A unitary Cuntz semigroup for C*-algebras of stable rank one
aa r X i v : . [ m a t h . OA ] D ec A UNITARY CUNTZ SEMIGROUP FOR C ∗ -ALGEBRAS OF STABLE RANK ONE LAURENT CANTIERA bstract . We introduce a new invariant for separable C ∗ -algebras of stable rank one that merges the Cuntzsemigroup information together with the K -group information. This semigroup, termed the Cu -semigroup,is constructed as equivalence classes of pairs consisting of a positive element in the stabilization of the given C ∗ -algebra together with a unitary element of the unitization of hereditary subalgebra generated by the givenpositive element. We show that the Cu -semigroup is a well-defined continuous functor from C ∗ to a suitablecodomain category that we write Cu ∼ . Furthermore, we compute the Cu -semigroup of some specific classesof C ∗ -algebras. Finally, in the course of our investigation, we show that we can recover functorially Cu, K and K ∗ : = K ⊕ K from Cu .
1. I ntroduction
The Elliott classification program aims to find a complete invariant for nuclear separable simple C ∗ -algebras. The original version of this invariant, written Ell( A ), is based on K-Theoretical informationtogether with tracial data. As up to now, adding up decades of research, this invariant has providedsatisfactory results for simple, separable, unital, nuclear, Z -stable C ∗ -algebras satisfying the UCT as-sumption. On the other hand, the Cuntz semigroup has recently appeared to be a key tool to recoverregular properties of a (not necessarily simple) C ∗ -algebra. As a matter of fact, it has been proved thatthe Cuntz semigroup of C ( T ) ⊗ A is naturally isomorphic to Ell( A ), for any unital, simple, separable,nuclear, Z -stable C ∗ -algebra A (see [1]).Classification of non-simple C*-algebras has had an important resurgence in the recent years. When-ever considering non-simple C ∗ -algebras, the Cuntz semigroup, written Cu, seems to be a good candidateitself for classification. For instance, it has been shown that the Cuntz semigroup classifies any (unital)inductive limits of NCCW 1-algebra whose K -group is trivial (see [17]). As a matter of fact, the Cuntzsemigroup entirely captures the lattice of ideals of any separable C ∗ -algebra A , since we have a naturalcomplete lattice isomorphism between Lat( A ) ≃ Lat(Cu( A )) (see [3, Proposition 5.1.10]). However, amain limitation of the Cuntz semigroup lies within the fact that it fails to capture any K informationwhatsoever.In this paper, we introduce a unitary version of the Cuntz semigroup, denoted by Cu , for separableand stable rank one C ∗ -algebras. This construction incorporates the K groups of the C ∗ -algebra and itsideals to overcome this lack of information in the original construction of the Cuntz semigroup. We hereestablish the basic functorial properties of this construction. More concretely we show that: The author was supported by MINECO through the grant BES-2016-077192 and partially supported by the projects MDM-2014-0445 and MTM-2017-83487 at the Centre de Recerca Matem`atica in Barcelona.
Date : December 9, 2020.
Key words and phrases.
Unitary Cuntz semigroup, K-Theory, C ∗ -algebras, Category Theory. The Cu -semigroup is a continuous functor from the category of separable C ∗ -algebras with stablerank one, that we denote C ∗ , to a certain subcategory of semigroups, written Cu ∼ , modeled after thecategory Cu of abstract Cuntz semigroups. Theorem 1.1.
The functor Cu : C ∗ −→ Cu ∼ is continuous. More precisely, given an inductive system ( A i , φ i j ) i ∈ I in C ∗ , then: Cu ∼ − lim −→ (Cu ( A i ) , Cu ( φ i j )) ≃ Cu ( C ∗ − lim −→ (( A i , φ i j ))) ≃ γ ∼ (W ∼ − lim −→ (W ( A i ) , W ( φ i j ))) . We then recover functorially the K ∗ -group from the Cu -semigroup as follows: Theorem 1.2.
The functor H ∗ : Cu ∼ u −→ AbGp u ( S , u ) (Gr( S c ) , S c , u ) α Gr( α c ) yields a natural isomorphism η ∗ : H ∗ ◦ Cu , u ≃ K ∗ . This paper is organized as follow: In a first part, we construct our invariant, for separable C ∗ -algebraswith stable rank one. We show that it is an ordered monoid that satisfies the Cuntz axioms. We then finda suitable category, called the category Cu ∼ , and prove that Cu is a well-defined continuous functor.Then, we give an alternative picture of our invariant, making use of the lattice of ideals of the C ∗ -algebra, in order to compute the Cu -semigroup of classes of C ∗ -algebras, such as the simple case, AF,AI and A T algebras. We also show that Cu does not pass through pullbacks.Finally, we explicitly define the notion of recovering an invariant from another and how one canrecover classifying results. We then see that we can recover Cu, K and also K ∗ from Cu , to concludethat Cu is a complete invariant for AH d algebras with real rank zero.We mention that this article is part of a twofold. The author has been investigating further on theunitary Cuntz semigroup in [8], studying its ideal structure and a recast of a more complete version ofthe Elliott invariant. Acknowledgements.
The author is indebted to Ramon Antoine and Francesc Perera for suggestingthe construction of such an invariant, as this work was part of my PhD. I am grateful for their patience andmany fruitful discussions on the Cuntz semigroup and details about the continuity of the Cu -semigroup.2. P reliminaries The Cuntz semigroup.
We recall some definitions and properties on the Cuntz semigroup of a C ∗ -algebra. More details can be found in [3], [4], [10], [18]. C ∗ -algebra) . Let A be a C ∗ -algebra. We denote by A + the set ofpositive elements. Let a and b be in A + . We say that a is Cuntz subequivalent to b , and we write a . Cu b ,if there exists a sequence ( x n ) n ∈ N in A such that a = lim n ∈ N x n bx ∗ n . After antisymmetrizing this relation, weget an equivalence relation over A + , called Cuntz equivalence, denoted by ∼ Cu .Let us write Cu( A ) : = ( A ⊗K ) + / ∼ Cu , that is, the set of Cuntz equivalence classes of positive elements of A ⊗ K . Given a ∈ ( A ⊗ K ) + , we write [ a ] for the Cuntz class of a . This set is equipped with an addition asfollows: let v and v be two isometries in the multiplier algebra of A ⊗K , such that v v ∗ + v v ∗ = M ( A ⊗K ) . UNITARY CUNTZ SEMIGROUP FOR C ∗ -ALGEBRAS OF STABLE RANK ONE 3 Consider the ∗ -isomorphism ψ : M ( A ⊗ K ) −→ A ⊗ K given by ψ ( a b ) = v av ∗ + v bv ∗ , and we write a ⊕ b : = ψ ( a b ). For any [ a ] , [ b ] in Cu( A ), we define [ a ] + [ b ] : = [ a ⊕ b ] and [ a ] ≤ [ b ] whenever a . Cu b .In this way Cu( A ) is a semigroup called the Cuntz semigroup of A .For any ∗ -homomorphism φ : A −→ B , one can define Cu( φ ) : Cu( A ) −→ Cu( B ), a semigroup map,by [ a ] [( φ ⊗ id K )( a )]. Hence, we get a functor from the category of C ∗ -algebras into a certainsubcategory of PoM, called the category Cu, that we describe next. Definition 2.2.
Let ( S , ≤ ) be an ordered semigroup. An auxiliary relation on S is a binary relation ≺ such that:(i) For any a , b ∈ S such that a ≺ b then a ≤ b .(ii) For any a , b , c , d ∈ S such that a ≤ b ≺ c ≤ d then a ≺ d . Cu ) . Let ( S , ≤ ) be a positively ordered semigroup. For any x , y in S , we say that x is way-below y and we write x ≪ y if, for any increasing sequence ( z n ) n ∈ N that has a supremum in S such that sup n ∈ N z n ≥ y , there exists k such that z k ≥ x . This is an auxiliary relation on S called the compact-containment relation . In particular x ≪ y implies x ≤ y and we say that x is a compact elementwhenever x ≪ x .We say that S is an abstract Cu-semigroup if it satisfies the Cuntz axioms:(O1): Every increasing sequence of elements in S has a supremum.(O2): For any x ∈ S , there exists a ≪ -increasing sequence ( x n ) n ∈ N in S such that sup n ∈ N x n = x .(O3): Addition and the compact containment relation are compatible.(O4): Addition and suprema of increasing sequences are compatible.A Cu -morphism between two Cu-semigroups S , T is a positively ordered monoid morphism that pre-serves the compact containment relation and suprema of increasing sequences.The Cuntz category, written Cu, is the subcategory of PoM whose objects are Cu-semigroups andmorphisms are Cu-morphisms. . Let S be a Cu-semigroup. We say that S is countably-based if there exists a countable subset B ⊆ S such that for any a , a ′ ∈ S such that a ′ ≪ a , then there exists b ∈ B such that a ′ ≤ b ≪ a . An element u ∈ S is called an order-unit of S if for any x ∈ S , thereexists n ∈ N such that x ≤ n . u . A countably-based Cu-semigroup has a largest element or, equivalently,it is singly-generated -for instance, by its largest element-. Let us also mention that if A is a separable C ∗ -algebra, then Cu( A ) is countably-based. In fact, its largest element, that we write ∞ A , can be explicitlyconstructed as ∞ A = sup n ∈ N n . [ s A ], where s A is any strictly positive element (or full) in A . A fortiori, [ s A ] isan order-unit of Cu( A ).A notion of ideals in the category Cu has been considered in several places, we refer the reader to [3,§5.1.6] for more details. We recall that for any countably-based Cu-semigroup and any x ∈ S , the idealgenerated by x is I x : = { y ∈ S such that y ≤ ∞ . x } . For any separable C ∗ -algebra A, I Cu( I ) defines alattice isomorphism between the lattice Lat( A ) of closed two-sided ideals of A and the lattice Lat(Cu( A ))of ideals of Cu( A ). In fact, a is a full element in I if and only if [ a ] is a full element in Cu( I ). And in thiscase, we have Cu( I a ) = I [ a ] , where I a = AaA . LAURENT CANTIER
The stable rank one context.
As mentioned before, we work with separable C ∗ -algebras with sta-ble rank one. In this context, Cuntz subequivalence of positive elements admits a nicer description easierto work with. Let us shortly explicit this alternative picture and we refer the reader to [13, Proposition4.3 - §6], [9, Proposition 1] and [14] for more details.Let A be a C ∗ -algebra. We recall that an open projection is a projection p ∈ A ∗∗ such that p belongsto the strong closure of the hereditary subalgebra A p : = pA ∗∗ p ∩ A of A . These open projections are inone-to-one correspondence with the hereditary subalgebras of A . For any positive element a of A , weshall call the support projection of a , the (unique) open projection p a ∈ A ∗∗ such that her a = A p a . In thecase that A is separable, we have that p a : = S OT − lim a / n .We also recall that two open projections p , q ∈ A ∗∗ are Peligrad-Zsid´o equivalent , and we write p ∼ PZ q if there exists a partial isometry v ∈ A ∗∗ such that p = v ∗ v , q = vv ∗ , vA p ⊆ A , A q v ∗ ⊆ A . We saythat p . PZ q if there exists an open projection p ′ ∈ A ∗∗ such that p ∼ PZ p ′ ≤ q ; see [14, Definition 1.1].Suppose now that A has stable rank one. Then a . Cu b if and only if there exists x ∈ A such that xx ∗ = a and x ∗ x ∈ her b . This is in turn equivalent to saying that p a . PZ p b . In this case, for any partialisometry α ∈ A ∗∗ that realizes the Peligrad-Zsid´o equivalence between p a and p b , we have an explicitinjection as follows: θ ab ,α : her a ֒ −→ her bd α ∗ d α The next proposition is similar to [13, Proposition 3.3] and [14, Theorem 1.4]. For the sake of com-pleteness we will give a proof in this slightly di ff erent picture. Proposition 2.5.
Let p be a support projection in A ∗∗ . Let a in A + be such that p = p a . Let α be a partialisometry in A ∗∗ such that p = αα ∗ . Set q : = α ∗ α and x : = a / α . Then p ∼ PZ q if and only if x belongs toA. In this case, q = p x ∗ x .Proof. The forward implication is coming from the definition of the Peligrad-Zsid´o equivalence itself.Conversely, let us suppose that x : = a / α belongs to A . Let d be in aAa . Then there exists δ d in A such that d = a δ d a . Now observe that α ∗ d = α ∗ a / a / δ d a belongs to A . We obtain that α ∗ aAa ⊆ A , andhence α ∗ aAa ⊆ A , that is, α ∗ A p ⊆ A . Now since p is a support projection and q = α ∗ p α , we deduce that q is a support projection and moreover α ∗ A p α = A q . Finally, observe that α A q = α A q α ∗ α = A p α and that( α ∗ A p ) ∗ = A p α , so α A q ⊆ A . We conclude that p ∼ PZ q and by construction q = p x ∗ x . (cid:3) Lemma 2.6.
