aa r X i v : . [ m a t h . OA ] J a n COVERING DIMENSION OF CUNTZ SEMIGROUPS
HANNES THIEL AND EDUARD VILALTA
Abstract.
We introduce a notion of covering dimension for Cuntz semigroupsof C ∗ -algebras. This dimension is always bounded by the nuclear dimension ofthe C ∗ -algebra, and for subhomogeneous C ∗ -algebras both dimensions agree.Cuntz semigroups of Z -stable C ∗ -algebras have dimension at most one.Further, the Cuntz semigroup of a simple, Z -stable C ∗ -algebra is zero-dimen-sional if and only if the C ∗ -algebra has real rank zero or is stably projectionless. Introduction
The Cuntz semigroup of a C ∗ -algebra is a powerful invariant in the structureand classification theory of C ∗ -algebras. We define a notion of covering dimensionfor Cuntz semigroups, thus introducing a second-level invariant for C ∗ -algebras;see Definition 3.1. More generally, we define covering dimension for abstract Cuntzsemigroups, usually called Cu-semigroups, as introduced in [CEI08] and extensivelystudied in [APT18, APT19, APT20, APRT18, APRT19].Our definition really captures a notion of covering dimension: for every compact,metrizable space X , the Cu-semigroup Lsc( X, N ) of lower-semicontinuous functions X → N = { , , , . . . , ∞} has dimension agreeing with the covering dimension of X ;see Example 3.4. More interestingly, we show that a similar result holds for Cuntzsemigroups of commutative C ∗ -algebras: Proposition A (4.3) . Let X be a compact, Hausdorff space. Then dim(Cu( C ( X ))) = dim( X ) . We prove the expected permanence properties: The covering dimension does notincrease when passing to ideals or quotients of a Cu-semigroup (Proposition 3.5);the covering dimension of a direct sum of Cu-semigroups is the maximum of the cov-ering dimensions of the summands (Proposition 3.5); and if S = lim −→ λ S λ is an induc-tive limit of Cu-semigroups, then dim(lim −→ λ S λ ) ≤ lim inf λ dim( S λ ) (Proposition 3.9).In Section 4, we study the connection between the dimension of the Cuntz semi-group of a C ∗ -algebra and the nuclear dimension [WZ10] of the C ∗ -algebra. Theorem B (4.1, 4.10) . Every C ∗ -algebra A satisfies dim(Cu( A )) ≤ dim nuc ( A ) .If A is subhomogeneous, then dim(Cu( A )) = dim nuc ( A ) . We note that dim(Cu( A )) can be strictly smaller than dim nuc ( A ). For example,the irrational rotation algebra A θ satisfies dim(Cu( A θ )) = 0 while dim nuc ( A θ ) = 1;see Example 4.11. Date : January 13, 2021.2010
Mathematics Subject Classification.
Primary 46L05, 46L85; Secondary 54F45, 55M10.
Key words and phrases. C ∗ -algebras, Cuntz semigroups, covering dimension.The first named author was partially supported by the Deutsche Forschungsgemeinschaft(DFG, German Research Foundation) under Germany’s Excellence Strategy EXC 2044-390685587(Mathematics M¨unster: Dynamics-Geometry-Structure) and by the ERC Consolidator Grant No.681207. The second named author was partially supported by MINECO (grant No. PRE2018-083419 and No. MTM2017-83487-P), and by the Comissionat per Universitats i Recerca de laGeneralitat de Catalunya (grant No. 2017SGR01725). The dimension of the Cuntz semigroup of a C ∗ -algebra A can also be computedin many situations of interest beyond the subhomogeneous case:(1) If A has real rank zero, then dim(Cu( A )) = 0; see Proposition 5.4.(2) If A is unital and of stable rank one, then dim(Cu( A )) = 0 if and only if A has real rank zero; see Corollary 5.8.(3) If A is W -stable , then dim(Cu( A )) = 0. If A is Z -stable , then we havedim(Cu( A )) ≤
1; see Proposition 3.22.(4) If A is purely infinite (not necessarily simple), then dim(Cu( A )) = 0; seeProposition 3.21.Our results allow us to compute the dimension of the Cuntz semigroup of manysimple C ∗ -algebras. In particular, by Corollary 5.9, if A is a separable, simple, Z -stable C ∗ -algebra, thendim(Cu( A )) = ( , if A has real rank zero or if A is stably projectionless1 , otherwise.This should be compared to the computation of the nuclear dimension of aseparable, simple C ∗ -algebra A as accomplished in [CET +
19, CE19]:dim nuc ( A ) = , if A is an AF-algebra1 , if A is nuclear, Z -stable, but not an AF-algebra ∞ , of A is nuclear and not Z -stable, or non-nuclearIt will be interesting to tackle the following problem: Problem C.
Compute the dimension of the Cuntz semigroups of simple C ∗ -alge-bras. In particular, what dimensions can occur (beyond zero and one)?In the last two sections, we thoroughly investigate the class of countably based,simple, weakly cancellative Cu-semigroups S satisfying (O5) and (O6) (which in-cludes the Cuntz semigroups of separable, simple C ∗ -algebras of stable rank one).We show that S is zero-dimensional if and only if S is algebraic (that is, the compactelements are sup-dense) or soft (that is, S contains no nonzero compact elements);see Lemma 7.1.To describe the structure of S in the second case, we introduce the class of ele-ments with thin boundary (see Definition 6.3), which turn out to play a similar roleto that of the compact elements in the algebraic case. We show that an element x has thin boundary if and only if it is complementable in the sense that for every y satisfying x ≪ y there exists z such that x + z = y ; see Theorem 6.12. Further, theelements with thin boundary form a cancellative monoid; see Theorem 6.13. Theorem D (7.8) . Let S be a countably based, simple, soft, weakly cancellative Cu -semigroup satisfying (O5) and (O6). Then S is zero-dimensional if and only ifthe elements with thin boundary are sup-dense. We finish Section 7 by briefly studying the relation between zero-dimensionality,almost divisibility and the Riesz interpolation property; see Proposition 7.13.2.
Preliminaries
Let a, b be two positive elements in a C ∗ -algebra A . Recall that a is said to be Cuntz subequivalent to b , in symbols a - b , if there exists a sequence ( r n ) n in A such that a = lim n r n br ∗ n . One writes a ∼ b if a - b and b - a , and denotes theclass of a ∈ A by [ a ]. A is W -stable if A ∼ = A ⊗ W , for the Jacelon-Razac algebra W A is Z -stable if A ∼ = A ⊗ Z , for the Jiang-Su algebra Z OVERING DIMENSION OF CUNTZ SEMIGROUPS 3
The Cuntz semigroup of A , denoted by Cu( A ), is defined as the quotient of( A ⊗ K ) + by the equivalence relation ∼ . Endowed with the addition induced by[ a ] + [ b ] = [( a b )] and the order induced by - , the Cuntz semigroup Cu( A ) becomesa positively ordered monoid. Given a pair of elements x, y in a partially ordered set, we say that x is way-below y , in symbols x ≪ y , if for any increasing sequence ( y n ) n for which thesupremum exists and is greater than y one can find n such that x ≤ y n .It was shown in [CEI08] that the Cuntz semigroup of any C ∗ -algebra satisfiesthe following properties:(O1) Every increasing sequence has a supremum.(O2) Every element can be written as the supremum of an ≪ -increasing sequence.(O3) Given x ′ ≪ x and y ′ ≪ y , we have x ′ + y ′ ≪ x + y .(O4) Given 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 ).In a more abstract setting, any positively ordered monoid satisfying (O1)-(O4)is called a Cu -semigroup .A map between two Cu-semigroups is called a generalized Cu -morphism if it isa positively ordered monoid homomorphism that preserves suprema of increasingsequences. We say that a generalized Cu-morphism is a Cu -morphism if it alsopreserves the way-below relation. Every *-homomorphism A → B between C ∗ -al-gebras induces a Cu-morphism Cu( A ) → Cu( B ); see [CEI08, Theorem 1].We denote by Cu the category whose objects are Cu-semigroups and whosemorphisms are Cu-morphisms.The reader is referred to [CEI08] and [APT18] for a further detailed exposition. In addition to (O1)-(O4), it was proved in [APT18, Proposition 4.6] and[Rob13] that the Cuntz semigroup of a C ∗ -algebra always satisfies the followingadditional properties:(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 v ≤ x, y and w ≤ x, z such that x ′ ≤ v + w .Axiom (O5) is often used with y = 0. In this case, it states that, given x ′ ≪ x ≤ z , there exists c such that x ′ + c ≤ z ≤ x + c .Recall that a Cu-semigroup is said to be weakly cancellative if x ≪ y whenever x + z ≪ y + z for some element z . Stable rank one C ∗ -algebras have weaklycancellative Cuntz semigroups by [RW10, Theorem 4.3]. A subset D ⊆ S in a Cu-semigroup S is said to be sup-dense if whenever x ′ , x ∈ S satisfy x ′ ≪ x , there exists y ∈ D with x ′ ≤ y ≪ x . Equivalently, everyelement in S is the supremum of an increasing sequence of elements in D .We say that a Cu-semigroup is countably based if it contains a countable sup-dense subset. Cuntz semigroups of separable C*-algebras are countably based (see,for example, [APS11]).3. Dimension of Cuntz semigroups
In this section we introduce a notion of covering dimension for Cu-semigroupsand study some of its main permanence properties while providing a variety ofexamples; see Proposition 3.5, Proposition 3.9 and Proposition 3.15.In Proposition 3.17 we investigate the relation between the dimension of a simpleCu-semigroup and its soft part, while in Proposition 3.20 we study how the dimen-sion behaves in the presence of certain R -multiplications. This result is then applied HANNES THIEL AND EDUARD VILALTA to the Cuntz semigroups of purely infinite, W -stable and Z -stable C ∗ -algebras; seeProposition 3.21 and Proposition 3.22. Definition 3.1.
Let S be a Cu-semigroup. Given n ∈ N , we write dim( S ) ≤ n if,whenever x ′ ≪ x ≪ y + . . . + y r in S , then there exist z j,k ∈ S for j = 1 , . . . , r and k = 0 , . . . , n such that:(i) z j,k ≪ y j for each j and k ;(ii) x ′ ≪ P j,k z j,k ;(iii) P rj =1 z j,k ≪ x for each k = 0 , . . . , n .We set dim( S ) = ∞ if there exists no n ∈ N with dim( S ) ≤ n . Otherwise, we letdim( S ) be the smallest n ∈ N such that dim( S ) ≤ n . We call dim( S ) the (covering)dimension of S . Remark 3.2.
Recall that the (covering) dimension dim( X ) of a topological space X is defined as the smallest n ∈ N such that every finite open cover of X admitsa finite open refinement V such that at most n + 1 distinct elements in V havenonempty intersection; see for example [Pea75, Definition 3.1.1, p.111].By [KW04, Proposition 1.5], a normal space X satisfies dim( X ) ≤ n if and onlyif every finite open cover of X admits a finite open refinement V that is ( n + 1) -colorable , that is, there is a decomposition V = V ⊔ . . . ⊔ V n such that the sets in V j are pairwise disjoint for j = 0 , . . . , n . (The sets in V j have color j , and sets ofthe same color are disjoint.)Definition 3.1 is modeled after the above characterization of covering dimensionin terms of colorable refinements. We interpret the expression ‘ x ≪ y + . . . + y r ’ assaying that x is ‘covered’ by { y , . . . , y r } . Then, condition (i) from Definition 3.1means that { z j,k } is a ‘refinement’ of { y , . . . , y r } ; condition (ii) means that { z j,k } is a cover of x ′ (which is an approximation of x ); and condition (iii) means that { z j,k } is ( n + 1)-colorable.In Definition 3.1, some of the ≪ -relations may be changed for ≤ . Lemma 3.3.
Let S be a Cu -semigroup and n ∈ N . Then we have dim( S ) ≤ n if and only if, whenever x ′ ≪ x ≪ y + . . . + y r in S , there exist z j,k ∈ S for j = 1 , . . . , r and k = 0 , . . . , n such that:(1) z j,k ≤ y j for each j and k ;(2) x ′ ≤ P j,k z j,k ;(3) P rj =1 z j,k ≤ x for each k = 0 , . . . , n .Proof. The forward implication is clear. To show the converse, let x ′ ≪ x ≪ y + . . . + y r in S . Choose s ′ , s, y ′ , . . . , y ′ r ∈ S such that x ′ ≪ s ′ ≪ s ≪ x ≪ y ′ + . . . + y ′ r , y ′ ≪ y , . . . , and y ′ r ≪ y r . Applying the assumption, we obtain elements z j,k for j = 1 , . . . , r and k =0 , . . . , n satisfying properties (1)-(3) for s ′ ≪ s ≪ y ′ + . . . + y ′ r . Then the sameelements satisfy (i)-(iii) in Definition 3.1 for x ′ ≪ x ≪ y + . . . + y r , thus verifyingdim( S ) ≤ n . (cid:3) Example 3.4.
