ : L → S be a Cu-morphism with d ( ϕ i , φ ) → φ (1 j ) = ϕ i (1 j ) forevery j ≤ r . Using the triangle inequality, we obtain d ( ϕ, φ ) ≤ d ( ϕ, ϕ i ) + d ( ϕ i , φ ) → . This shows that d ( ϕ, φ ) = 0 and, consequently, ϕ = φ . (cid:3) A local characterization for AI-algebras.
In order to ease the notation, in this subsectionwe will denote the Cuntz semigroup Lsc([0 , , N ) by L . Lemma 4.13.
Let S be a Cu -semigroup satisfying (O5) and weak cancellation. Let ( n i ) i be a sequencein N and consider a pair of sequences ϕ i : L n i → S and σ i +1 ,i : L n i → L n i +1 of Cu -morphisms. Let σ i,j denote the composition σ j,j − ◦ · · · ◦ σ i +1 ,i .If there exists a strictly decreasing sequence ( ǫ i ) i with d ( ϕ j σ j,i , ϕ j +1 σ j +1 ,i ) < ǫ i / j , and ϕ j +1 σ j +1 ,i (1 k ) = ϕ j σ j,i (1 k ) for each k ≤ n i , then we can find a Cu -morphism φ : lim( L n i , σ i +1 ,i ) → S such that its canonical morphisms φ i : L n i → S are the limits of the sequences ( ϕ j σ j,i ) j .Proof. By Theorem 4.12, each sequence ( ϕ j σ j,i ) j has a limit, which we denote by φ i : L n i → S .We will now see that φ i +1 σ i +1 ,i = φ i for each i . Thus, we will obtain a Cu-morphism φ : lim L n i → S ,as required.Fix i ∈ N and take any ǫ >
0. Let δ > σ i +1 ,i and distance ǫ . Since φ i is the limit of the sequence ( ϕ j σ j,i ) j , we can take j such that d ( φ i , ϕ j σ j,i ) < ǫ, and d ( φ i +1 , ϕ j σ j,i +1 ) < δ. We have d ( φ i , φ i +1 σ i +1 ,i ) ≤ d ( φ i , ϕ j σ j,i ) + d ( ϕ j σ j,i , φ i +1 σ i +1 ,i )= d ( φ i , ϕ j σ j,i ) + d ( ϕ j σ j,i +1 σ i +1 ,i , φ i +1 σ i +1 ,i ) < ǫ. Consequently, φ i = φ i +1 σ i +1 ,i for every i and this induces a morphism φ from the limit lim( L n i , σ i +1 ,i )to S . (cid:3) We are now ready to prove the main result of the paper, which gives a characterization of the Cuntzsemigroup of AI-algebras in terms of certain decompositions of suitable Cu-morphisms.
Definition 4.14.
We will say that a Cu-semigroup S is compactly bounded if every compactly con-tained element in S is bounded by a compact. That is, for every element a such that a ≪ b for some b ∈ S , there exists a compact element p ≥ a .The following proof combines some ideas from [She79, Theorem 3.1] and [Nad92, Chapter 12, Section3]. Theorem 4.15.
Let S be a countably based and compactly bounded Cu -semigroup satisfying weakcancellation and (O5). Then, S is Cu -isomorphic to the Cuntz semigroup of an AI-algebra if and onlyif for every Cu -morphism ϕ : Lsc([0 , , N ) r → S , for every finite subset F ⊂ Lsc([0 , , N ) r and every ǫ > , there exist s ≥ and Cu -morphisms θ : L r → L s , φ : L s → S , such that the diagram Lsc([0 , , N ) r ϕ / / θ (cid:15) (cid:15) S Lsc([0 , , N ) s φ ttttttttttt satisfies:(i) d ( φθ, ϕ ) < ǫ .(ii) For every x, x ′ , y ∈ F , we have θ ( x ) ≪ θ ( y ) whenever x ≪ x ′ and ϕ ( x ′ ) ≪ ϕ ( y ) .(iii) ϕ (1 j ) = φθ (1 j ) for every ≤ j ≤ r .Proof. Let S be isomorphic the Cuntz semigroup of an AI-algebra A , and let ϕ , F and ǫ be as in thestatement of the theorem. Since S ∼ = Cu( A ), we know by [CE08, Theorem 12.1] that this map lifts toa *-homomorphism g : B → A , where B is a direct sum of interval algebras and A is the limit of aninductive system ( B i , f i +1 ,i ) with B i a direct sum of interval algebras for each i .By Corollary 4.7, for every triple of elements x, x ′ , y ∈ F with x ≪ x ′ and ϕ ( x ′ ) ≪ ϕ ( y ), there exists ǫ ( x, x ′ , y ) such that whenever d ( ϕ, ψ ) < ǫ ( x, x ′ , y ) and ϕ (1 j ) = ψ (1 j ) for every j , we have ψ ( x ) ≪ ψ ( y ).Since the cardinality of F is finite, so is the number of way-below relations between its elements.This means that the number ǫ F = min( ǫ, , ǫ ( x, x ′ , y )) is strictly positive.Since B i is projective (see, for example, [EK86, Section 3]), there is a *-homomorphism h : B → B i such that k g ( x ) − f i h ( x ) k < ǫ F for every x ∈ B . Here, f i : B i → A is the canonical map. In particular,since ǫ F <
1, it follows that g ( p ) is Murray-Von Neumann equivalent to f i h ( p ) for every projection p ∈ B .Applying the functor Cu we obtain the following diagramLsc([0 , , N ) r ϕ / / Cu( h ) (cid:15) (cid:15) S Cu( B i ) Cu( f i ) sssssssssss Since the norm k g − f i h k is an upper bound for d ( ϕ, Cu( h ) Cu( f i )) (see, for example, [RS10, Lemma1]), we get d ( ϕ, Cu( f i ) Cu( h )) < ǫ F < ǫ . Moreover, since being Murray-Von Neumann equivalentimplies being Cuntz equivalent, one also gets ϕ (1 k ) = Cu( f i ) Cu( h )(1 k ) for every k ≤ r .By the choice of ǫ F , we also have that Cu( f i ) Cu( h )( x ) ≪ Cu( f i ) Cu( h )( y ) for every triple x, x ′ , y ∈ F with x ≪ x ′ and ϕ ( x ′ ) ≪ ϕ ( y ).Finally, note that Cu( f i ) : Cu( B i ) → S is the canonical morphism from Cu( B i ) to the limitlim Cu( B i ) ∼ = Cu( A ) ∼ = S . Thus, we know that Cu( f i ) Cu( h )( x ) ≪ Cu( f i ) Cu( h )( y ) if and only ifthere exists i ( x, y ) ≥ i with Cu( f i ( x,y ) ,i ) Cu( h )( x ) ≪ Cu( f i ( x,y ) ,i ) Cu( h )( y ).Since F is finite, so is the supremum j of all the i ( x, y )’s with x, y ∈ F . Setting θ := Cu( f j,i ) Cu( h )and φ := Cu( f j ), we have(i) d ( φθ, ϕ ) = d (Cu( f j ) Cu( f j,i ) Cu( h ) , ϕ ) = d (Cu( f i ) Cu( h ) , ϕ ) < ǫ .(ii) For every triple x, x ′ , y ∈ F such that x ≪ x ′ and ϕ ( x ′ ) ≪ ϕ ( y ), we getCu( f i ) Cu( h )( x ) ≪ Cu( f i ) Cu( h )( y ) . By our choice of j , it follows that θ ( x ) = Cu( f j,i ) Cu( h )( x ) ≪ Cu( f j,i ) Cu( h )( y ) = θ ( y ) . (iii) φθ (1 k ) = Cu( f i ) Cu( h )(1 k ) = ϕ (1 k ) for every k ≤ r .as required.We are now left to prove the other implication. LOCAL CHARACTERIZATION FOR THE CUNTZ SEMIGROUP OF AI-ALGEBRAS 15
Let s , s , · · · be a countable basis for S , where we may assume s i ∈ S ≪ for each i , and consider aCu-morphism ψ i : Lsc([0 , , N ) → S such that ψ i ( χ (0 , ) = s i . Such a morphism can always be foundby Theorem 3.1 and the fact that all compactly contained elements in S are bounded by a compact.Also, denote by ρ i : Lsc([0 , , N ) i → S the direct sum ρ i = ψ ⊕ · · · ⊕ ψ i .For every j ∈ N , fix a countable and ordered basis for L j . By ”the first i basic elements in L j ” wewill mean the first i elements of the fixed ordered basis of L j .The idea of the proof is as follows:We will first define inductively Cu-morphisms σ i +1 ,i : L n i → L n i +1 and ϕ i : L n i → S such that:(i)’ There exists a decreasing sequence of positive elements ( ǫ i ) tending to 0 such that, for every i ,there exists a Cu-morphism θ i : L n i − ⊕ L i → L n i with d ( ϕ i − ⊕ ρ i , ϕ i θ i ) < ǫ i .(ii)’ For every fixed i , we have d ( ϕ j σ j,i , ϕ j +1 σ j +1 ,i ) < ǫ i / j , and ϕ j +1 σ j +1 ,i (1 k ) = ϕ j σ j,i (1 k )for each k ≤ n i . Here, σ i,j denotes the composition σ j,j − ◦ · · · ◦ σ i +1 ,i .(iii)’ For every fixed k , we also have d ( ϕ j σ j,k θ k , ϕ j +1 σ j +1 ,k θ k ) < ǫ k / j .(iv)’ For each i , let F i be the finite set consisting of the images of the first i basic elements of L n r through σ i,r for each r ≤ i . Then, for every x, x ′ , y ∈ F i satisfying ϕ i ( x ′ ) ≪ ϕ i ( y ) with x ≪ x ′ ,we have σ i +1 ,i ( x ) ≪ σ i +1 ,i ( y ).Condition (ii)’ and Lemma 4.13 will provide a limit morphism φ : lim L n i → S with the canonicalmorphisms φ i : L n i → S being the limits of the sequences ( ϕ j σ j,i ) j .Conditions (i)’ and (iii)’ will imply that φ is surjective. Condition (iv)’ will be used to prove that φ is also an order embedding, thus showing the desired result.Set ϕ := ρ and take some fixed i ∈ N . Assume that for each k ≤ i − ǫ k , themorphisms σ k,k − , ϕ k and θ k and the sets F k − have been defined so that conditions (i)-(iv) above aresatisfied.For every k ≤ i −
1, let δ k be the distance given in Corollary 4.10 such that for any pair of morphisms ζ , ζ : L n i − → S at distance less than δ k , we have(4.1) d ( ζ σ i − ,k , ζ σ i − ,k ) < ǫ k / i , and d ( ζ σ i − ,k θ k , ζ σ i − ,k θ k ) < ǫ k / i . Set ǫ i := min ≤ k ≤ i − { δ k , ǫ k } >
0. As defined above, let F i − be the set that contains, for each r ≤ i −