Let A be a C ∗ -algebra with stable rank one and let a and b be contractions in A + suchthat a . Cu b. Let α and β be in A ∗∗ such that they both realize the Peligrad-Zsid´o subequivalence ofp a . PZ p b . For any u ∈ U (her a ∼ ) , we have [ θ ∼ ab ,α ( u )] K (her b ∼ ) = [ θ ∼ ab ,β ( u )] K (her b ∼ ) where θ ∼ ab ,α (resp θ ∼ ab ,β ) is the unitized morphism of θ ab ,α in Section 2.2.Proof. Since a and b are fixed elements, we shall write θ α instead of θ ab ,α (respectively θ β for θ ab ,β ).Consider the injections given by α and β as in Section 2.2. Define x : = a / α and y : = a / β . We have x , y ∈ A . We first consider elements of aAa and the result will follow by continuity. Rewrite θ α and θ β : θ α : aAa ֒ −→ bAb θ β : aAa ֒ −→ bAba δ a x ∗ a / δ a / x a δ a y ∗ a / δ a / y UNITARY CUNTZ SEMIGROUP FOR C ∗ -ALGEBRAS OF STABLE RANK ONE 5 Let u be a unitary element of her a ∼ . There exists a pair ( u , λ ) with u ∈ her a and λ ∈ T such that u = u + λ .Let 0 < ǫ <
2. Since her a = aAa , we can find δ ∈ A such that k u − a δ a k ≤ ǫ/
3. We write M : = k δ k and we set ǫ ′ : = ǫ/ (6 M ). On the one hand, observe that k a / k ≤ k a / δ a / k ≤ M .On the other hand, since a = xx ∗ = yy ∗ , by [9, Lemma 2] we know there exists a unitary element u ǫ of her b ∼ such that k y − xu ǫ k ≤ ǫ ′ (equivalently k u ∗ ǫ x ∗ − y ∗ k ≤ ǫ ′ ). Now, we compute: k u ∗ ǫ θ ∼ α ( a δ a + λ ) u ǫ − θ ∼ β ( a δ a + λ ) k = k u ∗ ǫ x ∗ a / δ a / xu ǫ − y ∗ a / δ a / y k≤ k u ∗ ǫ x ∗ a / δ a / xu ǫ − y ∗ a / δ a / xu ǫ k + k y ∗ a / δ a / xu ǫ − y ∗ a / δ a / y k≤ k u ∗ ǫ x ∗ − y ∗ k k a / δ a / k k xu ǫ k + k y − xu ǫ k k a / δ a / k k y ∗ k≤ ǫ ′ . M + ǫ ′ . M k u ∗ ǫ θ ∼ α ( a δ a + λ ) u ǫ − θ ∼ β ( a δ a + λ ) k ≤ ǫ/ . Combining the fact that u and a δ a + λ are close up to ǫ/ θ ∼ α and θ ∼ β , we concludethat k u ∗ ǫ θ ∼ α ( u ) u ǫ − θ ∼ β ( u ) k ≤ ǫ <
2. On the other hand, it is well-known that unitary elements that are closeenough (i.e. k u − v k <
2) are homotopic. We conclude that u ∗ ǫ θ ∼ α ( u ) u ǫ ∼ h θ ∼ β ( u ) and the result follows. (cid:3) We will use C ∗ to denote the category of separable C ∗ -algebras of stable rank one. Also, we denoteby Mon ≤ the category of ordered monoids, in contrast to the category of positively ordered monoids, thatwe write PoM. This will be reminded several times throughout the paper.3. T he Cu semigroup In this section, we define the invariant and establish its first properties. The unitary Cuntz semigroupconsists of classes of pairs of element ( a , u ), where a is a positive element of A ⊗ K and u is a unitary ofher a ∼ , under a suitable equivalence relation, written ∼ , that is built using the Cuntz subequivalence tocompare positive elements and using Lemma 2.6 to compare unitary elements. Our main result focuseson the continuity of our invariant. Also, we give an algebraic property that characterizes the notion ofreal rank zero.3.1. The . binary relation. Let A be a C ∗ -algebra with stable rank one . Let a , b ∈ A + and let u , v beunitary elements of her a ∼ and her b ∼ respectively. We say that ( a , u ) is unitarily Cuntz subequivalent to( b , v ), and we write ( a , u ) . ( b , v ) if a . Cu b [ θ ∼ ab ,α ( u )] = [ v ] in K (her b ∼ )where θ ab ,α is the injection given by a partial isometry α as constructed in Section 2.2. Lemma 3.1.
The relation . is reflexive and transitive.Proof. Reflexivity of . follows from the fact that . Cu is reflexive and that id her a = θ aa , p a .Now let a , b and c be in A + and let u a , u b and u c be unitary elements of her a ∼ , her b ∼ and her c ∼ respectively. Assume that ( a , u a ) . ( b , u b ) and ( b , u b ) . ( c , u c ). By hypothesis, we know that a . Cu b LAURENT CANTIER and b . Cu c . Since A has stable rank one, there exist x , y ∈ A such that a = xx ∗ , b = yy ∗ , x ∗ x ∈ her b and y ∗ y ∈ her c . Let us consider the polar decompositions of x and y . That is, x = a / α, y = b / β ,for some partial isometries α, β of A ∗∗ . Using Section 2.2, we get p a = αα ∗ ∼ PZ α ∗ α ≤ p b and also p b ∼ PZ β ∗ β ≤ p c . We set q a : = α ∗ α, q b : = β ∗ β . One can check that p a = γγ ∗ and that γ : = αβ is a partialisometry of A ∗∗ .Let us write z : = a / γ . Observe that zz ∗ = a and also z = x β . We hence compute that z ∗ z = β ∗ x ∗ x β ∈ her c . We deduce that zz ∗ = a and z ∗ z ∈ her c . By [4, Proposition 2.12] we may write x : = u ∗ ( x ∗ x ) / forsome element u of A . Since ( x ∗ x ) ∈ A p b and β ∗ A p b ⊆ A , we deduce that β ∗ x ∗ is in A , and hence z ∈ A .Using Proposition 2.5, we obtain that q c : = γ ∗ γ is the support projection of z ∗ z and is Peligrad-Zsid´oequivalent to p a . Finally, Lemma 2.6 tells us that θ ac ,γ : = θ bc ,β ◦ θ ab ,α is one of the morphisms describedin Section 2.2, from which the transitivity of . follows. (cid:3) Standard maps.
We have seen that for any unitary element u of her a ∼ and any partial isometry α ∈ A ∗∗ such that p a = αα ∗ , the K -class of θ ∼ ab ,α ( u ) does not depend on the α chosen. In the sequel,whenever a . Cu b , we will refer to the maps θ ∼ ab ,α as standard maps and will rewrite them as θ ab . Inparticular, whenever a ≤ b observe that the canonical inclusion map i is a standard map. Also, notice thatevery standard morphism between a and b gives rise to the same group morphism at the K -level, that wewill denote by χ ab . That is, χ ab : = K ( θ ab ) : K (her a ) −→ K (her b ).3.3. The Cu -semigroup. Let A be a C ∗ -algebra with stable rank one and let H ( A ) : = { ( a , u ) : a ∈ ( A ⊗ K ) + , u ∈ U (her a ∼ ) } . By antisymetrizing . , we define an equivalence relation on H ( A ) called the unitary Cuntz equivalence ,written ∼ . Now we define the unitary Cuntz semigroup of A , and we write Cu ( A ), as follows:Cu ( A ) : = H ( A ) / ∼ The equivalence class of an element ( a , u ) ∈ H ( A ) is denoted by [( a , u )].By the isomorphism ψ : M ( A ⊗K ) ≃ A ⊗K (see Paragraph 2.1), given any two elements ( a , u ) , ( b , v ) ∈ H ( A ), we know that a ⊕ b : = ψ ( a b ) is a positive element of A ⊗ K . Besides, ( her a
00 her b ) ⊆ her( a ⊕ b ) andhence u ⊕ v : = ( u v ) is a unitary element of her( a ⊕ b ) ∼ . Thus, for [( a , u )] , [( b , v )] ∈ Cu ( A ), we write[( a , u )] ≤ [( b , v )] if ( a , u ) . ( b , v ), and we set [( a , u )] + [( b , v )] : = [( a ⊕ b ) , ( u ⊕ v )].It is easy to check that (Cu ( A ) , + , ≤ ) defined is a partially ordered monoid whose neutral elementis [(0 A , C )]. The positive elements of Cu ( A ) are of the form ([ a , a ∈ ( A ⊗ K ) + and thusCu ( A ) is in general not positively ordered.We now show that (Cu ( A ) , ≤ ) satisfies the Cuntz axioms mentioned in Paragraph 2.1 as an orderedmonoid. Proposition 3.2.
Let A be a C ∗ -algebra with stable rank one. Let ( a n ) n be a sequence in A + such thata n . Cu a m , for any n ≤ m. Let a ∈ A + be any representative of sup n [ a n ] ∈ Cu( A ) obtained from axiom(O1). Then for any unitary element u ∈ her a ∼ , there exists a unitary element in her a ∼ n for some n ∈ N such that [( a n , u n )] ≤ [( a , u )] in Cu ( A ) . UNITARY CUNTZ SEMIGROUP FOR C ∗ -ALGEBRAS OF STABLE RANK ONE 7 Proof.
For any n ∈ N , consider b n : = ( a − / n ) + . It is well-known that ([ b n ]) n is a ≪ -increasing sequencein Cu( A ) whose supremum is [ a ]; see e.g [18, Proposition 2.61]. Also, it is not hard to check thatAbGp − lim −→ (K (her b n ) , χ b n b m ) ≃ (K (her a ) , χ b n a ). Since we are in the category AbGp, for any [ u ] ∈ K (her a ), we can find [ u n ] ∈ K (her b n ) such that χ b n a ([ u n ]) = [ u ]. Since A has stable rank one, then sodoes her b ∼ n . Hence using K -surjectivity, we can find a unitary element u n of her b ∼ n whose K -class is[ u n ]. On the other hand, ([ a m ]) m is an increasing sequence in Cu( A ) whose supremum is [ a ] and hencethere exists m ∈ N such that [ b n ] ≤ [ a m ] in Cu( A ). So we can consider the unitary element θ b n a m ( u n ) inher a ∼ m . By transitivity of . , we obtain that χ a m a ([ θ b n a m ( u n )]) = χ a m a ◦ χ b n a m ([ u n ]) = χ b n a ([ u n ]) = [ u ] andthe result follows. (cid:3) Lemma 3.3.
Let A be a C ∗ -algebra with stable rank one. Then any increasing sequence in Cu ( A ) has asupremum.Proof. Let ([( a n , u n )]) n ∈ N be an increasing sequence in Cu ( A ). Then ([ a n ]) n ∈ N is an increasing sequencein Cu( A ). By (O1) in Cu( A ), the sequence ([ a n ]) n ∈ N has a supremum [ a ] in Cu( A ). Now, let n ≤ m . Since[( a n , u n )] ≤ [( a m , u m )], we get that χ a n a m ([ u n ]) = [ u m ]. By transitivity of . , we obtain that χ a n a ([ u n ]) = χ a m a ([ u m ]) in K (her a ). Write [ u ] : = χ a n a ([ u n ]). We deduce that [( a , u )] ≥ [( a n , u n )] in Cu ( A ) for any n ∈ N .Let us check that [( a , u )] is in fact the supremum of the sequence. Let [( b , v )] ∈ Cu ( A ) such that[( b , v )] ≥ [( a n , u n )] for every n ∈ N . Since [ a ] = sup n ∈ N [ a n ], we have [ b ] ≥ [ a ] in Cu( A ). Using transitivityof . , the following diagram is commutative:K (her a ∼ n ) χ ana & & ◆◆◆◆◆◆◆◆◆◆ χ anb ) ) χ anam (cid:15) (cid:15) K (her a ∼ ) χ ab / / K (her b ∼ )K (her a ∼ m ) χ ama ♣♣♣♣♣♣♣♣♣♣♣ χ amb Hence for every n and m in N , we have χ a n b ([ u n ]) = χ a m b ([ u m ]) = χ ab ([ u ]) in K (her b ). We deduce that χ ab ([ u ]) = [ v ] in K (her b ) and hence [( a , u )] ≤ [( b , v )]. (cid:3) Proposition 3.4.
Let A be a C ∗ -algebra with stable rank one and let [( a , u )] , [( b , v )] ∈ Cu ( A ) . Then [( a , u )] ≪ [( b , v )] if and only if [ a ] ≪ [ b ] in Cu( A ) and χ ab ([ u ]) = [ v ] in K (her b ) .Proof. Suppose that [( a , u )] ≪ [( b , v )]. A fortiori [( a , u )] ≤ [( b , v )], so χ ab [ u ] = [ v ]. Now let ([ c n ]) n bean increasing sequence in Cu( A ) whose supremum [ c ] satisfies [ c ] ≥ [ b ]. Write w : = θ bc ( v ) and consider s : = [( c , w )] ∈ Cu ( A ). By Proposition 3.2, we know that there exists a unitary element w n of her c ∼ n forsome n ∈ N such that χ c n c ([ w n ]) = [ w ]. Now define s k : = [ c n + k , θ c n c n + k ( w n )], then ( s k ) k is an increasingsequence in Cu ( A ). By the description of suprema obtained in Lemma 3.3, we know that ( s k ) k admits s as a supremum. Further, s ≥ [( b , v )] and since [( a , u )] ≪ [( b , v )], we deduce that there exists k ∈ N suchthat [( a , u )] ≤ s k and hence that [ a ] ≤ [ c n + k ]. We conclude that [ a ] ≪ [ b ] in Cu( A ). LAURENT CANTIER
Conversely, let [( a , u )] , [( b , v )] ∈ Cu ( A ) such that [ a ] ≪ [ b ] in Cu( A ) and χ ab [ u ] = [ v ] in K (her b ).Let ([( c n , w n )]) n be an increasing sequence in Cu ( A ) that has a supremum in Cu ( A ), say [( c , w )]. Alsosuppose that [( b , v )] ≤ [( c , w )]. First, by transitivity of . , observe that χ ac ([ u ]) = χ bc ◦ χ ab ([ u ]) = [ w ] inK (her c ).Arguing as in the proof of [7, Lemma 4.3], since A has stable rank one, we can find a strictly decreasingsequence ( ǫ n ) n in R ∗ + and unitary elements ( u n ) n in ( A ⊗ K ) ∼ such thather( c − ǫ ) + ⊆ u (her( c − ǫ ) + ) u ∗ ⊆ ... ⊆ u n ... u (her( c n + − ǫ n + ) + ) u ∗ ... u ∗ n ⊆ ... and such that sup n [( c n − ǫ n ) + ] = [ c ] in Cu( A ). Hence, by Proposition 3.2 we can find a unitary element ˜ w k of (her( c k − ǫ k ) + ) ∼ such that χ ( c k − ǫ k ) + c k [ ˜ w k ] = [ w k ] in K (her c k ), for every k ∈ N . Now, using the sameargument as in the proof of Proposition 3.2, we observe thatAbGp − lim −→ (K (her( c n − ǫ n ) + ) , χ ( c n − ǫ n ) + ( c m − ǫ m ) + ) ≃ (K (her c ) , χ ( c n − ǫ n ) + c ) . On the other hand, since [ a ] ≪ [ b ] ≤ sup n [( c n − ǫ n ) + ], there exists l ∈ N such that [ a ] ≤ [( c l − ǫ l ) + ] inCu( A ). Without loss of generality, l ≥ k . Using transitivity of . again, we have that χ ( c l − ǫ l ) + c ([ ˜ w l ]) = χ c l c ◦ χ ( c l − ǫ l ) + c l ([ ˜ w l ]) = [ w ] = χ ac ([ u ]) = χ ( c l − ǫ l ) + c ◦ χ a ( c l − ǫ l ) + ([ u ]) in K (her c ). Since we are in the categoryAbGp, there exists some l ′ ≥ l such that χ ( c l − ǫ l ) + ( c l ′ − ǫ l ′ ) + ([ ˜ w l ]) = χ ( c l − ǫ l ) + ( c l ′ − ǫ l ′ ) + ◦ χ ( ac l − ǫ l ) + ([ u ]). Composingwith χ ( c l ′ − ǫ l ′ ) + c l ′ on both sides, we finally obtain that [ w l ′ ] = χ ac l ′ [ u ] and hence [( a , u )] ≤ [( c l ′ , w l ′ )], whichcompletes the proof. (cid:3) Corollary 3.5.