Let X be a compact, metrizable space. We use Lsc( X, N ) to denotethe set of functions f : X → N that are lower-semicontinuous, that is, for each n ∈ N the set f − ( { n, n + 1 , . . . , ∞} ) ⊆ X is open. We equip Lsc( X, N ) with pointwiseaddition and order. Then Lsc( X, N ) is Cu-semigroup. We havedim(Lsc( X, N )) = dim( X ) . We will omit the elaborate verification of the inequality ‘ ≤ ’ since it follows fromthe computation of dim( C ( X )); see Corollary 4.4. OVERING DIMENSION OF CUNTZ SEMIGROUPS 5
Let us prove the inequality ‘ ≥ ’. Set n := dim(Lsc( X, N )), which we may assumeto be finite. To verify that dim( X ) ≤ n , let U = { U , . . . , U r } be a finite open coverof X . We need to find a ( n + 1)-colourable, finite, open refinement of U .We use χ U to denote the characteristic function of a subset U ⊆ X . Then χ X ≪ χ X ≪ χ U + . . . + χ U r . Applying that dim(Lsc( X, N )) ≤ n , we obtain elements z j,k ∈ Lsc( X, N ) for j = 1 , . . . , r and k = 0 , . . . , n such that(i) z j,k ≪ χ U j for every j, k ;(ii) χ X ≪ P j,k z j,k ;(iii) P j z j,k ≪ χ X for every k .For each j and k , condition (i) implies that z j,k = χ V j,k for some open subset V j,k ⊆ U j . Condition (ii) implies that X is covered by the sets V j,k . Thus, the family V := { V j,k } is a finite, open refinement of U . For each k , condition (iii) impliesthat the sets V ,k , . . . , V r,k are pairwise disjoint. Thus, V is ( n + 1)-colourable, asdesired.Recall that an ideal I of a Cu-semigroup S is a downward-hereditary submonoidclosed under suprema of increasing sequences; see [APT18, Section 5].Given x, y ∈ S , we write x ≤ I y if there exists z ∈ I such that x ≤ y + z . We set x ∼ I y if x ≤ I y and y ≤ I x . The quotient S/ ∼ I endowed with the induced sumand order ≤ I is denoted by S/I .As shown in [APT18, Lemma 5.1.2],
S/I is a Cu-semigroup and the quotientmap S → S/I is a Cu-morphism.
Proposition 3.5.
Let S be a Cu -semigroup, and let I ⊆ S be an ideal. Then: dim( I ) ≤ dim( S ) , and dim( S/I ) ≤ dim( S ) . Proof.
Set n := dim( S ), which we may assume to be finite, since otherwise thereis nothing to prove. It is straightforward to show that dim( I ) ≤ n using that I is downward-hereditary. Given x ∈ S , we use [ x ] to denote its equivalence classin S/I .To verify dim(
S/I ) ≤ n , let [ u ] ≪ [ x ] ≪ [ y ] + . . . + [ y r ] in S/I . Then thereexists y r +1 ∈ I such that x ≤ y + . . . + y r + y r +1 in S . Using that the quotientmap S → S/I preserves suprema of increasing sequences, we can choose x ′′ , x ′ ∈ S such that x ′′ ≪ x ′ ≪ x, and [ u ] ≤ [ x ′′ ] . Applying the definition of dim( S ) ≤ n to x ′′ ≪ x ′ ≪ y + . . . + y r + y r +1 , we obtainelements z j,k ∈ S for j = 1 , . . . , r + 1 and k = 0 , . . . , n such that z j,k ≪ y j for every j, k , such that x ′′ ≪ P j,k z j,k , and such that P j z j,k ≪ x ′ for every k .Since y r +1 ∈ I , we have z r +1 ,k ∈ I and thus [ z r +1 ,k ] = 0 in S/I for k = 0 , . . . , n .Using also that the quotient map S → S/I is ≪ -preserving, we see that the elements[ z j,k ] for j = 1 , . . . , r and k = 0 , . . . , n have the desired properties. (cid:3) Problem 3.6.
Let S be a Cu-semigroup, and let I ⊆ S be an ideal. Can webound dim( S ) in terms of dim( I ) and dim( S/I )? In particular, do we always havedim( S ) ≤ dim( I ) + dim( S/I ) + 1?Given Cu-semigroups S and T , we use S ⊕ T to denote the Cartesian product S × T equipped with elementwise addition and order. It is straightforward to verifythat S ⊕ T is a Cu-semigroup and that S ⊕ T is both the product and coproductof S and T in the category Cu; see also [APT19, Proposition 3.10]. We omit thestraightforward proof of the next result. HANNES THIEL AND EDUARD VILALTA
Proposition 3.7.
Let S and T be Cu -semigroups. Then dim( S ⊕ T ) = max { dim( S ) , dim( T ) } . (Inductive limits) . By [APT18, Corollary 3.1.11], the category Cu has inductivelimits. (The sequential case was previously shown in [CEI08, Theorem 2].)To recall the construction, let (( S λ ) λ ∈ Λ , ( ϕ µ,λ ) λ ≤ µ in Λ ) be a directed systemin Cu, that is, Λ is a directed set, each S λ is a Cu-semigroup, and for λ ≤ µ in Λwe have a connecting Cu-morphism ϕ µ,λ : S λ → S µ such that ϕ λ,λ = id S λ for each λ ∈ Λ and ϕ ν,µ ◦ ϕ µ,λ = ϕ ν,λ for all λ ≤ µ ≤ ν in Λ.It is shown in [APT19, Theorem 2.9] that Cu is a full, reflective subcategory of amore algebraic category W defined in [APT19, Definition 2.5]. The inductive limitin Cu can therefore be constructed by applying the reflection functor W → Cu tothe inductive limit in W.Consider the equivalence relation ∼ on the disjoint union F λ S λ given by x λ ∼ x µ (for x λ ∈ S λ and x µ ∈ S µ ) if there exists ν ≥ λ, µ such that ϕ ν,λ ( x λ ) = ϕ ν,µ ( x µ ).The set of equivalence classes is the set-theoretic inductive limit, which we denoteby S alg . We write [ x λ ] for the equivalence class of x λ ∈ S λ .We define an addition + and a binary relation ≺ on S alg as follows: Given x λ ∈ S λ and x µ ∈ S µ , set[ x λ ] + [ x µ ] := [ ϕ ν,λ ( x λ ) + ϕ ν,µ ( x µ )]for any ν ≥ λ, µ . Further, set [ x λ ] ≺ [ x µ ] if there exists ν ≥ λ, µ such that ϕ ν,λ ( x λ ) ≪ ϕ ν,µ ( x µ ) in S ν . This gives S alg the structure of a W-semigroup, whichtogether with the natural maps S λ → S alg , x λ [ x λ ], is the inductive limit in W.The reflection of S alg in Cu is a Cu-semigroup S together with a (universal) W-morphism α : S alg → S . Using [APT18, Theorem 3.1.8], S and α are characterizedby the following conditions:(R1) α is an embedding in the sense that [ x λ ] ≺ [ x µ ] if (and only if) α ([ x λ ]) ≪ α ([ x µ ]), for any x λ ∈ S λ and x µ ∈ S µ ;(R2) α has dense image in the sense that for all x ′ , x ∈ S satisfying x ′ ≪ x thereexists x λ ∈ S λ such that x ′ ≪ α ([ x λ ]) ≪ x .It follows that a Cu-semigroup S together with Cu-morphisms ϕ λ : S λ → S for λ ∈ Λ is the inductive limit in Cu of the system (( S λ ) λ ∈ Λ , ( ϕ µ,λ ) λ ≤ µ in Λ ) if andonly if the following conditions are satisfied:(L0) we have ϕ µ ◦ ϕ µ,λ = ϕ λ for all λ ≤ µ in Λ;(L1) if x λ ∈ S λ and x µ ∈ S µ satisfy ϕ λ ( x λ ) ≪ ϕ µ ( x µ ), then there exists ν ≥ λ, µ such that ϕ ν,λ ( x λ ) ≪ ϕ ν,µ ( x µ );(L2) for all x ′ , x ∈ S satisfying x ′ ≪ x there exists x λ ∈ S λ such that x ′ ≪ ϕ λ ( x λ ) ≪ x . Proposition 3.9.
Let S = lim −→ λ ∈ Λ S λ be an inductive limit of Cu -semigroups. Then dim( S ) ≤ lim inf λ dim( S λ ) .Proof. Let ϕ λ : S λ → S be the Cu-morphisms into the inductive limit. We usethat S and the ϕ λ ’s satisfy (L0)-(L2) from Paragraph 3.8. Set n := lim inf λ dim( S λ ),which we may assume to be finite. To verify dim( S ) ≤ n , let x ′ ≪ x ≪ y + . . . + y r in S . Choose y ′ , . . . , y ′ r ∈ S such that x ≪ y ′ + . . . + y ′ r , y ′ ≪ y , . . . , and y ′ r ≪ y r . Using (L2), we obtain a λ ∈ S λ such that x ′ ≪ ϕ λ ( a λ ) ≪ x . Analogously, weobtain b λ k ∈ S λ k such that y ′ k ≪ ϕ λ k ( b λ k ) ≪ y k for k = 1 , . . . , r .Using that ϕ λ is a Cu-morphism, we obtain a ′ λ ∈ S λ such that x ′ ≪ ϕ λ ( a ′ λ ) ≪ ϕ λ ( a λ ) ≪ x, and a ′ λ ≪ a λ . OVERING DIMENSION OF CUNTZ SEMIGROUPS 7
Choose µ ∈ Λ such that µ ≥ λ, λ , . . . , λ r , and set a ′ := ϕ µ,λ ( a ′ λ ) , a := ϕ µ,λ ( a λ ) , b := ϕ µ,λ ( b λ ) , . . . , and b r := ϕ µ,λ r ( b λ r ) . Hence, ϕ µ ( a ) = ϕ µ ( ϕ µ,λ ( a λ )) = ϕ λ ( a λ ) ≪ x ≪ y ′ + . . . + y ′ r ≪ ϕ µ ( b + . . . + b r ) . Applying (L1), we obtain ν ≥ µ such that ϕ ν,µ ( a ) ≪ ϕ ν,µ ( b + . . . + b r ).Using that lim inf λ dim( S λ ) ≤ n , we may also assume that dim( S ν ) ≤ n . Apply-ing dim( S ν ) ≤ n to ϕ ν,µ ( a ′ ) ≪ ϕ ν,µ ( a ′ ) ≪ ϕ ν,µ ( b ) + . . . + ϕ ν,µ ( b r ) , we obtain elements z j,k ∈ S ν for j = 1 , . . . , r and k = 0 , · · · , n satisfying properties(i)-(iii) from Definition 3.1. It is now easy to check that the elements ϕ ν ( z j,k ) ∈ S have the desired properties to verify dim( S ) ≤ n . (cid:3) Proposition 3.10.
Given a C ∗ -algebra A and a (closed, two-sided) ideal I ⊆ A ,we have dim(Cu( I )) ≤ dim(Cu( A )) , and dim(Cu( A/I )) ≤ dim(Cu( A )) . Given C ∗ -algebras A and B , we have dim(Cu( A ⊕ B )) = max { dim(Cu( A )) , dim(Cu( B )) } . Given an inductive limit of C ∗ -algebras A = lim −→ λ A λ , we have dim(Cu( A )) ≤ lim inf λ dim(Cu( A λ )) . Proof.
The first statement follows from Proposition 3.5 using that Cu( I ) is natu-rally isomorphic to an ideal of Cu( A ), and that Cu( A/I ) is naturally isomorphicto Cu( A ) / Cu( I ); see [APT18, Section 5.1]. The second statement follows fromProposition 3.7 using that Cu( A ⊕ B ) is isomorphic to Cu( A ) ⊕ Cu( B ). Finally,the third statements follows from Proposition 3.9 and the fact that the Cuntz semi-group of an inductive limit of C ∗ -algebras is naturally isomorphic to the inductivelimit of the C ∗ -algebras; see [APT18, Corollary 3.2.9]. (cid:3) Example 3.11.
