Cones of traces arising from AF C*-algebras
aa r X i v : . [ m a t h . OA ] S e p CONES OF TRACES ARISING FROM AF C*-ALGEBRAS
MARK MOODIE AND LEONEL ROBERT
Abstract.
We characterize the topological non-cancellative cones that are ex-pressible as projective limits of finite powers of [0 , ∞ ]. These are also the cones oflower semicontinuous extended-valued traces on AF C*-algebras. Our main resultmay be regarded as a generalization of the fact that any Choquet simplex is aprojective limit of finite dimensional simplices. To obtain our main result, we firstestablish a duality between certain non-cancellative topological cones and Cuntzsemigroups with real multiplication. This duality extends the duality betweencompact convex sets and complete order unit vector spaces to a non-cancellativesetting. Introduction
By a theorem of Lazar and Lindenstrauss, any Choquet simplex can be expressedas a projective limit of finite dimensional simplices (see [17], [11]). This has impli-cations for C*-algebras: given a Choquet simplex K , there exists a simple, unital,approximately finite dimensional (AF) C*-algebra whose set of tracial states is iso-morphic to K ([5, 11]). In the investigations on the structure of a C*-algebra,another kind of trace is also of interest, namely, the lower semicontinuous traceswith values in [0 , ∞ ]. These traces form a non-cancellative topological cone. (Bycone we understand an abelian monoid endowed with a scalar multiplication bypositive scalars.) Our goal here is to characterize through intrinsic properties thetopological cones arising as the lower semicontinuous [0 , ∞ ]-valued traces on an AFC*-algebra. These are also the projective limits of cones of the form [0 , ∞ ] n , with n ∈ N , and also, the cones arising as the [0 , ∞ ]-valued monoid morphisms on thepositive elements of a dimension group.Let A be C*-algebra. Denote its cone of positive elements by A + . A map τ : A + → [0 , ∞ ] is called a trace if it is linear (additive, homogeneous, mapping0 to 0) and satisfies that τ ( x ∗ x ) = τ ( xx ∗ ) for all x ∈ A . We are interested in thelower semicontinuous traces. Let T ( A ) denote the cone of [0 , ∞ ]-valued lower semi-continuous traces on A + . By the results of [12], T ( A ) is a complete lattice whenendowed with the algebraic order, and addition in T ( A ) is distributive with respectto the lattice operations. Further, one can endow T ( A ) with a topology that islocally convex, compact and Hausdorff. We call an abstract topological cone withthese properties an extended Choquet cone (see Section 2).By an AF C*-algebra we understand an inductive limit, over a possibly uncount-able index set, of finite dimensional C*-algebras. Not every extended Choquet cone arises as the cone of lower semicontinuous traces on an AF C*-algebra. The requisiteadditional properties are sorted out in the theorem below.An element w in a cone is called idempotent if 2 w = w . Given a cone C , wedenote by Idem( C ) the set of idempotent elements of C . Theorem 1.1.
Let C be an extended Choquet cone (see Definition 2.1). The fol-lowing are equivalent: (i) C is isomorphic to T ( A ) for some AF C*-algebra A . (ii) C is isomorphic to Hom( G + , [0 , ∞ ]) for some dimension group ( G, G + ) .(Here Hom( G + , [0 , ∞ ]) denotes the set of monoid morphisms from G + to [0 , ∞ ] .) (iii) C is a projective limit of cones of the form [0 , ∞ ] n , n ∈ N . (iv) C has the following properties:(a) Idem( C ) is an algebraic lattice under the opposite algebraic order,(b) for each w ∈ Idem( C ) , the set { x ∈ C : x ≤ w } is connected.Moreover, if C is metrizable and satisfies (iv), then the C*-algebra in (i) may bechosen separable, the group G in (ii) may be be chosen countable, and the projectivelimit in (iii) may be chosen over a countable index set. We refer to property (a) in part (iv) as “having an abundance of compact idem-potents”. The fact that the primitive spectrum of an AF C*-algebra has a basisof compact open sets makes this condition necessary. We call property (b) “strongconnectedness”. The existence of a non-trivial trace on every simple ideal-quotientof an AF C*-algebra makes this condition necessary. In general, if a C*-algebra A is such that its primitive spectrum has a basis of compact open sets, and every sim-ple quotient I/J , where J ( I are ideals of A , has a non-zero densely finite trace,then T ( A ) has an abundance of compact idempotents and is strongly connected,i.e., properties (a) and (b) above hold. For example, if A has real rank zero, stablerank one, and is exact, then these conditions are met. Theorem 1.1 then asserts theexistence of an AF C*-algebra B such that T ( A ) ∼ = T ( B ).The crucial implication in Theorem 1.1 is (iv) implies (iii). A reasonable ap-proach to proving it is to first prove that (iv) implies (ii) by directly constructinga dimension group G from the cone C , very much in the spirit of the proof of theLazar-Lindenstrauss theorem obtained by Effros, Handelmann, and Shen in [11](which, unlike the proof in [17], also deals with non-metrizable Choquet simpleces).If the cone C is assumed to be finitely generated, then we indeed obtain a directconstruction of an ordered vector space with the Riesz property ( V, V + ) such thatHom( V + , [0 , ∞ ]) is isomorphic to C . This is done in the last section of the paper.In the general case, however, such an approach has eluded us.To prove Theorem 1.1 we first establish a duality between extended Choquet coneswith an abundance of compact idempotents and certain abstract Cuntz semigroups.Briefly stated, this duality works as follows: C Lsc σ ( C ) and S F ( S ) . That is, to an extended Choquet cone C with an abundance of compact idempotentsone assigns the Cu-cone Lsc σ ( C ) of lower semicontinuous linear functions f : C → ONES OF TRACES ARISING FROM AF C*-ALGEBRAS 3 [0 , ∞ ] with “ σ -compact support”. In the other direction, to a Cu-cone S with anabundance of compact ideals one assigns the cone of functionals F ( S ); see Section 5and Theorem 5.2. In the context of this duality, strong connectedness in C translatesinto the property of weak cancellation in Lsc σ ( C ). We then use this arrow reversingduality to turn the question of finding a projective limit representation for a coneinto one of finding an inductive limit representation for a Cu-cone. To achievethe latter, we follow the strategy of proof of the Effros-Handelmann-Shen theorem,adapted to the category at hand. The main technical complication here is the non-cancellative nature of Cu-cones, but this is adequately compensated by the abovementioned property of “weak cancellation” (dual to strong connectedness).A question that is closely related to the one addressed by Theorem 1.1 asks fora characterization of the lattices arising as (closed two-sided) ideal lattices of AFC*-algebras. For separable AF C*-algebras, this problem was solved by Bratteli andElliott in [7], and independently by Bergman in unpublished work: Any completealgebraic lattice with a countable set of compact elements is the lattice of closedtwo-sided ideals of a separable AF C*-algebra. A thorough discussion of this resultis given by Goodearl and Wehrung in [14]. The cardinality restriction on the setof compact elements is necessary, as demonstrated by examples of R˚uˇziˇcka andWehrung ([22], [24]). Now, the lattice of closed two-sided ideals of a C*-algebra A is in order reversing bijection with the lattice of idempotents of T ( A ) via theassignment I τ I , where τ I is the { , ∞} -valued trace vanishing on I + . Thus, therealization of a cone C in the form T ( A ) entails the realization of (Idem( C ) , ≤ op )as the ideal lattice of A . Curiously, no cardinality restriction is needed in Theorem1.1 above. This demonstrates that the examples of R˚uˇziˇcka and Wehrung are alsoexamples of algebraic lattices that cannot be realized as the lattice of idempotentsof a cone C satisfying any of the equivalent conditions of Theorem 1.1.This paper is organized as follows: In Section 2 we define extended Choquet conesand prove a number of background results on their structure. In Section 3 we goover three constructions—starting from a C*-algebra, a dimension group, and a Cu-semigroup—yielding extended Choquet cones that are strongly connected and havean abundance of compact idempotents. Sections 4 and 5 delve into spaces of linearfunctions on extended Choquet cones with an abundance of compact idempotents.In Theorem 5.2 we establish the above mentioned duality assigning to a cone C theCu-cone Lsc σ ( C ), and conversely to a Cu-cone S its cone of functionals F ( S ). InSection 6 we prove Theorem 1.1. In Section 7 we assume that the cone C is finitelygenerated. In this case we give a direct construction of an ordered vector space withthe Riesz property ( V, V + ) such that C ∼ = Hom( V + , [0 , ∞ ]). The vector space V isdescribed as R -valued functions on a certain spectrum of the cone C . Acknowledgement : The second author thanks Hannes Thiel for fruitful dis-cussions on the topic of topological cones and for sharing his unpublished notes[23].
MARK MOODIE AND LEONEL ROBERT Extended Choquet Cones
Algebraically ordered compact cones.
We call cone an abelian monoid( C, +) endowed with a scalar multiplication by positive real numbers (0 , ∞ ) × C → C such that(i) the map ( t, x ) tx is additive on both variables,(ii) s ( tx ) = ( st ) x for all s, t ∈ (0 , ∞ ) and x ∈ C ,(iii) 1 · x = x for all x ∈ C .We do not assume that the addition operation on C is cancellative. In fact, theprimary example of the cones that we investigate below is [0 , ∞ ] endowed with theobvious operations.The algebraic pre-order on C is defined as follows: x ≤ y if there exists z ∈ C such that x + z = y . We say that C is algebraically ordered if this pre-order is anorder.We call C a topological cone if it is endowed with a topology for which theoperations of addition and multiplication by positive scalars are jointly continuous. Definition 2.1.
An algebraically ordered topological cone C is called an extendedChoquet cone if (i) C is a lattice under the algebraic order, and the addition operation is dis-tributive over both ∧ and ∨ : x + ( y ∧ z ) = ( x + y ) ∧ ( x + z ) ,x + ( y ∨ z ) = ( x + y ) ∨ ( x + z ) , for all x, y, z ∈ C , (ii) the topology on C is compact, Hausdorff, and locally convex, i.e., it has abasis of open convex sets.Remark . It is a standard result that in a compact algebraically ordered monoidboth upward and downward directed sets converge to their supremum and infimum,respectively ([2, Proposition 3.1], [13, Proposition VI-1.3, p441]). We shall makefrequent use of this fact applied to extended Choquet cones. It readily follows fromthis and the existence of finite suprema and infima that extended Choquet conesare complete lattices.
Remark . By Wehrung’s [25, Theorem 3.11], the algebraic and order theoreticproperties of an extended Choquet cone may be summarized as saying that it is aninjective object in the category of positively ordered monoids.
Example . The set [0 , ∞ ] is an extended Choquet cone when endowed with thestandard operations of addition and scalar multiplication and the standard topology.More generally, the powers [0 , ∞ ], endowed with coordinatewise operations and theproduct topology are extended Choquet cones.Let C and D be extended Choquet cones. A map φ : C → D is a morphism in theextended Choquet cones category if φ is linear (additive, homogeneous with respectto scalar multiplication, and mapping 0 to 0) and continuous. ONES OF TRACES ARISING FROM AF C*-ALGEBRAS 5
Theorem 2.5.
The category of extended Choquet cones has projective limits.Proof.
Let { C i : i ∈ I } , { ϕ i,j : C i → C j : i, j ∈ I with j ≤ i } , be a projective systemof extended Choquet cones, where I is an upward directed set. Define C = (cid:26) ( x i ) i ∈ Y i ∈ I C i : x j = ϕ i,j ( x i ) for all i, j ∈ I with j ≤ i (cid:27) . Endow the product Q i ∈ I C i with coordinatewise operations, coordinatewise order,and with the product topology; endow C with the topological cone structure inducedby inclusion. Let π i : C → C i , i ∈ I , denote the projection maps. It follows fromwell known arguments that { C, π i | C : i ∈ I } is the projective limit of the system { C i , φ i,j : i, j ∈ I } as compact Hausdorff topological cones (cf. [10, Theorem 13]).Since for each i the topology of C i has a basis of open convex sets, the producttopology on Q i C i also has a basis of open convex sets. Further, since C is a convexsubset of Q i C i , the induced topology on C is locally convex as well.Let us now prove that C is a lattice. The proof runs along the same lines asthe one in [10, Theorem 13] for projective limits of Choquet simplices. We showthat C is closed under finite suprema; the argument for finite infima is similar. Let x = ( x i ) i and y = ( y i ) i be in C . Their coordinatewise supremum exists in Q i C i ,but does not necessarily belong to C . For each k ∈ I define z ( k ) ∈ Q i C i by( z ( k ) ) i = ( φ k,i ( x k ∨ y k ) if i ≤ k,x i ∨ y i otherwise . If k ′ ≥ k , then φ k ′ ,k ( x k ′ ∨ y k ′ ) ≥ φ k ′ ,k ( x k ′ ) = x k , and similarly φ k ′ ,k ( x k ′ ∨ y k ′ ) ≥ y k , whence φ ( x k ′ ∨ y k ′ ) ≥ x k ∨ y k . It follows that( z ( k ′ ) ) i = φ k ′ ,i ( x k ′ ∨ y k ′ ) ≥ φ k,i ( x k ∨ y k ) = ( z ( k ) ) i , for i ≤ k , while ( z ( k ′ ) ) i ≥ x i ∨ y i = ( z ( k ) ) i for i (cid:2) k . Thus, ( z ( k ) ) k ∈ I is an upward directed net. Set x ∨ y := lim k z ( k ) , whichis readily shown to belong to C . Then x ∨ y ≥ z ( k ) ≥ x, y for all k . Suppose that w = ( w i ) i ∈ C is such that w ≥ x, y . Then w i ≥ x i ∨ y i for all i , and further w i = ϕ k,i ( w k ) ≥ ϕ k,i ( x k ∨ y k ) . Hence, w ≥ z ( k ) for all k , and so w ≥ x ∨ y . This proves that x ∨ y is in fact thesupremum of x and y in C . MARK MOODIE AND LEONEL ROBERT
Let us prove distributivity of addition over ∨ . Let x, y, v ∈ C . Fix an index i .Then (( x + v ) ∨ ( y + v )) i = lim k φ k,i (( x k + v k ) ∨ ( y k + v k ))= lim k φ k,i (( x k ∨ y k ) + v k )= lim k φ k,i ( x k ∨ y k ) + v i = ( x ∨ y + v ) i , where we have used the distributivity of addition over ∨ on each coordinate and theconstruction of joins in C obtained above. Thus, ( x + v ) ∨ ( y + v ) = ( x ∨ y ) + v .Distributivity over ∧ is handled similarly. (cid:3) Lattice of idempotents.