1, the image through σ i − ,r of i − L n r .Let τ i − : L n i − → L n i − ⊕ L i be the canonical inclusion, and let F = τ i − ( F i − ).By our assumptions, we can find morphisms ϕ i , θ i such that the diagram L n i − ⊕ L i ϕ i − ⊕ ρ i / / θ i (cid:15) (cid:15) SL n i ϕ i ♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥ satisfies conditions (i)-(iii) in the statement of the theorem with distance ǫ i and finite set F .Define σ i,i − := θ i ◦ τ i − , and note that condition (i)’ is immediately satisfied. Also, condition (iv)’is satisfied by construction. L n i − τ i − / / σ i,i − ( ( ◗◗◗◗◗◗◗◗ ϕ i − % % ♦ ♠ ❥ ❣ ❞ ❜ ❴ ❭ ❩ ❲ ❚ ◗ ❖ ▲ L n i − ⊕ L i ϕ i − ⊕ ρ i / / θ i (cid:15) (cid:15) SL n i ϕ i ♥♥♥♥♥♥♥♥♥♥♥♥♥♥♥ Furthermore, since τ i − is an inclusion, we have by Definition 4.3 and ρ i τ i − = 0 that d ( ϕ i σ i,i − , ϕ i − ) = d ( ϕ i θ i τ i − , ( ϕ i − ⊕ ρ i ) τ i − ) ≤ d ( ϕ i θ i , ϕ i − ⊕ ρ i ) < ǫ i ≤ δ k for each k . By the choice of δ k made in (4.1), one gets d ( ϕ i σ i,k , ϕ i − σ i − ,k ) = d ( ϕ i σ i,i − σ i − ,k , ϕ i − σ i − ,k ) < ǫ k / i . Moreover, as ϕ i σ i,i − (1 j ) = ϕ i − (1 j ) for every j ≤ n i − , condition (ii)’ also holds. An analogousargument shows that (iii)’ holds.This finishes the inductive argument.By Lemma 4.13, condition (ii)’ induces a Cu-morphism φ : lim L n i → S with the canonical mor-phisms φ i : L n i → S being the limits of the sequences ( ϕ j σ j,i ) j .To see that φ is surjective, first note that, for every i ∈ N and for every ǫ >
0, there exists j ∈ N such that d ( φ i θ i , ϕ j σ j,i θ i ) < ǫ . This is due to Corollary 4.10 and because d ( φ i , ϕ j σ j,i ) tends to 0 as j tends to infinity.Thus, we get d ( ϕ i θ i , φ i θ i ) ≤ d ( ϕ i θ i , ϕ i +1 σ i +1 ,i θ i ) + · · · + d ( ϕ j − σ j − ,i θ i , ϕ j σ j,i θ i ) + d ( ϕ j σ j,i θ i , φ i θ i ) ≤ ǫ i i + · · · + ǫ i j − + ǫ ≤ ǫ i + ǫ, where we have used property (iii)’ to bound all but the last element.Since this holds for every ǫ , one gets d ( ϕ i θ i , φ i θ i ) ≤ ǫ i . In particular, the distance is strictlydecreasing on i .Now let s i be a basic element of S and let x ∈ S be such that x ≪ s i . We have x ≪ ψ i ( χ (0 , ) and,consequently, there exists s, t ∈ (0 ,
1] such that x ≪ ψ i ( χ ( t +2 s, ) ≪ ψ i ( χ ( t, ) ≪ ψ i ( χ (0 , ) . Note that, for every k > i , we have d ( ϕ k − ⊕ ρ k , φ k θ k ) ≤ d ( ϕ k − ⊕ ρ k , ϕ k θ k ) + d ( ϕ k θ k , φ k θ k ) < ǫ k + 2 ǫ k = 3 ǫ k , where in the previous bound we have used condition (i)’ and the inequality obtained above.Thus, there exists a large enough k so that d ( ϕ k − ⊕ ρ k , φ k θ k ) < s .Since k > i , we know that ( ϕ k − ⊕ ρ k ) η i = ψ i , where η i : L → L n k − ⊕ L k is the canonical inclusionin the ( n k − + i )-th L -summand.Thus, we have x ≪ ψ i ( χ ( t +2 s, ) = ( ϕ k − ⊕ ρ k ) η i ( χ ( t +2 s, ) ≪ φ k θ k η i ( χ ( t + s, ) ≪ ( ϕ k − ⊕ ρ k ) η i ( χ ( t, ) = ψ i ( χ ( t, ) . This shows that for every i and every x ≪ s i , there exist some k with x ≪ φ k ( l ) ≪ s i with l ∈ L n k .Consequently, φ is surjective.We will now prove that φ is an order-embedding. To do this, we will denote by [ a ] the elements inlim L n i coming from some block L n i of the direct limit.Take x, y ∈ lim L n i such that φ ( x ) ≤ φ ( y ). Let a ∈ L n s be a basic element with [ a ] ≪ x . Also, take[ z ] , [ z ′ ] such that [ a ] ≪ [ z ′ ] ≪ [ z ] ≪ x with z, z ′ ∈ L n s ′ basic elements as well. Finally, take b ∈ L n k abasic element such that [ b ] ≪ y and φ ([ z ]) ≪ φ ([ b ]) ≪ φ ( y ).We can assume that, for a large enough i , we have σ i,s ( a ) , σ i,k ( b ) , σ i,s ′ ( z ) , σ i,s ′ ( z ′ ) ∈ L n i with σ i,s ( a ) ≪ σ i,s ′ ( z ′ ) ≪ σ i,s ′ ( z ) , and φ i ( σ i,s ′ ( z )) ≪ φ i ( σ i,k ( b )) . Thus, since we have d ( φ i , ϕ j σ j,i ) →
0, it follows from Corollary 4.7 that ϕ j σ j,i ( σ i,s ′ ( z ′ )) ≪ ϕ j σ j,i ( σ i,k ( b ))for every sufficiently large j .Also, since z, z ′ , a, b are basic elements in their respective blocks, we can take j large enough so thatwe also have σ j,s ′ ( z ) , σ j,s ′ ( z ′ ) , σ j,s ( a ) , σ j,k ( b ) ∈ F j .Therefore, since σ j,s ( a ) ≪ σ j,s ′ ( z ′ ) and ϕ j ( σ j,s ′ ( z ′ )) ≪ ϕ j ( σ j,k ( b )), it follows from condition (iv)that σ j +1 ,j ( σ j,s ( a )) ≪ σ j +1 ,j ( σ j,k ( b )). That is to say, we have σ j +1 ,s ( a ) ≪ σ j +1 ,k ( b ) LOCAL CHARACTERIZATION FOR THE CUNTZ SEMIGROUP OF AI-ALGEBRAS 17 and thus [ a ] ≪ [ b ] ≪ y .Since x can be written as the supremum of an ≪ -increasing sequence ([ x n ]) with x n basic elements,it follows from the previous argument that [ x n ] ≪ y for every n . Taking the supremum, one gets x ≤ y as required.We now have S ∼ = lim i L n i , and the desired result follows from Theorem 2.5. (cid:3) The following result shows that we only need to focus on one ≪ -relation instead of working with afinite subset F and all of its ≪ -relations. Proposition 4.16.
Let S be a countably based and compactly bounded Cu -semigroup satisfying weakcancellation and (O5). Then, S is Cu -isomorphic to the Cuntz semigroup of an AI-algebra if and onlyif for every Cu -morphism ϕ : Lsc([0 , , N ) r → S , every ǫ > and every triple x, x ′ , y in Lsc([0 , , N ) r such that x ≪ x ′ with ϕ ( x ′ ) ≪ ϕ ( y ) , there exist Cu -morphisms θ, φ such that the diagram Lsc([0 , , N ) r ϕ / / θ (cid:15) (cid:15) S Lsc([0 , , N ) s φ ttttttttttt satisfies:(i) d ( φθ, ϕ ) < ǫ .(ii) θ ( x ) ≪ θ ( y ) .(iii) ϕ (1 j ) = φθ (1 j ) for every ≤ j ≤ r .Proof. One direction follows trivially from Theorem 4.15, so we only need to prove the other. To dothis, we will see that the conditions in Theorem 4.15 are satisfied.Let ϕ : Lsc([0 , , N ) r → S and define the following set R ϕ = { ( x, x ′ , y ) ∈ (Lsc([0 , , N ) r ) | x ≪ x ′ with ϕ ( x ′ ) ≪ ϕ ( y ) } . Let R = { ( x i , x ′ i , y i ) } be a finite subset of R ϕ (note that some x i ’s, x ′ i ’s and y j ’s might coincide).We will prove by induction on the cardinality of R that given any ϕ : Lsc([0 , , N ) r → S , any ǫ > R ⊂ R ϕ there exist morphisms θ, φ such that(i) d ( φθ, ϕ ) < ǫ .(ii) θ ( x ) ≪ θ ( y ) for every ( x, x ′ , y ) ∈ R .(iii) ϕ (1 j ) = φθ (1 j ) for every 1 ≤ j ≤ r .Note that, given any finite subset F ⊂ Lsc([0 , , N ) r , the set F ∩ R ϕ is finite. Thus, our resultwill follow from this fact.For n = 1, the results holds by assumption.Now assume that for every k ≤ n −
1, the desired result has been proven for every Cu-morphism ϕ : Lsc([0 , , N ) r → S , for every ǫ > R ⊂ R ϕ such that | R | = k . Fix anyCu-morphism ϕ : Lsc([0 , , N ) r → S , any ǫ > R ⊂ R ϕ with | R | = n .As before, write R = { ( x , x ′ , y ) , · · · , ( x n , x ′ n , y n ) } . By Corollary 4.7, there exist elements x ′′ i with x i ≪ x ′′ i ≪ x i and a bound δ > ψ : Lsc([0 , , N ) r → S with d ( ψ, ϕ ) < δ and ψ (1 j ) = φ (1 j ) for each j , we have ψ ( x ′′ i ) ≪ ψ ( y i ).Set ǫ ′ := min { ǫ/ , δ } >
0, and apply the induction hypothesis to ϕ , ǫ ′ and the set { ( x , x ′′ , y ) } .Thus, we get Cu-morphisms θ : Lsc([0 , , N ) r → Lsc([0 , , N ) t , and φ : Lsc([0 , , N ) t → S such that(i) d ( φ θ , ϕ ) < ǫ ′ .(ii) θ ( x ) ≪ θ ( y ).(iii) ϕ (1 j ) = φ θ (1 j ) for every 1 ≤ j ≤ r .In particular, note that (i) implies that ( θ ( x i ) , θ ( x ′′ i ) , θ ( y i )) ∈ R φ for every i ≤ n . Indeed, wehave φ θ ( x i ) ≪ φ θ ( x ′′ i ) and ϕ ( x ′′ i ) ≪ ϕ ( y i ). Since d ( φ θ , ϕ ) < ǫ ′ < δ , we get φ θ ( x ′′ i ) ≪ φ θ ( y i ).Therefore, we have θ ( x i ) ≪ θ ( x ′′ i ) , and φ θ ( x ′′ i ) ≪ φ θ ( y i ) , which shows that ( θ ( x i ) , θ ( x ′′ i ) , θ ( y i )) ∈ R φ as desired.Now take the finite subset R = { ( θ ( x i ) , θ ( x ′′ i ) , θ ( y i )) } ≤ i ≤ n , which has cardinality n −
1. Let ν be the bound given in Corollary 4.10 for the morphism θ and the constant ǫ/ φ , ν > R we get Cu-morphisms θ , φ with(i) d ( φ θ , φ ) < ν .(ii) θ θ ( x i ) ≪ θ θ ( y i ) for every 1 ≤ i ≤ n . For i = 1, this is because θ ( x ) ≪ θ ( y ).(iii) φ (1 j ) = φ θ (1 j ) for every 1 ≤ j ≤ t .Since d ( φ θ , φ ) < ν , we have by Corollary 4.10 that d ( φ θ θ , φ θ ) < ǫ/
2. In this situation, weget d ( ϕ, φ θ θ ) ≤ d ( ϕ, φ θ ) + d ( φ θ θ , φ θ ) < ǫ. Finally, given any 1 j ∈ Lsc([0 , , N ) r , write θ (1 j ) = k + · · · + k t t ∈ Lsc([0 , , N ) t . We have ϕ (1 j ) = φ θ (1 j ) = φ ( k + · · · + k t t ) = k φ (1 ) + · · · + k t φ (1 t )= k φ θ (1 ) + · · · + k t φ θ (1 t ) = φ θ θ (1 j ) . This shows that φ := φ and θ := θ θ satisfy the required properties. (cid:3) We will now reduce the hypothesis of the theorem further, by showing that we can discretizeProposition 4.16 and that one only needs to focus on a particular kind of elements x, x ′ , y . Moreexplicitly, we will later prove the following:Given any l ∈ N , let C Ml = { χ s ( i/l, } i,s ∪ { s } ⊂ Lsc([0 , , N ) M , where 1 s and χ s ( i/l, denote theunit and the element χ ( i/l, in the s -th summand respectively for every s ≤ M . Also, let B Ml be theadditive span of the elements in C Ml . Theorem 4.17.