Recall that Cu( C ) is naturally isomorphic to N := { , , , . . . , ∞} .We say that a Cu-semigroup S is simplicial if S ∼ = N k = N ⊕ . . . k ⊕ N for some k ≥
1. If A is a finite-dimensional C ∗ -algebra, then Cu( A ) is simplicial.It is easy to verify that dim( N ) = 0. By Proposition 3.7, we get dim( N k ) = 0for every k ≥
1. Thus, if S is an inductive limit of simplicial Cu-semigroups,then dim( S ) = 0 by Proposition 3.9. Further, it follows from Proposition 3.10 thatdim(Cu( A )) = 0 for every AF-algebra A . In Proposition 5.4, we will generalize thisto C ∗ -algebras of real rank zero (which include all AF-algebras).By applying the Cu-semigroup version of the Effros-Handelman-Shen theorem,[APT18, Corollary 5.5.13], it also follows that every countably-based, weakly can-cellative, unperforated, algebraic Cu-semigroup satisfying (O5) and (O6) is zero-dimensional. In Corollary 5.3, we will generalize this to weakly cancellative, alge-braic Cu-semigroups satisfying (O5) and (O6). Example 3.12.
Recall that a Cu-semigroup is said to be elementary if it is iso-morphic to { } , or if it is simple and contains a minimal nonzero element; see[APT18, Paragraph 5.1.16]. Typical examples of elementary Cu-semigroups are N and E k = { , , , . . . , k, ∞} for k ∈ N , where the sum of two elements in E k isdefined as ∞ if their usual sum would exceed k ; see [APT18, Paragraph 5.1.16].By [APT18, Proposition 5.1.19], these are the only elementary Cu-semigroups thatsatisfy (O5) and (O6). HANNES THIEL AND EDUARD VILALTA
It is easy to see that every elementary Cu-semigroup satisfying (O5) and (O6) iszero-dimensional. In Example 3.13 below, we show that this is no longer the casewithout (O5). To see that (O6) is also necessary, consider S := N ∪ { ′ } , with 1 ′ acompact element not comparable with 1 and such that 1 ′ + 1 ′ = 2 and 1 + k = 1 ′ + k for every k ∈ N \ { } . We claim that dim( S ) = ∞ .Assume, for the sake of contradiction, that dim( S ) ≤ n for some n ∈ N . Then,since 1 ′ ≪ ′ ≪ z ,k , z ,k ∈ S for k = 0 , . . . , n satisfying conditions (i)-(iii) from Definition 3.1. By condition (i), we have z j,k ≪ z j,k = 0 or z j,k = 1 for every j, k . By condition (ii), we have 1 ′ ≪ P j,k z j,k , and so there exist j ′ ∈ { , } and k ′ ∈ { , . . . , n } such that z j ′ ,k ′ = 1.However, by condition (iii), we have z j ′ ,k ′ ≪ ′ , which is a contradiction becausethe elements 1 and 1 ′ are not comparable. Example 3.13.
Let k, l ∈ N , and let E k and E l be the elementary Cu-semigroupsas in Example 3.12. Then the abstract bivariant Cu-semigroup J E k , E l K , as definedin [APT20], has dimension one whenever l > k and dimension zero otherwise.Indeed, by [APT20, Proposition 5.18], we know that J E k , E l K = { , r, . . . , l, ∞} with r = ⌈ ( l + 1) / ( k + 1) ⌉ . Thus, if l ≤ k , then J E k , E l K = E l , which is zero-dimensional by Example 3.12. Note that J E k , E l K is an elementary Cu-semigroupsatisfying (O6). Further, J E k , E l K satisfies (O5) if and only if l ≤ k .Let us now assume that l > k , that is r >
1. Then, even though r + 1 ≪ r + 1 ≪ r + r , one cannot find z , z ≪ r such that r + 1 = z + z . This shows thatdim( J E k , E l K ) = 0.To verify dim( J E k , E l K ) ≤
1, let x ≪ x ≪ y + . . . + y r in J E k , E l K . We mayassume that y j is nonzero for every j . If there exists i ∈ { , . . . , r } with x ≤ y i ,then z i, := x and z j,k := 0 for j = i or k = 1 have the desired properties.So we may assume that y j < x for every j . Let k be the least integer such that x ≤ y + . . . + y k . Define z j, := y j for every j < k and z j, := 0 for j ≥ k . Further,define z k, := y k and z j, := 0 for j = k . By choice of k , we have P j z j, ≪ x . Wealso have P j z j, = y k ≪ x . Finally, x ≪ P j z j, + P j z j, , as desired. Definition 3.14.
Let S and T be Cu-semigroups. We say that S is a retract of T if there exist a Cu-morphism ι : S → T and a generalized Cu-morphism σ : T → S such that σ ◦ ι = id S .Many properties of Cu-semigroups pass to retracts. In Lemma 7.12 we showthis for the Riesz interpolation property and for almost divisibility. The next resultshows that the dimension does not increase when passing to a retract. Proposition 3.15.
Let S and T be Cu -semigroups and assume that S is a retractof T . Then dim ( S ) ≤ dim( T ) .Proof. Let ι : S → T be a Cu-morphism, and let σ : T → S be a generalized Cu-morphism such that σ ◦ ι = id S . Set n := dim( T ), which we may assume to befinite. To verify the assumptions of Lemma 3.3, let x ′ ≪ x ≪ y + . . . + y r in S .Then ι ( x ′ ) ≪ ι ( x ) ≪ ι ( y ) + . . . + ι ( y r )in T . Using that dim( T ) ≤ n , we obtain elements z j,k in T satisfying conditions (i)-(iii) of Definition 3.1. Applying σ , we see that the elements σ ( z j,k ) satisfy conditions(1)-(3) in Lemma 3.3, from which the result follows. (cid:3) Given a simple Cu-semigroup S , let us now show that its sub-Cu-semigroup ofsoft elements S soft , as defined in Paragraph 6.1, is a retract of S . As we will see inProposition 3.17 below, such elements play an important role in the study of thedimension of S . OVERING DIMENSION OF CUNTZ SEMIGROUPS 9
Proposition 3.16.
Let S be a countably based, simple, weakly cancellative Cu -semigroup satisfying (O5) and (O6). Then S soft is a retract of S .Proof. By [APT18, Proposition 5.3.18], S soft is a Cu-semigroup. By [Thi20b,Proposition 2.9], for each x ∈ S there exists a (unique) maximal soft elementdominated by x and the map σ : S → S soft given by σ ( x ) := max (cid:8) x ′ ∈ S soft : x ′ ≤ x (cid:9) , for x ∈ S ,is a generalized Cu-morphism. Further, the inclusion ι : S soft → S is a Cu-morphismand the composition σ ◦ ι is the identity on S soft , as desired. (cid:3) Proposition 3.17.
Let S be a countably based, simple, weakly cancellative Cu -semigroup satisfying (O5) and (O6). Then dim( S soft ) ≤ dim( S ) ≤ dim( S soft ) + 1 . Proof.
The first inequality follows from Propositions 3.15 and 3.16. To show thesecond inequality, set n := dim( S soft ), which we may assume to be finite. If S iselementary, then dim( S ) = 0 as noted in Example 3.12. Thus, we may assume that S is nonelementary. By [APT18, Proposition 5.3.16], every nonzero element of S is either soft or compact. To verify dim( S ) ≤ n + 1, let x ′ ≪ x ≪ y + . . . + y r in S . We may assume that x and y are nonzero. If x is soft, then we let s ′ , s ∈ S be any pair of soft elements satisfying x ′ ≪ s ′ ≪ s ≪ x . If x is compact, then weapply Lemma 6.4 to obtain a nonzero element w ∈ S satisfying w ≤ x, y . Then x ≪ σ ( x ) + w , which allows us to choose soft elements s ′ ≪ s such that s ≪ σ ( x )and x ≪ s ′ + w . In both cases, we have s ′ ≪ s ≪ σ ( x ) ≤ σ ( y ) + . . . + σ ( y r )in S soft . Using that dim( S soft ) ≤ n , we obtain (soft) elements z j,k ∈ S for j =1 , . . . , r and k = 0 , . . . , n such that(i) z j,k ≪ σ ( y j ) (and thus, z j,k ≪ y j ) for each j and k ;(ii) s ′ ≪ P j,k z j,k ;(iii) P rj =1 z j,k ≪ s ≪ σ ( x ) (and thus, P rj =1 z j,k ≪ x ) for each k = 0 , . . . , n .If x is soft, then x ′ ≪ s ′ ≪ P j,k z j,k , which shows that the elements z j,k havethe desired properties. If x is compact, then set z ,n +1 := w and z j,n +1 := 0 for j = 2 , . . . , r . Then z j,n +1 ≪ y j for each j . Further, x ′ ≪ x ≪ s ′ + w ≤ n X k =0 r X j =1 z j,k + r X j =1 z j,n +1 = n +1 X k =0 r X j =1 z j,k . Lastly, P rj =1 z j,n +1 = w ≪ x , which shows that the elements z j,k have the desiredproperties. (cid:3) Remark 3.18.
Proposition 3.17 applies in particular to the Cuntz semigroups ofseparable, simple C ∗ -algebras of stable rank one (see [Rob13, Proposition 5.1.1]).More generally, Engbers showed in [Eng14] that for every separable, simple, stablyfinite C ∗ -algebra A , every compact element in Cu( A ) has a predecessor. The proofof Proposition 3.17 can be generalized to this situation and we obtaindim(Cu( A ) soft ) ≤ dim(Cu( A )) ≤ dim(Cu( A ) soft ) + 1 . Example 3.19.
Let Z = Cu( Z ), the Cuntz semigroup of the Jiang-Su algebra Z .Then, dim( Z ) = 1. Indeed, since Z is a simple, weakly cancellative Cu-semi-group satisfying (O5) that is neither algebraic nor soft, it follows from Lemma 7.1that dim( Z ) >
0. On the other hand, we have Z soft ∼ = [0 , ∞ ], and it is easy toverify that dim([0 , ∞ ]) = 0. Therefore, we have dim( Z ) ≤ dim([0 , ∞ ]) + 1 = 1 byProposition 3.17. A similar argument shows that dim( Z ′ ) = 1, where Z ′ is the Cu-semigroupconsidered in [APT18, Question 9(8)], that is, Z ′ := Z ∪ { ′′ } with 1 ′′ a compactelement not comparable with 1 and such that 1 ′′ + 1 ′′ = 2 and 1 + x = 1 ′′ + x forevery x ∈ Z \ { } .The notion of R -multiplication on a Cu-semigroup for a Cu-semiring R wasintroduced in [APT18, Definition 7.1.3]. Given a solid Cu-semiring R (such as { , ∞} , [0 , ∞ ] or Z ), any two R -multiplications on a Cu-semigroup are equal, andtherefore having an R -multiplication is a property; see [APT18, Remark 7.1.9].It was shown in [APT18, Theorem 7.2.2] that a Cu-semigroup has { , ∞} -multiplication if and only if every element in the semigroup is idempotent. By[APT18, Theorem 7.3.8], a Cu-semigroup has Z -multiplication if and only if it isalmost unperforated and almost divisible. By [APT18, Theorem 7.5.4], a Cu-sem-igroup has [0 , ∞ ]-multiplication if and only if it has Z -multiplication and everyelement in S is soft. Proposition 3.20.
Let S be a Cu -semigroup satisfying (O5) and (O6). Then:(1) If S has { , ∞} -multiplication, then dim( S ) = 0 .(2) If S has [0 , ∞ ] -multiplication, then dim( S ) = 0 .(3) If S has Z -multiplication, then dim( S ) ≤ .Proof. (1) Given elements x ′ ≪ x ≪ y + . . . + y r in a Cu-semigroup with { , ∞} -multiplication, apply (O6) to obtain elements z j ≤ x, y j such that x ′ ≤ z + . . . + z r . Using that every element in S is idempotent, one also has z + . . . + z r ≤ x + . . . r + x = rx = x. This shows that the elements z j satisfy the conditions in Lemma 3.3, as required.(2) Note that S is isomorphic to its realification S R by Theorem 7.5.4 and Propo-sition 7.5.9 in [APT18]. We can now use the decomposition property of S R provenin [Rob13, Theorem 4.1.1] to deduce that S is zero-dimensional.(3) Assume that S has Z -multiplication. By [APT18, Proposition 7.3.13], anelement x ∈ S is soft if and only if x = 1 ′ x (where 1 ′ denotes the soft one in Z ).Further, the Cu-semigroup S soft := 1 ′ S of soft elements in S is isomorphic to therealification of S ; see [APT18, Corollary 7.5.10]. Since the realification of S has[0 , ∞ ]-multiplication, we get dim( S soft ) = 0 by (2).To verify dim( S ) ≤
1, let x ′ ≪ x ≪ y + . . . + y r in S . Using that S has Z -multiplication, one gets x ′ ≪ x ≪ y + . . . + y r . Note that all elements in the previous expression belong to S soft . Since dim( S soft ) =0, we obtain (soft) elements z , . . . , z r ∈ S such that z j ≪ y j for each j , and suchthat x ′ ≪ z + . . . + z r ≪ x. Define z j, := z j and z j, := z j for j = 1 , . . . , r . We trivially have z j,k ≪ y j foreach j and k . Further, x ′ ≤ x ′ ≪ z + · · · + z r ) = X j,k z j,k , and X j z j,k ≪ x ≤ x. for each k = 0 ,
1, as desired. (cid:3)
OVERING DIMENSION OF CUNTZ SEMIGROUPS 11
Let A be a C ∗ -algebra. Then, we know from [APT18, Proposition 7.2.8] that A is purely infinite if and only if Cu( A ) has { , ∞} -multiplication. Proposition 3.21.