Let A be a C ∗ -algebra with stable rank one and let [( a , u )] ∈ Cu ( A ) . Then [( a , u )] iscompact if and only if [ a ] is compact in Cu( A ) . Theorem 3.6.
Let A be a C ∗ -algebra with stable rank one. Then (Cu ( A ) , ≤ ) satisfies axioms (O1), (O2),(O3), and (O4).Proof. (O1) follows from Lemma 3.3.(O2): Let s : = [( a , u )] ∈ Cu ( A ). We want to write s as the supremum of a ≪ -increasing sequence inCu ( A ). By (O2), we can find a ≪ -increasing sequence ([ a n ]) n in Cu( A ) such that sup n [ a n ] = [ a ]. Write a n any representative of [ a n ] in ( A ⊗K ) + . Using Proposition 3.2, we know that we can find a unitary element u n of her a ∼ n for some n ∈ N such that [( a n , u n )] ≤ [( a , u )]. Now we consider s k : = [( a n + k , θ a n a n + k ( u n ))], forany k ∈ N . Then, by Proposition 3.4 we deduce that ( s k ) k is a ≪ -increasing sequence in Cu ( A ). By thedescription of suprema obtained in Lemma 3.3, sup k s k = s .(O3): Let [( a , u )] ≪ [( b , v )] and [( a , u )] ≪ [( b , v )]. We already know that [( a , u )] + [( a , u )] ≤ [( b , v )] + [( b , v )] and that [ a ] + [ a ] ≪ [ b ] + [ b ] in Cu( A ). The conclusion follows fromProposition 3.4.(O4): Let ([( a n , u n )]) n ∈ N and ([( b n , v n )]) n ∈ N be two increasing sequences in Cu ( A ). Let [( a , u )] : = sup n ∈ N [( a n , u n )] and [( b , v )] : = sup n ∈ N [( b n , v n )]. Now we define ([( c n , w n )]) : = ([( a n , u n )]) + ([( b n , v n )]) for any n ∈ N . Since [ c n ] = [ a n ] + [ b n ] in Cu( A ) and Cu( A ) satisfies (O4), we have sup n ∈ N [ c n ] = [ a ⊕ b ]. Also, weknow that χ a n a ([ u n ]) = [ u ] and χ b n b ([ v n ]) = [ v ], and hence we obtain χ c n c ( u n ⊕ v n ) = u ⊕ v . We concludethat sup and + are compatible in Cu ( A ), using Lemma 3.3. (cid:3) UNITARY CUNTZ SEMIGROUP FOR C ∗ -ALGEBRAS OF STABLE RANK ONE 9 The Cu -semigroup as a functor. Now that we have proved that Cu ( A ) is a semigroup satisfyingthe Cuntz axioms, the aim is to define a functor Cu from the category C ∗ to a suitable category ofsemigroups as was done for the Cu-semigroup; see [3, Chapter 3], [10]. Since Cu ( A ) is usually notpositively ordered, we need to adjust the definition of the codomain category. In the sequel, we show thatCu : C ∗ −→ Cu ∼ is a well-defined functor that is continuous. Definition 3.7.
The unitary Cuntz category , written Cu ∼ is the subcategory of Mon ≤ whose objects areordered monoids satisfying the Cuntz axioms and such that 0 ≪ ≤ -morphismsthat respect suprema of increasing sequences and the compact-containment relation. Definition 3.8.
Let M ∈ Mon ≤ and let S ∈ Cu ∼ . We define their positive cones , that we write M + and S + respectively, as the subset of positive elements. Observe that M + ∈ PoM and S + ∈ Cu.
Lemma 3.9.
The category Cu (respectively PoM ) is a coreflective subcategory of Cu ∼ (respectively Mon ≤ ). More precisely, the assignment S −→ S + defines a coreflector ν + : Cu ∼ −→ Cu .Proof. Since Cu ∼ -morphisms respect ≤ , we deduce that ν + is a well-defined functor. Moreover, one cancheck that Hom Cu ∼ ( i ( S ) , T ) ≃ Hom Cu ( S , ν + ( T )) for any S ∈ Cu and T ∈ Cu ∼ . We get that the inclusionfunctor i : Cu ֒ −→ Cu ∼ is left adjoint to ν + , which implies that Cu is a full (obviously faithful) coreflectivesubcategory of Cu ∼ . (cid:3) Proposition 3.10.
Let ϕ : A −→ B a ∗ -homomorphism between A , B ∈ C ∗ . We denote by ϕ ∼ the unitizedmorphism between ( A ⊗ K ) ∼ −→ ( B ⊗ K ) ∼ . Then: Cu ( ϕ ) : Cu ( A ) −→ Cu ( B )[( a , u )] [( ϕ ( a ) , ϕ ∼ ( u ))] is a Cu ∼ -morphism.Proof. Let a ∈ A ⊗ K . The restriction ϕ | her a : her a −→ her ϕ ( a ) of ϕ gives us the following commutativesquare: her a (cid:15) (cid:15) ϕ / / her( ϕ ( a )) (cid:15) (cid:15) her a ∼ ϕ ∼ / / (her ϕ ( a )) ∼ Hence, ϕ ∼ ( u ) is a unitary element of (her ϕ ( a )) ∼ and we deduce that [( ϕ ( a ) , ϕ ∼ ( u ))] ∈ Cu ( B ). Let uscheck it does not depend on the representative ( a , u ) chosen. Let [( a , u )] , [( b , v )] ∈ Cu ( A ) such that[( a , u )] ≤ [( b , v )]. Then we get a . Cu b in A ⊗ K . Since ϕ is a ∗ -homomorphism, we deduce that ϕ ( a ) . Cu ϕ ( b ) in B ⊗ K . Further, using [15, Theorem 26.55], if α is a partial isometry of ( A ⊗ K ) ∗∗ that realizes one of our standard morphisms θ ab ,α (see Section 3.2) between her a and her b , then ϕ ∗∗ ( α )is a partial isometry of ( B ⊗ K ) ∗∗ that realizes θ ϕ ( a ) ϕ ( b ) ,ϕ ∗∗ ( α ) between her ϕ ( a ) and ϕ ( b ). It follows that thefollowing diagram is commutative: her a ∼ θ ab ,α / / ϕ ∼ (cid:15) (cid:15) her b ∼ ϕ ∼ (cid:15) (cid:15) (her ϕ ( a )) ∼ θ ϕ ( a ) ϕ ( b ) ,ϕ ∗∗ ( α ) / / (her ϕ ( b )) ∼ from which we deduce that θ ϕ ( a ) ϕ ( b ) ( ϕ ∼ ( u )) ∼ ϕ ∼ ( v ) and thus [( ϕ ( a ) , ϕ ∼ ( u ))] ≤ [( ϕ ( b ) , ϕ ∼ ( v ))]. So Cu ( ϕ )is indeed well-defined, respects ≤ and it is easy to check that Cu ( ϕ ) also respects addition. We concludethat Cu ( ϕ ) is a Mon ≤ -morphism.By Proposition 3.4, Cu ( ϕ ) preserves the compact containment relation. Finally, we leave to the readerto check that Cu ( ϕ ) preserves suprema of increasing sequences. (cid:3) Corollary 3.11.
The assignment A Cu ( A ) from C ∗ to Cu ∼ is a functor. It has been shown that the functor Cu from the category of C ∗ -algebras to Cu is arbitrarily continuous([3, Corollary 3.2.9]), generalizing the result of [10, Theorem 2] that established sequential continuity.We shall expect a similar result for the functor Cu since many C ∗ -algebras are built as inductive limits.In the sequel, we shall prove that Cu : C ∗ −→ Cu ∼ is a continuous functor, using an analogousprocess as in [3, Chapter 2 and 3] for the Cu-semigroup.To do so, we are going to consider a pre-completed version of Cu , that we will denote by W , to thenextend the result to Cu using Category Theory techniques. We first introduce categories analogous toPreW and W defined in [3, Chapter 2] that we shall call PreW ∼ and W ∼ (cf [3, §2.1.1]). Since that themain di ff erence of our context lies in the fact that the underlying monoids involved are not necessarilypositively ordered, most of the proofs from [3] remain valid. (We give additional details when needed.)3.5. The categories
PreW ∼ and W ∼ . Let S ∈ Mon ≤ and consider an auxiliary relation ≺ on S . For any s ∈ S we denote s ≺ : = { s ′ ∈ S | s ′ ≺ s } . Let us recall the W-axioms from [3, Definition 2.1.2]:(W1): For any s ∈ S , there exists a ≺ -increasing sequence ( s k ) k in s ≺ such that for any s ′ ∈ s ≺ , thereexists some k such that s ′ ≺ s k .(W2): For any s ∈ S , we have s = sup s ≺ .(W3): Addition and ≺ are compatible.(W4): For any s , t , x ∈ S such that x ≺ s + t , we can find s ′ , t ′ ∈ S such that s ′ ≺ s , t ′ ≺ t and x ≺ s ′ + t ′ .A PreW ∼ -semigroup is a pair ( S , ≺ ), where S ∈ Mon ≤ and ≺ is an auxiliary relation on S such that( S , ≺ ) satisfies axioms (W1)-(W3)-(W4) and such that 0 ≺
0. If moreover ( S , ≺ ) satisfies (W2), we say itis a W ∼ -semigroup .A W ∼ -morphism between any two S , T ∈ PreW ∼ is a Mon ≤ -morphism g : S −→ T that respects theauxiliary relation and satisfies the following W ∼ -continuity axiom :(M): For any s ∈ S and t ∈ T such that t ≺ g ( s ), there exists s ′ ∈ s ≺ such that t ≤ g ( s ′ ).Finally, we define the categories PreW ∼ and W ∼ whose objects are respectively PreW ∼ -semigroupsand W ∼ -semigroups and whose morphisms are W ∼ -morphisms.The category PreW ∼ has inductive limits. More precisely, let ( S i , ϕ i j ) i ∈ I be an inductive system inPreW ∼ and let S : = Mon ≤ − lim −→ ( S i , ϕ i j ). Then ( S , ≺ ) ≃ PreW ∼ − lim −→ ( S i , ϕ i j ), where ≺ is the followingauxiliary relation on S : s ≺ t in S if, ϕ ik ( s i ) ≺ ϕ jk ( t j ), where s i ∈ S i , t j ∈ S j are representatives of s , t respectively and k ≥ i , j . Finally, as in [3, Proposition 2.1.6] we easily deduce that W ∼ is a (full) reflectivesubcategory of PreW ∼ and we denote the explicit reflector obtained by µ ∼ : PreW ∼ −→ W ∼ . A direct con-sequence is that W ∼ has inductive limits. More particularly, W ∼ − lim −→ ( S i , ϕ i j ) = µ ∼ (PreW ∼ − lim −→ ( S i , ϕ i j )). UNITARY CUNTZ SEMIGROUP FOR C ∗ -ALGEBRAS OF STABLE RANK ONE 11 Now that we have a well-defined categorical setup, we define a pre-completed version of Cu andshow that it is continuous. More precisely, we build a functor from the category C ∗ loc to the category W ∼ ,termed W . First let us recall some defiitions and properties about C ∗ loc . We refer the reader to [3, §2.2]for more details.3.6. Local C ∗ -algebras. A local C ∗ -algebra A is an upward-directed union of C ∗ -algebras. That is, A = ∪ i A i where { A i } i is a family of complete ∗ -invariant subalgebras such that for any i , j , there exists k ≥ i , j such that A i ∪ A j ⊆ A k .If A is a local C ∗ , then so is M k ( A ) for any k ∈ N . In fact, M k ( A ) sits as upper-left corner inside M k ′ ( A ) for any k ′ ≥ k and we can picture any M k ( A ) as a corner of M ∞ ( A ) : = ∪ k M k ( A ), which is again alocal C ∗ -algebra. Observe that the completion of a local C ∗ -algebra A , that we write A , is a C ∗ -algebra.In particular, we have M k ( A ) ≃ M k ( A ) for any k ∈ N and M ∞ ( A ) ≃ A ⊗ K . Further A is closed underfunctional calculus. Moreover, for any local C ∗ -algebra A : = ∪ i A i , if each A i has stable rank one, thenby [16, Theorem 5.1], we get that A has stable rank one. We may abuse the language and say that A hasstable rank one.We now consider C ∗ loc , the category whose objects are separable local C ∗ -algebras that have stablerank one and morphisms are ∗ -homomorphisms. Obviously, C ∗ is a full subcategory of C ∗ loc . In fact, C ∗ is a reflective subcategory of C ∗ loc and the assignment A A defines a reflector from C ∗ loc to C ∗ that wedenote by γ .Finally, let ( A i , ϕ i j ) i ∈ I be an inductive system in C ∗ loc . As in [3, §2.2.8], we consider the algebraicinductive limit A alg : = F i ∈ I A i / ∼ with the pre-norm: k x k : = inf j {k ϕ i j ( x ) k} ), for x ∈ A i and we define: C ∗ loc − lim −→ ( A i , ϕ i j ) : = ( A alg / N , k k )where N : = { a ∈ A alg , k a k = } . Besides, ϕ i j induces a ∗ -homomorphism that we also write ϕ i j : M ∞ ( A i ) −→ M ∞ ( A j ) and we have C ∗ loc − lim −→ ( M ∞ ( A i ) , ϕ ∼ i j ) ≃ M ∞ ( C ∗ loc − lim −→ ( A i , ϕ i j )). See [3, §2.2.8].3.7. The precompleted unitary Cuntz semigroup W ( A ) . We briefly recall the definition of the pre-completed Cuntz semigroup W ( A ) of a C ∗ -algebra A and we refer the reader to [3, §2.2] for details. Infact, we give an equivalent definition that can be found in [3, Remark 3.2.4]; see also [3, Lemma 3.2.7].Let A ∈ C ∗ loc . We define W( A ) : = { [ a ] ∈ Cu( A ) | a ∈ M ∞ ( A ) + } . Obviously, (W( A ) , + , ≤ ) ∈ PoM as asubmonoid of Cu( A ). Given [ a ] , [ b ] ∈ W( A ), we write [ a ] ≺ [ b ] if a . Cu ( b − ǫ ) + in M ∞ ( A ) + for some ǫ >
0. We have that (W( A ) , ≺ ) ∈ W. (See [3, Proposition 2.2.5].)
Lemma 3.12.