Throughout this subsection C denotes an extendedChoquet cone.An element w ∈ C is called idempotent if 2 w = w . It follows, using that C is algebraically ordered, that tw = w for all t ∈ (0 , ∞ ]. We denote the set ofidempotents of C by Idem( C ). The set Idem( C ) is a sub-lattice of C : if w and w are idempotents then 2( w ∨ w ) = (2 w ∨ w ) = w ∨ w , where we have used that multiplication by 2 is an order isomorphism. Hence, w ∨ w is an idempotent. Similarly, w ∧ w is shown to be an idempotent. Moreover, w ∨ w = w + w , a fact easily established.In the lattice Idem( C ), we use the symbol ≫ to denote the way-below relationunder the opposite order. That is, w ≫ w if whenever inf i v i ≤ w for a decreasingnet ( v i ) i in Idem( C ), we have v i ≤ w for some i . We call w ∈ Idem( C ) a compactidempotent if w ≫ w . More explicitly, w is compact if whenever inf i v i ≤ w for adecreasing net ( v i ) i in Idem( C ), we have v i ≤ w for some i . Note: we only use thenotion of compact element in Idem( C ) in the sense just defined, i.e., applied to the opposite order .A complete lattice is called algebraic if each of its elements is a supremum ofcompact elements ([13, Definition I-4.2]). Definition 2.6.
We say that an extended Choquet cone C has an abundance ofcompact idempotents if (Idem( C ) , ≤ op ) is an algebraic lattice, i.e., every idempotentin C is an infimum of compact idempotents. Let x ∈ C . Consider the set { z ∈ C : x + z = x } . This set is closed under additionand also closed in the topology of C . It follows that it has a maximum element ǫ ( x ).Since 2 · ǫ ( x ) is also absorbed additively by x , we have ǫ ( x ) = 2 ǫ ( x ), i.e., ǫ ( x ) is anidempotent. We call ǫ ( x ) the support idempotent of x . Lemma 2.7. (Cf. [2, Lemma 3.2] ) Let x, y, z ∈ C . (i) ǫ ( x ) = lim n n x . (ii) If x + z ≤ y + z then x + ǫ ( z ) ≤ y + ǫ ( z ) . ONES OF TRACES ARISING FROM AF C*-ALGEBRAS 7
Proof. (i) Observe that w := lim n n x exists, since the infimum of a decreasingsequence is also its limit. It is also clear that 2 w = w , and that x + w = x . Let z ∈ C be such that x + z = x . Then x + nz = x , i.e., n x + z = n x , for all n ∈ N .Letting n → ∞ , we get that w + z = w , and in particular, w ≤ z . Thus, w is thelargest element absorbed by x , i.e. w = ǫ ( x ).(ii) This is [2, Lemma 3.2]. Here is the argument: We deduce, by induction, that nx + z ≤ ny + z for all n ∈ N . Hence, x + n z ≤ y + n z . Letting n → ∞ and using(i), we get x + ǫ ( z ) ≤ y + ǫ ( z ). (cid:3) Lemma 2.8.
Let K ⊆ C be closed and convex. Then the map x ǫ ( x ) attains amaximum on K .Proof. Let W = { ǫ ( x ) : x ∈ K } . Let x , x ∈ K , with ǫ ( x ) = w and ǫ ( x ) = w .Since K is convex, ( x + x ) / ∈ K . Since ǫ (cid:16) x + x (cid:17) = lim n n x + 12 n x = ǫ ( x ) + ǫ ( x ) , the set W is closed under addition. For each w ∈ W , let us choose x w ∈ K with ǫ ( x w ) = w . By compactness of K , the net ( x w ) w ∈ W has a convergent subnet. Say x h ( λ ) → x ∈ K , where h : Λ → W is increasing and with cofinal range. For each λ we have x h ( λ ′ ) + h ( λ ) = x h ( λ ′ ) for all λ ′ ≥ λ . Passing to the limit in λ ′ we get x + h ( λ ) = x . Since h ( λ ) ranges through a cofinal set in W , x + w = x for all w ∈ W .Thus, ǫ ( · ) attains its maximum on W at x . (cid:3) Lemma 2.9.
For each idempotent w ∈ C the set { x ∈ C : w ≫ ǫ ( x ) } is open.(Recall that ≫ is the way below relation in the lattice (Idem( C ) , ≤ op ) .)Proof. Let x ∈ C be such that w ≫ ǫ ( x ). By Lemma 2.8, for each closed convexneighborhood K of x , there exists x K ∈ K at which ǫ ( · ) attains its maximum. Bythe local convexity of C , the system of closed convex neighborhoods of x is downwarddirected. It follows that ( ǫ ( x K )) K is downward directed. Moreover, x K → x , sincethe topology is Hausdorff. We claim that ǫ ( x ) = inf K ǫ ( x K ), where K ranges throughall the closed convex neighborhoods of x . Proof: Set y = inf K ǫ ( x K ). We have y ≤ ǫ ( x K ) ≤ x K n for all K and n ∈ N . Passing to the limit, first in K and then in n ,we get that y ≤ ǫ ( x ). On the other hand, ǫ ( x ) ≤ ǫ ( x K ) for all K (since x ∈ K and ǫ attains its maximum on K at x K ). Thus, ǫ ( x ) ≤ y , proving our claim.We have w ≫ ǫ ( x ) = inf K ǫ ( x K ). Hence, there is K such that w ≫ ǫ ( x K ). So,there is a neighborhood of x all whose members belong to { z ∈ C : w ≫ ǫ ( z ) } . Thisshows that { z ∈ C : w ≫ ǫ ( z ) } is open. (cid:3) Cancellative subcones.
Fix an idempotent w ∈ C . Let C w = { x ∈ C : ǫ ( x ) = w } . Then C w is closed under sums, scalar multiplication by positive scalars, finite infima,and finite suprema. By Lemma 2.7 (ii), C w is also cancellative: x + z ≤ y + z impliesthat x ≤ y for all x, y, z ∈ C w . It follows that C w embeds in a vector space; namely,the abelian group of formal differences x − y , with x, y ∈ C w endowed with the uniquescalar multiplication extending the scalar multiplication on C w . Let V w denote the MARK MOODIE AND LEONEL ROBERT vector space of differences x − y , with x, y ∈ C w . Let η : C w × C w → V w be definedby η ( x, y ) = x − y . We endow C w with the topology that it receives as a subset of C . We endow V w with the quotient topology coming from the map η . Theorem 2.10.
Let w ∈ Idem( C ) be a compact idempotent. Then V w is a lo-cally convex topological vector space whose topology restriced to C w agrees with thetopology on C w . Moreover, either C w = { w } or C w has a compact base. Note: A subset B of a cone T is called a base if for each nonzero x ∈ T theintersection of (0 , ∞ ) · x with B is a singleton set. Proof.
Let us first show that the topology on C w is locally compact. Since w iscompact, the set { x ∈ C : w ≥ ǫ ( x ) } is open by Lemma 2.9. We then have that C w is the intersection of the closed set { x ∈ C : w ≤ x } and the open set { x : w ≥ ǫ ( x ) } .Hence, C w is locally compact in the induced topology.We can now apply [16, Theorem 5.3], which asserts that if C w is a locally compactcancellative cone, then indeed V w is a locally convex topological vector space whosetopology extends that of C w . Finally, by [1, Theorem II.2.6], a locally compactnontrivial cone in a locally convex topological space has a compact base. (cid:3) Strong connectedness.
Let v, w ∈ Idem( C ) be such that v ≤ w . Let’s saythat v is compact relative to w if v is a compact idempotent in the extended Choquetcone { x ∈ C : x ≤ w } . Put differently, if a downward directed net ( v i ) i in C satisfiesthat inf i v i ≤ v , then v i ∧ w ≤ v for some i . Theorem 2.11.
Let C be an extended Choquet cone. The following are equivalent: (i) For any two w , w ∈ Idem( C ) such that w ≤ w , w = w , and w iscompact relative to w , there exists x ∈ C such that w ≤ x ≤ w and x isnot an idempotent. (ii) The set { x ∈ C : x ≤ w } is connected for all w ∈ Idem( C ) .Moreover, if the above hold then the element x in (i) may always be chosen suchthat ǫ ( x ) = w .Proof. We show that the negations of (i) and (ii) are equivalent.Not (ii) ⇒ not (i): Suppose that { x ∈ C : x ≤ w } is disconnected for someidempotent w . Working in the cone { x ∈ C : x ≤ w } as the starting extendedChoquet cone, we may assume without loss of generality that w = ∞ (the largestelement of C ). Let U and V be open disjoint sets whose union is C . Assume that ∞ / ∈ U . Observe that totally ordered subsets of U have an upper bound: if ( x i ) i isa chain then x i → sup i x i , and since U is closed, sup x i ∈ U . By Zorn’s lemma, U contains a maximal element v . Since 2 v is connected to v by the path t tv with t ∈ [1 , v = v , i.e., v is an idempotent. Let’s show that v iscompact: Let ( v i ) i be a decreasing net of idempotents with infimum v . Suppose, forthe sake of contradiction, that v i = v for all i . Then v i ∈ U c for all i . Since U c isclosed and v i → v , v ∈ U c , which is a contradiction. Thus, v is compact. Let x ∈ C be such that v ≤ x ≤ ∞ . If ǫ ( x ) = ∞ , then x = ∞ . Suppose that ǫ ( x ) = v . Since x ONES OF TRACES ARISING FROM AF C*-ALGEBRAS 9 is connected to v by the path t tx , t ∈ (0 , x ∈ U . But v is maximalin U . Thus, x = v . This proves not (i).Not (i) ⇒ not (ii): Suppose that there exist w , w ∈ Idem( C ) such that w < w , w is relatively compact in w , and there is no non-idempotent x ∈ C such that w ≤ x ≤ w . By Zorn’s lemma, we can choose w minimal among the idempotentssuch that w ≤ w and w = w . Then(1) w ≤ x ≤ w ⇒ x ∈ { w , w } for all x ∈ C. Let us show that { x ∈ C : x ≤ w } is disconnected. Let U = { x ∈ C : x ≤ w } and U = { x ∈ C : x (cid:2) w } . These sets are clearly disjoint, non-empty ( w ∈ U and w ∈ U ), and cover { x ∈ C : x ≤ w } . It is also clear that U is open in C . Let’sconsider U . By (1), x ∈ U if and only if ǫ ( x ) ≤ w and x ≤ w . Further, since w is a compact idempotent in the extended Choquet cone { z ∈ C : z ≤ w } , the set U may be described as all x in the cone { z ∈ C : z ≤ w } such that w ≫ ǫ ( x ), wherethe relation ≫ is taken in the idempotent lattice of the cone { z ∈ C : z ≤ w } .Thus, by Lemma 2.9 applied in the extended Choquet cone { z ∈ C : z ≤ w } , theset U is (relatively) open in { x ∈ C : x ≤ w } .Finally, let us argue that x in (i) may be chosen such that ǫ ( x ) = w : Startingfrom w ≤ w , with w relatively compact in w , choose w ′ minimal element in { w ∈ Idem( C ) : w ≤ w ≤ w , w = w } , which exists by Zorn’s lemma. Let x ∈ C be a non-idempotent such that w ≤ x ≤ w ′ . Then ǫ ( x ) ∈ { w , w ′ } , but we cannothave ǫ ( x ) = w ′ , since this entails that x = w ′ . So ǫ ( x ) = w . (cid:3) Definition 2.12.