Let A be a purely infinite C ∗ -algebra. Then dim( A ) = 0 . Let W denote the Jacelon-Racak algebra. Given a C ∗ -algebra A , it follows from[APT18, Proposition 7.6.3] that Cu( A ⊗ W ) has [0 , ∞ ]-multiplication, and thatCu( A ⊗ Z ) has Z -multiplication. Proposition 3.22.
Let A be a C ∗ -algebra. Then dim(Cu( A ⊗ W )) = 0 , and dim(Cu( A ⊗ Z )) ≤ . In particular, Cuntz semigroups of W -stable C ∗ -algebras are zero-dimensional, andCuntz semigroups of Z -stable C ∗ -algebras have dimension at most one. Example 3.23.
Let X be a compact, metrizable space containing at least twopoints, and let S := Lsc( X, N ) ++ ∪ { } be the sub-Cu-semigroup of Lsc( X, N )consisting of strictly positive functions and 0. Then dim( S ) = ∞ .Indeed, assume for the sake of contradiction that dim( S ) ≤ n for some n ∈ N ,and take r > n . Since X contains at least two points, we can choose open subsets U ′ , U ⊂ X such that ∅ 6 = U ′ , U ′ ⊆ U, and U = X. Let χ U ′ and χ U denote the corresponding characteristic functions. Consider theelements x ′ := 1 + ( n + 1) χ U ′ and x := 1 + ( n + 1) χ U in S . Then, we have x ′ ≪ x ≪ r + 1 = 1 + . . . r +1 + 1 in S .Using that dim( S ) ≤ n , we obtain elements z j,k ∈ S for j = 1 , . . . , r + 1 and k =0 , . . . , n satisfying (i)-(iii) from Definition 3.1. By condition (i), we have z j,k ≪ z j,k = 0 or z j,k = 1 for each j, k .Given k ∈ { , . . . , n } , we have P j z j,k ≪ x by condition (iii), and thus all butpossibly one of the elements z ,k , . . . , z r,l are zero. Thus, P j z j,k ≤
1. Using thisat the last step, and using condition (ii) at the first step, we get x ′ ≪ X j,k z j,k = n X k =0 (cid:0) r X j =1 z j,k (cid:1) ≤ n + 1 , a contradiction.4. Commutative and subhomogeneous C ∗ -algebras In this section, we first prove that the dimension of the Cuntz semigroup of a C ∗ -algebra A is bounded by the nuclear dimension of A ; see Theorem 4.1. For everycompact, Hausdorff space X , we show that the dimension of the Cuntz semigroup of C ( X ) agrees with the dimension of X ; see Proposition 4.3. More generally, on theclass of subhomogeneous C ∗ -algebras, the dimension of the Cuntz semigroup agreeswith the topological dimension, which in turn is equal to the nuclear dimension;see Theorem 4.10. Theorem 4.1.
Let A be a C ∗ -algebra. Then dim(Cu( A )) ≤ dim nuc ( A ) .Proof. Set n := dim nuc ( A ), which we may assume to be finite. By [Rob11, Propo-sition 2.2], there exists an ultrafilter U on an index set Λ, and finite-dimensionalC*-algebras F λ,k for λ ∈ Λ and k = 0 , . . . , n , and completely positive, contractive(cpc.) order-zero maps ψ k : A → Q U F λ,k and ϕ k : Q U F λ,k → A U such that ι = n X k =0 ϕ k ◦ ψ k , where ι : A → A U denotes the natural inclusion map. To verify dim(Cu( A )) ≤ n , let x ′ , x, y , . . . , y r ∈ Cu( A ) satisfy x ′ ≪ x ≪ y + . . . + y r . A cpc. order-zero map α : C → D induces a generalized Cu-morphism ¯ α : Cu( C ) → Cu( D ); see, for example, [APT18, Paragraph 3.2.5].For each k ∈ { , . . . , n } , set x k := ¯ ψ k ( x ) ∈ Cu( Q U F λ,k ). We have¯ ι ( x ′ ) ≪ ¯ ι ( x ) = n X k =0 ¯ ϕ k ( ¯ ψ k ( x )) = n X k =0 ¯ ϕ k ( x k ) . Using that ¯ ϕ k preserves suprema of increasing sequences, we can choose anelement x ′ k ∈ Cu( Q U F λ,k ) such that x ′ k ≪ x k and¯ ι ( x ′ ) ≪ n X k =0 ¯ ϕ k ( x ′ k ) . Given k ∈ { , . . . , n } , we have x ′ k ≪ x k = ¯ ψ k ( x ) ≤ ¯ ψ k ( r X j =1 y j ) = r X j =1 ¯ ψ k ( y j ) . Since Q U F λ,k has real rank zero, we obtain z ,k , . . . , z r,k ∈ Cu( Q U F λ,k ) such that z j,k ≤ ¯ ψ k ( y j ) for j = 1 , . . . , r and x ′ k ≤ r X j =1 z j,k ≤ x k . We now consider the elements ¯ ϕ k ( z j,k ) ∈ Cu( A U ). For each j and k , we have¯ ϕ k ( z j,k ) ≤ ¯ ϕ k ( ¯ ψ k ( y j )) ≤ n X k ′ =0 ¯ ϕ k ′ ( ¯ ψ k ′ ( y j )) = ¯ ι ( y j ) . Further, we have¯ ι ( x ′ ) ≪ n X k =0 ¯ ϕ k ( x ′ k ) ≤ n X k =0 ¯ ϕ k ( r X j =1 z j,k ) = n X k =0 r X j =1 ¯ ϕ k ( z j,k ) . For each k ∈ { , . . . , n } , we also have r X j =1 ¯ ϕ k ( z j,k ) = ¯ ϕ k ( r X j =1 z j,k ) ≤ ¯ ϕ k ( x k ) = ¯ ϕ k ( ¯ ψ k ( x )) ≤ ¯ ι ( x ) . Since the classes of elements in S N ∈ N ( A U ⊗ M N ) + are sup-dense in Cu( A U ),there exist N ∈ N and positive elements c j,k ∈ A U ⊗ M N such that [ c j,k ] ≪ ¯ ϕ k ( z j,k )and ¯ ι ( x ′ ) ≪ P j,k [ c j,k ].We have A U = Q λ A/c U , where c U = { ( a λ ) λ ∈ Y λ A : lim λ →U k a λ k = 0 } . We let π : Q λ A → A U denote the quotient map.We have A U ⊗ M N ∼ = ( A ⊗ M N ) U . We also use π to denote its amplification tomatrix algebras. Choose positive elements c j,k,λ ∈ A ⊗ M N such that π (( c j,k,λ ) λ ) = c j,k . Then, for a sufficiently large λ , the elements [ c j,k,λ ] ∈ Cu( A ) satisfy theproperties of Lemma 3.3 for x ′ ≪ x ≪ y + · · · + y r , as desired. (cid:3) Lemma 4.2.
Let X be a compact, Hausdorff space. Then dim( X ) ≤ dim(Cu( C ( X ))) . OVERING DIMENSION OF CUNTZ SEMIGROUPS 13
Proof.
Set n := dim(Cu( C ( X ))), which we may assume to be finite. To verify thatdim( X ) ≤ n , let U = { U , . . . , U r } be a finite open cover of X . We need to find a( n + 1)-colourable, finite, open refinement of U ; see Remark 3.2.Since X is a normal space, we can find an open cover V = { V , . . . , V r } of X such that V j ⊆ U j for each j ; see for example [Pea75, Proposition 1.3.9, p.20]. Foreach j , by Urysohn’s lemma we obtain a continuous function f j : X → [0 ,
1] thattakes the value 1 on V j and that vanishes on X \ U j .We have 1 ≤ f + . . . + f r , and therefore[1] ≪ [1] ≤ [ f + . . . + f r ] ≤ [ f ] + . . . + [ f r ]in Cu( C ( X )). Using that dim(Cu( C ( X ))) ≤ n , we obtain elements z j,k ∈ Cu( C ( X ))for j = 1 , . . . , r and k = 0 , . . . , n satisfying (i)-(iii) in Definition 3.1.For each j and k , choose g j,k ∈ ( C ( X ) ⊗ K ) + such that z j,k = [ g j,k ]. Viewing g j,k as a positive, continuous function g j,k : X → K , we set W j,k := { x ∈ X : g j,k ( x ) = 0 } . Then W j,k is an open set. Condition (i) implies that g j,k = lim n h n f j h ∗ n for somesequence ( h n ) n in C ( X ) ⊗ K . Thus, g j,k ( x ) = 0 whenever f j ( x ) = 0, which showsthat W j,k ⊆ U j . Condition (ii) implies that X is covered by the sets W j,k . Thus,the family W := { W j,k } is a finite, open refinement of U .Let k ∈ { , . . . , n } . Given x ∈ X , it follows from condition (iii) that therank of g ,k ( x ) ⊕ . . . ⊕ g r,k ( x ) is at most one. This implies that at most one of g ,k ( x ) , . . . , g r,k ( x ) is nonzero. Thus, the sets W ,k , . . . , W r,k are pairwise disjoint.Hence, W is ( n + 1)-colourable, as desired. (cid:3) Proposition 4.3.
Let X be a compact, Hausdorff space. Then dim(Cu( C ( X ))) = dim( X ) . Proof.
The inequality ‘ ≥ ’ is shown in Lemma 4.2. By [WZ10, Proposition 2.4],we have dim( X ) = dim nuc ( C ( X )) if X is second-countable. By Theorem 4.8, thisalso holds for arbitrary compact, Hausdorff spaces. Thus, the inequality ‘ ≤ ’ followsfrom Theorem 4.1. (cid:3) Corollary 4.4.
Let X be a compact, metrizable space. Then dim(Lsc( X, N )) = dim( X ) Proof.
It is enough to see that Lsc( X, N ) is a retract of Cu( C ( X )), since the in-equality ’ ≥ ’ has already been proven in Example 3.4 and the inequality ’ ≤ ’ willfollow from Lemma 3.15 and Proposition 4.3.Thus, set S = Lsc( X, N ) and T = Cu( C ( X )). Define ι : Lsc( X, N ) → Cu( C ( X ))as the unique Cu-morphism mapping the characteristic function χ U to the class ofa positive function in C ( X ) with support U for every open subset U ⊂ X .Also, let σ : T → S be the generalized Cu-morphism mapping the class of anelement a ∈ C ( X ) ⊗ K to its rank function σ ( a ) : X → N , σ ( a )( x ) = rank( a ( x )).It is easy to check that σ ◦ ι = id S , as desired. (cid:3) Recall that the local dimension locdim( X ) of a topological space X is definedas the smallest n ∈ N such that every point in X has a closed neighborhood ofcovering dimension at most n ; see [Pea75, Definition 5.1.1, p.188]. If X is a locallycompact, Hausdorff space, thenlocdim( X ) = sup (cid:8) dim( K ) : K ⊆ X compact (cid:9) . If X is σ -compact, locally compact and Hausdorff, then locdim( X ) = dim( X ), butin general locdim( X ) can be strictly smaller than dim( X ). If X is locally compact,Hausdorff but not compact, then it follows from [Pea75, Proposition 3.5.6] that locdim( X ) agrees with the the dimension of αX , the one-point compactification of X . Theorem 4.5.
Let X be a locally compact, Hausdorff space. Then dim(Cu( C ( X ))) = locdim( X ) . Proof.