Let A ∈ C ∗ loc and let B : = A be its completion in C ∗ . Then, for any a ∈ A + we haveaAa = aBa.Proof. The direct inclusion is trivial. Now let x ∈ aBa . Then there exists a sequence ( b k ) k in B such that x = lim k ab k a . Furthermore, for any k ∈ N , there exists a sequence ( a k , i ) i in A such that b k = lim i a k , i . Wededuce that x = lim k a (lim i a k , i ) a = lim k lim i ( aa k , i a ). Thus x ∈ aAa . (cid:3) Definition 3.13.
Let A ∈ C ∗ loc and let B : = A be its completion in C ∗ . For a ∈ A + , we define the hereditarysubalgebra generated by a as her a : = aBa . We have now all the tools to define a precompleted version of Cu that we will denote by W ( A ), as asubmonoid of Cu ( A ). Definition 3.14.
Let A ∈ C ∗ loc . We define W ( A ) : = { [( a , u )] ∈ Cu ( A ) | a ∈ M ∞ ( A ) + } . Obviously,(W ( A ) , + , ≤ ) ∈ Mon ≤ as a submonoid of Cu ( A ). Now we equip W ( A ) with the following binaryrelation. Let [( a , u )] , [( b , v )] ∈ W ( A ), we say [( a , u )] ≺ [( b , v )] if: a . Cu ( b − ǫ ) + in M ∞ ( A ) + for some ǫ > . [ θ ab ( u )] = [ v ] in K (her b ∼ ) . Proposition 3.15.
Let A ∈ C ∗ loc . Let ( a n ) n be a sequence in A + such that a n . Cu a m for any n ≤ m. Alsowe suppose that [ a n ] n has a supremum in W ( A ) that we write [ a ] . Let a ∈ A + be any representative of sup n [ a n ] ∈ W( A ) . Then for any unitary element u ∈ her a ∼ , there exists a unitary element u n in her a ∼ n forsome n ∈ N such that [( a n , u n )] ≤ [( a , u )] in W ( A ) .Proof. Combine the fact that A has stable rank one, with Definition 3.13 and the result follows fromProposition 3.2. (cid:3) Proposition 3.16. (cf [3, Proposition 2.2.5] ). Let A ∈ C ∗ loc . The relation defined in Definition 3.14 is anauxiliary relation and ( W ( A ) , ≺ ) satisfies axioms (W1), (W2), (W3) and (W4). That is, ( W ( A ) , ≺ ) ∈ W ∼ .We may omit the reference to ≺ and simply write W ( A ) ∈ W ∼ .Proof. Let us check that ≺ is an auxiliary relation on W ( A ). If [( a , u )] ≤ [( b , v )] ≺ [( c , w )] ≤ [( d , z )],then we have χ ad ([ u ]) = [ z ] and a . Cu ( d − ǫ ) + for some ǫ > b . Cu ( c − δ ) + for some δ >
0. Thus,[( a , u )] ≺ [( d , z )].Now, given [( a , u )] ∈ W ( A ), use Proposition 3.15 to construct a sequence in W ( A ) where [(( a − / n ) + , u n )] ≺ [( a , u )] and in such a way that [( a , u )] = sup n [(( a − / n ) + , u n )]; see Lemma 3.3. Thus (W2)holds in W ( A ).If [( b , v )] ≺ [( a , u )], then, by Proposition 3.4, we have [( b , v )] ≪ [( a , u )] in Cu ( A ) and thus [( b , v )] ≤ [( a − / n ) + , u n ] for some n ∈ N . Hence (W1) holds. To check (W3) and (W4) is routine. (cid:3) Proposition 3.17.
Let ϕ : A −→ B be a ∗ -homomorphism between A , B ∈ C ∗ loc , and denote by ϕ itsextension to M ∞ ( A ) . We write ϕ : = γ ( ϕ ) and ϕ ∼ : M ∞ ( A ) ∼ −→ M ∞ ( B ) ∼ its unitization. Then the map: W ( ϕ ) : W ( A ) −→ W ( B )[( a , u )] [( ϕ ( a ) , ϕ ∼ ( u ))] is a W ∼ -morphism.Proof. Using the same argument as in Proposition 3.10, we easily deduce that W ( ϕ ) is a Mon ≤ -morphismthat respects ≺ . Further, we have to check that W ( ϕ ) satisfies the W ∼ -continuity axiom (see Section 3.5).Let us write f : = W ( ϕ ). Let x : = [( a , u )] ∈ W ( A ) and y : = [( b , v )] ∈ W ( B ) such that y ≺ f ( x ). Wehave to find x ′ ∈ W ( A ) such that x ′ ≺ x and y ≤ f ( x ′ ).We know that there exists k > b ] ≤ [( ϕ ( a ) − / k ) + ] in W( B ) and χ b ϕ ( a ) ([ v ]) = [ ϕ ∼ ( u )]in K (her ϕ ( a )). On the other hand, observe that ([( a − / n ) + ]) n is an increasing sequence in W( A )that has admits [ a ] as supremum in W( A ). Thus, by Proposition 3.15, we can find a unitary element UNITARY CUNTZ SEMIGROUP FOR C ∗ -ALGEBRAS OF STABLE RANK ONE 13 u n ∈ her(( a − / n ) + ) ∼ for some n ∈ N , such that [(( a − / n ) + , u n )] ≤ [( a , u )] in W ( A ). In particular,[(( ϕ ( a ) − / m ) + , ϕ ∼ ( θ ( a − / n ) + ( a − / m ) + ( u n )))] ≤ [( ϕ ( a ) , ϕ ∼ ( u ))] in W ( B ) for any m ≥ n . Now choose m > k , n ,we get the following: [ b ] ≤ [( ϕ ( a ) − / k ) + ] ≤ [( ϕ ( a ) − / m ) + ] in W( B ) . [ θ b ϕ ( a ) ( v )] = [ θ ( ϕ ( a ) − / n ) + ϕ ( a ) ( ϕ ∼ ( u n ))] in K (her ϕ ( a )) . By transitivity of . , we obtain:[ θ ( ϕ ( a ) − / m ) + ϕ ( a ) ◦ θ b ( ϕ ( a ) − / m ) + ( v )] = [ θ ( ϕ ( a ) − / m ) + ϕ ( a ) ◦ θ ( ϕ ( a ) − / n ) + ( ϕ ( a ) − / m ) + ( ϕ ∼ ( u n ))] in K (her ϕ ( a )) . Finally, since AbGp − lim −→ (K (her( ϕ ( a ) − / m ) + ) , χ ( ϕ ( a ) − / n ) + ( ϕ ( a ) − / m ) + ) ≃ (K (her a ) , χ ( ϕ ( a ) − / m ) + ϕ ( a ) ), weconclude that there exists l ≥ m such that: [ b ] ≤ [( ϕ ( a ) − / l ) + ] in W( B ) . [ θ b ( ϕ ( a ) − / l ) + ( v )] = [ θ ( ϕ ( a ) − / n ) + ( ϕ ( a ) − / l ) + ( ϕ ∼ ( u n ))] in K (her( ϕ ( a ) − / l ) + ) . Write x ′ : = [(( a − / l ) + , θ ( a − / n ) + ( a − / l ) + ( u n ))]. Then we already know that x ′ ≺ x in W ( A ) and the aboveexactly states that y ≤ f ( x ′ ) in W ( B ). (cid:3) Corollary 3.18.
The assignment A W ( A ) from C ∗ loc to W ∼ is a functor. Theorem 3.19.
The functor W : C ∗ loc −→ W ∼ is continuous.Proof. This proof is an adapted version of [3, Theorem 2.2.9]. Let ( A i , ϕ i j ) i ∈ I be an inductive systemin C ∗ loc and let ( A alg / N , ϕ i ∞ ) be its inductive limit. Without loss of generality, we can suppose that each A i ≃ M ∞ ( A i ); see Section 3.6. Thus, we may suppose that each element of W( A i ) is realized by a positiveelement of A i .Let σ i j : = W ( ϕ i j ) and consider the inductive system (W ( A i ) , ψ i j ) i ∈ I in PreW ∼ . We denote by ( S , σ i ∞ )its inductive limit in PreW ∼ . Observe that (W ( A alg / N ) , W ( ϕ i ∞ )) is a cocone for the inductive system.Hence from universal properties, we deduce that there exists a unique w : S −→ W ( A alg / N ) such thatfor all i ∈ I , the following diagram commutes:W ( A i ) σ i ∞ / / W ( ϕ i ∞ ) & & ▲▲▲▲▲▲▲▲▲▲ S w (cid:15) (cid:15) W ( A alg / N )Moreover we know that ( µ ∼ ( S ) , η S ) is the inductive limit in W ∼ of the inductive system above, where η S : S −→ µ ∼ ( S ). Hence, there exists a unique θ : µ ∼ ( S ) −→ W ( A alg / N ) such that the followingdiagram commutes: S η S / / w $ $ ■■■■■■■■■■ µ ∼ ( S ) θ (cid:15) (cid:15) W ( A alg / N ) Let us sum up the context with the following commutative diagram:W ( A i ) σ ij = W ( ϕ ij ) (cid:15) (cid:15) σ i ∞ " " ❊❊❊❊❊❊❊❊❊ W ( ϕ i ∞ ) ! ! S ∃ ! w / / η S ! ! ❈❈❈❈❈❈❈❈❈ W ( A alg / N ) µ ∼ ( S ) ∃ ! θ W ( A j ) σ j ∞ E E ☛☛☛☛☛☛☛☛☛☛☛☛☛☛☛☛ W ( ϕ j ∞ ) G G To complete the proof, let us show that θ is a W ∼ -isomorphism. Surjectivity: Let [( a , u )] ∈ W ( A alg / N ).Since a ∈ A alg / N , we know that there exists a k ∈ ( A k ) + such that ϕ k ∞ ( a k ) = a . Also, u is a unitary elementof her a ∼ = ϕ k ∞ ( a k )( A alg / N ) ϕ k ∞ ( a k ) ∼ . Now, observe that C ∗ − lim −→ j > k (her ϕ k j ( a k ) , ϕ jl ) ≃ (her a , ϕ j ∞ ). Hencefor any ǫ >
0, there exists j ≥ k and a unitary element u j of her ϕ k j ( a k ) ∼ such that k u − ϕ ∼ j ∞ ( u j ) k < ǫ . Inparticular, for ǫ <
2, we obtain a unitary element u j of her ϕ k j ( a k ) ∼ such that [ u ] = [ ϕ ∼ j ∞ ( u j )] in K (her a ).We compute that W ( ϕ j ∞ )([( ϕ k j ( a k ) , u j )]) = [( ϕ k ∞ ( a k ) , ϕ j ∞∼ ( u j ))] = [( a , u )].Thus, by the commutativity of the diagram above we obtain w ◦ σ j ∞ ([( ϕ k j ( a k ) , u j )]) = W ( ϕ j ∞ )([( ϕ k j ( a k ) , u j )]) = [( a , u )]as desired. We conclude that w is surjective and hence that θ is surjective.Injectivity: Let us show that for any s , t ∈ S such that w ( s ) ≤ w ( t ) then s ≤ t . In fact, it is su ffi cientto prove that for any s , t ∈ S such that w ( s ) ≤ w ( t ) then s ≺ ⊆ t ≺ . Indeed this implies that η S ( s ) ≤ η S ( t ),and since im( η S ) = µ ∼ ( S ), we are able to conclude that θ is order-embedding.Let s , t ∈ S such that w ( s ) ≤ w ( t ) and let s ′ ≺ s . Since the inductive limit is algebraic, there exists s k , s ′ k , t k ∈ W ( A k ) such that σ k ∞ ( s ′ k ) = s ′ , σ k ∞ ( s k ) = s and σ k ∞ ( t k ) = t and such that s ′ k ≺ s k in W ( A k ).Now choose a ′ , a , b ∈ ( A k ) + and unitary elements u ′ , u , v in the respective hereditary subalgebras suchthat s ′ k = [( a ′ , u ′ )] , s k = [( a , u )] and t k = [( b , v )]. We already know that [ a ′ ] ≺ [ a ] in W( A k ) and that[ θ a ′ a ( u ′ )] = [ u ] in K (her a ∼ ). On the other hand, since w ( s ) ≤ w ( t ), by the commutativity of thediagram, we deduce that: [ ϕ k ∞ ( a ′ )] ≺ [ ϕ k ∞ ( a )] ≤ [ ϕ k ∞ ( b )] in W( A alg / N ) . [ θ ϕ k ∞ ( a ′ ) ϕ k ∞ ( b ) ( ϕ k ∞∼ ( u ′ ))] = [ θ ϕ k ∞ ( a ′ ) ϕ k ∞ ( b ) ( ϕ k ∞∼ ( u ))] = [ ϕ k ∞∼ ( v )] in K (her ϕ k ∞ ( b )) . By the proof [3, Theorem 2.2.9], we deduce that there exists some j ≥ k and some δ > ϕ k j ( a ′ )] ≤ [( ϕ k j ( b ) − δ ) + ] in W( A j ) . Finally, since the inductive limits are algebraic, we conclude that there exists l ≥ k , j such that: [ ϕ kl ( a ′ )] ≤ [( ϕ kl ( b ) − δ ) + ] in W( A l ) . [ θ ϕ kl ( a ′ ) ϕ kl ( b ) ( ϕ kl ∼ ( u ′ ))] = [ ϕ kl ∼ ( v )] in K (her ϕ kl ( b )) . We conclude that σ kl ( s ′ k ) ≺ σ kl ( t k ), which ends the proof. (cid:3) UNITARY CUNTZ SEMIGROUP FOR C ∗ -ALGEBRAS OF STABLE RANK ONE 15 Continuity of the functor Cu . We now have all the tools to conclude that Cu : Cu ∼ −→ W ∼ is acontinuous functor, using the same techniques as in [3, Chapter 3].We recall that for any Mon ≤ -morphism f : M −→ N between two Cu ∼ -semigroups, the W ∼ -continuityaxiom is equivalent to preserving suprema of increasing sequences; cf [3, Lemma 3.1.4]. Also, with anargument similar to [3, Proposition 3.1.6], we easily deduce the following: Let ( S , ≺ ) be a PreW ∼ -semigroup. Then there exists a Cu ∼ -semigroup γ ∼ ( S ) together with a W ∼ -morphism α S : S −→ γ ∼ ( S )satisfying the following conditions:(i) The morphism α S is an ‘auxiliary-embedding’ in the sense that s ′ ≺ s whenever α ( s ′ ) ≪ α ( s ).(ii) The morphism α S has a ‘dense image’ in the sense that for any two t ′ , t ∈ γ ∼ ( S ) such that t ′ ≪ t there exists s ∈ S such that t ′ ≤ α ( s ) ≤ t .Arguing as in [3, Theorem 3.1.8], we deduce that Cu ∼ is a (full) reflective subcategory of PreW ∼ withreflector γ ∼ . In particular, Cu ∼ has inductive limits. Finally, observe that for any A ∈ C ∗ , the compact-containment relation on Cu ( A ) and the auxiliary relation on W ( A ⊗ K ) agree; see [3, Remark 3.2.4].Thus, we have that Cu ( A ) = W ( A ⊗ K ) as Cu ∼ -semigroups. Theorem 3.20.