Let C be an extended Choquet simplex. Let us say that C isstrongly connected if it satisfies either one of the equivalent properties listed in The-orem 2.11. Proposition 2.13. If C is a projective limit of extended Choquet cones of the form [0 , ∞ ] n , then C is strongly connected and has an abundance of compact idempotents.Proof. Suppose that C = lim ←−{ C i , φ i,j : i, j ∈ I } , where C i ∼ = [0 , ∞ ] n i for all i ∈ I .A projective limit of continua (compact Hausdorff connected spaces) is again acontinuum. Since each C i is a continuum, so is C . In particular, C is connected. If w ∈ Idem( C ), with w = ( w i ) i ∈ Q i C i , then { x ∈ C : x ≤ w } = lim ←−{ x ∈ C i : x ≤ w i } . Thus, the same argument shows that { x ∈ C : x ≤ w } is connected.The lattice of idempotent elements of C i is finite, whence algebraic under theopposite order, for all i . Further, by additivity and continuity, the maps φ i,j preservedirected infima and arbitrary suprema (i.e., directed suprema and arbitrary infimaunder the opposite order). That Idem( C ) is algebraic under the opposite ordercan then be deduced from the fact that a projective limit of algebraic lattices isagain an algebraic lattice, where the morphisms preserve directed suprema andarbitrary infima. Let us give a direct argument instead: Let w ∈ Idem( C ), with w = ( w i ) i ∈ Q i C i . For each index k ∈ I define w ( k ) ∈ Q i C i as the unique element in C such that ( w ( k ) ) i = sup { z ∈ C i : φ i,k ( z ) = w k } for all i ≥ k. It is not hard to show that ( w ( k ) ) k ∈ I is a decreasing net in Idem( C ) with infimum w .Moreover, from the compactness of w k ∈ Idem( C k ) we deduce that w ( k ) ∈ Idem( C )is compact for all k ∈ I . Thus, Idem( C ) is an algebraic lattice under the oppositeorder. (cid:3) Cones of traces and functionals
Here we review various constructions giving rise to extended Choquet cones.Let A be a C*-algebra. Let A + denote the cone of positive elements of A . A map τ : A + → [0 , ∞ ] is called a trace if it maps 0 to 0, it is additive, homogeneous withrespect to scalar multiplication, and satisfies τ ( x ∗ x ) = τ ( xx ∗ ) for all x ∈ A . The setof all lower semicontinuous traces on A is denoted by T ( A ). It is endowed with thepointwise operations of addition and scalar multiplication. T ( A ) is endowed withthe topology such that a net ( τ i ) i in T ( A ) converges to τ ∈ T ( A ) iflim sup τ i (( a − ǫ ) + ) ≤ τ ( a ) ≤ lim inf τ i ( a )for all a ∈ A + and ǫ >
0. By [12, Theorems 3.3 and 3.7], T ( A ) is an extendedChoquet cone. Proposition 3.1.
Let A be a C*-algebra. (i) If the primitive spectrum of A has a basis of compact open sets, then T ( A ) has an abundance of compact idempotents. In particular, this holds if A hasreal rank zero. (ii) Suppose that for all J ( I ⊆ A , closed two-sided ideals of A such that I/J hascompact primitive spectrum, there exists a non-zero lower semicontinuousdensely finite trace on
I/J . Then T ( A ) is strongly connected. In particular,this holds if A has stable rank one and is exact.Proof. (i) The lattice of closed two-sided ideals of A is in order reversing bijectionwith the lattice of idempotents of T ( A ) via the assignment I τ I , where τ I ( a ) := ( a ∈ I + , ∞ otherwise.On the other hand, the lattice of closed two-sided ideals of A is isomorphic to thelattice of open sets of the primitive spectrum of A ([19, Theorem 4.1.3]). Thus,the lattice of idempotents of T ( A ) is algebraic (under the opposite order) if andonly if the lattice of open sets of the primitive spectrum is algebraic. The latter isequivalent to the existence of a basis of compact open sets for the topology.(ii) Let us check that T ( A ) satisfies condition (i) of Theorem 2.11. Recall thatidempotents in T ( A ) have the form τ I , where I is a closed two-sided ideal. Let I and J be (closed, two-sided) ideals of A , with J ⊆ I , so that τ I ≤ τ J . The propertythat τ I is compact relative to τ J means that if ( I i ) i is an upward directed net ofideals such that J ⊆ I i ⊆ I for all i and I = S I i , then I = I i for some i . This, ONES OF TRACES ARISING FROM AF C*-ALGEBRAS 11 in turn, is equivalent to
I/J having compact primitive spectrum. By assumption,there exists τ ∈ T ( I/J ) that is densely finite and non-zero. Pre-composed with thequotient map π : I → I/J (which maps I + onto ( I/J ) + ), τ gives rise to a trace τ ◦ π ∈ T ( I ). Let ˜ τ be the extension of τ ◦ π to A + such that ˜ τ ( a ) = ∞ for all a ∈ A + \ I + . Then τ I ≤ ˜ τ ≤ τ J and ˜ τ is not an idempotent, as it attains values otherthan { , ∞} . This proves that T ( A ) is strongly connected.Suppose now that A has stable rank one and is exact. By the arguments fromthe previous paragraph, it suffices to show that if I/J is a non-trivial ideal-quotientwith compact primitive spectrum, then there is a nontrivial lower semicontinuousdensely finite trace on
I/J . Observe that
I/J has stable rank one and is exact, sinceboth properties pass to ideals and quotients. An exact C*-algebra of stable rankone with compact primitive spectrum always has a nonzero, densely finite lowersemicontinuous trace; see [20, Theorem 2.15]. (cid:3)
Let (
G, G + ) be a dimension group, i.e., an ordered abelian group that is unper-forated and has the Riesz refinement property. Let Hom( G + , [0 , ∞ ]) denote the setof all [0 , ∞ ]-valued monoid morphisms on G + (i.e., λ : G + → [0 , ∞ ] additive andmapping 0 to 0). Endow Hom( G + , [0 , ∞ ]) with pointwise cone operations and withthe topology of pointwise convergence. Proposition 3.2.
Let G be a dimension group. Then Hom( G + , [0 , ∞ ]) is an ex-tended Choquet cone that is strongly connected and has an abundance of compactidempotents.Proof. By [25, Theorem 2.33], Hom( G + , [0 , ∞ ]) is a complete positively orderedmonoid, which entails that it is a complete lattice and that addition distributesover ∧ and ∨ . The topology on Hom( G + , [0 , ∞ ]) is that induced by its inclusionin [0 , ∞ ] G + . Since the latter is compact and Hausdorff, so is Hom( G + , [0 , ∞ ]).Further, since Hom( G + , [0 , ∞ ]) is a convex subset of [0 , ∞ ] G + , the induced topologyis locally convex. Thus, Hom( G + , [0 , ∞ ]) is an extended Choquet cone. To see thatit is strongly connected and has an abundance of compact idempotents, we can firstexpress ( G, G + ) as an inductive limit of ( Z n , Z n + ) using the Effros-Handelmann-Shentheorem ([11, Theorem 2.2]), apply the functor Hom( · , [0 , ∞ ]) to this limit, and thenapply Proposition 2.13. We give a direct argument in the paragraphs below.A subgroup I ⊆ G is an order ideal if I + := G + ∩ I is a hereditary set and I = I + − I + . Idempotent elements of Hom( G + , [0 , ∞ ]) have the form λ I ( g ) = 0if g ∈ I + and λ I ( g ) = ∞ if g ∈ G + \ I + , for some ideal I . Moreover, the map I λ I is an order reversing bijection between the two lattices. It is well knownthat the lattice of ideals of an ordered group is algebraic. Thus, Hom( G + , [0 , ∞ ])has abundance of compact idempotents.Let us now prove strong connectedness. Let I, J ⊆ G be order ideals such that J ( I and λ I is compact relative to λ J . In this case, this means that I/J is finitely(thus, singly) generated. Thus, it has a finite nonzero functional λ : ( I/J ) + → [0 , ∞ )(e.g., by [11, Theorem 1.4]). As in the proof of Proposition 3.1 (ii), we define afunctional on all of G + by pre-composing λ with the quotient map I I/J and setting it equal to ∞ on G + \ I + . This produces a functional ˜ λ ∈ Hom( G + , [0 , ∞ ])such that λ I ≤ ˜ λ ≤ λ J and ˜ λ is not an idempotent. (cid:3) Yet another construction yielding an extended Choquet cone is the dual of a Cu-semigroup. Let us first briefly recall the definition of a Cu-semigroup. Let S be apositively ordered monoid. Given x, y ∈ S , let us write x ≪ y (read “ x is way below y ”) if whenever ( y n ) ∞ n =1 is an increasing sequence in S such that y ≤ sup n y n , thereexists n such that x ≤ y n .We call S a Cu-semigroup if it satisfies the following axioms:O1. For every increasing sequence ( x n ) n in S , the supremum sup n x n exists.O2. For every x ∈ S there exists a sequence ( x n ) n in S such that x n ≪ x n +1 forall n ∈ N and x = sup n x n .O3. If ( x n ) n and ( y n ) n are increasing sequences in S , then sup n ( x n + y n ) =sup n x n + sup n y n .O4. If x i ≪ y i for i = 1 ,
2, then x + x ≪ y + y .Observe that in our definition of the way-below relation above we only con-sider increasing sequences ( y n ) n , rather than increasing nets. In the context ofCu-semigroups we always use the symbol ≪ to indicate this sequential version ofthe way below relation.Two additional conditions that we often impose on Cu-semigroups are the follow-ing:O5. If x ′ ≪ x ≤ y then there exists z such that x ′ + z ≤ y ≤ x + z .O6. If x, y, z ∈ S are such that x ≤ y + z , then for every x ′ ≪ x there areelements y ′ , z ′ ∈ S such that x ′ ≤ y ′ + z ′ , y ′ ≤ x, y and z ′ ≤ x, z .An ordered monoid map λ : S → [0 , ∞ ] is called a functional on S if it preservesthe suprema of increasing sequences. The collection of all functionals on S , denotedby F ( S ), is a cone, with the cone operations defined pointwise. F ( S ) is endowedwith the topology such that a net ( λ i ) i ∈ I in F ( S ) converges to a functional λ iflim sup i λ i ( s ′ ) ≤ λ ( s ) ≤ lim inf i λ i ( s )for all s ′ ≪ s , in S . By [12, Theorem 4.8] and [21, Theorem 4.1.2], if S is aCu-semigroup satisfying O5 and O6, then F ( S ) is an extended Choquet cone. InSection 5 we address the problem of what conditions on S guarantee that F ( S ) hasan abundance of compact idempotents and is strongly connected.4. Functions on an extended Choquet cone
Throughout this section we let C denote an extended Choquet cone with anabundance of compact idempotents, i.e., such that the lattice (Idem( C ) , ≤ op ) isalgebraic.4.1. The spaces
Lsc( C ) and A( C ) . Let us denote by Lsc( C ) the set of all func-tions f : C → [0 , ∞ ] that are linear (additive, homogeneous with respect to scalarmultiplication, and mapping 0 to 0) and lower semicontinuous ( f − (( a, ∞ ]) is openfor any a ∈ [0 , ∞ )). The linearity of the functions in Lsc( C ) implies that they ONES OF TRACES ARISING FROM AF C*-ALGEBRAS 13 are also order preserving, for if x ≤ y in C , then y = x + z for some z , and so f ( y ) = f ( x ) + f ( z ) ≥ f ( x ). We endow Lsc( C ) with the operations of pointwiseaddition and scalar multiplication, and with the pointwise order. Lsc( C ) is thus anordered cone. Further, the pointwise supremum of functions in Lsc( C ) is again inLsc( C ); thus, Lsc( C ) is a directed complete ordered set (dcpo).Let us denote by Lsc σ ( C ) the subset of Lsc( C ) of functions f : C → [0 , ∞ ] forwhich the set f − (( a, ∞ ]) is σ -compact—in addition to being open—for all a ∈ [0 , ∞ ) (equivalently, for a = 1, by linearity.) We denote by A( C ) the functions inLsc( C ) that are continuous. Notice that A( C ) ⊆ Lsc σ ( C ), since f − (( a, ∞ ]) = [ n f − ([ a + 1 n , ∞ ]) , and the right side is a union of closed (hence, compact) subsets of C .Our goal is to show that every function in Lsc( C ) (Lsc σ ( C )) is the supremum ofan increasing net (sequence) of functions in A( C ). We achieve this in Theorem 4.4after a number of preparatory results.Given f ∈ Lsc( C ), define its support supp( f ) ∈ C assupp( f ) = sup { x ∈ C : f ( x ) = 0 } . Since f ( x ) = 0 ⇒ f (2 x ) = 0, it follows easily that supp( f ) is an idempotent of C .For each w ∈ Idem( C ), let χ w ( x ) = ( x ≤ w, ∞ otherwise.This is a function in Lsc( C ). Lemma 4.1.