Let S be a countably based and compactly bounded Cu -semigroup satisfying weakcancellation and (O5). Then, S is the Cuntz semigroup of an AI-algebra if and only if for every Cu -morphism ϕ : Lsc([0 , , N ) M → S , every l ∈ N and every triple of elements x, x ′ , y ∈ B Ml suchthat x ≪ x ′ with ϕ ( x ′ ) ≪ ϕ ( y ) , there exist Cu -morphisms θ, φ such that the diagram Lsc([0 , , N ) M ϕ / / θ (cid:15) (cid:15) S Lsc([0 , , N ) N φ sssssssssss satisfies:(i) ϕ ( χ s (( i +1) /l, ) ≪ φθ ( χ s ( i/l, ) and φθ ( χ s (( i +1) /l, ) ≪ ϕ ( χ s ( i/l, ) for every χ s ( i/l, ∈ C Ml .(ii) θ ( x ) ≪ θ ( y ) .(iii) ϕ (1 s ) = φθ (1 s ) for every s ≤ M . Example 4.18.
Let Z be the Jiang-Su algebra (as introduced in [JS99]) and let Z be its Cuntzsemigroup. It is known that Z is Cu-isomorphic to N ⊔ (0 , ∞ ], where the elements in N are compactand the ones in (0 , ∞ ] are not. For more details, see [APT18, Paragraph 7.3.2].We will show that Z does not satisfy the conditions in Theorem 4.17. This is trivial since Z is notan AI-algebra, but we give here an explicit proof.Let l ≥ ϕ : Lsc([0 , , N ) → Z be any Cu-morphism such that ϕ ( χ ( k/l, ) = 1 ′ − k/l and ϕ (1) = 1. At least one such morphism exists by Theorem 3.1, since (1 ′ − k/l ) k ⊂ Z is a rapidlydecreasing sequence bounded by the compact 1.We have that ϕ (1) = 1 ≪ / ϕ ( χ (1 / , ). Then, if Z were to satisfy the previous theorem,we would get morphisms θ : Lsc([0 , , N ) → Lsc([0 , , N ) N and φ : Lsc([0 , , N ) N → Z such that d ( φθ, ϕ ) < / θ (1) ≪ θ (3 χ (1 / , ) and φθ (1) = ϕ (1).Note that one also has φ ( θ (1) ∧
1) = ϕ (1) = 1, where the infimum is taken componentwise. Moreover,since θ (1) ≪ θ ( χ (1 / , ), we have θ (1) ∧ ≪ θ ( χ (1 / , ) ∧ θ (1) ∧ ⊂ supp( θ ( χ (1 / , ) ∧ θ ( χ (1 / , ) ∧ θ (1) ∧ φ ( θ ( χ (1 / , ) ∧
1) = φ ( θ (1) ∧
1) = ϕ (1) ≫ ϕ ( χ (0 , ) ≫ φθ ( χ (1 / , ) ≥ φ ( θ ( χ (1 / , ) ∧ , LOCAL CHARACTERIZATION FOR THE CUNTZ SEMIGROUP OF AI-ALGEBRAS 19 where in the second inequality we have used d ( ϕ, φθ ) < / ϕ ( χ (0 , ) = 1 ′ is compact, a contradiction.Recall from Definition 2.6 that an element in Lsc([0 , , N ) is said to be basic if it can be writtenas a finite sum of basic indicator functions. In particular, a basic element f satisfies min( f ) = f (0) ifand only if it can be written as a finite sum of elements of the form 1 , χ ( · , and χ ( · , · ) . Similarly, f isincreasing if and only if it can be written as a finite sum of elements of the form 1 and χ ( · , . Lemma 4.19.
Let x ≪ y be basic elements in Lsc([0 , , N ) . Then, there exists basic increasingelements a and d such that x + a ≪ d ≪ y + a .The same holds, component-wise, for elements x ≪ y in Lsc([0 , , N ) M ≪ for every M ∈ N .Proof. We will first prove the following claim.
Claim
Given a basic element x and a basic indicator y such that x ≪ y ≤ there exist basic increasingelements a, d such that x + a ≪ d ≪ y + a .Proof. Write x = P i χ U i and y = χ V with U i , V intervals. Note that, since x ≤
1, we may assume theintervals U i to be pairwise disjoint.If V = [0 ,
1] (i.e. y = 1), we can take a = 0 and d = y , so we assume otherwise.If V is of the form [0 , t ) for some t , let ǫ > ⊔ i U i ⋐ [0 , t − ǫ ). Set d = 1and a = χ ( t − ǫ, . One clearly has P i χ U i + a ≪ d ≪ χ V + a .If there exists some s so that V = ( s, ǫ > P i χ U i ≪ χ ( s + ǫ, . Set a = 0 and d = χ ( s + ǫ, .Finally, if V = ( s, t ) for some s < t , take ǫ, δ > ⊔ i U i ⋐ ( s + ǫ, , and ( ⊔ i U i ) ∩ ( t − ǫ,
1] = ∅ . Set a = ( t − ǫ,
1] and d = ( s + ǫ, P i χ U i + a ≪ d ≪ y + a as required. (cid:3) Given two basic elements x ≪ y in Lsc([0 , , N ), we know that these can be written as x = P i χ U i and y = P i χ V i with χ U i ≪ χ V i , where χ U i is a basic element (since U i may not be an interval) and χ V i is a basic indicator function.For every i , the previous claim gives us basic increasing elements a i , d i such that χ U i + a i ≪ d i ≪ χ V i + a i . Set a = P i a i and d = P i d i , where note that x + a ≪ d ≪ y + a . Since every sum of basicincreasing elements is again basic and increasing, the result follows.To see that the same result holds in Lsc([0 , , N ) M for every M , simply apply everything compo-nentwise. (cid:3) Proposition 4.20.
Let S be a countably based and compactly bounded Cu -semigroup satisfying (O5)and weak cancellation.If conditions (i)-(iii) of Proposition 4.16 are satisfied for every Cu -morphism ϕ , for every ǫ > and any triple of basic increasing elements a, a ′ , b such that a ≪ a ′ with ϕ ( a ′ ) ≪ ϕ ( b ) , then S is Cu -isomorphic to the Cuntz semigroup of an AI-algebra.Proof. Let ϕ : Lsc([0 , , N ) r → S be a Cu-morphism and take ǫ > x, x ′ , y be basic elements with x ≪ x ′ and ϕ ( x ′ ) ≪ ϕ ( y ). We will prove that conditions(i)-(iii) in Proposition 4.16 hold for these elements. Since basic elements are sup-dense, this will showthat S is Cu-isomorphic to the Cuntz semigroup of an AI-algebra.Take y ′ ≪ y with ϕ ( x ′ ) ≪ ϕ ( y ′ ). Now apply Lemma 4.19 to obtain basic increasing elements d, f, a, b such that x + a ≪ d ≪ x ′ + a, and y ′ + b ≪ f ≪ y + b, where by an increasing element we mean an element that is increasing in each component.Thus, there exists r, t > R t ( a ) , R r ( b ) are still basic increasing elements with x + a ≪ R ( d ) ≪ d ≤ x ′ + R t ( a ) , and y ′ + b ≪ f ≤ y + R r ( b )for some retraction R ( d ) of d . Thus, we have the following ϕ ( d + b ) ≤ ϕ ( x ′ + R t ( a ) + b ) ≤ ϕ ( y ′ + R t ( a ) + b ) ≪ ϕ ( f + R t ( a )) . Note that the elements R ( d ) + R r ( b ), d + b and f + R t ( a ) are a triple of basic increasing elementssuch that R ( d ) + R r ( b ) ≪ d + b, and ϕ ( d + b ) ≪ ϕ ( f + R t ( a )) . Thus, we get by assumption that there exist Cu-morphisms θ and φ satisfying conditions (i)-(iii) inProposition 4.16 for x = R ( d ) + R r ( b ) and y = f + R t ( a ). Using this at the second step, one obtains θ ( x + a + R r ( b )) ≪ θ ( R ( d ) + R r ( b )) ≪ θ ( f + R t ( a )) ≤ θ ( y + R r ( b ) + a ) . Applying weak cancellation to the previous inequality, we get θ ( x ) ≪ θ ( y ) as required. (cid:3) Proof. [of Theorem 4.17]The forward implication follows trivially from Proposition 4.16 by taking ǫ < /l and applyingLemma 4.5 to prove condition (i).To show the converse, we know by Proposition 4.20 that it is enough to show that conditions (i)-(iii)of Proposition 4.16 are satisfied for every morphism ϕ : Lsc([0 , , N ) M → S , for every ǫ > x, x ′ , y such that x ≪ x ′ with ϕ ( x ′ ) ≪ ϕ ( y ).Let l ∈ N be large enough such that 2 /l < ǫ and x, x ′ , y ∈ B Ml . By assumption, we obtain Cu-morphisms φ and θ satisfying conditions (i)-(iii) in Theorem 4.17. In particular, condition (i) statesthat ϕ ( χ s (( i +1) /l, ) ≪ φθ ( χ s ( i/l, ) , and φθ ( χ s (( i +1) /l, ) ≪ ϕ ( χ s ( i/l, )for every χ s ( i/l, ∈ C Ml .Thus, it follows from Lemma 4.8 that d ( ϕ, φθ ) < /l + 1 /l = 2 /l < ǫ . This shows that condition (i)of Proposition 4.16 is satisfied.Conditions (ii) and (iii) from Proposition 4.16 and Theorem 4.17 coincide. Using Proposition 4.20,we have that S is Cu-isomorphic to the Cuntz semigroup of an AI-algebra, as desired. (cid:3) An abstract characterization
The aim of this section is to provide an abstract characterization for the Cuntz semigroups of AI-algebras using Theorem 4.17. The property used in this characterization will be Property I, as definedbelow; see Definition 5.30. One could possibly use Theorem 4.17 to prove other, maybe simpler,characterizations; see Question 5.36.5.1.
The sets Ω n and X n . Let Ω n = {−∞ , , · · · , n, ∞} . For α, α ′ ∈ Ω n , we define α ′ ≺ α if α ′ = α = −∞ , or α ′ = α = ∞ or α ′ ≤ α with α ′ = α Let udiag(Ω n × Ω n ) be the subset of Ω n × Ω n consisting of the pairs ( α, β ) with α (cid:12) β . Define X n as the free abelian monoid on udiag(Ω n × Ω n ). In X n we denote the unit by (0 , w, ( α, β ) ∈ X n , write w ≺ ( α, β ) if and only if w = (0 ,
0) or else there exist elements ( α i , β i )in udiag(Ω n × Ω n ) such that α ≺ α ≺ β ≺ α ≺ · · · ≺ α m ≺ β m ≺ β and w = P ( α i , β i ). In particular, ( α ′ , β ′ ) ≺ ( α, β ) if and only if α ≺ α ′ ≺ β ′ ≺ β .Also, set (0 , ≺ (0 , w, v ∈ X n , we write w ≺ v if there exist (possibly repeated and zero) elements w i ∈ X n and ( α i , β i ) ∈ udiag(Ω n × Ω n ) ∪ { (0 , } such that w = P w i , v = P ( α i , β i ) and w i ≺ ( α i , β i )as above for every i .Note that ≺ is trivially compatible with addition. Lemma 5.1.