There exists a natural isomorphism γ ∼ ◦ W ≃ Cu ◦ γ , where γ is the reflector from C ∗ loc to C ∗ defined in Section 3.6. In particular, for any A ∈ C ∗ , there is a (natural) Cu ∼ -isomorphism between Cu ( A ) ≃ γ ∼ (W ( A )) .Proof. The aim of the proof is to show that (Cu ( γ ( A ) , W ( i )) is a Cu ∼ -completion of W ( A ) for any A ∈ C ∗ loc , where W ( i ) is built as follows:Let A ∈ C ∗ loc , write B : = M ∞ ( A ) ∈ C ∗ loc . Consider the canonical inclusion i : B ֒ −→ B ≃ A ⊗ K . Then i induces a W ∼ -morphism W ( i ) : W ( B ) −→ W ( B ). On the other hand, we know that W ( B ) = W ( A )and that W ( B ) ≃ Cu ( A ). Thus, we obtain a W ∼ -morphism W ( i ) : W ( A ) −→ Cu ( A ) (we use thesame notation). By the argument in [3, Theorem 3.1.8], we only have to check that W ( i ) is an auxiliary-embedding and that it has a dense image.Let s , s ′ ∈ W ( A ) such that W ( i )( s ′ ) ≪ W ( i )( s ′ ). We deduce that W ( i )( s ′ ) ≺ W ( i )( s ′ ). Also,observe that W ( i ) is in fact an order embedding (even more, it is the canonical injection). Thus, weconclude that s ≺ s ′ and hence W ( i ) is an ‘auxiliary-embedding’.Let t , t ′ ∈ Cu ( γ ( A )) such that t ′ ≪ t . Now pick a , a ′ ∈ ( γ ( A ) ⊗ K ) + and unitary elements u , u ′ inthe respective hereditary subalgebras of a , a ′ , such that t : = [( a , u )] and t ′ : = [( a ′ , u ′ )]. Then, we knowthat [ a ′ ] ≪ [ a ] in Cu( A ) and that χ a ′ a ([ u ′ ]) = [ u ]. Using the argument in [3, Lemma 3.2.7], there exists b ∈ M ∞ ( A ) + such that [ a ′ ] ≤ [ b ] ≤ [ a ] in Cu( A ). Now consider s : = [( b , θ a ′ b ( u ))] ∈ W ( A ), we get that t ′ ≤ W ( i )( s ) ≤ t in Cu ( A ). It follows that W ( i ) has a ‘dense image’ and hence that (W ( i ) , Cu ( γ ( A )))is a Cu ∼ -completion of W ( A ). (cid:3) Corollary 3.21.
The functor Cu : C ∗ −→ Cu ∼ is continuous. More precisely, given an inductive system ( A i , φ i j ) i ∈ I in C ∗ , then: Cu ∼ − lim −→ (Cu ( A i ) , Cu ( φ i j )) ≃ Cu ( C ∗ − lim −→ (( A i , φ i j ))) ≃ γ ∼ (W ∼ − lim −→ (W ( A i ) , W ( φ i j ))) . Algebraic Cu ∼ semigroups and Mon ≤ -completion. In this last subsection, we will briefly intro-duce algebraic Cu ∼ -semigroups in order to link the notion of real rank zero for a C ∗ -algebra A , that en-sures a lot of projections, with the notion of ‘density’ of compact elements in Cu ( A ). In fact, as compactelements of Cu ( A ) are entirely determined by to the ones of its positive cone Cu( A ) (see Corollary 3.5),all results from Cu( A ) will apply here. These can be found in [3, §5.5].Let S ∈ Cu ∼ . We denote by S c : = { s ∈ S | s ≪ s } . It is easily shown that S c ∈ Mon ≤ and that forany Cu ∼ -morphism f : S −→ T between S , T ∈ Cu ∼ , we have f ( S c ) ⊂ T c . Thus, f induces a Mon ≤ -morphism f c : = f | S c : S c −→ T c . Hence, alike ν + that recovers the positive cone of a Cu ∼ -semigroup (seeLemma 3.9), the assignment S −→ S c defines a functor ν c : Cu ∼ −→ Mon ≤ that recovers the compactelements of a Cu ∼ -semigroup. Proposition 3.22.
Let M ∈ Mon ≤ . Then ( M , ≤ ) ∈ W ∼ . We denote Cu ∼ ( M ) : = γ ∼ ( M , ≤ ) the Cu ∼ -completion of ( M , ≤ ) . Any Mon ≤ -morphism f : M −→ N between M , N ∈ Mon ≤ induces a Cu ∼ -morphism γ ∼ ( f ) : γ ∼ ( M ) −→ γ ∼ ( N ) . Thus we obtain a functor: Cu ∼ : Mon ≤ −→ Cu ∼ M Cu ∼ ( M ) f γ ∼ ( f ) Proof.
Observe that in the case where the auxiliary relation is the same as the order, the completionprocess corresponds to ‘adding’ suprema of ≤ -increasing sequences. Further, the induced morphism of f naturally sends suprema of ≤ -increasing sequences of Cu ∼ ( M ) to the ones in Cu ∼ ( N ). See [3, §5.5.3] (cid:3) As noticed in the above proof, we emphasize that for any submonoid M of a Cu ∼ -semigroup S , thecompletion γ ∼ ( M ) is the subset of S consisting of suprema (in S ) of any increasing sequence in M . Thatis γ ∼ ( M ) = { sup n m n , ( m n ) n ∈ M N } . Definition 3.23.
Let S ∈ Cu ∼ . We say that S is an algebraic Cu ∼ -semigroup if every element in S isthe supremum of an increasing sequence of compact elements, that is, an increasing sequence in S c . Wedenote by Cu ∼ alg the full subcategory of Cu ∼ consisting of algebraic Cu ∼ -semigroups (see [3, §5.5]). Proposition 3.24. (cf [3, Proposition 5.5.4] )(i) For any algebraic Cu ∼ -semigroup S , we have Cu ∼ ( S c ) ≃ S .(ii) Cu ∼ ( M ) is an algebraic Cu ∼ -semigroup for any M ∈ Mon ≤ . Proposition 3.25. (cf [10, Corollary 5] , [3, Remark 5.5.2] ). Whenever A has real rank zero, Cu( A ) is analgebraic Cu -semigroup. If moreover A has stable rank one, then the converse is true. Corollary 3.26.
Let A ∈ C ∗ . Then A has real rank zero if and only if Cu ( A ) ∈ Cu ∼ alg if and only if Cu( A ) ∈ Cu alg .Proof. Using the characterization of compacts elements of Cu ( A ) by compact elements of Cu( A ) as inCorollary 3.5, we get that Cu( A ) is algebraic if and only if Cu ( A ) is algebraic. (cid:3) UNITARY CUNTZ SEMIGROUP FOR C ∗ -ALGEBRAS OF STABLE RANK ONE 17 We end this section by observing that ν + and ν c satisfy the following: ν + ◦ ν c ≃ ν c ◦ ν + . Hence, wesometimes consider ν + , c : Cu ∼ −→ PoM as the composition of ν + and ν c . Naturally, for any S ∈ Cu ∼ , wedenote by S + , c : = ν + , c ( S ) the positively ordered monoid of positive compact elements of S .4. C omputations the Cu - semigroup This section is aiming to explicitly compute our invariant in some cases, such as simple C ∗ -algebras,AF algebras, and some A T , AI algebras. We first give another picture of the Cu -semigroup and itsmorphisms that uses the lattice of ideals of the C ∗ -algebra and makes these computations easier.4.1. Alternative picture of the invariant.
Let A ∈ C ∗ and let a ∈ ( A ⊗ K ) + . Recall that we write I a : = AaA the ideal generated by a and her a : = aAa the hereditary subalgebra generated by a . Then a isobviously a full element in I a and her a is a full hereditary subalgebra of I a . Since A is separable, then sois I a . Thus we can find a strictly positive element of I a , that we write s a . Since a ∈ her s a , we know that a . Cu s a . Observe that the canonical inclusion i : her a ֒ −→ her s a = I a is one of our standard morphisms(see Section 3.2). That is, in the notation of Section 3.2, χ as a = K ( i ).Furthermore, using [6, Theorem 2.8], we deduce that χ as a : K (her a ) ≃ K ( I a ) is in fact an abeliangroup isomorphism and χ as a ([ u ] K (her a ) ) = [ u ] K ( I a ) for any unitary element u ∈ her a ∼ . Proposition 4.1.
Let A ∈ C ∗ and let a , b ∈ ( A ⊗ K ) + be such that a . Cu b. Let s a , s b be strictly positiveelements of the ideals I a , I b respectively. Then the following diagram is commutative: U (her a ∼ ) / / θ ∼ ab (cid:15) (cid:15) K (her a ) χ ab (cid:15) (cid:15) ≃ χ asa / / K ( I a ) χ sasb (cid:15) (cid:15) U (her b ∼ ) / / K (her b ) ≃ χ bsb / / K ( I b ) In particular, for any other strictly positive element s a ′ of I a , we have her s a = her s a ′ and hence χ s a s a ′ = id K ( I a ) , which finally gives us χ as a = χ as a ′ .Proof. By definition, χ ab : = K ( θ ∼ ab ) and hence the left-square is commutative. Furthermore, by transi-tivity of . (see Section 3.1), we know that χ s a s b ◦ χ as a = χ as b = χ bs b ◦ χ ab . That is, the right square iscommutative, which ends the proof. (cid:3) Notation 4.2.
Let A ∈ C ∗ . Let a ∈ ( A ⊗ K ) + and let s A be any strictly positive element of I a . ByProposition 4.1, χ as a : K (her a ) ≃ K ( I a ) is a well-defined group isomorphism that does not depend onthe strictly positive element s a chosen. We write δ a : = χ as a .Let I , J ∈ Lat( A ) be ideals of A and let s I , s J be any strictly positive elements of I , J respectively.Suppose that I ⊆ J or, equivalently [ s I ] ≤ [ s J ] in Cu( A ). By Proposition 4.1, χ s I s J : K ( I ) −→ K ( J ) isa well-defined group morphism that does not depend on the strictly positive elements chosen. We write δ IJ : = χ s I s J .Observe that δ IJ = K ( i ), where i : I ֒ −→ J is the canonical inclusion. In particular, δ II = id K ( I ) . Proposition 4.3.
Let A ∈ C ∗ and let a , b ∈ ( A ⊗ K ) + such that [ a ] ≤ [ b ] in Cu( A ) . Let u , v be unitaryelements of her a ∼ , her b ∼ respectively. We write [ u ] : = [ u ] K (her a ) and [ v ] : = [ v ] K (her b ) . Then the followingare equivalent:(i) θ ∼ ab ( u ) ∼ h v in her b ∼ .(ii) χ ab ([ u ]) = [ v ] in K (her b ) .(iii) δ I a I b ( δ a ([ u ])) = δ b ([ v ]) in K ( I b ) , that is, δ I a I b ([ u ] K ( I a ) ) = [ v ] K ( I b ) .Proof. Since K ( θ ∼ ab ) = χ ab , we trivially obtain (i) is equivalent to (ii).Furthermore, by the right-square of the commutative diagram in Proposition 4.1, we know that δ I a I b ◦ δ a ([ u ]) = δ b ◦ χ ab ([ u ]). And since δ b is an isomorphism, we obtain that (ii) is equivalent to (iii). (cid:3) Corollary 4.4.