We have ∞ · f = χ supp( f ) , for all f ∈ Lsc( C ) . (Here ∞ · f :=sup n ∈ N nf .)Proof. The set { x ∈ C : f ( x ) = 0 } is upward directed and converges to itssupremum, i.e., to supp( f ). It follows, by the lower semicontinuity of f , that f (supp( f )) = 0.If x ≤ supp( f ), then f ( x ) ≤ f (supp( f )) = 0. Hence, ( ∞ · f )( x ) = 0. If on theother hand x (cid:2) supp( f ), then f ( x ) = 0, which implies that ( ∞ · f )( x ) = ∞ . Wehave thus shown that ∞ · f = χ supp( f ) . (cid:3) Let w ∈ C be an idempotent. Define A w ( C ) = { f ∈ A( C ) : supp( f ) = w } andA + ( C w ) = { f : C w → [0 , ∞ ) : f is continuous, linear, and f ( x ) = 0 ⇔ x = w } . (Recall that we have defined C w = { x ∈ C : ǫ ( x ) = w } .) Theorem 4.2. If f ∈ A( C ) then supp( f ) is a compact idempotent. Further, givena compact idempotent w ∈ Idem( C ) , the restriction map f f | C w is an orderedcone isomorphism from A w ( C ) to A + ( C w ) .Proof. Let f ∈ A( C ). We have already seen that supp( f ) is an idempotent. Toprove its compactness, let ( w i ) i ∈ I be a downward directed family of idempotents with infimum supp( f ). By the continuity of f , we have lim i f ( w i ) = f (supp( f )) = 0.But f ( w i ) ∈ { , ∞} for all i . Therefore, there exists i such that f ( w i ) = 0 for all i ≥ i . But supp( f ) is the largest element on which f vanishes. Hence, w i = supp( f )for all i ≥ i . Thus, supp( f ) is a compact idempotent.Now, fix a compact idempotent w . Let f ∈ A w ( C ). Clearly, f is continuous andlinear on C w , and f ( w ) = 0. Let x ∈ C w . If f ( x ) = 0, then x ≤ w , which impliesthat x = w . Thus, f ( x ) > x ∈ C w \{ w } . Suppose that f ( x ) = ∞ . Then f ( w ) = lim n f ( n x ) = ∞ , contradicting that w = supp( f ). Thus, f ( x ) < ∞ for all x ∈ C w . We have thus shown that f | C w ∈ A + ( C w ).It is clear that the restriction map A w ( C ) ∋ f f | C w ∈ A + ( C w ) is additive andorder preserving. Let us show that it is an order embedding. Let f, g ∈ A w ( C ) besuch that f | C w ≤ g | C w . Let x ∈ C . Suppose that x + w ∈ C w . Then f ( x ) = f ( x + w ) ≤ g ( x + w ) = g ( x ) . If, on the other hand, x + w / ∈ C w , then ǫ ( x + w ) > w . Hence, f ( x ) = f ( x + w ) ≥ f ( ǫ ( x + w )) = ∞ . We argue similarly that g ( x ) = ∞ . Thus, f ( x ) = g ( x ).Let us finally prove surjectivity. Suppose first that C w = { w } . Then A + ( C w )consists of the zero function only. Clearly then, χ w | C w = 0 and supp( χ w ) = w . Itremains to show that χ w is continuous. The set χ − w ( {∞} ) = { x ∈ C : x (cid:2) w } isopen. On the other hand, χ − w ( { } ) = { x ∈ C : x ≤ w } agrees with { x ∈ C : ǫ ( x ) ≤ w } (since we have assumed that C w = { w } ). The set { x ∈ C : ǫ ( x ) ≤ w } is open bythe compactness of w (Lemma 2.9). Thus, χ w is continuous.Suppose now that C w = { w } . Let ˜ f ∈ A + ( C w ). Define f : C → [0 , ∞ ] by f ( x ) = ( ˜ f ( x + w ) if x + w ∈ C w , ∞ otherwise.Observe that f | C w = ˜ f . Let us show that f ∈ A w ( C ). To show that supp( f ) = w ,note that f ( x ) = 0 ⇔ ˜ f ( x + w ) = 0 ⇔ x + w = w ⇔ x ≤ w. Thus, w is the largest element on which f vanishes, i.e., w = supp( f ). We leavethe not difficult verification that f is linear to the reader. Let us show that f iscontinuous. Let ( x i ) i be a net in C with x i → x . Suppose first that x + w ∈ C w ,i.e, ǫ ( x ) ≤ w . Since the set { y ∈ C : ǫ ( y ) ≤ w } is open (Lemma 2.9), ǫ ( x i ) ≤ w forlarge enough i . Therefore,lim i f ( x i ) = lim i ˜ f ( x i + w ) = ˜ f ( x + w ) = f ( x ) . Now suppose that x + w / ∈ C w , in which case f ( x ) = ∞ . To show that lim i f ( x i ) = ∞ , we may assume that x i ∈ C w for all i (otherwise f ( x i ) = ∞ by definition).Observe also that x i = w for large enough i . Let us thus assume that x i ∈ C w \{ w } for all i . Since w is a compact idempotent, C ω has a compact base K ⊆ C w \ { w } (Theorem 2.10). Write x i = t i ˜ x i with ˜ x i ∈ K and t i > i . Passing to aconvergent subnet and relabelling, assume that ˜ x i → y ∈ K and t i → t ∈ [0 , ∞ ]. If ONES OF TRACES ARISING FROM AF C*-ALGEBRAS 15 t < ∞ , then x = lim i t i ˜ x i = ty ∈ C w , contradicting our assumption that x + w / ∈ C w .Hence t = ∞ . Let δ > f on the compact set K . Then f ( x i ) = ˜ f ( x i ) = t i f (˜ x i ) ≥ t i δ. Hence f ( x i ) → ∞ , thus showing the continuity of f at x . (cid:3) We will need the following theorem from [2]:
Theorem 4.3 ([2, Theorem 3.5]) . Given f, g ∈ Lsc( C ) there exists f ∧ g and further, f ∧ sup i f i = sup i ( f ∧ f i ) , for any upward directed set ( f i ) i ∈ I in Lsc( C ) . Recall that throughout this section C denotes an extended Choquet cone with anabundance of compact idempotents. Theorem 4.4.
Every function in
Lsc( C ) is the supremum of an upward directedfamily of functions in A( C ) , and every function in Lsc σ ( C ) is the supremum of anincreasing sequence in A( C ) .Proof. Let f ∈ Lsc( C ) and set w = supp( f ). We first consider the case that w iscompact and then deal with the general case.Assume that w is compact. If C w = { w } , then f = χ w . Further, χ w is continuous,as shown in the proof of Theorem 4.2. Suppose that C w = { w } . Consider therestriction of f to C w . By [1, Corollary I.1.4], f | C w is the supremum of an increasingnet (˜ h i ) i of linear continuous functions ˜ h i : C w → R . Since f | C w is strictly positiveon C w \{ w } , it is separated from 0 on any compact base of C w . It follows that thefunctions ˜ h i are eventually strictly positive on C w \{ w } . Indeed, the sets U i,δ =˜ h − i (( δ, ∞ ]) ∩ C w , where i ∈ I and δ >
0, form an upward directed open cover of C w \{ w } . Thus, for some δ > i ∈ I , ˜ h i is greater than δ on a (fixed) compactbase of C w for all i ≥ i . Let us thus assume that ˜ h i ∈ A + ( C w ) for all i . By Theorem4.2, each ˜ h i has a unique continuous extension to an h i ∈ A w ( C ). Further, ( h i ) i isalso an increasing net. We claim that f = sup i h i . Let us first show that h i ≤ f forall i . Let x ∈ C be such that f ( x ) < ∞ . Then0 = lim n n f ( x ) = f ( ǫ ( x )) = 0 . Hence, ǫ ( x ) ≤ w , i.e., x + w ∈ C w . We thus have that h i ( x ) = ˜ h i ( x + w ) ≤ f ( x + w ) = f ( x ) . Hence, h i ≤ f for all i . Set h = sup i h i . Clearly h ≤ f . If ǫ ( x ) ≤ w then h ( x ) = h ( x + w ) = sup i h i ( x + w ) = f ( x ) . If, on the other hand, ǫ ( x ) (cid:2) w , then h i ( x ) = ∞ for all i and h ( x ) = ∞ = f ( x ).Thus, h = f .Let us now consider the case when w is not compact. Define H = { h ∈ A( C ) : h ≤ (1 − ǫ ) f for some ǫ > } . Let us show that H is upward directed and has pointwise supremum f . Let h , h ∈ H . Set v = supp( h ) and v = supp( h ), which are compact idempotents, byTheorem 4.2, and satisfy that w ≤ v , v . Set v = v ∧ v , which is also compactand such that w ≤ v . Set g = f ∧ χ v , which exists by Theorem 4.3. Since scalarmultiplication by a non-negative scalar is an order isomorphism on C , we have tg = ( tf ) ∧ χ v . Letting t → ∞ and using Theorem 4.3, we get ∞ · g = χ w ∧ χ v = χ v .Thus, supp( g ) = v (Lemma 4.1). Let ǫ > h , h ≤ (1 − ǫ ) f . Then, h , h ≤ (1 − ǫ ) g . Since we have already established the case of compact supportidempotent, there exists an increasing net ( g i ) i in A v ( C ) such that g = sup i g i . By[12, Proposition 5.1], h , h ≪ (1 − ǫ/ g in the directed complete ordered set Lsc( C )(see also the definition of the relation ⊳ in the next section). Thus, there exists i such that h , h ≤ (1 − ǫ/ g i . Now h = (1 − ǫ/ g i belongs to H and satisfiesthat h , h ≤ h . This shows that H is upward directed.Let us show that f is the pointwise supremum of the functions in H . It sufficesto show that f is the supremum of functions in A( C ), as we can then easily arrangefor the 1 − ǫ separation. Choose a decreasing net of compact idempotents ( v i ) i with w = inf v i (recall that C has an abundance of compact idempotents). For each fixed i , f ∧ χ v i has support idempotent v i , which is compact. Thus, as demonstratedabove, f ∧ χ v i is the supremum of an increasing net in A( C ). But f = sup i f ∧ χ v i (Theorem 4.3). It follows that f is the pointwise supremum of functions in A( C ).Finally, suppose that f ∈ Lsc σ ( C ), and let us show that there is a countable setin H with pointwise supremum f . For each h ∈ H , let U h = h − ((1 , ∞ ]). The sets( U h ) h ∈ H form an open cover of f − ((1 , ∞ ]). Since the latter is σ -compact, we canchoose a countable set H ′ ⊆ H such that ( U h ) h ∈ H ′ is also a cover of f − ((1 , ∞ ]).Observe that for each x ∈ C , f ( x ) > h ( x ) > h ∈ H ′ . Itfollows, by the homogeneity with respect to scalar multiplication of these functions,that sup h ∈ H ′ h ( x ) = f ( x ) for all x ∈ C . Now using that H is upward directed wecan construct an increasing sequence with supremum f . (cid:3) Theorem 4.5.
Let C be a metrizable extended Choquet cone with an abundance ofcompact idempotents. Then there exists a countable subset of A( C ) such that everyfunction in Lsc( C ) is the supremum of an increasing sequence of functions in thisset.Proof. Let us first argue that the set of compact idempotents is countable. Let( U i ) ∞ i =1 be a countable basis for the topology of C . Let w ∈ Idem c ( C ) be a compactidempotent. Since { x ∈ C : w ≤ x } is an open set, by Lemma 2.9, there exists U i such that w ∈ U i ⊆ { x ∈ C : w ≤ x } . Clearly then w = inf U i . Thus, the set ofcompact idempotents embeds in the countable set { inf U i : i = 1 , , . . . } .Now fix a compact idempotent w . Recall that A w ( C ) is isomorphic to the coneA + ( C w ) of positive linear functions on the cone C w . Suppose that C w = { w } . Let K denote a compact base of C w , which exists by Theorem 2.10, and is metrizablesince C is metrizable by assumption. Then A + ( C w ) is separable in the metricinduced by the uniform norm on K , since it embeds in C ( K ), which is separable.Let ˜ B w ⊆ A + ( C w ) be a countable dense subset. It is not hard now to express any ONES OF TRACES ARISING FROM AF C*-ALGEBRAS 17 function in A + ( C w ) as the supremum of an increasing sequence in ˜ B w . Indeed, itsuffices to show that for any ǫ > f ∈ A + ( C w ), there exists g ∈ ˜ B w such that(1 − ǫ ) f ≤ g ≤ f . Keeping in mind that f is separated from 0 on K , we can choose g ∈ ˜ B w such that (cid:13)(cid:13)(cid:13) (1 − ǫ f | K − g | K (cid:13)(cid:13)(cid:13) ∞ < ǫ x ∈ K | f ( x ) | . Then g is as desired. Let B w ⊆ A w ( C ) be the set mapping bijectively onto ˜ B w ⊆ A + ( C w ) via the restriction map. By Theorem 4.2, every function in A w ( C ) is thesupremum of an increasing sequence in B w . If, on the other hand, C w = { w } , thenA w ( C ) = { χ w } . In this case we set B w = { χ w } .Let B = S w B w , where w ranges through the set of compact idempotents, and B w is as in the previous paragraph. Observe that B is countable. Let us showthat every function in f ∈ Lsc( C ) is the supremum of an increasing sequence in B . Observe that Lsc( C ) = Lsc σ ( C ), since all open subsets of a compact metricspace are σ -compact. Thus, f = sup n h n , where ( h n ) ∞ n =1 is an increasing sequencein A( C ). The sequence h ′ n = (1 − n ) h n is also increasing, with supremum f , and h ′ n ≪ h ′ n +1 in the directed complete ordered set Lsc( C ) (see [12, Proposition 5.1]and also the definition of the relation ⊳ in the next section). Say h ′ n +1 ∈ A w n ( C ) forsome compact idempotent w n . Since h ′ n +1 is the supremum of a sequence in B w n ,we can choose g n ∈ B w n such that h ′ n ≤ g n ≤ h ′ n +1 . Then ( g n ) ∞ n =1 is an increasingsequence in B with supremum f . (cid:3) Duality with Cu-cones
By a Cu-cone we understand a Cu-semigroup S that is also a cone, i.e., it isendowed with a scalar multiplication by (0 , ∞ ) compatible with the monoid structureof S ; see Section 2. Further we ask that(1) t ≤ t and s ≤ s imply t s ≤ t s for all t , t ∈ (0 , ∞ ) and s , s ∈ S ,(2) sup n t n s n = (sup n t n )(sup n s n ) where ( t n ) ∞ n =1 and ( s n ) ∞ n =1 are increasingsequences in (0 , ∞ ) and S , respectively.Cu-cones are called Cu-semigroups with real multiplication in [21]. They are alsoCu-semimodules over the Cu-semiring [0 , ∞ ], in the sense of [4].In this section we prove a duality between extended Choquet cones with an abun-dance of compact idempotents and certain Cu-cones. Throughout this section, S denotes a Cu-cone satisfying the axioms O5 and O6, so that F ( S ) is an extendedChoquet cone.Let us recall the relation ⊳ in Lsc( C ) defined in [12]: Given f, g ∈ Lsc( C ), wewrite f ⊳ g if f ≤ (1 − ε ) g for some ε > f is continuous at each x ∈ C suchthat g ( x ) < ∞ . By [12, Proposition 5.1], f ⊳ g implies that f is way below g in thedcpo Lsc( C ), meaning that for any upward directed net ( g i ) i such that g ≤ sup g i ,there exists i such that f ≤ g i . Lemma 5.1. (Cf. [21, Lemma 3.3.2] ) Let f, g ∈ Lsc( C ) be such that f ⊳ g . Thenhere exists h ∈ Lsc( C ) such that f + h = g and h ≥ ǫg for some ǫ > . Moreover, if f, g ∈ Lsc σ ( C ) , then h may be chosen in Lsc σ ( C ) , and if f, g ∈ A( C ) , then h maybe chosen in A( C ) .Proof. Define h : C → [0 , ∞ ] by h ( x ) = ( g ( x ) − f ( x ) if g ( x ) < ∞ , ∞ otherwise . Then f + h = g . The linearity of h follows from a straightforward analysis. Since f ⊳ g , there exists ǫ > f ≤ (1 − ǫ ) g . Then g ( y ) − f ( y ) ≥ ǫg ( y ) whenever g ( y ) < ∞ , while if g ( y ) = ∞ then g ( y ) = ∞ = h ( y ). This establishes that h ≥ ǫg .The proof of [21, Lemma 3.3.2] establishes the lower semicontinuity of h . Let usrecall it here: Let ( x i ) i be a net in C such that x i → x . Suppose first that g ( x ) < ∞ .Then f ( x ) < ∞ , and by the continuity of f at x , f ( x i ) < ∞ for large enough i .Then, lim inf i h ( x i ) ≥ lim inf i g ( x i ) − f ( x i ) ≥ g ( x ) − f ( x ) = h ( x ) . Suppose now that g ( x ) = ∞ , so that h ( x ) = ∞ . Since h ≥ ǫg ,lim inf i h ( x i ) ≥ ǫ lim inf i g ( x i ) ≥ ǫg ( x ) = ∞ , thus showing lower semicontinuity at x .Assume now that f, g ∈ Lsc σ ( C ). It is not difficult to show that h ( x ) > g ( x ) > /ǫ or g ( x ) > r and f ( x ) ≤ r for some r ∈ Q . Thus, h − ((1 , ∞ ]) = g − ((1 /ǫ, ∞ ]) ∪ [ r ∈ Q g − ((1 + r, ∞ ]) ∩ f − ([0 , r ]) . The right side is σ -compact. Hence, h ∈ Lsc σ ( C ).Assume now that f, g ∈ A( C ). Continuity at x ∈ C such that h ( x ) = ∞ followsautomatically from lower semicontinuity. Let x ∈ C be such that h ( x ) < ∞ , i.e., g ( x ) < ∞ . If x i → x then g ( x i ) < ∞ and f ( x i ) < ∞ for large enough i . Then h ( x i ) = g ( x i ) − f ( x i ) → g ( x ) − f ( x ) = h ( x ) , where we used the continuity of g and f . Thus, h is continuous at x . (cid:3) By an ideal of a Cu-cone we understand a subcone that is closed under thesuprema of increasing sequence. There is an order reversing bijection between theideals of S and the idempotents of F ( S ): I λ I ( x ) := ( x ∈ I ∞ otherwise,where I ranges through the ideals of S .Let us say that a Cu-cone S has an abundance of compact ideals if the lattice ofideals of S is algebraic, i.e., every ideal of S is a supremum of compact ideals. ONES OF TRACES ARISING FROM AF C*-ALGEBRAS 19
Theorem 5.2.