Let n ∈ N and let w, v, ( α, β ) be elements in X n . We have(i) If w = P ( γ j , δ j ) ≺ ( α, β ) , there exists at most one γ j with α = γ j = −∞ (resp. at most one δ j with β = δ j = ∞ ).(ii) w + ( −∞ , ∞ ) ≺ ( −∞ , ∞ ) if and only if w = (0 , .(iii) w + ( −∞ , ∞ ) ≺ v + ( −∞ , ∞ ) if and only if w ≺ v . LOCAL CHARACTERIZATION FOR THE CUNTZ SEMIGROUP OF AI-ALGEBRAS 21
Proof.
For (i), we may assume that w = P j ≤ m ( γ j , δ j ) ≺ ( α, β ) with α ≺ γ ≺ · · · ≺ δ m ≺ β and( γ j , δ j ) = (0 ,
0) for every j . If some γ j is such that γ j = −∞ , we have α = γ = δ = · · · = γ j = −∞ . However, we know that γ = δ . This implies that j = 1 and, consequently, that γ j is unique.An analogous argument proves that if δ j = ∞ for some j , we must have j = m .In particular this implies that, whenever w ≺ ( α, β ), w has a summand of the form ( −∞ , ∞ ) if andonly if w = ( −∞ , ∞ ) = ( α, β ).Thus, it follows that w + ( −∞ , ∞ ) ≺ ( −∞ , ∞ ) if and only if w = (0 , v ′ = v + ( −∞ , ∞ ) and write v ′ = P i ≤ k ( α i , β i ) and w = P i ≤ k w i in sucha way that w + ( −∞ , ∞ ) ≺ ( α , β ) and w i ≺ ( α i , β i ) for i ≥ ≺ we have w = (0 ,
0) and ( α , β ) = ( −∞ , ∞ ).Therefore, we get v = P i ≥ ( α i , β i ).Using that ≺ is trivially compatible with addition, it follows that w = X i ≥ w i = X i ≥ w i ≺ X i ≥ v i = v as required. (cid:3) Lemma 5.2.
With the above notation, ≺ is a transitive and antisymmetric relation.Proof. First note that, given any two pairs w ≺ v and w ≺ v , one clearly has w + w ≺ v + v .Thus, let w ≺ v and v ≺ u , which by definition means that w = P w i , v = P ( α i , β i ) = P j v j and u = P j ( γ j , δ j ) such that w i ≺ ( α i , β i ) , and v j ≺ ( γ j , δ j )for every i, j , and where we may assume ( α i , β i ) , ( γ j , δ j ) non-zero for every i, j .Also, for every w i = 0, write w i = P k ( ζ k,i , η k,i ). Define min( w i ) and max( w i ) as the minimum ofthe ζ k,i and the maximum of the η k,i with respect to ≺ respectively. Note that such elements exist bythe definition of w i ≺ ( α i , β i ).For every j , let I j be such that P i ∈ I j ( α i , β i ) = v j , and define ˜ I j = { i ∈ I j | w i = 0 } . Using our firstobservation, we get X i ∈ ˜ I j w i = X i ∈ I j w i ≺ X i ∈ I j ( α i , β i ) ≺ ( γ j , δ j ) . Note that if I j is empty, so is ˜ I j , and so P i ∈ I j w i = (0 , ≺ ( γ j , δ j ). Else, if I j is not empty, thereexists an ordering i , · · · , i r of ˜ I j such that γ j ≺ α i ≺ min( w i ) ≺ ζ k,i ≺ η k,i ≺ max( w i ) ≺ β i ≺ · · · ≺ α i r ≺ min( w i r ) ≺ ζ k,i r ≺ η k,i r ≺ max( w i r ) ≺ β i r ≺ δ j Thus, given ˜ w j = P i ∈ I j w i , we have ˜ w j ≺ ( γ j , δ j ). Since w = P ˜ w j , it follows that w ≺ u . Thisshows that ≺ is transitive.To see that ≺ is antisymmetric first note that, by transitivity and (i)-(ii) in Lemma 5.1, givenelements w = ( α, β ) and v ∈ X n , we have w ≺ v ≺ w if and only if w = v = ( −∞ , ∞ ) or if w = v = 0.Let w ∈ X n and let m be the number of non-zero summands of w . We will now prove by inductionon m that w ≺ v ≺ w if and only if w = v = m ( −∞ , ∞ ). Thus, assume that for a fixed m we haveproven that w ≺ v ≺ w implies w = v = ( m − −∞ , ∞ ) for any w having m − v .Take w = P i ≤ m ( α i , β i ) and v = P j ≤ k ( γ j , δ j ), where we assume that all summands are non-zero.Then, set i = 1 and find j such that ( α i , β i ) ≺ ( γ j , δ j ). This can be done because w ≺ v .Now let i be such that ( α i , β i ) ≺ ( γ j , δ j ) ≺ ( α i , β i ) , which exists because v ≺ w . Note that, if i = i , we would have( α i , β i ) ≺ ( γ j , δ j ) ≺ ( α i , β i ) . This in turn would imply ( α i , β i ) = ( γ j , δ j ) = ( −∞ , ∞ ). By Lemma 5.1 (iii), we could cancelthis summand from w and v and apply induction to conclude that w = v = m ( −∞ , ∞ ).Thus, we assume i = i . Following this construction, one obtains an ordering i , · · · , i n of { , · · · , n } and pairwise different integers j , · · · , j n such that( α i , β i ) ≺ ( γ j , δ j ) ≺ · · · ≺ ( α i n , β i n ) ≺ ( γ j n , δ j n ) . Since v ≺ w , there must exists some s ≤ n such that ( γ j n , δ j n ) ≺ ( α s , β s ) and, since i , · · · , i n is anordering of { , · · · , n } , we must have s = j l for some l ≤ n .This implies ( α i l , β i l ) ≺ ( γ j l , δ j l ) ≺ ( α i l , β i l )and, by the same argument as before, we get w = v = m ( −∞ , ∞ ) as desired. (cid:3) In X n , we write w ≅ v if w = v or else there exist z ∈ X n and α ≺ γ ≺ β ≺ δ in Ω n such that w = z + ( α, β ) + ( γ, δ ) , and v = z + ( α, δ ) + ( γ, β )or v = z + ( α, β ) + ( γ, δ ) , and w = z + ( α, δ ) + ( γ, β ) . Note that, with z = 0, this yields ( α, β ) + ( γ, δ ) ≅ ( α, δ ) + ( γ, β ).Given two elements w, v ∈ X n , we write w ≃ v if and only if there exists w , · · · , w m ∈ X n suchthat w ≅ w ≅ · · · ≅ w m ≅ v .This construction tries to mimic the relation defined in Remark 2.7. Lemma 5.3. ≃ is an equivalence relation compatible with addition.Proof. First note that ≃ is transitive, symmetric and reflexive by construction, so we only need toprove that it is compatible with addition.To see this, first take w ≅ v with w = v , where we may assume that there exist z ∈ X n and α ≺ γ ≺ β ≺ δ in Ω n such that w = z + ( α, β ) + ( γ, δ ) , and v = z + ( α, δ ) + ( γ, β ) . Thus, given any w ′ ∈ X n , we have w + w ′ = ( z + w ′ ) + ( α, β ) + ( γ, δ ) , and v + w ′ = ( z + w ′ ) + ( α, δ ) + ( γ, β ) , which shows w + w ′ ≅ v + w ′ . Trivially, if w = v , we also have w + w ′ ≅ v + w ′ for any w ′ .Therefore, given w ≅ v and w ′ ≅ v ′ , we get w + w ′ ≅ v + w ′ ≅ v + v ′ , which implies w + w ′ ≃ v + v ′ .Now let w ≃ v and w ′ ≃ v ′ . Then, there exist w , · · · , w m and w ′ , · · · , w ′ m ′ such that w ≅ w ≅ · · · ≅ w m ≅ v, and w ′ ≅ w ′ ≅ · · · ≅ w ′ m ′ ≅ v ′ . Since w ≅ v whenever w = v , we may assume m = m ′ . Thus, we have w + w ′ ≅ w + w ′ ≅ · · · ≅ w m + w ′ m ≅ v + v ′ , which implies w + w ′ ≃ v + v ′ as required. (cid:3) Chainable subsets.
Given a Cu-semigroup S with weak cancellation, we will say that an ad-ditive map F : X n → S is an I-morphism if F ( q ) ≪ F ( t ) whenever q ≺ t and F ( q ) = F ( t ) whenever q ≃ t .Note that this implies F ((0 , F (( −∞ , ∞ )) = F (( −∞ , ∞ )) ≪ F (( −∞ , ∞ )) and, by weakcancellation, F ((0 , Definition 5.4.
Let H be a subsemigroup of a Cu-semigroup S , and let e be a compact element in H . We say that H is an ( n, e )-chainable subset if there exists a surjective I-morphism F : X n → S with F ( X n ) ⊂ H satisfying the following properties:(i) F (( −∞ , ∞ )) = e (ii) F (( α, β )) + F (( β, ∞ )) ≤ F (( α, ∞ ))(iii) For every m ≥ F ′ : X mn → S such that F ′ (( mα, mβ )) = F (( α, β ))for every α, β ∈ Ω n with α ≺ β . LOCAL CHARACTERIZATION FOR THE CUNTZ SEMIGROUP OF AI-ALGEBRAS 23
Remark 5.5.
Given a Cu-morphism ϕ : S → T and an ( n, e )-chainable subsemigroup H of S , it isclear that ϕ ( H ) is a ( n, ϕ ( e ))-chainable subset of T . Remark 5.6.
Let H be a subsemigroup of S . Then, H is ( n, e )-chainable if and only if H is theadditive span of a finite subset H , containing and bounded by e , such that there exists a surjectivemap F : udiag(Ω n × Ω n ) ∪ { (0 , } → H satisfying the following properties:(i) F ( −∞ , ∞ ) = e .(ii) F (( α, β )) + F (( β, ∞ )) ≤ F (( α, ∞ )).(iii) F (( α, β )) + F (( γ, δ )) = F (( α, δ )) + F (( γ, β )) whenever α ≺ γ ≺ β ≺ δ .(iv) P mi =1 F (( α i , β i )) ≪ F (( α, β )) whenever P mi =1 ( α i , β i ) ≺ ( α, b ).(v) For every m ≥ F ′ : X mn → S such that F ′ (( mα, mβ )) = F (( α, β ))for every α, β ∈ Ω n .5.2.1. Examples.
Example 5.7.
The additive span of a compact element e (i.e. the set of finite multiples of e ) is( n, e )-chainable for every n . Proof.
Given any n ≥
0, one can define the additive map F : X n → H as F (( α, β )) = e if β = ∞ and F (( α, β )) = 0 otherwise.Using Lemma 5.1 (i), it is easy to see that, if w ≺ v , then the number of summands of the form( α, ∞ ) of w must be less than or equal to that of v . This shows F ( w ) ≪ F ( v ) whenever w ≺ v .Also, given α ≺ γ ≺ β ≺ δ in Ω n , either δ = ∞ , in which case F (( α, β ) + ( γ, δ )) = 0 = F (( α, δ ) + ( γ, β )) , or δ = ∞ and F (( α, β ) + ( γ, δ )) = e = F (( α, δ ) + ( γ, β )) . This implies F ( w ) = F ( v ) whenever w ≃ v and, consequently, that F is an I-morphism.Note that properties (i) and (ii) of Definition 5.4 follow by construction and that, for property (iii),we can simply define F ′ : X nm → H analogously to F . (cid:3) Let S = Lsc([0 , , N ), and let L n be the additive span of { , χ ( i/n,j/n ) , χ ( i/n, , χ [0 ,j/n ) } i,j . We willnow show that L n is ( n, Notation 5.8.
Given two elements α ≺ β in Ω n , we denote by ( α/n, β/n ) the interval( α/n, β/n ) ∩ [0 , , α/n, ∞ /n ) corresponds to ( α/n,
1] whenever α = −∞ and ( −∞ /n, ∞ /n ) correspondsto [0 , Proposition 5.9.