Let A ∈ C ∗ and let [( a , u )] , [( b , v )] ∈ Cu ( A ) . Then [( a , u )] ≤ [( b , v )] in Cu ( A ) if and onlyif [ a ] ≤ [ b ] in Cu( A ) δ I a I b ([ u ] K ( I a ) ) = [ v ] K ( I b ) in K ( I b ) where δ I a I b is as in Proposition 4.1. We will now use all the above to get a new picture of the Cu -semigroup and its elements. Definition 4.5.
Let A ∈ C ∗ and let I ∈ Lat( A ) be an ideal of A. We recall that Cu( I ) is an ideal of Cu( A ).We also recall that for x ∈ Cu( A ), we write I x : = { y ∈ Cu( A ) such that y ≤ ∞ . x } the ideal of Cu( A )generated by x .Define Cu f ( I ) : = { [ a ] ∈ Cu( A ) | I a = I } . Equivalently, Cu f ( I ) : = { x ∈ Cu( A ) | I x = Cu( I ) } . In otherwords, Cu f ( I ) consists of the elements of Cu( A ) that are full in Cu( I ).For notational purposes, we will indistinguishably use I a or I [ a ] , refering to one or the other; seeParagraph 2.4. For instance, we might consider objects such as δ I x I y or K ( I x ), where x , y ∈ Cu( A ), whenwe really mean δ I a I b or K ( I a ), where a , b ∈ ( A ⊗ K ) + are representatives of x , y respectively. Definition 4.6.
Let A ∈ C ∗ . Let us consider S : = G I ∈ Lat( A ) Cu f ( I ) × K ( I ) . We equip S with addition and order as follows: For any ( x , k ) ∈ Cu f ( I x ) × K ( I x ) and ( y , l ) ∈ Cu f ( I y ) × K ( I y ), then ( x , k ) ≤ ( y , l ) if: x ≤ y and δ I x I y ( k ) = l . ( x , k ) + ( y , l ) = ( x + y , δ I x I x + y ( k ) + δ I y I x + y ( l )) . Lemma 4.7.
Let S be a Cu ∼ -semigroup and let T be a Mon ≤ . Let f : S −→ T be a
Mon ≤ -isomorphism.Then, T is a Cu ∼ -semigroup and f is a Cu ∼ -isomorphism. A fortiori, S ≃ T as Cu ∼ -semigroups.Proof. Let ( t k ) k be an increasing sequence in T . Since f is a surjective order-embedding, we can findan increasing sequence ( s k ) k in S such that f ( s k ) = t k for all k . We easily deduce that f (sup k s k ) ≥ t k forany k ∈ N . Now, if t ≥ t k for all k ∈ N , then since there exists s ∈ S such that f ( s ) = t and f is anorder-embedding, we have that s ≥ s k for any k and thus t = f ( s ) ≥ f (sup k s k ). Thus T satisfies (O1) andmoreover f preserves suprema of increasing sequences. UNITARY CUNTZ SEMIGROUP FOR C ∗ -ALGEBRAS OF STABLE RANK ONE 19 Now let x , y ∈ S be such that x ≪ y . Let ( t k ) k be an increasing sequence in T such that f ( y ) ≤ sup k t k .Since f is a surjective order-embedding, we know that there exists an increasing sequence ( s k ) k in S suchthat f ( s k ) = t k for any k ∈ N . Let s : = sup k s k . Since s ≥ s k for any k , then f ( s ) ≥ f ( s k ) = t k and passingto suprema, we deduce that f ( s ) ≥ f ( y ). Again, f is an order-embedding, so we deduce that s ≥ y in S .Now, since x ≪ y , there exists n ∈ N such that x ≤ s n , which implies f ( x ) ≤ f ( s n ) = t n . We conclude that f ( x ) ≪ f ( y ). From this, (O2) follows easily and hence f preserves the compact-containment relation.Axioms (O3) and (O4) are routine as well as the final conclusion. (cid:3) Theorem 4.8.
Let A ∈ C ∗ and let ( S , + , ≤ ) be the object defined in Definition 4.6. Then ( S , + , ≤ ) is a Cu ∼ -semigroup and the following map is a Cu ∼ -isomorphism: ξ : Cu ( A ) −→ S [( a , u )] ([ a ] , δ a ([ u ])) where [ a ] : = [ a ] Cu( A ) and [ u ] : = [ u ] K (her a ) .Proof. By Notation 4.2 and Definition 4.5, the map Cu ( A ) −→ F I ∈ Lat( A ) Cu f ( I ) × K ( I ) is well-defined.Further, by construction addition and order are well-defined in S . Now let a ∈ ( A ⊗ K ) + . Since A hasstable rank one, then so has her a . Hence, by K -surjectivity, we know that any element of K (her a )lifts to a unitary in her a ∼ and that any two of those lifts are homotopic. Also δ a is an isomorphismand obviously any two representatives of x in ( A ⊗ K ) + are Cuntz equivalent. Thus for any ( x , k ) ∈ Cu( A ) × K ( I x ), there exist a ∈ ( A ⊗ K ) + and u ∈ U (her a ∼ ) such that [ a ] = x and δ a [ u ] = k . Moreoverfor any other lift ( a ′ , u ′ ), by construction, gives us [( a ′ , u ′ )] = [( a , u )]. So we conclude that ξ is a setbijection.Now, using Proposition 4.3 and Corollary 4.4, we know that [( a , u )] ≤ [( b , v )] if and only if ξ ([( a , u )]) ≤ ξ ([( b , v )]). Moreover, using Proposition 4.1, we have ξ ([( a , u )] + [( b , v )]) = ξ ([( a , u )]) + ξ ([( b , v )]). In theend, we have ξ is a Mon ≤ -isomorphism. We finally conclude that S is a Cu ∼ -semigroup and that ξ is aCu ∼ -isomorphism using Lemma 4.7. (cid:3) In this new picture, the positive elements of Cu ( A ) can be identified with { ( x , , x ∈ Cu( A ) } (seeLemma 3.9). In other words, Cu ( A ) + ≃ Cu( A ) as Cu-semigroups. We will end this part by describingmorphisms from Cu ( A ) to Cu ( B ) in this new viewpoint of our invariant. Lemma 4.9.
Let A , B ∈ C ∗ . Let I ∈ Lat( A ) and let φ : A −→ B be a ∗ -homomorphism. Write J : = B φ ( I ) Bthe smallest ideal of B containing φ ( I ) . Also write α : = Cu ( φ ) , α : = Cu( φ ) and α I : = K ( φ | I ) , where φ | I : I φ −→ J.(i) For any x ∈ Cu f ( I ) , we have α ( x ) ∈ Cu f ( J ) . That is, I α ( x ) = Cu( J ) is the smallest ideal of Cu( B ) containing α (Cu( I )) .(ii) For any ( x , k ) with x ∈ Cu f ( I ) and k ∈ K ( I ) , we have α ( ξ − ( x , k )) = ( α ( x ) , α I ( k )) , where ξ A , ξ B are the Cu ∼ -isomorphism as in Theorem 4.8 for A , B respectively.Proof.
By functoriality of Cu and Paragraph 2.4, we know that Cu( J ) is the smallest ideal of Cu( B ) thatcontains α (Cu( I )). Now let x ∈ Cu f ( I ). Then α ( x ) ∈ α (Cu( I )). Hence I α ( x ) ⊆ Cu( J ). However, since x is full in Cu( I ), we have α (Cu( I )) ⊆ I α ( x ) . By minimality of Cu( J ) we deduce that I α ( x ) = Cu( J ), thatis, α ( x ) ∈ Cu f ( J ), which proves (i).(ii) Let ( x , k ) be an element of Cu ( A ), where x ∈ Cu( A ) and k ∈ K ( I x ). Let ( a , u ) be a representativeof ( x , k ), that is, ξ ([( a , u )]) = ( x , k ) as in Theorem 4.8. That is, [ a ] = x in Cu( A ) and δ a ([ u ] K (her a ) ) = [ u ] K ( I a ) = k . We know that α ( ξ − ( x , k )) = α ([ a , u ]) = [( φ ( a )) , φ ∼ ( u ))] = ([ φ ( a )] Cu( B ) , δ φ ( a ) ([ φ ∼ ( u )] K (her φ ( a ) ∼ ) )) = ([ φ ( a )] Cu( B ) , [ φ ∼ ( u )] K ( I ∼ φ ( a ) ) )Hence α ( ξ − ( x , k )) = ( α ( x ) , α I ( k )) as desired. (cid:3) Notation 4.10.
Whenever convenient, and many times in the future, we will describe elements of Cu ( A )as a couple ( x , k ) where x ∈ Cu( A ) and k ∈ K ( I x ). Again, we may describe morphisms α : = Cu ( φ )from Cu ( A ) to Cu ( B ), whenever convenient, as couples α : = ( α , { α I } I ∈ Lat( A ) ), where α : = Cu( φ ) and α I : = K ( φ | I ).We now compute the Cu -semigroup in some specific settings. In the process, we will remind thereader about lower semicontinuous functions which play a key role in the computation of Cu-semigroupsof certain C ∗ -algebras. We will also recall well-know constructions, such as UHF C ∗ -algebras, orNCCW 1, among others. . Let X be a topological space and S be a Cu-semigroup. Let f : X −→ S be a map. We say that f is lower semicontinuous if for any s ∈ S , the set { t ∈ X : s ≪ f ( t ) } is open in X . We write Lsc( X , S ) the set of lower-semicontinuous functions from X to S .Also, we recall that if A is a separable C ∗ -algebra of stable rank one such that K ( I ) = A and X is a locally compact Hausdor ff space that is second countable and of covering dimensionone, then Cu( C ( X ) ⊗ A ) ≃ Lsc( X , Cu( A )); see [2, Theorem 3.4].Finally, U I | U defines a one-to-one correspondence between the open subsets of X , that we write O ( X ), and the ideals of Lsc( X , N ). Note that for any f ∈ Lsc( X , N ), I f : = I supp( f ) , where supp( f ) : = { x ∈ X | f ( x ) , } is an open set of X .4.2. The simple case.Proposition 4.12.
Let A be a simple C ∗ -algebra. Then Cu ( A ) can be described in terms of Cu( A ) and K ( A ) as follows: Cu ( A ) ≃ −→ (Cu( A ) ∗ × K ( A )) ⊔ { } ( x , k ) if x = x , k ) otherwiseProof. Since A is simple, we know that Lat( A ) = { , A } . Therefore, in the description of the Cu -semigroup of Notation 4.10, we have Cu f ( { } ) = { } and Cu f ( A ) = Cu( A ) ∗ . The result follows. (cid:3) UNITARY CUNTZ SEMIGROUP FOR C ∗ -ALGEBRAS OF STABLE RANK ONE 21 The case of no K -obstructions.Definition 4.13. We say that a C ∗ -algebra A has no K -obtructions , if A has stable rank one and K ( I ) istrivial for any I ∈ Lat( A ). Proposition 4.14.
Let A be a C ∗ -algebra with no K -obstructions. Then Cu ( A ) ≃ Cu( A ) . In particular,for any AF algebra A, Cu ( A ) ≃ Cu( A ) .Proof. By assumption, we know that K ( I ) is trivial for any I ∈ Lat( A ). Therefore, using again thedescription of the Cu -semigroup of Notation 4.10, we have Cu ( A ) ≃ Cu( A ) ×{ } . The result follows. (cid:3) and A T algebras: The case of C ([0 , and C ( T ) . We here compute the Cu -semigroup of theinterval algebra and the circle algebra. Also, using the continuity of Cu we give an explicit computationof the Cu -semigroup of AI-algebras (respectively A T -algebras), constructed as the tensor product of theinterval algebra with any UHF algebra of infinite type (respectively the circle algebra). Notation 4.15.
Let X be the interval or the circle and let f ∈ Lsc( X , N ). The open set supp f can be(uniquely) write as a countable disjoint union of open arcs of X . That is, supp f = n f S i = U i , for some n f ∈ N , where U i are pairwise disjoint open arcs of X . Also we choose the following convention: − ⊕ Z = ⊕ Z = { } . The C ([0 , case.Lemma 4.16. Let I ∈ Lat( C ([0 , . Consider U : = supp( ∞ I ) , the unique open set of [0 , that corre-sponds to I. We have:(i) Cu f ( I ) ≃ Lsc( U , N ∗ ) .(ii) Cu f ( I ) × K ( I ) ≃ Lsc( U , N ∗ ) × ( m U ⊕ Z ) , where m U : = n (1 | U ) − (1 U (0) + U (1)) .Proof. We know that Cu( I ) = I | U ≃ Lsc( U , N ) and we obtain that Cu f ( I ) ≃ Lsc( U , N ∗ ). Let us writesupp 1 | U = n (1 | U ) S i = U i as in Notation 4.15. The result follows by observing that open arcs of [0 ,
1] are] a , b [ [0 ,
1] ] a ,
1] [0 , a [ ∅ and the K groups of continuous map over these open arcs are respectively Z { } { } { } { } (cid:3) Theorem 4.17.
Let V : = [0 , and V : = ]0 , . Then:(i) Cu ( C ([0 , ≃ F U ∈O ([0 , Lsc( U , N ∗ ) × ( m U ⊕ Z ) ≃ Cu ( C (]0 , ⊔ ( F i = , Lsc( V i , N ∗ ) × { } ) ⊔ Lsc([0 , , N ∗ ) × { } . (ii) Cu ( C ([0 , c ≃ ( { n . | [0 , } n ∈ N ) × { } .Proof. (i) Combine Theorem 4.8 with Lemma 4.16 and Paragraph 4.11.(ii) From Corollary 3.5, we know that ( x , k ) ∈ Cu ( C ([0 , x iscompact in Lsc([0 , , N ) if and only if x is constant on [0 , (cid:3) The C ( T ) case.Lemma 4.18. Let I ∈ Lat( C ( T )) . Consider U : = supp( ∞ I ) , the unique open set of T that corresponds toI. We have:(i) Cu f ( I ) ≃ Lsc( U , N ∗ ) .(ii) Cu f ( I ) × K ( I ) ≃ Lsc( U , N ∗ ) × ( n U ⊕ Z ) , where n U : = n (1 | U ) .Proof. We know that Cu( I ) = I | U ≃ Lsc( U , N ) and we obtain that Cu f ( I ) ≃ Lsc( U , N ∗ ). Let us writesupp 1 | U = n (1 | U ) S i = U i as in Notation 4.15. The result follows by observing that open arcs of of T are] a , b [ T ∅ and the K groups of continuous map over these open arcs are respectively Z Z { } (cid:3) Theorem 4.19.