Let K ⊆ X be a compact subset. Then C ( K ) is a quotient of C ( X ). UsingProposition 4.3 at the first step and Proposition 3.10 at the second step, we getdim( K ) = dim(Cu( C ( K ))) ≤ dim(Cu( C ( X ))) . It follows that locdim( X ) ≤ dim(Cu( C ( X ))).Conversely, we use that C ( X ) is an ideal in C ( αX ). Applying Proposition 3.10at the first step, and using Proposition 4.3 and dim( αX ) = locdim( X ) at the secondstep, we get dim(Cu( C ( X ))) ≤ dim(Cu( C ( αX ))) = locdim( X ) . This show the converse inequality and finishes the proof. (cid:3)
Let d ∈ N with d ≥
1. Recall that a C ∗ -algebra A is said to be d -(sub)homo-geneous if every irreducible representation of A has dimension (at most) d . Fur-ther, A is (sub)homogeneous if it is d -(sub)homogeneous for some d . If A is d -subhomogeneous, then so is every sub- C ∗ -algebra of A .Let us briefly recall the main structure theorems for (sub)homogeneous C ∗ -al-gebras. For details, we refer to [Bla06, Sections IV.1.4, IV.1.7]. Given a locallytrivial M d ( C )-bundle over a locally compact, Hausdorff space X , the algebra ofsections vanishing at infinity is a d -homogeneous C ∗ -algebra with primitive idealspace homeomorphic to X . Moreover, every homogeneous C ∗ -algebra arises thisway.Let A be a d -subhomogeneous C ∗ -algebra. For each k ≥
2, let I ≥ k ⊆ A be theset of elements a ∈ A such that π ( a ) = 0 for every irreducible representation π of A of dimension at most k −
1. Set I ≥ = A . Then { } = I ≥ d +1 ⊆ I ≥ d ⊆ . . . ⊆ I ≥ ⊆ I ≥ = A is an increasing chain of (closed, two-sided) ideals of A . For each k ≥
1, thecanonical k -homogeneous ideal-quotient (that is, an ideal of a quotient) of A is A k := I ≥ k /I ≥ k +1 . Note that A k = { } for k ≥ d + 1.For each k ≥
1, we have a short exact sequence0 → A k +1 → A/I ≥ k +1 → A/I ≥ k → . In particular,
A/I ≥ is an extension of A/I ≥ = A by A . Then A/I ≥ is anextension of A/I ≥ by A , and so on. Finally, A is an extension of A/I ≥ d − by A d .Thus, every subhomogeneous C ∗ -algebra is obtained as a finite successive extensionof homogeneous C ∗ -algebras.In [BP09], Brown and Pedersen introduced the topological dimension for certain C ∗ -algebras, including all type I C ∗ -algebras. We only recall the definition forsubhomogeneous C ∗ -algebras. First, if A is homogeneous, then its primitive idealspace Prim( A ) is locally compact and Hausdorff, and then the topological dimensionof A is defined as topdim( A ) := locdim(Prim( A )).If A is subhomogeneous, then the topological dimension of A is defined asthe maximum of the topological dimensions of the canonical homogeneous ideal-quotients:topdim( A ) := max k =1 ,...,d topdim( A k ) = max k =1 ,...,d locdim(Prim( A k )) . OVERING DIMENSION OF CUNTZ SEMIGROUPS 15
Given a C ∗ -algebra A , we use Sub sep ( A ) to denote the collection of separablesub- C ∗ -algebras of A . A family S ⊆
Sub sep ( A ) is said to be σ -complete if for everycountable, directed subfamily T ⊆ S we have S { B : B ∈ T } ∈ S . Further, a family S ⊆
Sub sep ( A ) is said to be cofinal if for every B ∈ Sub sep ( A ) there exists B ∈ S with B ⊆ B . Proposition 4.7.
Let n ∈ N . Then for every subhomogeneous C ∗ -algebra A satis-fying topdim( A ) ≤ n , the set (cid:8) B ∈ Sub sep ( A ) : topdim( B ) ≤ n (cid:9) is σ -complete and cofinal.Proof. We will use the following facts. The first is a consequences of [Thi20a,Proposition 3.5], the second follows from [BP09, Proposition 2.2].
Fact 1:
Given a homogeneous C ∗ -algebra B with locdim( B ) ≤ n , the collection (cid:8) C ∈ Sub sep ( B ) : topdim( C ) ≤ n (cid:9) is σ -complete and cofinal. Fact 2: If B is subhomogeneous and I ⊆ B is an ideal, then topdim( B ) = max { topdim( I ) , topdim( B/I ) } . We prove the result for d -subhomogeneous C ∗ -algebras by induction over d .First, note that a C ∗ -algebra is 1-subhomogeneous if and only if it is 1-homogeneous(if and only if it is commutative). In this case, the result follows directly from Fact 1.Next, let d ≥ d -subhomogeneous C ∗ -algebra. Let A be ( d + 1)-subhomogeneous. We need to show that the set S := { B ∈ Sub sep ( A ) : topdim( B ) ≤ n } is σ -complete and cofinal.To verify that S is σ -complete, let T ⊆ S be a countable, directed family. Set C := S { B : B ∈ T } . Then C is a separable C ∗ -algebra that is approximated bythe sub- C ∗ -algebras B ⊆ C for B ∈ T that each satisfy topdim( B ) ≤ n . By [Thi13,Proposition 8], we have topdim( C ) ≤ n . Thus, C ∈ S , as desired.Next, we verify that S is cofinal. Set I := I ≥ d +1 ⊆ A , the ideal of all elementsin A that vanish under all irreducible representations of dimension at most d . Then I is ( d + 1)-homogeneous and A/I is d -subhomogeneous. By Fact 2, we havetopdim( I ) ≤ n and topdim( A/I ) ≤ n . By Fact 1 and by the assumption of theinduction, the collections T := (cid:8) C ∈ Sub sep ( I ) : topdim( C ) ≤ n (cid:9) , T := (cid:8) D ∈ Sub sep ( A/I ) : topdim( D ) ≤ n (cid:9) , are σ -complete and cofinal. By [Thi20a, Lemma 3.2], it follows that the families S := (cid:8) B ∈ Sub sep ( A ) : topdim( B ∩ I ) ≤ n (cid:9) , S := (cid:8) B ∈ Sub sep ( A ) : topdim( B/ ( B ∩ I )) ≤ n (cid:9) , are σ -complete and cofinal. Then S ∩ S is σ -complete and cofinal as well. Given B ∈ S ∩ S , it follows from Fact 2 thattopdim( B ) = max { topdim( B ∩ I ) , topdim( B/ ( B ∩ I )) } ≤ n. Thus, S ∩ S ⊆ S . Since S ∩ S is cofinal, so is S . (cid:3) We deduce a result that is probably known to the experts, but which does notappear in the literature so far. The equality of the topological dimension and thedecomposition rank dr( A ) of a separable subhomogeneous C ∗ -algebra A was shownin [Win04]. Theorem 4.8.
Let A be a subhomogeneous C ∗ -algebra. Then dim nuc ( A ) = dr( A ) = topdim( A ) . Proof.
As noted in [WZ10, Remarks 2.2(ii)], the inequality dim nuc ( B ) ≤ dr( B )holds for every C ∗ -algebra B . To verify dr( A ) ≤ topdim( A ), set n := topdim( A ).We may assume that n is finite. By Proposition 4.7, the family S := (cid:8) B ∈ Sub sep ( A ) : topdim( B ) ≤ n (cid:9) is cofinal. Each B ∈ S is a separable, subhomogeneous C ∗ -algebra, whence we canapply [Win04, Theorem 1.6] to deduce that dr( B ) = topdim( B ) ≤ n . Thus, A is approximated by the collection S consisting of C ∗ -algebras with decompositionrank at most n . It is straightforward to verify that this implies dr( A ) ≤ n .To verify topdim( A ) ≤ dim nuc ( A ), set m := dim nuc ( A ), which we may assumeto be finite. It follows from [WZ10, Proposition 2.6] that the family T := (cid:8) B ∈ Sub sep ( A ) : dim nuc ( B ) ≤ m (cid:9) is cofinal. Let B ∈ T . Then B is a separable, subhomogeneous C ∗ -algebra.For each k ≥
1, let B k be the canonical k -homogeneous ideal-quotient of B ; seeParagraph 4.6. Using [WZ10, Corollary 2.10] at the first step, and using that the nu-clear dimension does not increase when passing to ideals ([WZ10, Proposition 2.5])or quotients ([WZ10, Proposition 2.3(iv)]) at the second step, we gettopdim( B k ) = dim nuc ( B k ) ≤ dim nuc ( B ) ≤ m. Using that B is obtained as a successive extension of B by B , and then by B , and so on, it follows from [BP09, Proposition 2.2] (see Fact 2 in the proof ofProposition 4.7) that topdim( B ) ≤ m . Thus, A is approximated by the collec-tion T consisting of C ∗ -algebras with topological dimension at most m . By [Thi13,Proposition 8], we get topdim( A ) ≤ m , as desired. (cid:3) Lemma 4.9.
Let A be a homogeneous C ∗ -algebra. Then dim nuc ( A ) ≤ dim(Cu( A )) . Proof.
Let d ≥ A is d -homogeneous. Set X := Prim( A ), which is locallycompact and Hausdorff. Then topdim( A ) = locdim( X ), and we need to show thatlocdim( X ) ≤ dim(Cu( A )).Let x ∈ X . Since A is the algebra of sections vanishing at infinity of a locallytrivial M d ( C )-bundle over X , there exists a compact neighborhood Y of x overwhich the bundle is trivial. Let I ⊆ A be the ideal of all sections in A that vanishon X \ Y . Then A/I is the algebra of sections of the trivial M d ( C )-bundle over Y , and so A/I ∼ = C ( Y ) ⊗ M d . Using Lemma 4.2 at the first step, using that C ( Y )and C ( Y ) ⊗ M d have isomorphic Cuntz semigroup at the second step, and usingProposition 3.10 at the last step, we getdim( Y ) ≤ dim(Cu( C ( Y ))) = dim(Cu( C ( Y ) ⊗ M d )) ≤ dim(Cu( A )) . Thus, every point in X has a closed neighborhood of dimension at most dim(Cu( A )),whence locdim( X ) ≤ dim(Cu( A )), as desired. (cid:3) Theorem 4.10.
Let A be a subhomogeneous C ∗ -algebra. Then dim(Cu( A )) = dim nuc ( A ) = dr( A ) = topdim( A ) . Proof.
The second and third equalities are shown in Theorem 4.8. By Theorem 4.1,the inequality dim(Cu( A )) ≤ dim nuc ( A ) holds in general. It remains to verify thattopdim( A ) ≤ dim(Cu( A )). OVERING DIMENSION OF CUNTZ SEMIGROUPS 17
For each k ≥
1, let A k be the canonical k -homogeneous ideal-quotient of A asin Paragraph 4.6. Using Lemma 4.9 at the first step, and using Proposition 3.10 atthe second step, we gettopdim( A k ) ≤ dim(Cu( A k )) ≤ dim(Cu( A )) . Consequently, topdim( A ) = max k ≥ topdim( A k ) ≤ dim(Cu( A )) , as desired. (cid:3) Example 4.11.
There are many examples showing that Theorem 4.10 does nothold for all C ∗ -algebras. In Proposition 5.4, we will show that every C ∗ -algebra A of real rank zero satisfies dim(Cu( A )) = 0. On the other hand, a separable C ∗ -algebra A satisfies dim nuc ( A ) = 0 if and only if A is an AF-algebra; see [WZ10,Remarks 2.2(iii)]. Thus, every separable C ∗ -algebra A of real rank zero that is notan AF-algebra is an example where dim(Cu( A )) is strictly smaller than dim nuc ( A ).More extremely, every non-nuclear C ∗ -algebra A of real rank zero, such as B ( ℓ ( N )),satisfies dim(Cu( A )) = 0 while dim nuc ( A ) = ∞ . Another example is the irrationalrotation algebra A θ , which satisfies dim(Cu( A θ )) = 0 while dim nuc ( A θ ) = 1.5. Algebraic, zero-dimensional Cuntz semigroups
In this section we begin our systematic study of zero-dimensional Cu-semigroups.After giving a useful characterization of zero-dimensionality (Lemma 5.1), we pro-vide a sufficient criterion: A Cu-semigroup is zero-dimensional whenever it containsa sup-dense subsemigroup that satisfies the Riesz decomposition property with re-spect to the pre-order induced by the way-below relation; see Proposition 5.2. Wededuce that the Cuntz semigroup of every C ∗ -algebra of real rank zero is zero-dimensional; see Proposition 5.4. Conversely, we show that every unital C ∗ -algebraof stable rank one and with zero-dimensional Cuntz semigroup has real rank zero;Corollary 5.8.We also show that every weakly cancellative, zero-dimensional Cu-semigroup sat-isfying (O5) contains a largest algebraic ideal, which contains all compact elements;see Corollary 5.6. In Section 7, we study certain zero-dimensional Cu-semigroupsthat contain no compact elements. Lemma 5.1.
Let S be a Cu -semigroup. Then dim( S ) = 0 if and only if, whenever x ′ ≪ x ≪ y + y in S , there exist z , z ∈ S such that z ≪ y , z ≪ y , and x ′ ≪ z + z ≪ x. Proof.