Let S = Lsc([0 , , N ) and let n ∈ N . Then, L n is ( n, -chainable.Proof. Following Notation 5.8, define the additive map F : X n → L n as F (( α, β )) = χ ( α/n,β/n ) . Wewill now check that F is an I-morphism:First, note that given α, γ ≺ β, δ with F (( α, β ) + ( γ, δ )) = F (( α, δ ) + ( γ, β )) if and only if( α/n, β/n ) ∪ ( γ/n, δ/n ) = ( α/n, δ/n ) ∪ ( γ/n, β/n ) , ( α/n, β/n ) ∩ ( γ/n, δ/n ) = ( α/n, δ/n ) ∩ ( γ/n, β/n ) . Using that F is additive, one can check that this is equivalent to ( α, β ) + ( γ, δ ) ≅ ( α, δ ) + ( γ, β ).Thus, using the additivity of F once again, we get F ( q ) = F ( t ) whenever q ≃ t .Now, let P mi =1 ( α i , β i ) ≺ ( α, β ), which by definition implies that, after a possible reordering, we have α ≺ α ≺ β ≺ α ≺ · · · ≺ α m ≺ β m ≺ β . This implies ∪ ( α i /n, β i /n ) ⋐ ( α/n, β/n ) and ( α i /n, β i /n ) ∩ ( α j /n, β j /n ) = ∅ for every pair i = j . Thus, one gets F m X i =1 ( α i , β i ) ! = m X i =1 χ ( α i /n,β i /n ) ≪ χ ( α/n,β/n ) = F (( α, β )) , which shows that F ( q ) ≪ F ( t ) whenever q ≺ t . Indeed, we know by definition that q ≺ t if and onlyif q = P i ≤ k w i and t = P i ≤ k ( α i , β i ) with w i ≺ ( α i , β i ) for every i . By our previous argument we have F ( w i ) ≪ v i and, since F is additive, it follows that F ( q ) ≪ F ( t ).Finally, note that conditions (i)(for e = 1) and (ii) in Definition 5.4 are satisfied by construction,and that for every m ≥ F ′ : X nm → L nm analogously to F . This shows that L n is ( n, (cid:3) Our aim now is to show that the I-morphism F defined in the previous proposition satisfies F ( q ) = F ( t ) if and only if q ≃ t . Lemma 5.10.
Let
U, V be open sets of [0 , such that U = ⊔ i ≤ N ( α i /n, β i /n ) , and V = ⊔ j ≤ M ( γ j /n, δ j /n ) with α i , β i , γ j , δ j ∈ Ω n and the convention of Notation 5.8.Then, given the finite number of elements (( ǫ k , ζ k )) k , (( η l , θ l )) l ∈ X n such that U ∪ V = ⊔ k ( ǫ k /n, ζ k /n ) , and U ∩ V = ⊔ l ( η l /n, θ l /n ) , we have P i ( α i , β i ) + P j ( γ j , δ j ) ≃ P k ( ǫ k , ζ k ) + P l ( η l , θ l ) .Proof. We will prove the lemma by induction on M , the number of connected components of V .Thus, first assume that M = 1. If U ∩ V = ∅ , U ⊂ V or V ⊂ U , the result follows trivially (and weactually get an equality instead of ≃ ).If none of the above cases happens, note that there is at most one pair ( α s , β s ) such that α s ≺ γ ≺ β s ≺ δ and at most one pair ( α r , β r ) such that γ ≺ α r ≺ δ ≺ β r . We will prove the result assuming that both such pairs exist, since the remaining cases can bechecked analogously. Also, without loss of generality, we can assume that our first pair is ( α , β ). Welet the other pair be ( α r , β r ).Then, by the definition of ≃ and since it is compatible with addition (see Lemma 5.3), we have( α , β ) + ( γ , δ ) + ( α r , β r ) ≃ ( α , δ ) + ( γ , β ) + ( α r , β r ) ≃ ( α , β r ) + ( α r , δ ) + ( γ , β )Thus, we get N X i =1 ( α i , β i ) + ( γ , δ ) ≃ X i>r ( α i , β i ) + ( α , β r ) ! + X
Apply now the induction hypothesis to ⊔ k ( ǫ k /n, ζ k /n ) and ( γ M /n, δ M /n ). Thus, we obtain elements(( ǫ ′ k ′ , ζ ′ k ′ )) k ′ , (( η ′ l ′ , θ ′ l ′ )) l ′ satisfying (cid:16) ⊔ k ( ǫ k /n, ζ k /n ) (cid:17) ∪ ( γ M /n, δ M /n ) = ⊔ k ′ ( ǫ ′ k ′ /n, ζ ′ k ′ /n ) , (cid:16) ⊔ k ( ǫ k /n, ζ k /n ) (cid:17) ∩ ( γ M /n, δ M /n ) = ⊔ l ′ ( η ′ l ′ /n, θ ′ l ′ /n ) , and ( γ M , δ M ) + P ( ǫ k , ζ k ) ≃ P ( ǫ ′ k ′ , ζ ′ k ′ ) + P ( η ′ l ′ , θ ′ l ′ ), which we will refer to as relation (2).Therefore, we get U ∪ V = (cid:16) ⊔ k ( ǫ k /n, ζ k /n ) (cid:17) ∪ ( γ M /n, δ M /n ) = ⊔ k ′ ( ǫ ′ k ′ /n, ζ ′ k ′ /n )and U ∩ V = (cid:16) ⊔ l ( η l /n, θ l /n ) (cid:17) ⊔ (cid:16) U ∩ ( γ M /n, δ M /n ) (cid:17) = (cid:16) ⊔ l ( η l /n, θ l /n ) (cid:17) ⊔ (cid:16) ( U ∪ V ′ ) ∩ ( γ M /n, δ M /n ) (cid:17) = (cid:16) ⊔ l ( η l /n, θ l /n ) (cid:17) ⊔ (cid:16) ⊔ l ′ ( η ′ l ′ /n, θ ′ l ′ /n ) (cid:17) . Now using relations (1), (2) and the fact that ≃ is compatible with addition, we get: X i ( α i , β i ) + X j With the above notation, let q ∈ X n and f = F ( q ) . Then, q ≃ q f .In particular, since ≃ is transitive, one has F ( q ) = F ( t ) if and only if q ≃ t .Proof. Let q = P i ≤ m ( ǫ i , ζ i ) and f = F ( q ). As above, let α i,k , β i,k ∈ Ω n be such that { f ≥ k } = ⊔ i ( α i,k /n, β i,k /n ) . for every k . We will prove the result by induction on m , the amount of summands of q :For m = 1, the result is trivial since q = q f . Thus, fix some m and assume that the result has beenproven for every element with m − q = P i ≤ m ( ǫ i , ζ i ) and, since ≃ is compatiblewith addition (see Lemma 5.3), note that q ≃ q F ( q ′ ) + ( ǫ , ζ ) , where q ′ = P v k ≃ w + w + u + P k> v k .Repeating this process a finite number of times, one gets q ≃ q F ( q ′ ) + ( ǫ , ζ ) ≃ w + u + X k ≥ v k ≃ w + w + u + X k ≥ v k ≃ · · · ≃ w + · · · + w k +1 = q f as desired. (cid:3) Remark 5.12. With the notation of Proposition 5.9, given q, t with F ( q ) ≪ F ( t ), we do not neces-sarily have that q ≃ q ′ ≺ t ′ ≃ t .Consider, for example, the elements q = (1 , 2) + (2 , 3) and t = (0 , 4) in X . Clearly, we have F ( q ) ≪ F ( t ) but it is not true that q ≃ q ′ ≺ t ′ ≃ t .Following the previous remark, we will now define a subset L n of L n such that, whenever q, t ∈ X n satisfy F ( q ) ≪ F ( t ), we have q ≃ q ′ ≺ t ′ ≃ t .First, we note the following: Remark 5.13. Let χ U , χ V ∈ L n with U, V intervals (i.e. of the form ( α/n, β/n ) for some α, β ∈ Ω n )and let 0 < ǫ < / n . It is easy to check that R ǫ ( U ∪ V ) = R ǫ ( U ) ∪ R ǫ ( V ) and R ǫ ( U ∩ V ) = R ǫ ( U ) ∩ R ǫ ( V ), where R ǫ ( · ) denotes the ǫ -retraction of unions of intervals as defined in Notation 2.8.Thus, given any n > f ∈ L n with f = P χ U i , we have that ∪ | I | = k ∩ i ∈ I R ǫ ( U i ) = R ǫ ( ∪ | I | = k ∩ i ∈ I U i ) = R ǫ ( { f ≥ k } ) , and, consequently, X i χ R ǫ ( U i ) = X k χ ∪ | I | = k ∩ i ∈ I R ǫ ( U i ) = X k χ R ǫ ( { f ≥ k } ) This implies that, for every 0 < ǫ < / n , the function R ǫ ( f ) = P χ R ǫ ( U i ) does not depend on theexpression of f as P χ U i . Note, however, that R ǫ ( f ) may no longer belong to L n .Also note that, for every f, g ∈ L n , we have R ǫ ( f ) + R ǫ ( g ) = R ǫ ( f + g ). Definition 5.14. For every n > 0, define L n as the subset of L n consisting of the functions f ∈ L n such that, for every k ≤ sup( f ), the connected components (i.e. intervals) of the open subset { f ≥ k } are at pairwise distance at least 2 /n . Lemma 5.15. Let f be an element in L n . For every rational < ǫ < / n , there exists some m ∈ N such that R ǫ ( f ) ∈ L m .Proof. Since f ∈ L n , we know that { f ≥ k } is a finite disjoint union of open intervals for every k ≤ sup( f ). Thus, given any ǫ > 0, the ǫ -retractions of these intervals are at pairwise distance at least2 ǫ .If, additionally, ǫ < / n and rational, we can write ǫ = m /m for some m and m > 1. Set m = nm > n , and note that R ǫ ( f ) ∈ L m .Further, the connected components of { R ǫ ( f ) ≥ k } are at pairwise distance at least 4 ǫ = 4 m /m > /m , as required. (cid:3) Lemma 5.16. Let f, g ∈ L n . Then, f ≪ g if and only if q f ≺ q g , where q f , q g are the elementsdefined in Lemma 5.11.Proof. Let f, g ∈ L n and write, for every k ≤ max { sup( f ) , sup( g ) } , { f ≥ k } = ⊔ i ∈ I k ( α i,k /n, β i,k /n ) , and { g ≥ k } = ⊔ j ≤ J k ( γ j,k /n, δ j,k /n )with I k , J k finite sets and α i,k , β i,k , γ j,k , δ j,k elements of X n . LOCAL CHARACTERIZATION FOR THE CUNTZ SEMIGROUP OF AI-ALGEBRAS 27 Assume that f ≪ g , which implies (see, for instance, [Vil21, Lemma 4.19]) that { f ≥ k } ⋐ { g ≥ k } for every k . Thus, for every fixed k there exists a partition ⊔ j ∈ J k B j,k = I k such that ⊔ i ∈ B j,k ( α i,k /n, β i,k /n ) ⋐ ( γ j,k /n, δ j,k /n )for every j ∈ J k .This implies that there is an ordering i , · · · , i | B j,k | of B j,k such that γ j,k ≺ α i ,k ≺ β i ,k ≤ α i ,k ≺ · · · ≺ β i | Bj,k |− ,k ≤ α i | Bj,k | ,k ≺ β i | Bj,k | ,k ≺ δ j,k . Additionally, since f ∈ L n , we know that we cannot have β i s ,k = α i s +1 ,k for any s < | B j,k | . Thus,one gets γ j,k ≺ α i ,k ≺ β i ,k ≺ α i ,k ≺ · · · ≺ β i | Bj,k |− ,k ≺ α i | Bj,k | ,k ≺ β i | Bj,k | ,k ≺ δ j,k and, consequently, X i ∈ B j,k ( α i,k , β i,k ) ≺ ( γ j,k , δ j,k ) . Since ≺ is compatible with addition (see Lemma 5.2), we get X i ∈ I k ( α i,k , β i,k ) = X j ∈ J k X i ∈ B j,k ( α i,k , β i,k ) ≺ X j ∈ J k ( γ j,k , δ j,k )and thus q f ≺ q g as desired.The other implication follows from Lemma 5.11 and the fact that the map F : X n → L n defined inProposition 5.9 is an I-morphism. (cid:3) Corollary 5.17. Let n ∈ N and take q, t ∈ X n such that F ( q ) , F ( t ) ∈ L n . Then, F ( q ) ≪ F ( t ) if andonly if q ≃ q f ≺ q t ≃ t . Properties of chainable subsets. Throughout this subsubsection, we will denote by H an ( n, e )-chainable subsemigroup of a Cu-semigroup S and F will be its associated I-morphism. Definition 5.18. Given any m ≥ 1, let F : X m → L m be the I-morphism defined in Proposition 5.9.We will denote by ρ m the Cu-morphism ρ m : Lsc([0 , , N ) → S such that ρ m ( χ ( α/nm, ) = F ′ (( α, ∞ ))for every α ∈ Ω nm . Here, F ′ : X nm → S denotes the map given by (iii) in Definition 5.4.Observe that ρ m as defined above exists by Theorem 3.1, as the sequence ( F ′ (( α, ∞ ))) α is boundedby e and is ≪ -decreasing. This is because ( α, ∞ ) ≺ ( α − , ∞ ) for every α .Note that ρ m depends on F ′ (and not only on m ) but, since this will not be used in the notes, weomit it in order to ease the notation.Note that, using (iii) in Definition 5.4, one has ρ m ( χ ( α/n, ) = ρ m ( χ ( mα/mn, ) = F ′ (( mα, ∞ )) = F (( α, ∞ ))for every m > α ∈ Ω n . Lemma 5.19. Let ǫ > and let m ≥ ǫ − . Then, for any α, β ≤ n we have(i) ρ m ( χ ( α/n + ǫ, ) ≪ F (( α, ∞ )) ≪ ρ m ( χ ( α/n − ǫ, ) , trivially.(ii) ρ m ( χ ( α/n + ǫ,β/n − ǫ ) ) ≪ F (( α, β )) ≪ ρ m ( χ ( α/n − ǫ,β/n + ǫ ) ) .(iii) ρ m ( χ [0 ,β/n − ǫ ) ) ≪ F (( −∞ , β )) ≪ ρ m ( χ [0 ,β/n + ǫ ) ) Proof. We only prove (ii), since (i) is trivial and (iii) follows similarly:We know from (ii) in Definition 5.4, and the observation prior to this lemma applied to ( β, ∞ ), that F (( α, β )) + ρ m ( χ ( β/n, ) ≤ ρ m ( χ ( α/n, ). Thus, one gets F (( α, β )) + ρ m ( χ ( β/n, ) ≤ ρ m ( χ ( α/n, ) ≪ ρ m ( χ ( α/n − ǫ, ) ≤ ρ m ( χ ( α/n − ǫ,β/n + ǫ ) ) + ρ m ( χ ( β/n, )for any ǫ > F (( α, β )) ≪ ρ m ( χ ( α/n − ǫ,β/n + ǫ ) ). To see the other inequality, note that, since( mα, mβ ) + ( mβ − n, ∞ ) ≃ ( mα, ∞ ) + ( mβ − n, mβ ) , one has F (( α, β )) + F ′ (( mβ − n, ∞ )) = F ′ (( mα, mβ )) + F ′ (( mβ − n, ∞ ))= F ′ (( mα, ∞ )) + F ′ ( mβ − n, mβ ) ≥ F ′ (( mα, ∞ )) = F (( α, ∞ )) . Using the fact that ρ m is a Cu-morphism at the first step and the above inequality at the thirdstep, we get ρ m ( χ ( α/n +1 /m,β/n − /m ) ) + ρ m ( χ ( β/n − /m, ) ≪ ρ m ( χ ( α/n, ) = F (( α, ∞ )) ≤ F (( α, β )) + ρ m ( χ ( β/n − /m, ) . Applying weak cancellation, one gets F (( α, β )) ≫ ρ m ( χ ( α/n +1 /m,β/n − /m ) ) ≥ ρ m ( χ ( α/n + ǫ,β/n − ǫ ) ) , since ǫ > /m . (cid:3) Let f ∈ X n and write f as P ki =1 ( α i , β i ). Also, consider g = P ki =1 χ U i ∈ Lsc([0 , , N ) with U i =( α i /n, β i /n ). Then, the previous lemma shows that for every ǫ > m such that ρ m ( R ǫ ( g )) ≪ F ( f ) ≪ ρ m ( N ǫ ( g )), where R ǫ ( g ) and N ǫ ( g ) denote the functions P ki =1 χ R ǫ ( U i ) and P ki =1 χ N ǫ ( U i ) respectively. Recall that R ǫ ( a, b ) = ( a + ǫ, b − ǫ ) and N ǫ ( a, b ) = ( a − ǫ, b + ǫ ). Lemma 5.20. Let S be a Cu -semigroup and H an ( n, e ) -chainable subset of S with associated I-morphism F : X n → H . Then, given any finite family { q i ≺ t i } i =1 , ··· ,N in X n , there exist functions f i , g i ∈ Lsc([0 , , N ) for i ≤ N such that(i) f i ≪ g i for all i .(ii) ρ m ( f i ) ≪ F ( q i ) ≪ ρ m ( g i ) ≪ F ( t i ) for all i and for every sufficiently large m .(iii) f i = f j (resp. f i = g j , g i = g j ) whenever q i = q j (resp. q i = t j , t i = t j ) for i, j ≤ N .In fact, given the I-morphism G : X n → Lsc([0 , , N ) defined in Proposition 5.9, one can take f i = R ǫ ( G ( q i )) and g i = R ǫ ( G ( t i )) for ǫ < / n .Proof. For every i , write q i = P j ≤ M,k ≤ K ( α ij,k , β ij,k ) and t i = P j ≤ M ( α ij , β ij ) such that α ij ≺ α ij, ≺ β ij, ≺ · · · ≺ β ij,K ≺ β ij for every j , where it is understood that K depends on i and j , and M depends on i (see the definitionsbefore Lemma 5.1).Define the open subsets U ij,k = ( α ij,k /n, β ij,k /n ) and U ij = ( α ij /n, β ij /n ). Let G : X n → Lsc([0 , , N )be the I-morphism obtained in Proposition 5.9, which is defined as G ( α, β ) = χ ( α/n,β/n ) .Since X k ( α ij,k , β ij,k ) ≺ ( α ij , β ij ) , we can apply G to obtain X k χ U ij,k ≪ χ U ij for every i, j .Set ǫ < / n , and consider V ij = R ǫ ( U ij ), the ǫ -retraction of U ij . Note that, for every α ≺ α ′ ≺ β ′ ≺ β in Ω n , we have ( α ′ /n − ǫ, β ′ /n + ǫ ) ⋐ ( α/n + ǫ, β/n − ǫ ) in [0 , X k χ U ij,k ≪ X k χ N ǫ ( U ij,k ) ≪ χ V ij for every i, j . LOCAL CHARACTERIZATION FOR THE CUNTZ SEMIGROUP OF AI-ALGEBRAS 29 Now set V ij,k = R ǫ ( U ij,k ) and let m ≥ n . Then, using Lemma 5.19 at the first, second, and laststeps, and the fact that ρ m is a Cu-morphism at the third, we get X k ρ m ( χ V ij,k ) ≪ X k F (( α ij,k , β ij,k )) ≪ X k ρ m ( χ N ǫ ( U ij,k ) ) ≪ ρ m ( χ V ij ) ≪ F (( α ij , β ij )) . Define f i = P j ≤ M,k ≤ K χ V ij,k and g i = P j ≤ M χ V ij . By adding on j in the previous inequality, onegets ρ m ( f i ) ≪ F ( q i ) ≪ ρ m ( g i ) ≪ F ( t i ) , which is condition (ii).By the comments above, we also have P k χ V ij,k ≪ P k χ U ij,k ≪ χ V ij , which implies condition (i) inthe same fashion.Condition (iii) follows by construction, since we have applied the same retraction to all the elements. (cid:3) Remark 5.21. Given P i ( α i , β i ) ≃ P j ( γ j , δ j ) in X n , we have X i χ ( α i /n + ǫ,β i /n − ǫ ) = X j χ ( γ j /n + ǫ,δ j /n − ǫ ) in Lsc([0 , , N ) for every positive ǫ < / n .Indeed, P i ( α i β i ) ≃ P j ( γ j , δ j ) implies P i χ ( α i /n,β i /n ) = P j χ ( γ j /n,δ j /n ) in Lsc([0 , , N ) by Proposition 5.9.Since R ǫ ( f ) is well defined for any f ∈ L n , we get X i χ ( α i /n + ǫ,β i /n − ǫ ) = R ǫ X i χ ( α i /n,β i /n ) ! = R ǫ X j ( γ j , δ j ) = X j χ ( γ j /n + ǫ,δ j /n − ǫ ) . Properties I and I . In this last section we introduce property I , which provides a characteriza-tion for when a Cu-semigroup is Cu-isomorphic to an inductive limit of the form lim n Lsc([0 , , N ). Wealso introduce property I, which leads to a characterization of when a Cu-semigroup is Cu-isomorphicto the Cuntz semigroup of an AI-algebra.5.3.1. Property I . Definition 5.22. We say that a Cu-semigroup S has the reduced I property if, given any sequence0 = z l ≪ z l − ≪ z l − ≪ · · · ≪ z ≪ z −∞ = p ≪ p and r ≪ t such that r = P i ∈ I z i and t = P j ∈ J z j with I, J multisets of {−∞ , , · · · , l } , there exists k ∈ N and an ( n, e )-chainable subsemigroup H ⊂ S with associated I-morphism F and p = ke such that:(i) There exists a sequence a l − ≺ · · · ≺ a ≺ k ( −∞ , ∞ ) in X n such that z i ≪ F ( a i − ) ≪ z i − forevery i .(ii) There exist a, b ∈ X n such that P i ∈ I a i ≃ a ≺ b ≃ P j ∈ J a j .Recall that given any l, M ∈ N , the subset C Ml of Lsc([0 , , N ) M is defined as C Ml = { χ s ( i/l, } i,s ∪{ s } , where 1 s and χ s ( i/l, denote the unit and the element χ ( i/l, in the s -th summand respectively.Also recall that we denote by B Ml the additive span of the elements in C Ml . Proposition 5.23. Let ϕ : Lsc([0 , , N ) → S be a Cu-morphism, and assume that S has the reduced I property. Then, for every l ∈ N and x, x ′ , y ∈ B l with x ≪ x ′ and ϕ ( x ′ ) ≪ ϕ ( y ) , there exist Cu -morphisms θ : Lsc([0 , , N ) → Lsc([0 , , N ) and φ : Lsc([0 , , N ) → S satisfying the conditions inTheorem 4.17. That is to say, we have(i) ϕ ( χ (( i +1) /l, ) ≪ φθ ( χ ( i/l, ) and φθ ( χ (( i +1) /l, ) ≪ ϕ ( χ ( i/l, ) for every i < l .(ii) θ ( x ) ≪ θ ( y ) .