We have the following:(i) Cu ( C ( T )) ≃ F U ∈O ( T ) Lsc( U , N ∗ ) × n U ⊕ Z ≃ Cu ( C (]0 , ⊔ Lsc( T , N ∗ ) × Z . (ii) Cu ( C ( T )) c ≃ ( { n . | T } n ∈ N ) × Z .Proof. (i) Combine Theorem 4.8 with Lemma 4.16 and Paragraph 4.11.(ii) From Corollary 3.5, we know that ( x , k ) ∈ Cu ( C ( T )) is a compact element if and only if x iscompact in Lsc( T , N ) if and only if x is constant on T . (cid:3) Now that we have computed the Cu -semigroup of the interval algebra and the circle algebra, we areable to obtain the Cu -semigroup of any AI and A T algebra, using Corollary 3.21. Actually, we will nextcompute a concrete example of an an A T algebra that is constructed as C ( T ) ⊗ UHF (respectively an AIalgebra that can be constructed as C ([0 , ⊗ UHF).Let q be a supernatural number and consider M q the UHF algebra associated to q . Consider anysequence of prime numbers ( q n ) n such that q = Q n ∈ N q n . Write ( A n , φ nm ) n the inductive system associatedto ( q n ) n . Now consider the following A T algebra: A : = lim −→ n ( C ( T ) ⊗ A n , id ⊗ φ nm ). In fact, A ≃ C ( T ) ⊗ M q .(Similar construction and computations can be done for the interval). Theorem 4.20.
Let M q be a UHF algebra and let V : = [0 , and V : = ]0 , . Then:(i) Cu ( C ( T ) ⊗ M q ) ≃ F U ∈O ( T ) Lsc( U , Cu( M q ) ∗ ) × ( n U ⊕ K ( M q )) .In particular, for any UHF algebra of infinite type M p ∞ , we get:(i) Cu ( C ( T ) ⊗ M q ) ≃ F U ∈O ( T ) Lsc( U , N [ p ] ∗ ⊔ ]0 , ∞ ]) × ( n U ⊕ Z [ p ]) .Proof. We will only compute the circle case as the interval case is done similarly. Since UHF algebrasare simple, we know that all ideals of C ( T ) ⊗ M q are of the form C ( U ) ⊗ M q for some U ∈ O ( T ). Hence,using K¨unneth formula (see [5, Theorem 23.1.3]), we obtain that K ( C ( U ) ⊗ M q ) ≃ ( n U ⊕ Z ) ⊗ K ( M q ) ≃ UNITARY CUNTZ SEMIGROUP FOR C ∗ -ALGEBRAS OF STABLE RANK ONE 23 n U ⊕ K ( M q ). On the other hand, by [2, Theorem 3.4], we compute that Cu( C ( U ) ⊗ M q ) ≃ Lsc( U , Cu( M q )).The result follows from Theorem 4.8. (cid:3) The
NCCW 1 complexes.
In this subsection, we will be interested in a more general class: theNCCW 1 complexes. We refer the reader to [17] for a classification of some of these C ∗ -algebras.Let E , F be finite dimensional C ∗ -algebras and let φ , φ : E −→ be two ∗ -homomorphisms. We definea non-commutative CW complex of dimension 1 , written NCCW 1, as the following pullback: A / / (cid:15) (cid:15) C ([0 , , F ) ( ev , ev ) (cid:15) (cid:15) E ( φ ,φ ) / / F ⊕ F We write such a pullback as A : = A ( E , F , φ , φ ) and refer to the class of inductive limits of finite directsums of NCCW 1 as NCCW 1 algebras . This class contains the AF, AI, A T and AH d algebras (see e.g[11],[12]). We finally recall that any A ∈ NCCW 1 has stable rank one.As in the work done in [2] to compute the Cu-semigroup of NCCW 1, we would like to use the pull-back structure of (non simple) NCCW 1 complexes to compute their Cu -semigroup. However, knowingthe explicit computation of C ([0 , C ( T ), we deduce that a priori pullbacks do not pass through Cu .First, observe that the circle algebra can be written as follows: C ( T ) ≃ A ( C , C , id , id ). Now considerthe pullback (in the category Mon ≤ ):Cu ( C ([0 , ⊕ Cu ( C ⊕ C ) Cu ( C ) : = { ( s , t ) ∈ Cu ( C ([0 , ⊕ Cu ( C ) | (Cu ( ev ) , Cu ( ev ))( s ) = ( t , t ) }≃ { ( x , k ) ∈ (Lsc([0 , , N ) × K ( I x )) | x (0) = x (1) } Cu ( C ([0 , ⊕ Cu ( C ⊕ C ) Cu ( C ) ≃ Cu ( C (]0 , ⊔ Lsc( T , N ∗ ) × { } . It is clear that there is no Mon ≤ -isomorphism between Lsc( T , N ∗ ) × { } and Lsc( T , N ∗ ) × Z , since theupper is an upward-directed Mon ≤ whereas the latter is not, and hence there is no Mon ≤ -isomorphismbetween Cu ( C ( T )) and Cu ( C ([0 , ⊕ Cu ( C ⊕ C ) Cu ( C ).5. R elation of Cu with existing K-T heoretical invariants
The aim of this section is to recover functorially existing invariants. We have already seen that thepositive cone of Cu ( A ) is isomorphic to Cu( A ). Our first step is to capture the K group information. Tothat end, we define a well-behaved set of maximal elements of a Cu ∼ -semigroup S , written S max , and weprove that Cu ( A ) max is isomorphic to K ( A ). Subsequently, we recover functorially Cu, K and finallythe K ∗ group. As before, we shall assume that A is a separable C ∗ -algebra with has stable rank one anddenote the category of such C ∗ -algebras by C ∗ .5.1. An abelian group of maximal elements: ν max .Definition 5.1. Let S be a Cu ∼ -semigroup. We say that S is positively directed if, for any x ∈ S , thereexists p x ∈ S such that x + p x ≥ Lemma 5.2.
Let A ∈ C ∗ . Then Cu ( A ) is positively directed.Proof. Let A ∈ C ∗ . Using the picture as in Notation 4.10 consider ( x , k ) ∈ Cu ( A ), where x ∈ Cu( A ) and k ∈ K ( I x ), we deduce that ( x , k ) + ( x , − k ) = (2 x , ≥
0, and so Cu ( A ) is positively directed. (cid:3) Definition 5.3.
Let S be a Cu ∼ -semigroup. We define S max : = { x ∈ S | if y ≥ x , then y = x } . Proposition 5.4.
Let S be a countably-based positively directed Cu ∼ -semigroup. Then S max is a non-empty absorbing abelian group in S whose neutral element e S max is positive.Proof. By assumption, for any x ∈ S , there exists at least one element p x ∈ S , such that x + p x ≥
0. Wewill first show that S max is closed under addition.Let y , z be elements in S max and let x ∈ S be such that x ≥ y + z . We first have x + p z ≥ y + z + p z ≥ y and x + p y ≥ z + y + p y ≥ z , which gives us the following equalities: x + p z = y + z + p z = y and x + p y = z + y + p y = z . Obviously x ≤ x + p z + z = x + p z + x + p y = y + z and since x ≥ y + z , we have x = y + z which tells us that S max is closed under addition.Now, let us show that the neutral element is positive: for any z ∈ S max and any p z ∈ S such that z + p z ≥
0, we have z + p z ∈ S max . Let x ∈ S be such that x ≥ z + p z . We know that for any y ∈ S max , y + z + p z = y . In particular, 2 z + p z = z . Also, x + z ≥ z + p z = z . Hence x + z = z . Finallycompute that x ≤ x + z + p z = z + p z . Therefore x = z + p z , that is, z + p z ∈ S max . Further, for any y , z elements of S max , we have y + p y + z + p z ≥ z + p z , y + p y , which by what we have just proved gives us y + p y = y + p y + z + p z = z + p z . Hence, the positive element e S max : = y + p y belongs to S max and isindependant of y and p y . If z ∈ S max , since e S max ≥ z + e max ≥ z and we obtain z + e S max = z . Thus wehave that S max is an abelian monoid with a neutral element e S max which is positive.We already know that z + (2 p z + z ) = e S max for any z ∈ S max . Let us show that 2 p z + z belongs to S max for any z ∈ S max and any p z ∈ S . Let x ≥ p z + z . Then x + z ≥ e S max , hence x + z = e S max . On the otherhand, x ≤ x + z + p z = e S max + p z = p z + z . Therefore 2 p z + z belongs to S max and is the (unique) inverseof z , which finishes the proof that S max is an abelian group.Also, observe that ν + ( S ) (see Lemma 3.9) is a countably-based Cu-semigroup. Therefore it has amaximal element which ensure us the existence of a maximal positive element in S and a fortiori that S max is a non-empty abelian group.Lastly, let x ∈ S and let p ∈ S max , we know there exists y ∈ S such that x + y ≥
0. Hence x + y + p ≥ p .Let z ∈ S be such that z ≥ x + p . we have z + y ≥ x + y + p = p and hence z + y = p . Now since x + y ≥ z ≥ x + p = x + z + y ≥ z which gives us z = x + p , that is, x + p ∈ S max for any x ∈ S and p ∈ S max . (cid:3) In the context of Proposition 5.4, e S max is the only positive element of S max , and the only positivemaximal element of S . More precisely, e S max : = y + p y , where y is any element of S max and p y anyelement of S such that y + p y ≥
0. Also, the inverse of y is 2 p y + y .We will see later that whenever A is separable, Cu ( A ) max ≃ K ( A ) with neutral element ( ∞ Cu( A ) , K ( A ) ). Proposition 5.5.
Let α : S −→ T be a Cu ∼ -morphism between countably-based positively directed Cu ∼ -semigroups S , T . Let α max : = α + e T max . Then α max is a AbGp -morphism from S max to T max . UNITARY CUNTZ SEMIGROUP FOR C ∗ -ALGEBRAS OF STABLE RANK ONE 25 Proof.
Let us first show that α max is a group morphism. For any s ∈ S max , we know that ( α ( s ) + e T max ) ∈ T max . Now, since α is a Cu ∼ -morphism, we have α max ( s ) + α max ( s ) = α ( s ) + α ( s ) + e T max = α ( s + s ) + e T max = α max ( s + s ), for any s , s elements of S max .Let us now show that ν max satisfies the functor properties. Trivially, ν max ( id ) = id . Let α : S −→ T and β : T −→ R be two Cu ∼ -morphisms. Let s ∈ S max . Then: β max ◦ α max ( s ) = β ( α ( s ) + e T max ) + e R max = ( β ◦ α ) max ( s )Hence ν max ( β ◦ α ) = ν max ( β ) ◦ ν max ( α ). (cid:3) As with ν + and ν c , we define a functor ν max that recovers the maximal elements of a positively directedCu ∼ -semigroup as follows: ν max : Cu ∼ −→ AbGp S S max α α max In order to be thoroughly defined as a functor, ν max should have a full subcategory of Cu ∼ consisting ofcountably-based and positively directed Cu ∼ -semigroup as domain, that we also denote Cu ∼ . Observethat Cu ( C ∗ ) belongs to the latter full subcategory.5.2. Link with Cu and K . Recall that for S ∈ Cu ∼ countably-based positively directed, we have S + ∈ Cu and that S max ∈ AbGp; see Proposition 5.4. In fact, both categories Cu and AbGp can be seen assubcategories of Cu ∼ -by defining an order as the equality for the case of groups-. Therefore, in whatfollows, we consider ν + and ν max as functors with codomain Cu ∼ . Definition 5.6.
Let S be a countably-based and positively directed Cu ∼ -semigroup. Let us define twoCu ∼ -morphisms that link S to S + on the one hand and to S max on the other hand, as follows: i : S + ⊆ ֒ −→ S j : S ։ S max s s s s + e S max In the next theorem, we use the picture of the Cu -semigroup described in Notation 4.10. Theorem 5.7.
Let A ∈ C ∗ . We have the following natural isomorphisms in Cu and AbGp respectively: Cu ( A ) + ≃ Cu( A ) Cu ( A ) max ≃ K ( A )( x , x ( ∞ A , k ) kIn fact, we have the following natural isomorphisms: ν + ◦ Cu ≃ Cu and ν max ◦ Cu ≃ K .Proof. We know that any positive element of Cu ( A ) is of the form ( x ,
0) for some x ∈ Cu( A ) and that ∞ A : = [ s A ⊗K ] = sup n ∈ N n . [ s A ] is the largest element of Cu( A ). We also know that any maximal element ofCu ( A ) is of the form ( ∞ A , k ) for some k ∈ K ( A ). Hence we easily get the two canonical isomorphismsof the statement. Now consider let φ : A −→ B a ∗ -homomorphism, let ( x , ∈ Cu ( A ) + and let( ∞ A , k ) ∈ Cu ( A ) max . We have that Cu ( φ ) + ( x , = (Cu( φ )( x ) ,
0) and that Cu ( φ ) max ( ∞ A , k ) = (Cu( φ )( ∞ A ) , Cu ( φ ) A ( k )) + ( ∞ B , = ( ∞ B , δ I φ ( ∞ A ) B ◦ Cu ( φ ) A ( k )) = ( ∞ B , K ( φ )( k )) . This exactly gives us thatCu ( A ) + Cu ( φ ) + (cid:15) (cid:15) ≃ / / Cu( A ) Cu( φ ) (cid:15) (cid:15) Cu ( A ) max Cu ( φ ) max (cid:15) (cid:15) ≃ / / K ( A ) K ( φ ) (cid:15) (cid:15) Cu ( B ) + ≃ / / Cu( B ) Cu ( B ) max ≃ / / K ( B )are commutative squares. (cid:3) Recovering an invariant.