The forward implication is clear, so we are left to prove the converse. Given r ≥ x ′ ≪ x ≪ y + . . . + y r in S , we need to find z , . . . , z r ∈ S such that z ≪ y , . . . , z r ≪ y r , and x ′ ≪ z + . . . + z r ≪ x. We prove this by induction on r . The case r = 1 is clear and the case r = 2 holdsby assumption.Thus, let r > r −
1. Given x ′ ≪ x ≪ y + . . . + y r , apply the assumption to x ′ ≪ x ≪ ( y + . . . + y r − ) + y r to obtain u , u ∈ S such that u ≪ y + . . . + y r − , u ≪ y r , and x ′ ≪ u + u ≪ x. Choose u ′ such that u ′ ≪ u and x ′ ≪ u ′ + u . Applying the inductionhypothesis to u ′ ≪ u ≪ y + . . . + y r − , we obtain z , . . . , z r − ∈ S such that z ≪ y , . . . , z r − ≪ y r − , and u ′ ≪ z + . . . + z r − ≪ u . Set z r := u . Then z , . . . , z r have the desired properties. (cid:3) Recall that a semigroup S with a pre-order ≺ is said to satisfy the Riesz decom-position property if whenever x, y, z ∈ S satisfy x ≺ y + z , then there exist e, f ∈ S such that x = e + f , e ≺ y and f ≺ z . Proposition 5.2.
Let S be a Cu -semigroup, and let D ⊆ S be a sup-dense sub-semigroup such that D satisfies the Riesz decomposition property for the pre-orderinduced by ≪ . Then dim( S ) = 0 .Proof. To verify the condition in Lemma 5.1, let x ′ ≪ x ≪ y + y in S . Usingthat D is sup-dense, we find ˜ x, ˜ y ˜ y ∈ D such that x ′ ≪ ˜ x ≪ x ≤ ˜ y + ˜ y , ˜ y ≪ y , and ˜ y ≪ y . Then ˜ x ≪ ˜ y + ˜ y . Using that D satisfies the Riesz decomposition property, weobtain x , x ∈ D such that˜ x = x + x , x ≪ ˜ y , and x ≪ ˜ y . Then x and x have the desired properties to verify the condition of Lemma 5.1. (cid:3) Recall that a Cu-semigroup is said to be algebraic if its compact elements aresup-dense; see [APT18, Section 5.5].
Corollary 5.3.
Let S be a weakly cancellative, algebraic Cu -semigroup satisfying(O5) and (O6). Then dim( S ) = 0 .Proof. Set D := { x ∈ S : x ≪ x } , the semigroup of compact elements. Byassumption, D is sup-dense. By [APT18, Corollary 5.5.10], D satisfies the Rieszdecomposition property. Hence, dim( S ) = 0 by Proposition 5.2. (cid:3) Proposition 5.4.
Let A be a C ∗ -algebra of real rank zero. Then dim(Cu( A )) = 0 .Proof. Let C ⊆ Cu( A ) be the set of Cuntz equivalence classes of projections in A ⊗ K . Then C is a submonoid consisting of compact elements, and using that A has real rank zero it follows that C ⊆ Cu( A ) is sup-dense.To verify that C satisfies the Riesz decomposition property, let x, y, z ∈ C satisfy x ≪ y + z . Choose projections p, q, r ∈ A ⊗ K such that x = [ p ], y = [ q ] and z = [ r ]. Then p is Cuntz subequivalent to q ⊕ r . Since among projections Cuntzsubequivalence agrees with Murray-von Neumann subequivalence, we obtain that p is Murray-von Neumann subequivalent to q ⊕ r .By [BP91, Corollary 3.3], A ⊗K has real rank zero. Hence, it follows from [Zha90,Theorem 1.1] that the Murray-von Neumann semigroup of projections satisfies theRiesz decomposition property. Thus, there exist projections q ′ ≤ q and r ′ ≤ r suchthat p is Murray-von Neumann equivalent to q ′ ⊕ r ′ . Using at the first step thatMurray-von Neumann equivalence is stronger than Cuntz equivalence, we get x = [ p ] = [ q ′ ] + [ r ′ ] , [ q ′ ] ≪ [ q ] = y, and [ r ′ ] ≪ [ r ] = z. Now it follows from Proposition 5.2 that dim(Cu( A )) = 0. (cid:3) Lemma 5.5.
Let S be a weakly cancellative Cu -semigroup satisfying (O5) and dim( S ) = 0 . Let c ∈ S be compact. Then the ideal generated by c is algebraic. OVERING DIMENSION OF CUNTZ SEMIGROUPS 19
Proof.
Let I be the ideal generated by c . Note that x ∈ S belongs to I if and onlyif x ≤ ∞ c . To verify that I is algebraic, let x ′ , x ∈ I satisfy x ′ ≪ x . We need tofind a compact element z such that x ′ ≪ z ≪ x .Choose x ′′ ∈ S such that x ′ ≪ x ′′ ≪ x . Then x ′′ ≪ x ≤ ∞ c , which allows us tochoose n ∈ N such that x ′′ ≤ nc . Applying (O5) to x ′ ≪ x ′′ ≤ nc , we obtain y ∈ S such that x ′ + y ≤ nc ≤ x ′′ + y. Using that dim( S ) = 0 for nc ≪ nc ≪ x ′′ + y , we obtain z , z ∈ S such that nc = z + z , z ≪ x ′′ , and z ≪ y. By weak cancellation, z and z are compact. We now have x ′ + y ≪ nc = z + z ≤ z + y. Using weak cancellation, we get x ′ ≪ z . Thus, z has the desired properties. (cid:3) Corollary 5.6.
Let S be a weakly cancellative Cu -semigroup satisfying (O5) and dim( S ) = 0 . Then S contains a largest algebraic ideal, which agrees with the idealgenerated by all compact elements of S . Proposition 5.7.
Let S be a weakly cancellative Cu -semigroup satisfying (O5).Then the following are equivalent:(1) We have dim( S ) = 0 , and the compact elements of S are full (that is, thereis no proper ideal of S containing all compact elements);(2) S is algebraic and satisfies (O6).Proof. Assuming (1), it follows from Lemma 5.5 that S is algebraic. Further, it isclear that dim( S ) = 0 implies that S satisfies (O6). Conversely, assuming (2), itfollows from Corollary 5.3 that dim( S ) = 0, and since S is algebraic it is clear thatcompact elements of S are full. (cid:3) Corollary 5.8.
Let A be a unital C ∗ -algebra of stable rank one. Then we have dim(Cu( A )) = 0 if and only if A has real rank zero.Proof. If A has real rank zero, then dim(Cu( A )) = 0 by Proposition 5.4. (Thisimplication does not require stable rank one.) Conversely, assume that A hasstable rank one and dim(Cu( A )) = 0. Since A is unital, the compact elements inCu( A ) are full. Thus, by Proposition 5.7, Cu( A ) is algebraic. Now it follow from[CEI08, Corollary 5] that A has real rank zero. (cid:3) Corollary 5.9.
Let A be a separable, simple, Z -stable C ∗ -algebra. Then we have dim(Cu( A )) ≤ . Moreover, dim(Cu( A )) = 0 if and only if A has real rank zero orif A is stably projectionless.Proof. It follows from [Rør02, Theorem 4.1.10] that A is either purely infinite orstably finite. Thus, we can distinguish three cases: A is either purely infinite orstably projectionless, or stably finite and not stably projectionless.The first statement follows from Proposition 3.22. To show the forward impli-cation of the second statement, assume that dim(Cu( A )) = 0. We need to showthat A has real rank zero or is stable projectionless. First, if A is purely infi-nite, then A has real rank zero; see [Bla06, Proposition V.3.2.12]. Second, if A isstably projectionless, then there is nothing to show. Third, we consider the casethat A is stably finite and not stably projectionless. Let p ∈ A ⊗ K be a nonzeroprojection. Then p ( A ⊗ K ) p is a separable, unital, simple, stably finite, Z -stable C ∗ -algebra and therefore has stable rank one by [Rør04, Theorem 6.7]. Since A and p ( A ⊗ K ) p are stably isomorphic, they have isomorphic Cuntz semigroups. Thus,dim(Cu( p ( A ⊗ K ) p )) = 0, and we deduce from Corollary 5.8 that p ( A ⊗ K ) p has real rank zero. By [BP91, Corollary 2.8 and 3.3], a C ∗ -algebra has real rank zeroif and only if its stabiliation does. Thus, A has real rank zero.To show the backward implication of the second statement, assume that A hasreal rank zero or is stably projectionless. We need to show that dim(Cu( A )) = 0.If A has real rank zero, this follow from Proposition 5.4. Let us consider the casethat A is stably projectionless. Then Cu( A ) contains no nonzero compact elementsby [BC09]. Thus, Cu( A ) is soft and has Z -multiplication, which by [APT18, The-orem 7.5.4] implies that Cu( A ) has [0 , ∞ ]-multiplication. Hence, dim(Cu( A )) = 0by Proposition 3.20. (cid:3) Thin boundary and complementable elements
In this section, we study soft elements in simple Cu-semigroups that behave verysimilar to compact elements: the elements with thin boundary (Definition 6.3), andthe complementable elements (Definition 6.9). If S is a simple, stably finite, softCu-semigroup satisfying (O5) and (O6) (for example, the Cuntz semigroup of asimple, stably projectionless C ∗ -algebra; see Proposition 6.2), then every elementwith thin boundary is complementable; see Corollary 6.11. The converse holds if S is also weakly cancellative (for example, the Cuntz semigroup of a simple, stablyprojectionless C ∗ -algebra of stable rank one); see Theorem 6.12.In Section 7, we will show that zero-dimensionality of certain simple Cu-semi-groups is characterized by sup-denseness of the elements with thin boundary. We say that a simple Cu-semigroup S is stably finite if for all x, z ∈ S , wehave that x + z ≪ z implies x = 0. Using that S is simple, one can show that thisdefinition is equivalent to the one given in [APT18, Paragraph 5.2.2]. We note thatevery simple, weakly cancellative Cu-semigroup is stably finite.Let S be a simple, stably finite Cu-semigroup satisfying (O5). Recall that anelement x ∈ S is compact if x ≪ x . We say that x ∈ S is soft if x = 0 or if x = 0 and for every x ′ ∈ S satisfying x ′ ≪ x there exists a nonzero t ∈ S such that x ′ + t ≪ x . (Using [APT18, Proposition 5.3.8], one sees that this is equivalent tothe original definition.) We say that S is soft if every element in S is soft.We let S c and S soft denote the set of compact and soft elements in S , respec-tively. We also set S × soft := S soft \ { } . It is easy to see that S c , S soft and S × soft aresubmonoids of S . Further, S × soft is absorbing in the sense that x + y belongs to S × soft whenever x or y does; see [APT18, Theorem 5.3.11].By [APT18, Proposition 5.3.16], every element in S is either compact, or nonzeroand soft. Hence, S can be decomposed as S = S × soft ⊔ S c . Proposition 6.2.
Let A be a simple, stably projectionless C ∗ -algebra. Then Cu( A ) is a simple, stably finite, soft Cu -semigroup satisfying (O5) and (O6).Proof. The Cuntz semigroup Cu( A ) is simple and satisfies (O5) and (O6) since itis the Cuntz semigroup of a simple C ∗ -algebra (see, for example, [APT18, Corol-lary 5.1.12]). As A is stably projectionless, Cu( A ) has no nonzero compact elementsby [BC09].It is easy to check that a simple Cu-semigroup is stably finite if and only if ∞ is not compact or if S is zero. Therefore, the Cuntz semigroup of a stablyprojectionless C ∗ -algebra is always stably finite.By [APT18, Proposition 5.3.16] we have Cu( A ) × = Cu( A ) × soft as desired. (cid:3) Definition 6.3.
Let S be a simple Cu-semigroup. We say that an element x ∈ S has thin boundary if x ≪ x + t for every nonzero t ∈ S . We let S tb denote the setof elements in S with thin boundary. OVERING DIMENSION OF CUNTZ SEMIGROUPS 21
Note that every compact element has thin boundary, but the converse is nottrue: In [0 , ∞ ] every element has thin boundary, but only 0 is compact.We will repeatedly use the following result. Lemma 6.4.
Let S be a simple, nonelementary Cu -semigroup satisfying (O5) and(O6). Let u , u ∈ S be nonzero. Then there exists a nonzero w ∈ S such that w ≪ u , u .Proof. This follows by combining [APT18, Lemma 5.1.18] and [Rob13, Proposi-tion 5.2.1]. For the convenience of the reader, we include the simple argument.First, choose nonzero elements u ′′ , u ′ ∈ S such that u ′′ ≪ u ′ ≪ u . Since S issimple and u = 0, we have u ′ ≪ u ≤ ∞ = ∞ u , which allows us to choose n ≥ u ′ ≤ nu .Applying (O6) to u ′′ ≪ u ′ ≤ u + . . . n + u , we obtain z , . . . , z n ∈ S such that u ′′ ≪ z + . . . + z n , and z , . . . , z n ≪ u ′ , u . Since u ′′ is nonzero, there is j ∈ { , . . . , n } such that v := z j is nonzero. Then v ≪ u , u .Since S is nonelementary, v is not a minimal nonzero element. Thus, we canchoose a nonzero v ′ ∈ S with v ′ ≤ v and v ′ = v . Choose a nonzero v ′′ ∈ S with v ′′ ≪ v ′ . Applying (O5) to v ′′ ≪ v ′ ≤ v , we obtain c ∈ S such that v ′′ + c ≤ v ≤ v ′ + c. Since v ′ = v , we have c = 0. Applying the first part of the argument to the nonzeroelements v ′′ and c , we obtain w ∈ S such that 0 = w ≪ v ′′ , c . Then w has thedesired properties. (cid:3) Lemma 6.5.