(iii) ϕ (1) = φθ (1) .Proof. Given a Cu-morphism ϕ : Lsc([0 , , N ) → S and a fixed l ∈ N , consider the elements x i = ϕ ( χ ( i/l, ) and set x −∞ = ϕ (1). Also, set r = ϕ ( x ′ ) and t = ϕ ( y ).Let I, J be the multisets of {−∞ , , · · · , l } such that x ′ = P i ∈ I χ ( i/l, and y = P j ∈ J χ ( j/l, , whereit is understood that χ ( −∞ /l, = 1. Thus, we can write r, t as r = P i ∈ I x i and t = P j ∈ J x j . Let z i = ϕ ( χ ( i/ l, ) for every i ≤ l , and let p = x −∞ = ϕ (1), so that we have0 = x l = z l ≪ z l − ≪ · · · ≪ z = x ≪ x −∞ = p. Since z i = x i for every i ≤ l , we also get r = P i ∈ I z i and t = P j ∈ J z j .By the reduced I property applied to { z i } ≤ i ≤ l and r ≪ t , there exist k ∈ N , a compact element e such that p = ke , an ( n, e )-chainable subsemigroup H and an associated I-morphism F with asequence a l ≺ · · · ≺ a in X n such that z i ≪ F ( a i − ) ≪ z i − . By condition (ii) in Definition 5.22, there also exist a, b ∈ X n with P i ∈ I a i ≃ a ≺ b ≃ P j ∈ J a j .Note, in particular, that x i = z i ≪ F ( a i − ) ≪ F ( a i − ) ≪ z i − = x i − for every i .Consider the finite family of ≺ -relations { a i ≺ a i − } i ∪ { a ≺ k ( −∞ , ∞ ) } ∪ { a ≺ b } . By Lemma 5.20, if G : X n → Lsc([0 , , N ) is the I-morphism defined in Proposition 5.9, there existsa large enough m ∈ N such that the functions f i = R ǫ ( G ( a i )), f = R ǫ ( G ( a )) and g = R ǫ ( G ( b )) satisfy f i ≪ f i − ≪ k ,ρ m ( f i ) ≪ F ( a i ) ≪ ρ m ( f i − ) ≪ F ( a i − ) ,f ≪ g,ρ m ( f ) ≪ F ( a ) ≪ ρ m ( g ) ≪ F ( b )for every i , where recall that ρ m is the Cu-morphism defined in Definition 5.18.Furthermore, since P i ∈ I a i ≃ a , we get P G ( a i ) = G ( a ). By the argument in Remark 5.21 weget P R ǫ ( G ( a i )) = R ǫ ( G ( a )), that is, P i ∈ I f i = f . Similarly, we also have P j ∈ J f j = g .Thus, for every i ≤ n , one gets x i ≪ F ( a i − ) ≪ ρ m ( f i − ) ≪ F ( a i − ) ≪ x i − . Define the Cu-morphism θ : Lsc([0 , , N ) → Lsc([0 , , N ) as θ ( χ ( i/l, ) = f i and θ (1) = k 1. Thiscan be done by Theorem 3.1 because { f i } i is a decreasing sequence bounded by k θ ( x ) ≪ θ ( x ′ ) = X i ∈ I f i = f ≪ g = X j ∈ J f j = θ ( y ) , and thus condition (ii) of the proposition is satisfied.Set φ = ρ m , so that we get φθ (1) = kρ m (1) = ke = p = ϕ (1). By construction, one also has ϕ ( χ ( i/l, ) = x i ≪ F ( a i − ) ≪ ρ m ( f i − )= φ ( θ ( χ (( i − /l, )) ≪ F ( a i − ) ≪ x i − = ϕ ( χ (( i − /l, ) , which implies condition (i) in the proposition. (cid:3) We now strengthen the previous property. Definition 5.24. We say that S satisfies property I if, given any finite number of decreasing sequencesof the form 0 = z l,s ≪ z l − ,s ≪ · · · ≪ z ,s ≪ z −∞ ,s = p s ≪ p s , s ≤ M , and any pair r ≪ t such that r = P s ≤ M P i ∈ I z i,s and t = P s ≤ M P j ∈ J z j,s with I, J (possibly empty) multisets of {−∞ , , · · · , l } ,there exists an ( n, e )-chainable subsemigroup H with associated I-morphism F :(i) For each s there exists k s ∈ N such that p s = k s e .(ii) There exist sequences a l − ,s ≺ · · · ≺ a ,s ≺ k s ( −∞ , ∞ ) in X n such that z i,s ≪ F ( a i − ,s ) ≪ z i − ,s for every i, s .(iii) There exist a, b ∈ X n such that P s ≤ M,i ∈ I a i,s ≃ a ≺ b ≃ P s ≤ M,j ∈ J a j,s . Lemma 5.25. Property I goes through inductive limits. LOCAL CHARACTERIZATION FOR THE CUNTZ SEMIGROUP OF AI-ALGEBRAS 31 Proof. Let S = lim m S m be an inductive limit of Cu-semigroups S m satisfying property I , and let z i,s , p s , r, t be as in Definition 5.24. Given f ∈ S m , we will denote by [ f ] the image of f through thelimit morphism from S m to S .Then, there exists a large enough m and elements g i,s , f i,s , e s ∈ S m such that: g i,s ≪ f i,s ≪ g i − ,s ≪ f i − ,s ≪ e s ≪ e s ,z i,s ≪ [ g i − ,s ] ≪ [ f i − ,s ] ≪ z i − ,s , p s = [ e s ] , X s X i ∈ I f i,s ≪ X s X j ∈ J f j,s , X s X i ∈ I [ f i,s ] ≪ r ≪ X s X j ∈ J [ f j,s ] ≪ t. Since S m satisfies property I , we can find an ( n, e )-chainable subset H with e s = k s e such that:(i) There exist sequences a l − ,s ≺ b l − ,s ≺ a l − ,s ≺ · · · ≺ a ,s ≺ k s ( −∞ , ∞ ) in X n such that g i,s ≪ F ( a i,s ) ≪ f i,s ≪ F ( b i,s ) ≪ g i − ,s in S m for every i, s .(ii) There exist a, b ∈ X n such that P s ≤ M,i ∈ I a i,s ≃ a ≺ b ≃ P s ≤ M,j ∈ J a j,s .Therefore, one gets f i,s ≪ [ F ( a i − ,s )] ≪ f i − ,s for every i, s and p s = k s [ e ] for every s .By Remark 5.5, [ H ] is an ( n, [ e ])-chainable subset of S , and it satisfies the required properties bythe previous considerations. (cid:3) Example 5.26. Let S be an inductive limit of the form S = lim m S m , with S m = N for every m .Then S satisfies property I .To prove this, it is enough to show by Lemma 5.25 that N satisfies property I .Thus, let z i,s , p s , r, t be as in Definition 5.24, and note that there exist positive integers k i,s ≤ k i − ,s such that z i,s = k i,s i, s . There also exist integers k s such that p s = k s H = N , which is the additive span of 1. Then, it follows from Example 5.7 that H is ( n, n . Take n = 1 and let F : X → H be as in Example 5.7, where note that condition(i) in Definition 5.24 is satisfied by construction.Consider the elements a i,s = k i − ,s ( −∞ , ∞ ), whose image through F is k i − ,s 1. This impliescondition (ii) in Definition 5.24.Also, since P s ≤ M,i ∈ I k i,s ≤ P s ≤ M,j ∈ J k j,s , we clearly have X s ≤ M,i ∈ I k i,s ( −∞ , ∞ ) ≺ X s ≤ M,j ∈ J k j,s ( −∞ , ∞ ) . Letting a = P s ≤ M,i ∈ I k i,s ( −∞ , ∞ ) and b = P s ≤ M,j ∈ J k j,s ( −∞ , ∞ ), condition (iii) also follows. Example 5.27. Every Cu-semigroup S of the form S = lim m S m , with S m = Lsc([0 , , N ) for every m , satisfies property I .As in Example 5.26, by Lemma 5.25 we only need to prove that Lsc([0 , , N ) satisfies property I .Thus, take sequences0 = z l,s ≪ z l − ,s ≪ · · · ≪ z ,s ≪ z −∞ ,s = p s ≪ p s , s ≤ M and a pair r ≪ t with r = P s ≤ M,i ∈ I z i,s and t = P s ≤ M,j ∈ J z j,s in Lsc([0 , , N ).Let L m be the subsets of Lsc([0 , , N ) defined above Notation 5.8. Then, since ∪ m L m is dense inLsc([0 , , N ), there exist decreasing sequences f l − ,s ≪ · · · ≪ f ,s in L m such that z i,s ≪ f i − ,s ≪ z i − ,s and P s ≤ M,i ∈ I f i,s ≪ P s ≤ M,j ∈ J f j,s . Also take k s ∈ N such that k s p s .Take a positive rational ǫ < / n small enough so that z i,s ≪ R ǫ ( f i − ,s ) ≪ z i − ,s and X s ≤ M,i ∈ I R ǫ ( f i,s ) = R ǫ X s ≤ M,i ∈ I f i,s ≪ R ǫ X s ≤ M,j ∈ J f j,s = X s ≤ M,j ∈ J R ǫ ( f j,s ) , where the equalities follow from Remark 5.13.Define g i,s = R ǫ ( f i,s ) and note that, by Lemma 5.15, there exists some n ∈ N such that g i,s , X s ≤ M,i ∈ I g i,s , X s ≤ M,j ∈ J g j,s ∈ L n . Recall from Proposition 5.9 that L n is ( n, F : X n → L n be the I-morphismobtained in its proof, that is, the I-morphism induced by F (( α, β )) = χ ( α/n,β/n ) .Set G I = P s ≤ M,i ∈ I g i,s and G J = P s ≤ M,j ∈ J g j,s , so that G I ≪ G J . By Lemma 5.16, the elements q g i,s , q G I , q G J ∈ X n satisfy q g i,s ≺ q g i − ,s , and q G I ≺ q G J . Set a i,s = q g i,s , a = q G I and b = q G J . By Lemma 5.11, F ( a i,s ) = g i,s , F ( a ) = G I and F ( b ) = G J .Using this, we now have F ( a ) = P s ≤ M,i ∈ I F ( a i,s ), F ( b ) = P s ≤ M,j ∈ J F ( a j,s ) and z i,s ≪ g i − ,s = F ( a i − ,s ) ≪ z i − ,s for every i, s . Since F ( P s,i a i,s ) = G I and F ( P s,j a j,s ) = G J , it follows fromLemma 5.11 that P s,i a i,s ≃ a and, by Lemma 5.16, that P s,j a j,s ≺ b . Therefore, we have X s ≤ M,i ∈ I a i,s ≃ a ≺ b ≃ X s ≤ M,j ∈ J a j,s , which finishes the proof. Example 5.28. The Cu-semigroup Lsc([0 , , N ) ⊕ Lsc([0 , , N ) does not satisfy property I . To seethis, simply consider the sequences (0 , ≪ (1 , 0) and (0 , ≪ (0 , 1) and set r = t = (0 , , , N ) ⊕ Lsc([0 , , N ) were to satisfy property I , in particular it would have a compactelement e such that (1 , 0) = k e and (0 , 1) = k e for some k , k ∈ N . Clearly, this is not possible.Note that the same argument works for N ⊕ N . Theorem 5.29. Let S be a Cu -semigroup. Then, S is Cu -isomorphic to an inductive limit of theform lim Lsc([0 , , N ) if and only if S is countably based, compactly bounded (see Definition 4.14) andit satisfies (O5), (O6), weak cancellation, and property I .Proof. The proof of one implication is analogous to the proof of Proposition 5.23, so we omit it.The other implication is the previous example, together with the well known fact that (O5), (O6),weak cancellation and being countably based go through inductive limits (see, for example, [APT18,Chapter 4]). Clearly, being compactly bounded also goes through inductive limits, which finishes theproof. (cid:3) Property I . Definition 5.30. We say that a Cu-semigroup S satisfies property I if, given any finite number ofdecreasing sequences of the form 0 = z l,s ≪ z l − ,s ≪ · · · ≪ z ,s ≪ z −∞ ,s = p s ≪ p s , 1 ≤ s ≤ M , andany pair r ≪ t with r = P s ≤ M,i ∈ I z i,s and t = P s ≤ M,j ∈ J z j,s , there exist finitely many subsemigroups H k ⊂ S such that each H k is ( n k , e k )-chainable with associated I-morphism F k and:(i) Each compact p s can be written as a linear combination of { e k } k , p s = P k m k,s e k .