We will now define the categorical notion of ‘recovering’ a functor. Thisallows us to check whether information and classification results of an invariant can be recovered fromanother one. To that end, we introduce the notion of weakly-complete invariant: an isomorphism at thelevel of the codomain category implies an isomorphism at the level of C ∗ -algebras without knowing itactually corresponds to a lift. Definition 5.8.
Let C , D be arbitrary categories and let I : C ∗ −→ C and J : C ∗ −→ D be (covariant)functors. Let H : D −→ C be a functor such that there exists a natural isomorphism η : H ◦ J ≃ I . Thenwe say we can recover I from J through H . Theorem 5.9.
Let C , D be arbitrary categories and let I : C ∗ −→ C and J : C ∗ −→ D be (covariant)functors. Suppose that there exists a functor H : D −→ C such that we recover I from J through H.(i) If I is a complete invariant for C ∗ I , then J is a weakly-complete invariant for C ∗ I .(ii) If I classifies homomorphisms from C ∗ to C ∗ , then J weakly classifies homomorphisms from C ∗ toC ∗ .If moreover H is faithful, then J is a complete invariant for C ∗ I and J classifies homomorphisms fromC ∗ to C ∗ . In this case, we say that we can fully recover I from J through H.Proof.
Let I , J and H be functors as in the theorem.(i) Suppose that I is a complete invariant for C ∗ I . Take any two C ∗ -algebras A , B ∈ C ∗ I . If there existsan isomorphism α : J ( A ) ≃ J ( B ), by functoriality, we get an isomorphism H ( α ) : H ◦ J ( A ) ≃ H ◦ J ( B ).Using the natural isomorphism H ◦ J ≃ I , we know that H ( α ) gives us an isomorphism β : I ( A ) ≃ I ( B ).By hypothesis, we can lift β to an isomorphism in the category C ∗ . That is, there exists a ∗ -isomorphism φ : A ≃ B such that I ( φ ) = β . We have just shown that J is weakly-complete for C ∗ I .Suppose now that H is faithful. Then the natural isomorphism exactly gives us that H ◦ J ( φ ) = H ( α ).Now since H is faithful, we conclude that J ( φ ) = α . That is, J is a complete invariant for C ∗ I .(ii) Suppose that I classifies homomorphisms from A to B . Let α : J ( A ) −→ J ( B ) be any morphism in D . If φ, ψ : A −→ B are ∗ -homomorphisms such that J ( φ ) = J ( ψ ) = α , then composing with H , we get H ◦ J ( φ ) = H ◦ J ( ψ ) = H ( α ). Thus, I ( φ ) = I ( ψ ), which gives us, by hypothesis, that φ ∼ aue ψ . Hence J weakly classifies homomorphisms from A to B . UNITARY CUNTZ SEMIGROUP FOR C ∗ -ALGEBRAS OF STABLE RANK ONE 27 Finally if H is faithful, then for any α : J ( A ) ≃ J ( B ), using again the natural isomorphism H ◦ J ≃ I ,we obtain: For any lift φ : A −→ B of β : I ( A ) −→ I ( B ), where β is the morphism obtained from H ( α ) asin the proof of (i) above, we have H ◦ J ( φ ) = H ( α ). Since H is faithful, we get that α = J ( φ ), from whichwe deduce that J classifies homomorphisms from A to B . (cid:3) We illustrate all the above with the following results:
Proposition 5.10.
By Theorem 5.7, we can recover Cu and K from Cu through ν + and ν max respectively.As to be expected, neither ν + nor ν max are faithful functors.Proof. Use the natural isomorphisms of Theorem 5.7. (cid:3)
Corollary 5.11.
Let φ, ψ : A −→ B be two ∗ -homomorphism. If Cu ( φ ) = Cu ( ψ ) then Cu( φ ) = Cu( ψ ) and K ( ψ ) = K ( φ ) . Recovering the K ∗ invariant. We now study a concrete use of Theorem 5.9 to recover existingclassifying functors from Cu , and in the process, recall some classification results that have been ob-tained in the past. Here, we give some insight on K ∗ : = K ⊕ K . Although notations might slightly di ff er,all of this can be found in [11] and [12].An approximately homogeneous dimensional algebra, written AH d algebra, is an inductive limitof finite direct sums of the form M n ( I q ) and M n ( C ( X )), where I q : = { f ∈ M q ( C ([0 , f (0) , f (1) ∈ C q } is the Elliott-Thomsen dimension-drop interval algebra and X is one of the fol-lowing finite connected CW complexes: {∗} , T , [0 , ⊆ AI , A T ⊆ AH d ⊆ NCCW 1.The category of ordered groups with order-unit, written AbGp u , is the category whose objects areordered groups with order-unit and morphisms are ordered group morphisms that preserve the order-unit. Definition 5.12. (cf [12, Definition 1.2.1]) Let A be a (unital) C ∗ -algebra. We define K ∗ ( A ) : = K ( A ) ⊕ K ( A ). We also define K ∗ ( A ) + : = { ([ p ] K ( A ) , [ v ] K ( A ) ) } ⊆ K ( A ) ⊕ K ( A ), where p is a projection in A ⊗ K and v is a unitary in the corner p ( A ⊗ K ) p . Notice that we look at the K class of v in A , that is,[ v + (1 − p )] K ( A ) . Finally, we define 1 K ∗ ( A ) : = ([1 A ] K , K ). Proposition 5.13. (cf [12, §1.2.2] ) Let A , B ∈ C ∗ . Then (K ∗ ( A ) , K ∗ ( A ) + ) is an ordered group and K ∗ ( A ) ∈ K ∗ ( A ) + is an order-unit of K ∗ ( A ) . Thus, (K ∗ ( A ) , K ∗ ( A ) + , K ∗ ( A ) ) ∈ AbGp u . Moreover, any ∗ -homomorphism φ : A −→ B induces an ordered group morphism K ( φ ) ⊕ K ( φ ) : K ∗ ( A ) −→ K ∗ ( B ) thatpreserves the order-unit. Thus, we obtain a covariant functor K ∗ : AH d −→ AbGp u . We do not give a proof of the above, but we remind the reader that whenever a C ∗ -algebra A hasstable rank one -which is the case of any AH d algebra-, then the monoid V ( A ) has cancellation and henceK ( A ) + can be identified with V ( A ) and thus (K ( A ) , V ( A )) is an ordered group.We also recall that in the stable rank one case, the Murray-von Neumann equivalence and the Cuntzequivalence agree on the projections of A ⊗ K and that V ( A ) ≃ Cu( A ) c . That is, any compact element ofCu( A ) is the class of some projection of A ⊗ K .We now recall two notable classification results by means of K ∗ that catch our interest: Theorem 5.14. ( [12, Corollary 4.9] , [11, Theorem 7.3 - Theorem 7.4] )(i) The functor K ∗ is a complete invariant for (unital) AH d algebras of real rank zero.(ii) Let A , B be (unital) A T algebras of real rank zero and let α : K ∗ ( A ) −→ K ∗ ( B ) be a scaled orderedgroup morphism. Then there exists a unique ∗ -homomorphism (up to approximate unitary equivalence) φ : A −→ B such that K ∗ ( φ ) = α . The aim now is to recover K ∗ from Cu and thus show that Cu contains more information than K ∗ .For that purpose, we first define the category of Cu ∼ -semigroups with order-unit, that we denote by Cu ∼ u .Further, we create a functor H ∗ : Cu ∼ u −→ AbGp u such that H ∗ ◦ Cu ≃ K ∗ as functors. Moreover,restricting to an adequate subcategory of Cu ∼ u , we will see that H ∗ is faithful. Definition 5.15.
Let S be a Cu ∼ -semigroup. We say that S has weak cancellation if x + z ≪ y + z implies x ≤ y for x , y , z ∈ S . We say that S has cancellation of compact elements if x + z ≤ y + z implies x ≤ y for any x , y ∈ S and z ∈ S c .The following property is proved using the same argument as in [17, Proposition 2.1.3]. Proposition 5.16.
Let A ∈ C ∗ . Then Cu ( A ) has weak cancellation and a fortiori Cu ( A ) has cancellationof compact elements. Definition 5.17.
Let S be a countably-based positively directed Cu ∼ -semigroup. Suppose that S hascancellation of compact elements. Also suppose that S + admits a compact order-unit.We say that ( S , u ) is a Cu ∼ -semigroup with compact order-unit . Now, a Cu ∼ -morphism between twoCu ∼ -semigroups with compact order-unit ( S , u ) , ( T , v ) is a Cu ∼ -morphism α : S −→ T such that α ( u ) ≤ v .We define the category of Cu ∼ -semigroups with compact order-unit, denoted Cu ∼ u , as the categorywhose objects are Cu ∼ -semigroups with order-unit and morphisms are Cu ∼ -morphisms that preserve theorder-unit. Lemma 5.18.
The assignment Cu , u : C ∗ −→ Cu ∼ u A (Cu ( A ) , ([1 A ] , φ Cu ( φ ) from the category of unital separable C ∗ -algebras of stable rank one, denoted by C ∗ , to the category Cu ∼ u is a covariant functor.Proof. We know that Cu ( A ) + has cancellation of compact elements. Further, we know that ([1 A ] ,
0) isa compact order-unit of Cu ( A ) + , so it easily follows that Cu , u ( A ) ∈ Cu ∼ u . Finally, it is trivial to see thatCu ( φ )([1 A ]) ≤ [1 B ], which ends the proof. (cid:3) Lemma 5.19.
The assignment H ∗ : Cu ∼ u −→ AbGp u ( S , u ) (Gr( S c ) , S c , u ) α Gr( α c ) from the category Cu ∼ u to the category AbGp u is a covariant functor.Moreover, if we restrict the domain of H ∗ to the category of algebraic Cu ∼ u -semigroups with compactorder-unit, denoted by Cu ∼ u , alg , then H ∗ becomes a faithful functor. UNITARY CUNTZ SEMIGROUP FOR C ∗ -ALGEBRAS OF STABLE RANK ONE 29 Proof.
Let ( S , u ) ∈ Cu ∼ u . By Proposition 5.16, we know that S c is a monoid with cancellation andhence, using the Grothendieck construction, one can check that (Gr( S c ) , S c , u ) is an ordered group withorder-unit. Now let α : S −→ T be a Cu ∼ u -morphism between two Cu ∼ -semigroups with order-unit( S , u ) , ( T , v ). By functoriality of ν c , it follows that α c : S c −→ T c is a Mon ≤ -morphism, and hencethat Gr( α c ) : Gr( S c ) −→ Gr( T c ) is a group morphism such that Gr( α c )( S c ) ⊆ T c . Finally, using that α ( u ) ≤ α ( v ), we obtain Gr( α c )( u ) ≤ v . We conclude that H ∗ is a well-defined functor.Now, we have to show that if we restrict the domain of H ∗ to Cu ∼ sc , alg , then H ∗ becomes faithful. Let α, β : ( S , u ) −→ ( T , v ) be two scaled Cu ∼ -morphisms between ( S , u ) , ( T , v ) ∈ Cu ∼ sc , alg such that H ∗ ( α ) = H ∗ ( β ). In particular, α c = β c , and since we are in the category of algebraic Cu ∼ -semigroups, any elementis supremum of increasing sequences of compact elements. Thus any morphism is entirely determinedby its restriction to compact elements. One can conclude that α = β and the proof is complete. (cid:3) Theorem 5.20.
The functor H ∗ : Cu ∼ u −→ AbGp u yields a natural isomorphism η ∗ : H ∗ ◦ Cu , u ≃ K ∗ .Proof. First we prove that K ∗ ( A ) + ≃ Cu ( A ) c as monoids and the result will follow from the Grothendieckconstruction.We know that Cu ( A ) c is a monoid. Now consider [( a , u )] ∈ Cu ( A ) c . By Corollary 3.5, we knowthat [ a ] is a compact element of Cu( A ). Besides, since A has stable rank one, we know that we can finda projection p ∈ A ⊗ K such that [ p ] = [ a ] in Cu( A ). So without loss of generality, we now describecompact elements of Cu ( A ) as classes [( p , u )] where p is projection in A ⊗ K and u is a unitary elementin her p .On the other hand, by Theorem 5.7, we have Cu ( A ) max ≃ K ( A ), where the AbGp-isomorphism isgiven by [( s A ⊗K , u )] [ u ], where s A ⊗K is any strictly positive element of A ⊗ K . Combined withDefinition 5.6, we get a monoid morphism j : Cu ( A ) −→ K ( A ). Now set: α : Cu ( A ) c −→ K ∗ ( A ) + [( p , u )] ([ p ] , j ([ p , u ]))It is routine to check that α is monoid morphism. Further, observe that j ([ p , u ]) = δ I p A ([ u ]) for any[( p , u )] ∈ Cu ( A ) c , where δ I p A : K (her p ) K ( i ) −→ K ( A ) (see Notation 4.2). Thus, j ([ p , u ]) = [ u + (1 − p )] K ( A ) . Now, since A has stable rank one, ∼ MvN and ∼ Cu agree on projections. It is now clear that α isan isomorphism and hence Cu ( A ) c ≃ K ∗ ( A ) + as monoids. From the Grothendieck construction, one cancheck that (K ∗ ( A ) , K ∗ ( A ) + ) ≃ (Gr(Cu ( A ) c ) , Cu ( A ) c ) as ordered groups. Finally, it is routine to checkthat [(1 A , A )] is a compact order-unit for Cu ( A ) (a fortiori, an order-unit for (Gr(Cu ( A ) c ) , Cu ( A ) c ))and that α ([(1 A , A )]) = K ∗ ( A ) .We conclude that for any A ∈ C ∗ , there exists a natural ordered group isomorphism η ∗ A : H ∗ ◦ Cu , u ( A ) ≃ (K ∗ ( A ) , K ∗ ( A ) + , K ∗ ( A ) ) that preserves the order-unit and hence there exists a natural isomor-phism η ∗ : H ∗ ◦ Cu , u ≃ K ∗ . (cid:3) Corollary 5.21.
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