Let S be a simple Cu -semigroup satisfying (O5) and (O6). Then S tb is a submonoid.Proof. This is clear if S is elementary, since then every element in S way-belowanother is compact and therefore S tb = S ; see [APT18, Proposition 5.1.19].We now assume that S is nonelementary. Let x, y ∈ S tb . To verify that x + y has thin boundary, let t ∈ S be nonzero. By Lemma 6.4, there is a nonzero element s such that 2 s ≤ t . This implies x + y ≪ x + s + y + s ≤ x + y + t, as required. (cid:3) Lemma 6.6.
Let S be a simple, weakly cancellative Cu -semigroup satisfying (O5).Let x, y, z ∈ S satisfy x + z ≤ y + z . Assume that x, y are soft, and that z has thinboundary. Then x ≤ y .Proof. If x = 0 the result is trivial, so we may assume otherwise.Let x ′ ∈ S satisfy x ′ ≪ x . Choose x ′′ ∈ S such that x ′ ≪ x ′′ ≪ x . Since x isnonzero and soft, there exists a nonzero t ∈ S with x ′′ + t ≤ x . Hence, x ′ + z ≪ x ′′ + ( z + t ) ≤ x + z ≤ y + z. Using weak cancellation, we get x ′ ≪ y .Since this holds for every x ′ way-below x , we get x ≤ y . (cid:3) Lemma 6.7.
Let S be a simple, weakly cancellative Cu -semigroup. Let x, y ∈ S such that x + y has thin boundary. Then x and y have thin boundary.Proof. To show that x has thin boundary, let t ∈ S be nonzero. Then x + y ≪ ( x + y ) + t = ( x + t ) + y, which, by weak cancellation, implies that x ≪ x + t , as desired. Analogously, oneshows that y has thin boundary. (cid:3) Lemma 6.8.
Let S be a simple, stably finite Cu -semigroup satisfying (O5). Let x ∈ S have thin boundary, and let s, t ∈ S satisfy s ≪ t . Assume that t is nonzeroand soft. Then x + s ≪ x + t .Proof. Choose t ′ ∈ S such that s ≪ t ′ ≪ t . Since t is nonzero and soft, there existsa nonzero c ∈ S such that t ′ + c ≤ t . Then x + s ≪ ( x + c ) + t ′ ≤ x + t, as desired. (cid:3) Definition 6.9.
Let S be a simple, soft Cu-semigroup. We say that x ∈ S is complementable if for every y ∈ S satisfying x ≪ y there exists z ∈ S such that x + z = y .The next result implies that elements with thin boundary are complementable;see Corollary 6.11. Proposition 6.10.
Let S be a simple, stably finite Cu -semigroup satisfying (O5)and (O6). Let x, y ∈ S satisfy x ≪ y . Assume that x has thin boundary and that y is soft. Then there exists z ∈ S such that x + z = y .Proof. Applying [APT18, Proposition 5.1.19], the result is clear if S is elementary.Thus, we may assume that S is nonelementary. The result is also clear if x = 0, sowe may assume that x = 0. Step 1:
We construct an increasing sequence ( y n ) n with supremum y and x ≤ y , and a sequence ( s n ) n of nonzero elements such that y n + s n ≪ y n +1 for every n ∈ N . First, let (¯ y n ) n be any ≪ -increasing sequence in S with supremum y . Set y := x .Since S is simple and stably finite, it follows from [APT18, Proposition 5.3.18] thatthere exists a soft element y ′ such that y ≪ y ′ ≪ y . Since y ′ is nonzero and soft,one can find a non-zero element s such that y + s ≤ y ′ .Using that y + s and ¯ y are way-below y , choose y such that y + s ≪ y , ¯ y ≪ y , and y ≪ y. Then y ≪ y , and we can apply the previous argument once again to obtain s = 0 such that y + s ≪ y . Using that y + s and ¯ y are way-below y , we obtain y such that y + s ≪ y , ¯ y ≪ y , and y ≪ y. Continuing this way, we obtain the desired sequences ( y n ) n and ( s n ) n . Step 2:
We construct a sequence ( r n ) n of nonzero elements such that (6.1) 2 r n +1 ≪ r n , s n +1 , and y n + r n + r n +1 ≪ y n +1 for every n ∈ N . Applying Lemma 6.4 for s , we obtain a nonzero r ∈ S such that 2 r ≪ s .Then, applying Lemma 6.4 for r and s , we obtain a nonzero r ∈ S such that2 r ≪ r , s . Continuing this way, we obtain a sequence ( r n ) n such that 2 r n +1 ≪ r n , s n +1 for every n ∈ N .For each n ∈ N , we have y n + r n + r n +1 ≤ y n + 2 r n ≤ y n + s n ≪ y n +1 , which shows that ( r n ) n has the desired properties. Step 3:
We construct an ≪ -increasing sequence ( w n ) n and a sequence ( v n ) n such that x + r n +1 + v n ≤ y n ≤ x + r n + v n , w n ≪ r n + v n , v n +1 , and y n − ≤ x + w n OVERING DIMENSION OF CUNTZ SEMIGROUPS 23 for every n ≥ . To start, using Lemma 6.8 at the first step, we have x + r ≪ x + r ≤ y + s ≤ y . Applying (O5), we obtain v ∈ S such that x + r + v ≤ y ≤ x + r + v . Using that y ≪ y , we can choose w ∈ S such that y ≤ x + w , and w ≪ r + v . Next, let n ≥
1, and assume that we have chosen v n and w n . Using for the firstinequality that x + r n +1 + v n ≤ y n and (6.1), we have x + r n +1 + r n + v n ≤ y n +1 , x + r n +2 ≪ x + r n +1 , and w n ≪ r n + v n . Applying (O5), we obtain v n +1 ∈ S such that x + r n +2 + v n +1 ≤ y n +1 ≤ x + r n +1 + v n +1 , and w n ≪ v n +1 . Using that y n ≪ y n +1 and w n ≪ v n +1 ≤ r n +1 + v n +1 , we obtain w n +1 ∈ S suchthat y n ≤ x + w n +1 , and w n ≪ w n +1 ≪ r n +1 + v n +1 . Now, the sequence ( w n ) n is increasing, which allows us to set z := sup n w n . Forevery n ≥
1, we have x + w n ≤ x + v n +1 ≤ y n +2 ≤ y and therefore x + z ≤ y . Further, for every n ≥
1, we have y n ≤ x + w n +1 ≤ x + z and therefore y ≤ x + z . This implies x + z = y . (cid:3) Corollary 6.11.
Let S be a simple, soft, stably finite Cu -semigroup satisfying (O5)and (O6). Then every element in S with thin boundary is complementable. If we additionally assume that S is weakly cancellative, then the converse ofCorollary 6.11 also holds: Theorem 6.12.
Let S be a simple, soft, weakly cancellative Cu -semigroup satisfy-ing (O5) and (O6), and let x ∈ S satisfy x ≪ ∞ . Then x has thin boundary if andonly if x is complementable.Proof. The forwards implication follows from Corollary 6.11. To show the back-wards implication, assume that x is complementable. To verify that x has thinboundary, let t ∈ S be nonzero. Choose a nonzero element t ′ ∈ S with t ′ ≪ t .Then x ≪ ∞ = ∞ t ′ , which allows us to choose n ≥ x ≤ nt ′ . Choose t , . . . , t n ∈ S such that t ′ ≪ t ≪ t ≪ . . . ≪ t n ≪ t. Set y := t + . . . + t n . Then x ≤ nt ′ ≪ y . Since x is complementable, we obtain z ∈ S such that x + z = y .Note that y = t + . . . + t n ≪ t + . . . + t ≤ y + t, and therefore x + z = y ≪ y + t = x + z + t. By weak cancellation, we obtain x ≪ x + t , as desired. (cid:3) Theorem 6.13.
Let S be a simple, soft, weakly cancellative Cu -semigroup satisfy-ing (O5) and (O6). Then S tb is a cancellative monoid. Further, x, y ∈ S tb satisfy x ≪ y if and only if there exists z ∈ S × tb with x + z = y . Proof.
By Lemma 6.5 and 6.6, S tb is a cancellative monoid. Let x, y ∈ S tb . If x ≪ y , then by Theorem 6.12 there exists z ∈ S such that x + z = y . Since y isnot compact, we have z = 0. Further, by Lemma 6.7, we have z ∈ S tb . Conversely,if z ∈ S is nonzero such that x + z = y , then x ≪ x + z = y by definition. (cid:3) Simple, zero-dimensional Cuntz semigroups
In this section, we study countably based, simple, weakly cancellative Cu-semi-groups S that satisfy (O5) and (O6) (for example the Cuntz semigroups of separable,simple C ∗ -algebras of stable rank one). First, we prove a dichotomy: If S is zero-dimensional, then S is either algebraic or soft; see Lemma 7.1. Conversely, if S isalgebraic, then S is automatically zero-dimensional by Corollary 5.3. On the otherhand, if S is soft, then S is zero-dimensional if and only if the elements with thinboundary are sup-dense; see Theorem 7.8. We deduce that S is zero dimensional ifand only if S is the retract of a simple, algebraic Cu-semigroup; see Theorem 7.10.This should be compared with Corollary 5.9, where we showed that a separable,simple, Z -stable C ∗ -algebra has zero-dimensional Cuntz semigroup if and only if A has real rank zero or A is stably projectionless. Lemma 7.1.
Let S be a simple, weakly cancellative Cu -semigroup satisfying (O5).Assume that dim( S ) = 0 and S = { } . Then, S is either algebraic or soft.Proof. Assume that S is not soft. Then there exists a nonzero compact element in S , which by Lemma 5.5 implies that S is algebraic. (cid:3) Lemma 7.2.
Let S be a simple, weakly cancellative Cu -semigroup satisfying (O5)and (O6). Assume that S tb is sup-dense. Let x, y, z ∈ S satisfy x ≪ y + z , andassume that x has thin boundary and z is soft. Then there exist v, w ∈ S tb suchthat x = v + w, v ≪ y, and w ≪ z. Proof. If S is elementary, then it follows from [APT18, Proposition 5.1.19] that S = { } . Thus, we may assume that S is nonelementary. We may also assumethat z is nonzero, since otherwise v = x and w = 0 trivially satisfy the requiredconditions.Choose z ′ ∈ S such that x ≪ y + z ′ , and z ′ ≪ z. Using that z is nonzero and soft, we obtain a nonzero t ∈ S such that z ′ + t ≪ z .Since x has thin boundary, we have x ≪ x + t , which allows us to choose x ′ ∈ S such that x ′ ≪ x ≪ x ′ + t. Since S tb is sup-dense, we may assume that x ′ has thin boundary.Applying (O6) to x ′ ≪ x ≪ y + z ′ , we obtain e, f ∈ S such that x ′ ≪ e + f, e ≪ x, y, and f ≪ x, z ′ . Since S tb is sup-dense, we may assume that e has thin boundary.By Corollary 6.11, e is complementable. Thus, we obtain c ∈ S such that e + c = x . Then e + c = x ≪ x ′ + t ≤ e + f + t. By weak cancellation, we get c ≪ f + t and therefore c ≪ f + t ≤ z ′ + t ≪ z. By Lemma 6.7, e and c have thin boundary. Hence, v := e and w := c have thedesired properties. (cid:3) OVERING DIMENSION OF CUNTZ SEMIGROUPS 25
Proposition 7.3.
Let S be a simple, soft, weakly cancellative Cu -semigroup sat-isfying (O5) and (O6). Assume that S tb is sup-dense. Then S tb is a simple,cancellative refinement monoid and dim( S ) = 0 .Proof. By Theorem 6.13, S tb is a cancellative monoid such that x, y ∈ S tb satisfy x ≪ y if and only if there exists z ∈ S × tb with x + z = y . This implies that x, y ∈ S tb satisfy x ≤ alg y if and only if x = y or x ≪ y .It follows from Lemma 7.2 that S tb satisfies the Riesz decomposition propertyfor the pre-order induced by ≪ . Hence, dim( S ) = 0 by Proposition 5.2.Since S tb is a cancellative monoid, to show that it is a refinement monoid itsuffices to show that it satisfies the Riesz decomposition property for the algebraicpartial order ≤ alg . Let x, y, z ∈ S tb satisfy x ≤ alg y + z . We need to find y ′ , z ′ ∈ S tb such that x = y ′ + z ′ , y ′ ≤ alg y and z ′ ≤ alg z . We either have x = y + z or x ≪ y + z .In the first case, y ′ := y and z ′ := z have the desired properties. In the secondcase, we apply Lemma 7.2 to obtain y ′ , z ′ ∈ S tb such that x = y ′ + z ′ , y ′ ≪ y and z ′ ≪ z . Then y ′ ≤ alg y and z ′ ≤ alg z , which shows that y ′ and z ′ have the desiredproperties. Using that S is simple, it easily follows that S tb is a simple monoid. (cid:3) Example 7.4.