(ii) There exist sequences a ( k ) l − ,s ≺ · · · ≺ a ( k )0 ,s ≺ m k,s ( −∞ , ∞ ) in X n k such that z i,s ≪ X k F k ( a ( k ) i − ,s ) ≪ z i − ,s for every i, s .(iii) For each k , there exist a k , b k ∈ X n k such that X s ≤ M,i ∈ I a ( k ) i,s ≃ a k ≺ b k ≃ X s ≤ M,j ∈ J a ( k ) j,s . Remark 5.31. If a Cu-semigroup satisfies property I , then it clearly also satisfies property I. LOCAL CHARACTERIZATION FOR THE CUNTZ SEMIGROUP OF AI-ALGEBRAS 33 Lemma 5.32. Property I goes through inductive limits.Proof. We follow the proof of Lemma 5.25:Let S = lim S m with S m Cu-semigroups satisfying property I.As in Definition 5.30, consider decreasing sequences of the form0 = z l,s ≪ z l − ,s ≪ · · · ≪ z ,s ≪ z −∞ ,s = p s ≪ p s , ≤ s ≤ M, and a pair of elements r ≪ t with r = P s ≤ M,i ∈ I z i,s and t = P s ≤ M,j ∈ J z j,s .Then, for a large enough m , there exist elements f i,s , g i,s , q s ∈ S m such that g i,s ≪ f i,s ≪ g i − ,s ≪ f i − ,s ≪ q s ≪ q s ,z i,s ≪ [ g i − ,s ] ≪ [ f i − ,s ] ≪ z i − ,s , p s = [ q s ] , X s X i ∈ I f i,s ≪ X s X j ∈ J f j,s , X s X i ∈ I [ f i,s ] ≪ r ≪ X s X j ∈ J [ f j,s ] ≪ t, where as in Lemma 5.25 [ f ] denotes the image of f through the limit morphism from S m to S .Since S m satisfies property I, we can find a ( n k , e k )-chainable subsets H k such that:(i) Each compact q s can be written as a linear combination of { e k } k , q s = P k m k,s e k .(ii) There exist sequences a ( k ) l − ,s ≺ b ( k ) l − ,s ≺ a ( k ) l − ,s ≺ · · · ≺ a ( k )0 ,s ≺ m k,s ( −∞ , ∞ ) in X n k such that g i,s ≪ X k F k ( a ( k ) i,s ) ≪ f i,s ≪ X k F k ( b ( k ) i,s ) ≪ g i − ,s in S m for every i, s .(iii) There exist a k , b k ∈ X n k such that P s ≤ M,i ∈ I a ( k ) i,s ≃ a ≺ b ≃ P s ≤ M,j ∈ J a ( k ) j,s .It is now easy to check that the semigroups [ H k ], the elements a ( k ) i,s and the compacts [ e k ] satisfythe desired properties for our original pair of elements and sequences.Recall that [ H k ] is an ( n k , [ e k ])-chainable subsemigroup of S by Remark 5.5. (cid:3) The following lemma exemplifies one of the differences between property I and property I, since,from Example 5.28, we know that property I is not preserved under direct sums. Lemma 5.33. If S, T are Cu -semigroups satisfying property I , their direct sum S ⊕ T also satisfiesproperty I .Proof. Given a finite number of decreasing sequences in S ⊕ T of the form0 = z l,s ≪ z l − ,s ≪ · · · ≪ z ,s ≪ z −∞ ,s = p s ≪ p s , ≤ s ≤ M, and a pair r ≪ t with r = P s ≤ M,i ∈ I z i,s and t = P s ≤ M,j ∈ J z j,s , write z i,s = ( v i,s , w i,s ) with v i,s ∈ S and w i,s ∈ T . Also set q s ∈ S c and t s ∈ T c such that p s = ( q s , t s ).Thus, one gets the following decreasing sequences0 = v l,s ≪ v l − ,s ≪ · · · ≪ v ,s ≪ v −∞ ,s = q s ≪ q s , ≤ s ≤ M, w l,s ≪ w l − ,s ≪ · · · ≪ w ,s ≪ w −∞ ,s = t s ≪ t s , ≤ s ≤ M, in S and T respectively.Using that S satisfies property I , we obtain ( n k , e k )-chainable subsemigroups H k of S and elements a ( k ) i,s ∈ X n k satisfying the conditions in Definition 5.30 for the above sequences in S and the pair P s ≤ M,i ∈ I v i,s , P s ≤ M,j ∈ J v j,s .Similarly, we find ( m l , f l )-chainable subsemigroups U l of T and elements b ( l ) i,s ∈ X m l satisfying theconditions in Definition 5.30 for the sequences in T and the pair P s ≤ M,i ∈ I w i,s , P s ≤ M,j ∈ J w j,s .It is easy to check that H k ⊕ ⊕ T l are ( n k , ( e k , m l , (0 , f l )) chainable subsemigroupsof S ⊕ T respectively. Moreover, it is also clear that these subsemigroups with the elements a ( k ) i,s ∈ X n k and b ( l ) i,s ∈ X m l satisfy the conditions of property I for our original sequences and pairs of elements. (cid:3) Example 5.34. (1) Let S be the Cuntz semigroup of an AF-algebra. Then S satisfies property I.(2) The Cuntz semigroup of an AI-algebra satisfies property I.Indeed, by combining Lemma 5.32 and Lemma 5.33 it suffices to show that N and Lsc([0 , , N )satisfy property I. This follows from Examples 5.26 and 5.27, as both semigroups satisfy property I . Theorem 5.35. Let S be a Cu -semigroup. Then, S is Cu -isomorphic to the Cuntz semigroup of anAI-algebra if and only if S is countably based, compactly bounded and it satisfies (O5), (O6), weakcancellation, and property I .Proof. If S is Cu-isomorphic to the Cuntz semigroup of an AI-algebra, then it is well known that S iscountably based, compactly bounded and satisfies (O5), (O6), and weak cancellation.It follows from Example 5.34 that S satisfies property I.Conversely, assume that S is countably based, compactly bounded and satisfies (O5), (O6), weakcancellation, and property I. We will use Theorem 4.17 to show that S is the Cuntz semigroup of anAI-algebra.More explicitly, given a morphism ϕ : Lsc([0 , , N ) M → S , an integer l ∈ N and a triple x, x ′ , y ∈ B Ml such that x ≪ x ′ with ϕ ( x ′ ) ≪ ϕ ( y ), we will construct Cu-morphisms θ : Lsc([0 , , N ) M → Lsc([0 , , N ) N , and φ : Lsc([0 , , N ) N → S such that the following properties are satisfied:(i) ϕ ( χ s (( i +1) /l, ) ≪ φθ ( χ s ( i/l, ) and φθ ( χ s (( i +1) /l, ) ≪ ϕ ( χ s ( i/l, ) for every χ s ( i/l, ∈ C Ml .(ii) θ ( x ) ≪ θ ( y ).(iii) ϕ (1 s ) = φθ (1 s ) for every s ≤ M .We generalize the proof of Proposition 5.23:For every i ≤ n and s ≤ M , consider the elements x i,s = ϕ ( χ s ( i/l, ) and set x −∞ ,s = ϕ (1 s ). Also,set r = ϕ ( x ′ ) and t = ϕ ( y ).Let I, J be the multisets of {−∞ , , · · · , l } such that x ′ = X s ≤ M,i ∈ I χ s ( i/l, , and y = X s ≤ M,j ∈ J χ s ( j/l, , where it is understood that χ s ( −∞ /l, = 1 s . Thus, we can write r, t as r = P s ≤ M,i ∈ I x i,s and t = P j ≤ M,j ∈ J x j,s .Let z i,s = ϕ ( χ s ( i/ l, ) for every i ≤ l , and let p s = x −∞ ,s = ϕ (1 s ), so that we have0 = x l,s = z l,s ≪ z l − ,s ≪ · · · ≪ z ,s = x ,s ≪ x −∞ ,s = p s . Since z i,s = x i,s for every i ≤ l , we also get r = P s ≤ M,i ∈ I z i,s and t = P s ≤ M,j ∈ J z j,s .Apply property I to ( z i,s ) i,s and r ≪ t to obtain ( n k , e k )-chainable subsemigroup H k and sequences a ( k )2 l,s ≺ · · · ≺ a ( k )0 ,s in X n k satisfying conditions (i)-(iii) in Definition 5.30. In particular, note that x i,s = z i,s ≪ X k F k ( a ( k )2 i − ,s ) ≪ z i − ,s ≪ X k F k ( a ( k )2( i − ,s ) ≪ z i − ,s = x i − ,s . Now fix k < ∞ and consider the finite family of ≺ -relations { a ( k ) i,s ≺ a ( k ) i − ,s } s,i ∪ { a ( k )0 ,s ≺ m k,s ( −∞ , ∞ ) } s ∪ { a k ≺ b k } given by property I. LOCAL CHARACTERIZATION FOR THE CUNTZ SEMIGROUP OF AI-ALGEBRAS 35 By using the same argument as in Proposition 5.23, we obtain an integer N k and elements f ( k ) i,s , f k and g k in Lsc([0 , , N ) such that f ( k ) i,s ≪ f ( k ) i − ,s ≪ m k,s ,ρ N k ( f ( k ) i,s ) ≪ F k ( a ( k ) i,s ) ≪ ρ N k ( f ( k ) i − ,s ) ≪ F k ( a ( k ) i − ,s ) ,f k ≪ g k ) ,ρ N k ( f k ) ≪ F k ( a k ) ≪ ρ N k ( g k ) ≪ F k ( b k ) , where the Cu-morphism ρ N k : Lsc([0 , , N ) → S is as defined in Definition 5.18 for H k . Recall, inparticular, that ρ N k (1) = e k .Combining Lemma 5.20 and Remark 5.21, we have X ( i,s ) ∈ I f ( k )2 i,s = f k ≪ g k = X ( j,s ) ∈ J f ( k )2 j,s . Let φ = ⊕ k ρ N k : Lsc([0 , , N ) N → S , and define f i,s = P k f ( k ) i,s k and m s = P k m k,s k , where given h ∈ Lsc([0 , , N ) we denote by h k the function in Lsc([0 , , N ) N whose k -th component is h and therest are null.Note that f i,s ≪ f i − ,s ≪ m s for each i and s . Further, we have by construction that φ (1 k ) = e k .By (ii) of property I, we get x i,s ≪ φ ( f i − ,s ) ≪ x i − ,s for every i, s .For each s ≤ M , define θ s : Lsc([0 , , N ) → Lsc([0 , , N ) N as the Cu-morphism that sends χ ( i/l, to f i,s and 1 to m s . As mentioned, θ s exists by Theorem 3.1, as { f i,s } i is a ≪ -decreasing sequencebounded by 1 m s .Let θ = ⊕ s ≤ M θ s . We have θ ( x ) ≪ θ ( x ′ ) = X ( i,s ) ∈ I f (2 i,s ) = X k f k ≪ X k g k = X ( j,s ) ∈ I f (2 j,s ) = θ ( y ) , which implies condition (ii) in Theorem 4.17.Also, for every fixed s , we get ϕ (1 s ) = p s = X k m k,s e k = X k m k,s φ (1 k ) = φ ( m s ) = φ ( θ (1)) , which is condition (iii) in Theorem 4.17.To see condition (i) simply note that, since x i,s ≪ φ ( f i − ,s ) ≪ x i − ,s , we have by definition that ϕ ( χ s ( i/l, ) = x i,s ≪ φ ( θ ( χ ( i/l, )) ≪ x i − ,s = ϕ ( χ s (( i − /l, )as required. 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Math. (2012), 259–342. E. Vilalta, Departament de Matem`atiques, Universitat Aut`onoma de Barcelona, 08193 Bellaterra,Barcelona, Spain Email address ::