Let S be a Cu-cone satisfying O5 and O6 and having an abundanceof compact ideals. Then F ( S ) is an extended Choquet cone with an abundance ofcompact idempotents. Moreover, S ∼ = Lsc σ ( F ( S )) via the assignment S ∋ s ˆ s ∈ Lsc σ ( F ( S )) , where ˆ s ( λ ) := λ ( s ) for all λ ∈ F ( S ) .Let C be an extended Choquet cone with an abundance of compact idempotents.Then Lsc σ ( C ) is a Cu-cone satisfying O5 and O6 and having an abundance of com-pact ideals. Moreover, C ∼ = F (Lsc σ ( C )) via the assignment C ∋ x ˆ x ∈ F (Lsc σ ( C )) , where ˆ x ( f ) := f ( x ) for all f ∈ Lsc σ ( C ) .Proof. As recalled in Section 3, by the results of [21], F ( S ) is an extended Choquetcone. The bijection between the ideals of S and the idempotents of F ( S ) trans-lates the abundance of compact ideals of S directly into the abundance of compactidempotents of F ( S ). By [21, Theorem 3.2.1], the mapping S ∋ s ˆ s ∈ Lsc( F ( S ))is an isomorphism of the Cu-cone S onto the space of functions f ∈ Lsc( F ( S )) ex-pressible as the pointwise supremum of an increasing sequence ( h n ) ∞ n =1 in Lsc( F ( S ))such that h n ⊳ h n +1 for all n . The set of all such functions is denoted by L ( F ( S ))in [21]. Let us show that, under our present assumptions, L ( F ( S )) = Lsc σ ( F ( S )).Let f ∈ Lsc( F ( S )) be such that f = sup h n , where h n ⊳ h n +1 for all n . We have h − n ((1 , ∞ ]) ⊆ f − ((1 , ∞ ]) for all n ([12, Proposition 5.1]). Hence, f − ((1 , ∞ ]) = [ n h − n ((1 , ∞ ]) . Thus, f ∈ Lsc σ ( F ( S )). Suppose, on the other hand, that f ∈ Lsc σ ( F ( S )). Then, byTheorem 4.4, there exists an increasing sequence ( h n ) ∞ n =1 in A( F ( S )) with supremum f . Clearly, h ′ n = (1 − n ) h n is also increasing, has supremum f , and h ′ n ⊳ h ′ n +1 forall n . Hence, f ∈ L ( F ( S )).Let’s turn now to the second part of the theorem. Let C be an extended Choquetcone with an abundance of compact idempotents. Let us show that Lsc σ ( C ) satis-fies all axioms O1-O6 (Section 3). Let us show first that Lsc σ ( C ) is closed underthe suprema of increasing sequences: Let f = sup n f n , with ( f n ) ∞ n =1 an increasingsequence in Lsc σ ( C ). Then f − ((1 , ∞ ]) = S ∞ n =1 f − n ((1 , ∞ ]). Since the sets on theright side are σ -compact, so is the left side. Thus, f ∈ Lsc σ ( C ).Let f ∈ Lsc σ ( C ), and let ( h n ) ∞ n =1 be an increasing in A( C ) with supremum f .Then h ′ n = (1 − n ) h n has supremum f and h ′ n ≪ h ′ n +1 for all n (since h ′ n ⊳ h ′ n +1 ).This proves O2. Axiom O3 follows at once from the fact that suprema in Lsc σ ( C )are taken pointwise. Suppose that f ≪ g and f ≪ g . Choose h , h ∈ A( C )such that f i ≤ h i ⊳ g i for i = 1 ,
2. Then f + f ≤ h + h ⊳ g + g , from which wededuce O4. Let’s prove O5: Suppose that f ′ , f, g ∈ Lsc σ ( C ) are such that f ′ ≪ f ≤ g . Choose h ∈ A( C ) such that f ′ ≤ h ⊳ f . By Lemma 5.1, there exists h ′ ∈ Lsc σ ( C ) such that h + h ′ = g . Then, f ′ + h ′ ≤ g ≤ f + h ′ , proving O5.Let us prove O6. We prove the stronger property that Lsc σ ( C ) is inf-semilatticeordered, i.e., pairwise infima exist and addition distributes over infima. Recall that,by the results of [2], Lsc( C ) is inf-semilattice ordered (see Theorem 4.3). Let us showthat if f, g ∈ Lsc σ ( C ), then f ∧ g is also in Lsc σ ( C ). By [2, Lemma 3.4], for every x ∈ C there exist x , x ∈ C , with x + x = x , such that ( f ∧ g )( x ) = f ( x ) + g ( x ).It is then clear that( f ∧ g ) − ((1 , ∞ ]) = [ a ,a ∈ Q ,a + a > f − (( a , ∞ ]) ∪ g − (( a , ∞ ]) . Since the right side is a σ -compact set, f ∧ g ∈ Lsc σ ( C ). To verify O6, suppose that f ≤ g + g , with f, g , g ∈ Lsc σ ( C ). Then, using the distributivity of addition over ∧ , f ≤ g + g ∧ f , which proves O6.Finally, let us prove that C ∋ x ˆ x ∈ F (Lsc σ ( C )) is an isomorphism of extendedChoquet cones. We consider injectivity first: Let x, y ∈ C be such that f ( x ) = f ( y )for all f ∈ Lsc σ ( C ). Choose f ∈ A( C ). Passing to the limit as n → ∞ in f ( n x ) = f ( n y ) we deduce that f ( ǫ ( x )) = f ( ǫ ( y )) for all f ∈ A( C ). Since every functionin Lsc( C ) is the supremum of a directed net of functions in A( C ), we have that f ( ǫ ( x )) = f ( ǫ ( y )) for all f ∈ Lsc( C ). Now choosing f = χ w , for w ∈ Idem( C ),we conclude that ǫ ( x ) = ǫ ( y ), i.e., x and y have the same support idempotent.Set w = ǫ ( x ) = ǫ ( y ). Choose a compact idempotent v such that w ≤ v . Then x + v, y + v ∈ C v , and f ( x + y ) = f ( y + v ) for all f ∈ A( C ). By Theorem 4.2, f ( x + v ) = f ( y + v ) for all f ∈ A + ( C v ). Recall that C v has a compact base andembeds in a locally convex Hausdorff vector space V v (Theorem 2.10). We have f ( x + v ) = f ( y + v ) for all f ∈ A + ( C v ) − A + ( C v ). But A + ( C v ) − A + ( C v ) consists ofall the affine functions on C v that vanish at the origin. Thus, f ( x + v ) = f ( y + v )for all such functions, and in particular, for all continuous functionals on V v . Sincethe weak topology on V v is Hausforff, x + v = y + v . Passing to the infimum overall compact idempotents v such that w ≤ v , and using that C has an abundance ofcompact idempotents, we conclude that x = x + w = y + w = y . Thus, the map x ˆ x is injective.Let us prove continuity of the map x ˆ x . Let ( x i ) i be a net in C with x i → x .Let f ′ , f ∈ Lsc σ ( C ), with f ′ ≪ f . By the lower semicontinuity of f , we haveˆ x ( f ) = f ( x ) ≤ lim inf i f ( x i ) = lim inf i ˆ x i ( f ) . Choose h ∈ A( C ) such that f ′ ≤ h ≤ f , which is possible since f is supremum ofan increasing sequence in A( C ). Thenlim sup i ˆ x i ( f ′ ) ≤ lim sup i ˆ x i ( h ) = lim sup i h ( x i ) = h ( x ) ≤ f ( x ) = ˆ x ( f ) . This shows that ˆ x i → ˆ x in the topology of F (Lsc σ ( C )).Let us prove surjectivity of the map x ˆ x . (Linearity is straightforward; con-tinuity of the inverse is automatic from the fact that the cones are compact and ONES OF TRACES ARISING FROM AF C*-ALGEBRAS 21
Hausdorff.) The range of the map x ˆ x is a compact subcone of F (Lsc σ ( C ))that separates elements of Lsc σ ( C ) and contains 0. By the separation theorem [3,Corollary 4.6], it must be all of F (Lsc σ ( C )). (cid:3) Let S be a Cu-cone. We say that S has weak cancellation if x + z ≪ y + z implies x ≪ y for all x, y, z ∈ S . Lemma 5.3.
Let C be an extended Choquet cone. Let h, h ′ , g ∈ Lsc( C ) be such that h ⊳ g + h ′ and h ′ ⊳ h . Then supp( g + h ′ ) is relatively compact in supp( g ) .Proof. Set w = supp( g + h ′ ) and w = supp( g ). Let ( v i ) i be a downward directednet of idempotents with V i v i ≤ w . Then the functions ( χ v i ) i form an upwarddirected net such that g + h ′ ≤ χ w ≤ sup i χ v i . Since h ⊳ g + h ′ , there exists i suchthat h ≤ χ v i . We have that g + h ′ ≤ g + h ≤ χ w + χ v i = χ w ∧ v i . Hence, w ∧ v i ≤ w , which proves the lemma. (cid:3) Theorem 5.4.
Let C be an extended Choquet cone with an abundance of compactidempotents. Then C is strongly connected if and only if Lsc σ ( C ) has weak cancel-lation.Proof. Suppose first that C is strongly connected. Let f, g, h ∈ Lsc σ ( C ) be suchthat f + h ≪ g + h . Choose ⊳ -increasing sequences ( g n ) ∞ n =1 and ( h n ) ∞ n =1 in A( C )such that g = sup n g n and h = sup n h n . Then f + h ≪ g m + h m for some m . Wewill be done once we have shown that f ≤ g m .Let x ∈ C . If g m ( x ) = ∞ , then indeed f ( x ) ≤ ∞ = g m ( x ). Suppose that g m ( x ) < ∞ . If h m ( x ) < ∞ , then we can cancel h m ( x ) in f ( x ) + h m ( x ) ≤ g m ( x ) + h m ( x )to obtain the desired f ( x ) ≤ g m ( x ). It thus suffices to show that g m ( x ) < ∞ implies h m ( x ) < ∞ , i.e., that supp( g m ) ≤ supp( h m ). Let w = supp( g m + h m )and w = supp( g m ). Then w ≤ w and w is relatively compact in w , by theprevious lemma. Suppose for the sake of contradiction that w = w . By strongconnectedness, there exists x ∈ C such that w ≤ x ≤ w , with ǫ ( x ) = w and x = w . Then, h ( x ) ≤ g m ( x ) + h m ( x )= h m ( x ) ≤ (1 − δ ) h ( x ) , for some δ >
0. Hence, h ( x ) ∈ { , ∞} . If h ( x ) = 0, then h m ( x ) = g m ( x ) = 0, whileif h ( x ) = ∞ , then g m ( x ) + h m ( x ) ≥ h ( x ) = ∞ . In either case, we get a contradictionwith 0 < ( g m + h m )( x ) < ∞ , which holds by Theorem 4.2. Hence, w = w . Wethus have that supp( g m ) = supp( g m + h m ) ≤ supp( h m ).Suppose conversely that Lsc σ ( C ) has weak cancellation. Let w ≤ w be idempo-tents in C , with w relatively compact in w , and w = w . Further, using Zorn’slemma, choose w minimal such that w = w and w is relatively compact in w .Suppose for the sake of contradiction that w ≤ x ≤ w implies x ∈ { w , w } . Let D = { x ∈ C : x ≤ w } . Then D is an extended Choquet cone and w is a compactidempotent in D . Further, D w = { w } . So, as shown in the course of the proof of Theorem 4.2, χ w | D is continuous on D . Let ( h i ) i ∈ A( C ) be an upward directednet with supremum χ w . Since χ w | D ⊳ χ w | D , there exists i such that χ w | D ≤ h i | D .It follows that χ w ≤ h i + χ w (as functions on C ). Fix an index j ≥ i . Then3 h j ⊳ χ w ≤ h i + χ w . Now let ( l k ) k be an upward directed net in A( C ) with supremum χ w . Then thereexists an index k such that 3 h j ≤ h i + l k . Observe that h i ⊳ h k . By weak cancellationin Lsc σ ( C ), we conclude that h j ≤ l k . (Note: we have used weak cancellation in theform f + h ≤ g + h ′ and h ′ ≪ h imply f ≤ g .) Thus, h j ≤ χ w for all j ≥ i , implyingthat χ w ≤ χ w . This contradicts that w = w . (cid:3) In the following section we will make use of the following form of Riesz decompo-sition:
Theorem 5.5.