Let Z be the Cuntz semigroup of the Jiang-Su algebra Z . Thenevery element of Z has thin boundary, yet Z is neither algebraic nor soft, andtherefore Z is not zero-dimensional. (We have dim( Z ) = 1 by Example 3.19.)This show that Proposition 7.3 does not hold without assuming that S is soft.Next, we prove the converse of Proposition 7.3: Zero-dimensionality implies that S tb is sup-dense. We start with a crucial technical result. Lemma 7.5.
Let S be a weakly cancellative Cu -semigroup satisfying (O5), and let x ′ , x ′′ , x, e, t ∈ S satisfy x ′ ≪ x ′′ , and x ′′ + t ≤ x ≤ e ≪ e + t. Assume that dim( S ) = 0 . Then there exists y such that x ′ ≪ y ≪ x, and y ≪ y + t. Proof.
Applying (O5) to x ′ ≪ x ′′ ≤ e , we obtain c ∈ S such that x ′ + c ≤ e ≤ x ′′ + c. Then e ≪ e + t ≤ x ′′ + c + t. Using that dim( S ) = 0, we obtain u, v ∈ S such that u ≪ x ′′ , v ≪ c + t, and e ≪ u + v ≪ e + t. Then x ′ + c ≤ e ≪ u + v ≤ u + c + t, Using weak cancellation, we get x ′ ≪ u + t . Further, we have u + v ≪ e + t ≤ u + v + t and therefore u ≪ u + t by weak cancellation.Choose t ′ ∈ S such that t ′ ≪ t, x ′ ≪ u + t ′ , and u ≪ u + t ′ . Set y := u + t ′ . Then x ′ ≪ u + t ′ = y, and y = u + t ′ ≪ x ′′ + t ≤ x. Using that u ≪ u + t ′ and t ′ ≪ t , we get y = u + t ′ ≪ u + t ′ + t = y + t, which shows that y has the desired properties. (cid:3) Lemma 7.6.
Let S be a countably based, simple, soft, weakly cancellative Cu -sem-igroup satisfying (O5) and (O6).Assume that for every x ′ , x, t ∈ S satisfying x ′ ≪ x and t = 0 there exists y ∈ S such that x ′ ≪ y ≪ x, y ≪ y + t. Then for every x ′ , x ∈ S satisfying x ′ ≪ x there exists y ∈ S with thin boundarysuch that x ′ ≪ y ≪ x .Proof. Using that S is countably based, we can choose a sequence ( t n ) n ∈ N of nonzeroelements such that for every nonzero t ∈ S there exists n with t n ≤ t .To prove the statement, let x ′ , x ∈ S satisfy x ′ ≪ x . By assumption, we canchoose y ∈ S with x ′ ≪ y ≪ x and y ≪ y + t . Choose y ′ such that x ′ ≪ y ′ ≪ y ≪ x, y ≪ y ′ + t . Next, applying the assumption for y ′ , y , t , we obtain y such that y ′ ≪ y ≪ y and y ≪ y + t . Then choose y ′ such that y ′ ≪ y ′ ≪ y ≪ y , y ≪ y ′ + t . Inductively, choose y ′ n and y n such that x ′ ≪ y ′ ≪ . . . ≪ y ′ n ≪ y n ≪ . . . ≪ y ≪ x, y n ≪ y ′ n + t n . Set y := sup n y ′ n . Then x ′ ≪ y ′ ≤ y ≤ y ≪ x . To show that y has thinboundary, let t ∈ S be nonzero. By choice of ( t n ) n , there exists n such that t n ≤ t .Then y ≤ y n ≪ y ′ n + t n ≤ y + t n ≤ y + t, as desired. (cid:3) Proposition 7.7.
Let S be a countably based, simple, soft, weakly cancellative Cu -semigroup satisfying (O5) and (O6). Assume that dim( S ) = 0 . Then S tb issup-dense, that is, the elements with thin boundary form a basis.Proof. We verify the assumption of Lemma 7.6, which then proves the statement.Let x ′ , x, t ∈ S satisfy x ′ ≪ x and t = 0. We need to find y ∈ S such that x ′ ≪ y ≪ x, y ≪ y + t. If x ′ = 0, then set y := 0. Thus, we may assume from now on that x ′ is nonzero.Choose x ′′ , u ∈ S such that x ′ ≪ x ′′ ≪ u ≪ x. Since u is nonzero and soft, we obtain a nonzero element s ∈ S such that x ′′ + s ≪ u .By Lemma 6.4, there exists a nonzero r ∈ S with r ≤ s, t .Choose a nonzero r ′ ∈ S such that r ′ ≪ r . Then u ≪ ∞ = ∞ r ′ , which allowsus to choose n ≥ u ≤ nr ′ . Choose r , . . . , r n ∈ S such that r ′ ≪ r ≪ r ≪ . . . ≪ r n ≪ r. Set e := r + . . . + r n . As in the proof of Theorem 6.12, we obtain e ≪ e + r , andconsequently e ≪ e + t . Further, we have x ′ ≪ x ′′ , x ′′ + r ≤ x ′′ + s ≪ u ≤ nr ′ ≤ e ≪ e + t. Applying Lemma 7.5, we obtain y ∈ S such that x ′ ≪ y ≪ u, and y ≪ y + t, Now y has the desired properties. (cid:3) Theorem 7.8.
Let S be a countably based, simple, soft, weakly cancellative Cu -semigroup satisfying (O5) and (O6). Then the following are equivalent: OVERING DIMENSION OF CUNTZ SEMIGROUPS 27 (1) dim( S ) = 0 ;(2) the elements with thin boundary are sup-dense;(3) there exists a countably based, simple, algebraic, weakly cancellative Cu -semigroup T satisfying (O5) and (O6) such that S ∼ = T soft .Proof. By Proposition 7.7, (1) implies (2). Conversely, (2) implies (1) by Proposition 7.3.To show that (3) implies (1), let T be as in (3) such that S ∼ = T soft . UsingProposition 3.17 at the second step, and using Corollary 5.3 at the last step, we getdim( S ) = dim( T soft ) ≤ dim( T ) = 0 . Finally, assuming (2) let us verify (3). By Proposition 7.3, S tb is a simple,cancellative refinement monoid. Using that S is countably based and that S tb issup-dense, we can choose a countable subset M ⊆ S tb that is sup-dense.By successively adding elements to M we can construct a countable refinementsubmonoid M ⊆ S tb such that the algebraic order on M agrees with the restrictionof the algebraic order on S tb to M , that is, ( M, ≤ alg ) → ( S tb , ≤ alg ) is an order-embedding. Set T := Cu( M, ≤ alg ), the sequential round ideal completion of M with respect to the algebraic partial order; see [APT18, Section 5.5]. Then T is acountably based, algebraic Cu-semigroup. Using that M is a cancellative monoidthat is algebraically ordered and that satisfies the Riesz decomposition property, itfollows from [APT18, Proposition 5.5.8] that T is weakly cancelaltive and satisfies(O5) and (O6). Using that M is a simple monoid, it follows that T is simple.Recall that a subset I ⊆ M is an interval if I is downward hereditary and upwarddirected. Since M is countable, we can identify T with the set of intervals in M ,ordered by inclusion. The compact elements in T are precisely the intervals { y ∈ M : y ≤ alg x } for x ∈ M . Thus, the nonzero soft elements in T are precisely theintervals that do not contain a largest element. Using that every upward directedset in a countably based Cu-semigroup has a supremum, we can define α : T soft → S by α ( I ) := sup I, for every (soft) interval I ⊆ M . It is now straightforward to verify that α is anisomorphism. (cid:3) Remark 7.9.
There is no canonical choice for the algebraic Cu-semigroup T inTheorem 7.8(3). Take for example S = [0 , ∞ ]. For every supernatural number q satisfying q = q = 1, we consider the UHF-algebra M q of infinite type, andset R q := Cu( M q ); see [APT18, Section 7.4]. Then R q is a countably based,simple, algebraic, weakly cancellative Cu-semigroup satisfying (O5) and (O6), and( R q ) soft ∼ = [0 , ∞ ].Given a countably based, simple, soft, weakly cancellative Cu-semigroup S sat-isfying (O5) and (O6), we can consider T := Cu( S soft , ≤ alg ), which is a simple,algebraic, weakly cancellative Cu-semigroup satisfying (O5) and (O6) such that S ∼ = T soft . However, T is not countably based since every basis of T contains allcompact elements of T and so has at least the cardinality of S tb .Recall the notion of a retract from Definition 3.14. Theorem 7.10.
Let S be a countably based, simple, weakly cancellative Cu -sem-igroup satisfying (O5) and (O6). Then S is zero-dimensional if and only if S isa retract of a countably based, simple, algebraic, weakly cancellative Cu -semigroupsatisfying (O5) and (O6).Proof. By Lemma 7.1, S is either algebraic or soft. In the first case, we consider S as a retract of itself. In the second case, the result follows from Theorem 7.8 andProposition 3.16. (cid:3) Question 7.11.
Is every zero-dimensional, weakly cancellative Cu-semigroup sat-isfying (O5) a retract of a weakly cancellative, algebraic Cu-semigroup satisfying(O5) and (O6)?Recall that a partially ordered set M has the Riesz interpolation property if forall x , x , y , y ∈ M satisfying x j ≤ y k for all j, k ∈ { , } , there exists z ∈ M such that x j ≤ z ≤ y k for all j, k ∈ { , } . By [APRT18, Theorem 3.5], Cuntzsemigroups of stable rank one C ∗ -algebras have the Riesz interpolation property.Recall that a Cu-semigroup S is said to be almost divisible if for all n ∈ N and x ′ , x ∈ S satisfying x ′ ≪ x there exists y ∈ S such that ny ≤ x and x ′ ≤ ( n + 1) y ;see [APT18, Definition 7.3.4]. Lemma 7.12.
Let S be a retract of a Cu -semigroup T . Then, if T is almostdivisible, so is S . Further, if T has the Riesz interpolation property, then so does S .Proof. Let ι : S → T be a Cu-morphism, and let σ : T → S be a generalized Cu-morphism with σ ◦ ι = id S .First, assume that T has the Riesz interpolation property. Let x , x , y , y ∈ S satisfy x j ≤ y k for all j, k ∈ { , } . Then ι ( x j ) ≤ ι ( y k ) in T for all j, k ∈ { , } . Byassumption, there is z ∈ T such that ι ( x j ) ≤ z ≤ ι ( y k ) and thus x j ≤ σ ( z ) ≤ y k forall j, k ∈ { , } . Thus, σ ( z ) has the desired properties.Next, assume that T is almost divisible. Let n ∈ N and let x ′ , x ∈ S satisfy x ′ ≪ x . Then ι ( x ′ ) ≪ ι ( x ) in T . By assumption, there exists y ∈ T such that ny ≤ ι ( x ) and ι ( x ′ ) ≤ ( n + 1) y . Then nσ ( y ) ≤ x and x ′ ≤ ( n + 1) σ ( y ). (cid:3) Proposition 7.13.
Let S be a zero-dimensional, countably based, simple, weaklycancellative, nonelementary Cu -semigroup satisfying (O5). Then S satisfies theRiesz interpolation property and is almost divisible.Proof. By Theorem 7.10, there exists a countably based, simple, algebraic, weaklycancellative Cu-semigroup T satisfying (O5) and (O6) such that S is a retract of T .Then T c is a simple, cancellative refinement monoid, and therefore T c has the Rieszinterpolation property. Hence, T has the Riesz interpolation property by [APT18,Proposition 5.5.8(3)].Since S is nonelementary, it follows from [APGPSM10, Theorem 6.7] that T c isweakly divisible, that is, for every x ∈ T c there exist y, z ∈ T c such that x = 2 y +3 z .This implies that T is almost divisible.Now the result follows from Lemma 7.12. (cid:3) Question 7.14.
Let S be a countably based, simple, weakly cancellative Cu-sem-igroup satisfying (O5) and (O6). Assume that S is almost divisible and has theRiesz interpolation property. Is S zero-dimensional? Question 7.15.
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Email address : [email protected] URL : E. Vilalta, Departament de Matem`atiques, Universitat Aut`onoma de Barcelona,08193 Bellaterra, Barcelona, Spain
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