Let C be an extended Choquet cone that is strongly connected and hasan abundance of compact idempotents. Let f, g , g ∈ A( C ) be such that f ⊳ g + g .Then there exist f , f ∈ A( C ) such that f = f + f , f ⊳ g , and f ⊳ g .Proof. Let ǫ > f ≤ (1 − ε ) g +(1 − ε ) g . Then, using the distributivityof addition over ∧ , f ≤ f ∧ ((1 − ε ) g ) + (1 − ε ) g = (1 − ε )(( f ∧ g ) + g ) . Thus, f ⊳ ( f ∧ g ) + g (recall that f is continuous). By Theorem 4.4, f ∧ g is thesupremum of a net of functions in A( C ). Thus, there exists h ∈ A( C ) such that f ⊳ h + g and h ⊳ ( f ∧ g ). By Lemma 5.1, we can find l ∈ A( C ) such that f = h + l .Then h + l ⊳ h + g . By weak cancellation in Lsc σ ( C ) (Theorem 5.4), we have that l ⊳ g . Setting f = h and f = l yields the desired result. (cid:3) Proof of Theorem 1.1
Throughout this section C denotes an extended Choquet cone that is stronglyconnected and has an abundance of compact idempotents.6.1. The triangle lemma.
To prove Theorem 1.1 we follow a strategy similar tothe proof of the Effros-Handelman-Shen theorem ([11]). The key step in this proofis establishing a “triangle lemma”, Theorem 6.3 below.
Lemma 6.1.
A linear map φ : [0 , ∞ ] → Lsc σ ( C ) is a Cu-morphism if and only if φ ( ∞ ) = ∞ · φ (1) and φ (1) ∈ A( C ) .Proof. Suppose that φ is a Cu-morphism. That φ ( ∞ ) = ∞ · φ (1) follows at oncefrom φ being supremum preserving and additive. Set f = φ (1). To prove thecontinuity of f , it suffices to show that it is upper semicontinuous, since it is alreadylower semicontinuous by assumption. Fix ǫ >
0. Since 1 − ǫ ≪ , ∞ ], we have(1 − ǫ ) f ≪ f in Lsc σ ( C ). Choose g ∈ A( C ) such that (1 − ǫ ) f ≤ g ≤ f . Let x i → x be a convergent net in C . Then,(1 − ǫ ) lim sup i f ( x i ) ≤ lim sup g ( x i ) = g ( x ) ≤ f ( x ) . Letting ǫ →
0, we get that lim sup f ( x i ) ≤ f ( x ). Thus, f is upper semicontinuous. ONES OF TRACES ARISING FROM AF C*-ALGEBRAS 23
Conversely, suppose that φ (1) ∈ A( C ) and φ ( ∞ ) = ∞ · φ (1). Observe that if f ∈ A( C ) then αf ⊳ βf for all scalars 0 ≤ α < β ≤ ∞ . Hence, φ ( α ) ≪ φ ( β ) inLsc σ ( C ) whenever α ≪ β in [0 , ∞ ], i.e., φ preserves the way below relation. Therest of the properties of φ are readily verified. (cid:3) The core of the proof of Theorem 6.3 (the “triangle lemma”) is contained in thefollowing lemma:
Lemma 6.2.
Let φ : [0 , ∞ ] n → Lsc σ ( C ) be a Cu-morphism. Let x, y ∈ [0 , ∞ ) n ∩ Z n be such that φ ( x ) ≪ φ ( y ) . Then there exist N ∈ N and Cu-morphisms [0 , ∞ ] n Q −→ [0 , ∞ ] N ψ −→ Lsc σ ( C ) , such that ψQ = φ and Qx ≤ Qy . Moreover, Q maps [0 , ∞ ) n ∩ Z n to [0 , ∞ ) N ∩ Z N Proof.
Let x = ( x , . . . , x n ), y = ( y , . . . , y n ), and φ be as in the statement ofthe lemma. Let ( E i ) ni =1 denote the canonical basis of [0 , ∞ ] n . Set f i = φ ( E i ) for i = 1 , . . . , n , which belong to A( C ) by Lemma 6.1. Let M = max i | x i − y i | , n = { i : x i − y i = M } , n = { i : y i − x i = M } . Let us define the degree of the triple ( φ, x, y ), denoted deg( φ, x, y ), as the vector(
M, n , n , n ). We order the degrees lexicographically. We will prove the lemmaby induction on the degree of the triple ( φ, x, y ). Let us first deal with the case n = 1, i.e., the domain of φ is [0 , ∞ ]. Since [0 , ∞ ] is totally ordered, either x ≤ y or y < x . In the first case, setting Q the identity and φ = ψ gives the result. If y < x ,then φ ( y ) ≪ φ ( x ), which, together with φ ( x ) ≪ φ ( y ), implies that φ ( x ) = φ ( y ) isa compact element in A( C ). The only compact element in A( C ) is 0, for if f ≪ f ,then f ≪ (1 − ǫ ) f for some ǫ >
0, and so f = 0 by weak cancellation (Theorem 5.4).Thus, φ ( x ) = 0, which in turn implies that φ = 0. We can then choose Q and ψ tobe the 0 maps.Suppose now that φ , x , y are as in the lemma, and that the lemma holds for alltriples ( φ ′ , x ′ , y ′ ) with smaller degree. If x ≤ y , then we can choose Q the identitymap, φ = ψ , and we are done. Let us thus assume that x (cid:2) y . If x i = y i forsome index i , then we can write x = x i E i + ˜ x and y = x i E i + ˜ y , where ˜ x, ˜ y belong to S := span( E i ) i = i ∼ = [0 , ∞ ] n − . By weak cancellation, φ ( x ) ≪ φ ( y ) impliesthat φ (˜ x ) ≪ φ (˜ y ). Since ˜ x, ˜ y belong to a space of smaller dimension, the degree of( φ | S , ˜ x, ˜ y ) is smaller than that of ( φ, x, y ) ( M, n , n have not increased, while n hasdecreased). By the induction hypothesis, there exist maps ˜ Q : S → [0 , ∞ ] N and˜ ψ : [0 , ∞ ] N → Lsc σ ( C ) such that ˜ Q ˜ x ≤ ˜ Q ˜ y and φ | S = ˜ ψ ˜ Q . Define Q : [0 , ∞ ] n → [0 , ∞ ] N +1 as the extension of ˜ Q such that QE i = E N +1 . Extend ˜ ψ to [0 , ∞ ] N +1 setting ψ ( E N +1 ) = f i . Then φ = ψQ and Qx = ˜ Q ˜ x + x i E N +1 ≤ ˜ Q ˜ y + y i E N +1 = Qy, thus again completing the induction step.We assume in the sequel that x i = y i , i.e., either x i < y i or x i > y i , for all i = 1 , . . . , n . Let I = { i : x i > y i } and J = { j : y j > x j } . Let M = max i ∈ I x i − y i and M = max j ∈ J y j − x j . Then M = max( M , M ). We break-up the rest of theproof into two cases. Case M ≥ M . Using weak cancellation in n X i =1 x i f i = φ ( x ) ≪ φ ( y ) = n X i =1 y i f i we get X i ∈ I ( x i − y i ) f i ≪ X j ∈ J ( y j − x j ) f j . Let i ∈ I be such that x i − y i = M . From the last inequality we deduce that M f i ≪ X j ∈ J M f j , and since M ≤ M , we get f i ≪ P j ∈ J f j . By the Riesz decomposition propertyin A( C ) (Theorem 5.5), there exist g j , h j ∈ A( C ), with j ∈ J , such that f i = X j ∈ J g j and f j = g j + h j for all j ∈ J. Let N = n + | J | −
1, and let us label the canonical generators of [0 , ∞ ] N with theset { E i : i = 1 , . . . , n, i = i } ∪ { G j : j ∈ J } . Define Q : [0 , ∞ ] n → [0 , ∞ ] N asfollows: Q E i = E i if i ∈ I \{ i } ,Q E i = X j ∈ J G j ,Q E j = E j + G j if j ∈ J, and extend Q to a Cu-cone morphism on [0 , ∞ ] n . Next, define a Cu-cone morphism ψ : [0 , ∞ ] N → Lsc σ ( C ) on the same generators as follows: ψ ( E i ) = f i , if i ∈ I \{ i } ,ψ ( E j ) = h j , if j ∈ J,ψ ( G j ) = g j , if j ∈ J. It is easily checked that ψ Q = φ and that Q maps [0 , ∞ ] n ∩ Z n to [0 , ∞ ] N ∩ [0 , ∞ ] N . Also, Q x = X i ∈ I \{ i } x i E i + X j ∈ J x i G j + X j ∈ J x j ( E j + G j )= X i = i x i E i + X j ∈ J ( x i + x j ) G j . Similarly, Q y = X i = i y i E i + X j ∈ J ( y i + y j ) G j . ONES OF TRACES ARISING FROM AF C*-ALGEBRAS 25
We claim that deg( ψ , Q x, Q y ) < deg( φ, x, y ). Indeed, the maximum of the dif-ferences of the coordinates ( M above) has not gotten larger. Moreover, the numberof times that M is attained ( n above) is smaller, since we have removed the coor-dinate i and added new coordinates for which( x i + x j ) − ( y i + y j ) = M + x j − y j ∈ [0 , M − . By induction, the lemma holds for ( ψ , Q x, Q y ). Thus, there exist Cu-morphisms Q : [0 , ∞ ] N → [0 , ∞ ] N and ψ : [0 , ∞ ] N → Lsc σ ( C ) such that Q Q x ≤ Q Q y and ψ = ψ Q . Setting Q = Q Q and ψ = ψ , we get the desired result. Case M > M . This case is handled similarly to the previous case, though witha few added complications. Observe first that M ≥ M ≥
1; otherwise x ≤ y ). Choose ǫ > φ ( x ) ≪ (1 − ǫ ) φ ( y ). If necessary, make ǫ smaller, sothat we also have ǫ < min { x i , y j : x i = 0 , y j = 0 } . Notice that this implies that(2) x i > (1 − ǫ ) y i ⇔ x i > y i , for i = 1 , , . . . , n,x i < (1 − ǫ ) y i ⇔ x i < y i , for i = 1 , , . . . , n. Let h ∈ A( C ) be such that h + φ ( x ) = (1 − ǫ ) φ ( y ), which exists by Lemma5.1. Enlarge the domain of φ to [0 , ∞ ] n +1 , labelling the new generator by H (=(0 , . . . , , φ ( H ) = h . We then have (1 − ǫ ) φ ( y ) ≪ h + φ ( x ), i.e., n X i =1 (1 − ǫ ) y i f i ≪ h + n X i =1 x i f i . Using weak cancellation and the inequalities (2) we can move terms around to get X j ∈ J ((1 − ǫ ) y j − x j ) f j ≪ h + X i ∈ I ( x i − (1 − ǫ ) y i ) f i . Let j ∈ J be such that y j − x j = M . Then((1 − ǫ ) y j − x j ) f j ≪ h + X i ∈ I ( x i − (1 − ǫ ) y i ) f i . By our choice of ǫ , we have the inequalities(1 − ǫ ) y j − x j ≥ M −
12 and x i − (1 − ǫ ) y i ≤ M + 12 for all i. Hence, ( M −
12 ) f j ≪ h + X i ∈ I ( M + 12 ) f i . Further, M + ≤ M − (since M > M ) and M − > M ≥ f j ≪ h + X i ∈ I f i . By the Riesz decomposition property in A( C ) (Theorem 5.5), f j = h ′ + P i ∈ I g i for some h ′ ≪ h and g i ≪ f i , with i ∈ I . Let us choose h ′′ , h i ∈ A( C ) such that h = h ′ + h ′′ and f i = g i + h i for all i ∈ I (Lemma 5.1). Label the canonical generatorsof the Cu-cone [0 , ∞ ] N , where N = n + | I | + 1, with the set { E j : j = 1 , . . . , n, j = j } ∪ { G i : i ∈ I } ∪ { H, H ′ } . Define a Cu-cone morphism Q : [0 , ∞ ] n +1 → [0 , ∞ ] N as follows: Q E j = E j for j ∈ J \{ j } ,Q E j = H ′ + X i ∈ I G i ,Q E i = E i + G i for i ∈ I,Q H = H + H ′ , Next, define a Cu-cone map ψ : [0 , ∞ ] N → Lsc σ ( C ) by ψ E j = f j for j ∈ J \{ j } ψ E i = h i , for i ∈ I,ψ G i = g i , for i ∈ I,ψ H = h ′′ and ψ H ′ = h ′ . Now ψ Q E j = f j for j ∈ J \{ j } , and ψ Q E j = ψ H ′ + X i ∈ I G i ! = h ′ + X i ∈ I g i = f j . Also, ψ Q E i = ψ ( E i + G i ) = h i + g i = f i , for i ∈ I. Finally, ψ Q H = h ′ + h ′′ = h . Thus, we have checked that ψ Q = φ . Clearly, Q maps integer valued vectors to integer valued vectors.Let us examine the degree of ( ψ , Q ( x + H ) , Q y ). We have that Q ( x + H ) = X j ∈ J \{ j } x j E j + X i ∈ I x j G i + x j H ′ + X i ∈ I x i ( E i + G i ) + ( H + H ′ )= X j = j x j E j + X i ∈ I ( x j + x i ) G i + H + ( x j + 1) H ′ . Similarly, we compute that Q y = X j = j y j E j + X i ∈ I ( y j + y i ) G i + H + y j H ′ . We claim that the deg( ψ , Q ( x + H ) , Q y ) < deg( φ, x, y ). To show this we checkthat for the pair ( Q ( x + H ) , Q y ) we have that:(1) the maximum coordinates difference for the indices i such that x i > y i (number M above) is strictly less than M , ONES OF TRACES ARISING FROM AF C*-ALGEBRAS 27 (2) the maximum coordinates difference for the indices where y j > x j is at most M ,(3) the number of indices for which M is attained (number n above) has de-creased relative to the pair ( x, y ).The first two points are straightforward to check. The last point follows from thefact that we have removed the coordinate j , and that for the new coordinates thatwe have added we have( y j + y i ) − ( x j + x i ) = M + ( y i − x i ) ∈ [0 , M − ,y j − ( x j + 1) = M − < M . Observe that( ψ Q )( x + H ) = h + φ ( x ) = (1 − ǫ ) φ ( y ) ≪ φ ( y ) = ψ Q y. Hence, by the induction hypothesis, there exist Q and ψ such that ψ = ψ Q and Q Q ( x + H ) ≤ Q Q y . Then Q = Q Q and ψ = ψ are as desired, thuscompleting the step of the induction. (cid:3) Theorem 6.3.
Let φ : [0 , ∞ ] n → Lsc σ ( C ) be a Cu-morphism. Let F ⊂ [0 , ∞ ) n be afinite set. Then there exist N ∈ N and Cu-morphisms [0 , ∞ ] n Q −→ [0 , ∞ ] N ψ −→ Lsc σ ( C ) , such that ψQ = φ , φx ≪ φy = ⇒ Qx ≪ Qy for all x, y ∈ F, and Q maps [0 , ∞ ] n ∩ Z n to [0 , ∞ ] N ∩ Z N .Proof. We start by noting that given elements x = ( x i ) ni =1 and y = ( y i ) ni =1 in [0 , ∞ ] n ,we have x ≪ y if and only if x i < y i or x i = y i = 0 for all i = 1 , . . . , n .Suppose first that F = { x, y } ⊆ [0 , ∞ ) n and that φ ( x ) ≪ φ ( y ). Choose ε > ε ) φ ( x ) ≪ (1 − ε ) φ ( y ). Choose x ′ , y ′ ∈ [0 , ∞ ) n ∩ Q n such that x ≪ x ′ ≤ (1 + ε ) x and (1 − ε ) y ≤ y ′ ≪ y . Then φ ( x ′ ) ≪ φ ( y ′ ). Let m ∈ N be such that mx ′ , my ′ ∈ [0 , ∞ ) n ∩ Z n . By Lemma 6.2, there exist Q, ψ such that φ = ψQ and Q ( mx ′ ) ≤ Q ( my ′ ), i.e., Qx ′ ≤ Qy ′ . Then Qx ≪ Qx ′ ≤ Qy ′ ≪ Qy.
Lemma 6.2 also guarantees that Q maps integer valued vectors to integer valuedvectors. Thus, Q and ψ are as desired.To deal with an arbitrary finite set F ⊆ [0 , ∞ ) n , choose x, y ∈ F such that φ ( x ) ≪ φ ( y ) and obtain Q , ψ such that φ = ψ Q and Q x ≪ Q y . Set F = Q F and apply the same argument to a new pair x ′ , y ′ ∈ F to obtain maps Q , ψ .Continue inductively until all pairs have been exhausted. Set Q = Q k · · · Q and ψ = ψ k . (cid:3) Building the limit.Theorem 6.4.
Let C be an extended Choquet cone that is strongly connected andhas an abundance of compact idempotents. Then Lsc σ ( C ) is an inductive limit in theCu-category of an inductive system of Cu-cones of the form [0 , ∞ ] n , n ∈ N , and withCu-morphisms that map integer valued vectors to integer valued vectors. Moreover,if C is metrizable, then this inductive system can be chosen over a countable indexset.Proof. For each n = 1 , , . . . , choose an increasing sequence ( A ( n ) k ) ∞ k =1 of finite subsetsof [0 , ∞ ) n with dense union in [0 , ∞ ] n .We will construct an inductive system of Cu-cones { S F , φ G,F } , where F, G rangethrough the finite subsets of A( C ), such that S F ∼ = [0 , ∞ ] n F for all F . We alsoconstruct Cu-morphisms ψ F : S F → Lsc σ ( C ) for all F , finite subset of A( C ), makingthe overall diagram commutative. We follow closely the presentation of the proofof the Effros-Shen-Handelmann theorem given in [14], adapted to the category ofCu-cones.For each f ∈ A( C ), define S { f } = [0 , ∞ ] and ψ { f } : [0 , ∞ ] → Lsc σ ( C ) as theCu-morphism such that ψ { f } (1) = f . Fix a finite set F ⊆ A( C ). Suppose that wehave defined S G and ψ G for all proper subsets G of F . Set S F := Q G S G , where G ranges though all proper subsets of F . Define φ F : S F → Lsc σ ( C ) as φ F (( s G ) G ) = X G ψ G ( s G ) . Next, we construct Q : S F → S F and ψ : S F → Lsc σ ( C ) using Theorem 6.3. Hereis how: For each G , proper subset of F , let n G be such that S G ∼ = [0 , ∞ ] n G . Let A = Q G A ( n G ) k , where k = | F | and where G ranges through all proper subsets of F .Then A is a finite subset of S F . Let us apply Theorem 6.3 to φ F and the set A ,in order to obtain maps Q : S F → S F ∼ = [0 , ∞ ] n F and ψ : S F → Lsc σ ( C ) such that φ F = ψQ and φ F ( x ) ≪ φ F ( y ) ⇒ Qx ≪ Qy for all x, y ∈ A. Set ψ F = ψ , and for each proper subset G of F , define φ G,F : S G → S F as thecomposition of the embedding of S G in S F with the map Q : S G ֒ → S F Q → S F . Observe that φ G,F maps [0 , ∞ ] n G ∩ Z n G to [0 , ∞ ] n F ∩ Z n F , as both Q and S G ֒ → S F map integer valued vectors to integer valued vectors. Continuing in this waywe obtain an inductive system { S F , φ G,F } , indexed by the finite subsets of A( C ),and maps ψ F : S F → Lsc σ ( C ) for all F . By construction, the overall diagram iscommutative. To show that Lsc σ ( C ) is the inductive limit in the Cu-category ofthis inductive system, we must check that(1) every element in Lsc σ ( C ) is supremum of an increasing sequence containedin the union of the ranges of the maps ψ F , ONES OF TRACES ARISING FROM AF C*-ALGEBRAS 29 (2) for each finite set F (index of the system) and elements x ′ , x, y ∈ S F suchthat x ′ ≪ x and ψ G ( x ) ≤ ψ G ( y ) in Lsc σ ( C ), there exists F ′ ⊃ F such that φ F,F ′ ( x ′ ) ≪ φ F,F ′ ( y ).Let’s check the first property. By construction, if F = { f } then f is contained inthe range of ψ F . Examining the construction of ψ F for arbitrary F , it becomes clearthat F is contained in the range of ψ F . Thus, as F ranges through all finite subsetsof A( C ), the union of the ranges of the maps ψ F contains A( C ). Moreover, byTheorem 4.4, every function in Lsc σ ( C ) is the supremum of an increasing sequencein A( C ).Suppose that x ′ , x, y ∈ S F are such that ψ F ( x ) ≤ ψ F ( y ) and x ′ ≪ x . Then x ′ ∈ [0 , ∞ ) n F and ψ F ( x ′ ) ≪ ψ F ( y ). Choose y ′ ≪ y and x ′ ≪ x ′′ ≪ x such that ψ F ( x ′′ ) ≪ ψ F ( y ′ ). Next, choose v, w ∈ A ( n F ) k for some k , such that x ′ ≪ u ≪ x ′′ and y ′ ≪ v ≪ y . Observe then that ψ F ( u ) ≪ ψ F ( v ). Let F ′ ⊂ A( C ) be a finiteset such that F ⊂ F ′ and | F ′ | ≥ k . Then, by our construction of the inductivesystem, we have that φ F,F ′ ( u ) ≪ φ F,F ′ ( v ). This implies that φ F,F ′ ( x ′ ) ≪ φ F,F ′ ( y ),thus proving the second property of an inductive limit.Let us address the second part of the theorem. Suppose that C is metrizable.By Theorem 4.5, there exists a countable set B ⊆ A( C ) such that every function inLsc σ ( C ) is the supremum of an increasing sequence in B . The construction of theinductive limit for Lsc σ ( C ) in the preceding paragraphs can be repeated mutatismutandis, letting the index set of the inductive limit be the set of finite subsetsof B , rather than the finite subsets of A( C ). The resulting inductive limit is thusindexed by a countable set. (cid:3) We are now ready to proof Theorem 1.1 from the introduction.
Proof of Theorem 1.1. (i) ⇒ (iv): An AF C*-algebra has real rank zero, stable rankone, and is exact (these properties hold for finite dimensional C*-algebras and arepassed on to their inductive limits). Thus, (i) implies (iv) by Proposition 3.1.(iv) ⇒ (iii): Suppose that we have (iv). By Theorem 6.4, Lsc σ ( C ) is an inductivelimit in the Cu-category of Cu-cones of the form [0 , ∞ ] n , with n ∈ N . We have F ([0 , ∞ ] n ) ∼ = [0 , ∞ ] n via the map F ([0 , ∞ ] n ) ∋ λ ( λ ( E ) , . . . , λ ( E n )) ∈ [0 , ∞ ] n , where E , . . . , E n are the canonical generators of [0 , ∞ ] n . Applying the functor F ( · )to the inductive system with limit Lsc σ ( C ) we obtain a projective system in thecategory of extended Choquet cones where each cone is isomorphic to [0 , ∞ ] n forsome n . By the continuity of the functor F ( · ) ([12, Theorem 4.8]), and the fact that F (Lsc σ ( C )) ∼ = C (Theorem 5.2), we get (iii).(iii) ⇒ (ii): Suppose that we have (iii). Say C = lim ←− i ∈ I ([0 , ∞ ] n i , α i,j ). Observethat α i,j maps [0 , ∞ ) n i to [0 , ∞ ) n j . Indeed, the support idempotent of an elementin [0 , ∞ ) n i is 0. By continuity of α i,j , the same holds for the image of these elements;thus, they belong to [0 , ∞ ) n j . It follows then that α i,j is given by multiplication bya matrix M i,j with non-negative finite entries: α i,j ( v ) = M i,j v for all v ∈ [0 , ∞ ] n i (in M i,j v we regard v as a column vector and use the rule 0 ·∞ = 0). The transpose ma-trix M ti,j can then be regarded as a map from R n j to R n i . Let us form an inductivesystem of dimension groups whose objects are R n i , endowed with the coordinatewiseorder, with i ∈ I , and with maps M ti,j : R n j → R n i . This inductive system of dimen-sion groups gives rise to the original system after applying the functor Hom( · , [0 , ∞ ])to it, and making the isomorphism identifications Hom( R n i + , [0 , ∞ ]) ∼ = [0 , ∞ ] n i . Let G be its limit in the category of dimension groups ( G is in fact a vector space). Bythe continuity of the functor Hom( · , [0 , ∞ ]), we have Hom( G + , [0 , ∞ ]) ∼ = C . Thus,(iii) implies (ii).(ii) ⇒ (i): By Elliott’s theorem, there exists an AF C*-algebra A whose Murray-vonNeumann monoid of projections V ( A ) is isomorphic to G + . The result now followsfrom the fact, well known to experts, that T ( A ) ∼ = Hom( V ( A ) , [0 , ∞ ]) for an AF A (where Hom( V ( A ) , [0 , ∞ ]) denotes the cone of monoid morphisms). Let us sketcha proof of this fact here: Since AF C*-algebras are exact, we have by Haagerup’stheorem that 2-quasitraces on A , and on the ideals of A , are traces. We applyhere the version due to Blanchard and Kirchberg that includes densely finite lowersemicontinuous 2-quasitraces; see [6, Remark 2.29 (i)]. Thus, T ( A ) = QT ( A ), where QT ( A ) denotes the cone of lower semicontinuous [0 , ∞ ]-valued 2-quasitraces on A .Further, by [12, Theorem 4.4], QT ( A ) ∼ = F (Cu( A )) for any C*-algebra A . Thus, wemust show that F (Cu( A )) ∼ = Hom( V ( A ) , [0 , ∞ ]) when A is an AF C*-algebra. LetCu c ( A ) denote the submonoid of Cu( A ) of compact elements, i.e., of elements e ∈ Cu( A ) such that e ≪ e . By [8, Theorem 3.5] of Brown and Ciuperca, for stably finite A the map from V ( A ) to Cu( A ) assigning to a Murray-von Neumann class [ p ] MvN the Cuntz class [ p ] Cu ∈ Cu( A ) is a monoid isomorphism with Cu c ( A ). This holds inparticular for A AF. Thus, we must show that F (Cu( A )) ∼ = Hom(Cu c ( A ) , [0 , ∞ ]).This isomorphism is given by the restriction map. Indeed, since A has real rank zeroand stable rank one, every element of Cu( A ) is supremum of an increasing sequenceof compact elements ([9, Corollary 5]). This shows that λ λ | Cu c ( A ) is injective. Toprove surjectivity, suppose that we have a monoid morphism τ : Cu c ( A ) → [0 , ∞ ].Define λ ( x ) = sup { τ ( e ) : e ≤ x, e ∈ Cu c ( A ) } . Then λ is readily shown to be a functional on Cu( A ) that extends τ . Finally, fromthe definition of the topology on F (Cu( A )) it is evident that a convergent net ( λ i ) i in F (Cu( A )) converges pointwise on compact elements of Cu( A ). This shows that themap λ λ | Cu c ( A ) is continuous. Since it is a bijection between compact Hausdorffspaces, its inverse is also continuous. In summary, we have the following chain ofextended Choquet cones isomorphisms when A is AF: T ( A ) = QT ( A ) ∼ = F (Cu( A )) ∼ = Hom(Cu c ( A ) , [0 , ∞ ]) ∼ = Hom( V ( A ) , [0 , ∞ ]) . Finally, suppose that C is metrizable and satisfies (iv). Then, in the proof of(iv) ⇒ (iii) above, Theorem 6.4 allows us to start with an inductive limit for Lsc σ ( C )over a countable index set. Applying the functor F ( · ), we get a projective limitfor C over a countable index set. Moreover, the Cu-morphisms in the inductivesystem of Theorem 6.4 map integer valued vectors to integer valued vectors. Thus, ONES OF TRACES ARISING FROM AF C*-ALGEBRAS 31 the matrices M i,j implementing these morphisms have nonnegative integer entries.Thus, in the proof of (iii) ⇒ (ii) we start with C = lim ←− i ∈ I ([0 , ∞ ] n i , α i,j ), where α i,j isimplemented by a matrix with nonnegative integer entries. We can thus construct aninductive system ( Z n i , M i,j ) i,j ∈ I , in the category of dimension groups, whose limitis a countable dimension group G such that Hom( G + , [0 , ∞ ]) ∼ = C , as desired. (cid:3) Finitely generated cones
A cone C is called finitely generated if there exists a finite set X ⊆ C suchthat for every x ∈ C we have x = P ni =1 α i x i for some α i ∈ (0 , ∞ ) and x i ∈ X .In this section we give a direct construction of an ordered vector space (over R )( V, V + ) with the Riesz property and such that Hom( V + , [0 , ∞ ]) is isomorphic toa given finitely generated, strongly connected, extended Choquet cone C . HereHom( V + , [0 , ∞ ]) denotes the monoid morphisms from V + to [0 , ∞ ]. These mapsare automatically homogeneous with respect to scalar multiplication; thus, they arealso cone morphisms. Lemma 7.1.
Let C be a finitely generated extended Choquet cone. Then Idem( C ) is finite and for each w ∈ Idem( C ) the sub-cone C w is either isomorphic to { } or to [0 , ∞ ) n w for some n w ∈ N . (Recall that we have defined C w = { x ∈ C : ǫ ( x ) = w } .)Proof. Let Z be a finite set that generates C . Let w ∈ C be an idempotent, andwrite w = P ni =1 α i x i , with x i ∈ Z and α i ∈ (0 , ∞ ). Multiplying both sides by ascalar δ > δ →
0, we get that w is the sum of supportidempotents of elements in Z . It follows that Idem( C ) is finite.Next, let w ∈ Idem( C ). Define Z w = { x + w : x ∈ Z and ǫ ( x ) ≤ w } , which is afinite subset of C w . We claim that Z w generates C w as a cone. Indeed, let x ∈ C w and write x = P ni =1 α i x i , with x i ∈ Z and α i ∈ (0 , ∞ ). Adding w on both sideswe get x = P ni =1 α i ( x i + w ). Since ǫ ( x i ) ≤ ǫ ( x ) = w , the elements x i + w are in Z w . If Z w = { w } then C w is isomorphic to { } . Suppose that Z w = { w } . Since w is a compact idempotent, C w has a compact base K which is a Choquet simplex(Theorem 2.10). Further, K is finitely generated (by the set (0 , ∞ ) · Z w ∩ K ). Hence, K has finitely many extreme points, which in turn implies that C w ∼ = [0 , ∞ ) n w forsome n w ∈ N . (cid:3) For the remainder of this section we assume that C is a finitely generated, stronglyconnected, extended Choquet cone. Thus, each idempotent w ∈ Idem( C ) is compactand, by strong connectedness, C w = { w } for all w = ∞ (here ∞ denotes the largestelement in C ).Let w ∈ Idem( C ) and x ∈ C w . If z ∈ C is such that z + w = x , we call z andextension of x . The set of extensions of x is downward directed: if z and z areextensions of x , then so is z ∧ z . Consider the element ˜ x = inf { z ∈ C : z + w = x } .By the continuity of addition, ˜ x is also an extension of x , which we call the minimumextension. Lemma 7.2.
Let w ∈ Idem( C ) . Let x ∈ C w \{ w } be an element generating anextreme ray in C w , and let ˜ x denote the minimum extension of x . (i) ˜ x generates an extreme ray in C ǫ (˜ x ) . (ii) If y, z ∈ C are such that y + z = ˜ x , then either y ≤ z or z ≤ y .Proof. Set v = ǫ (˜ x ).(i) Let y, z ∈ C v be such that y + z = ˜ x . Adding w on both sides we get( y + w ) + ( z + w ) = x . Since y + w, z + w ∈ C w , and x generates an extreme rayin C w , both y + w and z + w are either positive scalar multiples of x or equal to w .Assume that y + w = w and z + w = x . The latter says that z is an extension of x .Hence y + z = ˜ x ≤ z in C v . By cancellation in C v (Lemma 2.7), we get y = v and z = ˜ x . Suppose on the other hand that y + w = αx and z + w = βx for positivescalars α, β such that α + β = 1. Then y/α and z/β are extensions of x . We deducethat α ˜ x ≤ y and β ˜ x ≤ z . Hence, α ˜ x + z ≤ y + z = ˜ x = α ˜ x + β ˜ x. By cancellation in C v , z ≤ β ˜ x , and so z = β ˜ x . Similarly, y = α ˜ x . Thus, ˜ x generatesan extreme ray in C v .(ii) The argument is similar to the one used in (ii). After arriving at y + w = αx and z + w = βx , we assume without loss of generality that α ≤ ≤ β . Using againthat ˜ x is the minimum extension of x , we get z ≥ ˜ x/ ≥ y/ z/
2, and applyingLemma 2.7 (ii), we arrive at z/ ≥ y/ (cid:3) Remark . The property of ˜ x in Lemma 7.2 (ii) says that ˜ x is an irreducibleelement of the cone C in the sense defined by Thiel in [23].Next, we construct a suitable set of generators of C . For each w ∈ Idem( C ), let X w denote the set of minimal extensions of all elements x ∈ C w \{ w } that generatean extreme ray in C w . Consider the set S w ∈ Idem( C ) X w , which is closed underscalar multiplication. We form a set X by picking a representative from each ray in S w ∈ Idem( C ) X w . Proposition 7.4.
Let X ⊆ C be as described in the paragraph above. Each y ∈ C has a unique representation of the form y = n X i =1 α i x i + w, where x i ∈ X and α i ∈ (0 , ∞ ) for all i , and w ∈ Idem( C ) is such that ǫ ( x i ) ≤ w but x i (cid:2) w for all i .Proof. Let y ∈ C , and set w = ǫ ( y ). If y = w then its representation is simply y = w . Suppose that y = w . In C w , express y as a sum of elements that lie inextreme rays (Lemma 7.1). By the construction of X , these elements have the form α i ( x i + w ), with x i ∈ X and α i ∈ (0 , ∞ ). We thus have that y = n X i =1 α i ( x i + w ) = n X i =1 α i x i + w. We have x i + w ∈ C w \{ w } for all i ; equivalently, ǫ ( x i ) ≤ w and x i (cid:2) w for all i .Thus, this is the desired representation. ONES OF TRACES ARISING FROM AF C*-ALGEBRAS 33
To prove uniqueness of the representation, suppose that y = X i ∈ I α i x i + w = X j ∈ J β j x j + w ′ . Since ǫ ( x i ) ≤ w for all i , the support of y is w . Thus, w = w ′ . We can now rewritethe equation above as y = X i ∈ I α i ( x i + w ) = X j ∈ J β j ( x j + w ) . This equation occurs in C w ∼ = [0 , ∞ ) n w . Further, x i + w and x j + w generate extremerays of C w for all i, j . It follows that I = J and that the two representations arethe same up to relabeling of the terms. (cid:3) Constructing the vector space.
We continue to denote by X the subset of C defined in the previous subsection. For each w ∈ Idem( C ), define O w = { x ∈ X : x (cid:2) w } . Lemma 7.5.
Let w , w ∈ Idem( C ) . Then (i) O w ∪ O w = O w ∧ w . (ii) O w ∩ O w = O w + w . (iii) O w ⊆ O w if and only if w ≥ w .Proof. (i) It is more straightforward to work with the complements of the sets: x / ∈ O w ∧ w if and only if x ≤ w ∧ w , if and only if x ≤ w and x ≤ w , i.e., x / ∈ O w and x / ∈ O w .(ii) Again, we work with complements. Let’s show that O cw + w ⊆ O cw ∪ O cw (the opposite inclusion is clear). Let x ∈ O cw + w , i.e., x ≤ w + w . Choose z such that x ∧ w + z = x . Recall that the elements of X are minimal extensions ofnon-idempotent elements that generate an extreme ray. Thus, by Lemma 7.2 (ii),either x ∧ w ≤ z or z ≤ x ∧ w . If z ≤ x ∧ w , then x = x ∧ w + z ≤ x ∧ w ) ≤ w . Hence x ∈ O cw , and we are done. Suppose that x ∧ w ≤ z . It follows that2( x ∧ w ) ≤ x . Now repeat the same argument with x and w . We are doneunless we also have that 2( x ∧ w ) ≤ x . In this case, adding the inequalities we get2( x ∧ w ) + 2( x ∧ w ) ≤ x , i.e., x ∧ w + x ∧ w ≤ x . But x ≤ x ∧ w + x ∧ w (since x ≤ w + w ). Hence, x = x ∧ w + x ∧ w . Applying Lemma 7.2 (ii) again we getthat either x ≤ x ∧ w ) ≤ w or x ≤ x ∧ w ) ≤ w . Hence, x ∈ O cw ∪ O cw , asdesired.(iii) Suppose that O w ⊆ O w . By (i), O w ∧ w = O w ∪ O w = O w . Assume, forthe sake of contradiction, that w ∧ w = w . Since C is strongly connected, thereexists x ∈ C w ∧ w \ { w ∧ w } such that x ≤ w . We can choose x in an extremeray of C w ∧ w , since the set of all x ∈ C w ∧ w such that x ≤ w is a face. Considerthe minimum extension ˜ x of x . Adjusting x by a scalar multiple, we may assume that ˜ x ∈ X . Now ˜ x ≤ w , i.e, ˜ x / ∈ O w . But we cannot have ˜ x ≤ w ∧ w , since thiswould imply that x = ˜ x + w ∧ w = w ∧ w . Thus, x ∈ O w ∧ w . This contradicts that O w ∧ w = O w . (cid:3) Let w ∈ Idem( C ). Define P w = { x ∈ O w : ǫ ( x ) ≤ w } , e P w = P w ∪ O cw = { x ∈ X : ǫ ( x ) ≤ w } . Observe that if y ∈ C , and y = P ni =1 α i x i + w is the representation of y describedin Proposition 7.4, then x i ∈ P w for all i, ≤ i ≤ n . Lemma 7.6.
Let w , w ∈ Idem( C ) . The following statements hold: (i) e P w ∧ w = e P w ∩ ˜ P w . (ii) If w (cid:3) w then P w \ O w = ∅ .Proof. (i) This is straightforward: ǫ ( x ) ≤ w and ǫ ( x ) ≤ w if and only if ǫ ( x ) ≤ w ∧ w .(ii) Suppose that w w . Let w = w + w . By Lemma 7.5 (ii), O w ∩ O w = O w . Also w ≤ w and w = w . Since C is strongly connected, there exists y ∈ C w \ { w } such that w ≤ y ≤ w . Choose y on an extreme ray (alwayspossible, since the set of all y ∈ C w such that y ≤ w is a face) and adjust it by ascalar so that its minimum extension ˜ y belongs to X . Since ˜ y + w ∈ C w \{ w } , wehave that ˜ y (cid:2) w and ǫ (˜ y ) ≤ w . That is, ˜ y ∈ P w . Since ˜ y ≤ w , we also have that˜ y ∈ O cw ⊆ O cw . We have thus obtained an element ˜ y ∈ P w \ O w . (cid:3) Let us say that a function f : X → R is positive provided that there exists w ∈ Idem( C ) such that f ( x ) = 0 for x / ∈ O w and f ( x ) > x ∈ P w . We call w thesupport of f and denote it by supp( f ). Lemma 7.7.
The support of a positive function is unique. Further, if f, g : X → R are positive then supp( f + g ) = supp( f ) ∧ supp( g ) .Proof. Let w , w ∈ Idem( C ) be both supports of f . Suppose that w = w , andwithout loss of generality, that w w . Then there exists x ∈ P w ∩ O cw (byLemma 7.6). On one hand, x ∈ P w implies that f ( x ) >
0. On the other hand, x ∈ O cw implies that f ( x ) = 0, a contradiction. Thus w = w , whereby provingthe first part of the lemma.To prove the second part, assume that f and g are positive functions on X , andset v = supp( f ) and w = supp( g ). Clearly f + g vanishes on O cv ∩ O cw = O cv ∧ w . Let x ∈ P v ∧ w . Then, by Lemma 7.6 (i), x ∈ ˜ P v ∩ ˜ P w . Thus, x is in one of the followingsets: P v ∩ P w , P ν ∩ O cw , or P w ∩ O cν . In all cases we see that ( f + g )( x ) >
0. Indeed,if x ∈ P ν ∩ P w then f ( x ) , g ( x ) >
0; if x ∈ P ν ∩ O cw then f ( x ) > g ( x ) = 0; if x ∈ P w ∩ O cν then f ( x ) = 0 and g ( x ) >
0. Therefore supp( f + g ) = v ∧ w . (cid:3) Let us denote by V C the vector space of R -valued functions on X and by V + C theset of positive functions in V C . ONES OF TRACES ARISING FROM AF C*-ALGEBRAS 35
Theorem 7.8.
The pair ( V C , V + C ) is an ordered vector space having the Riesz in-terpolation property.Proof. By the previous lemma, V + C is closed under addition. Clearly, V + C is closedunder multiplication by positive scalars. Since the pointwise strictly positive func-tions belong to V + C and span V C , we have V + C − V + C = V C . Also, V + C ∩ − V + C = { } ,for if f and − f are positive then, by the previous lemma,supp( f ) ≥ supp( f + − f ) = supp(0) = ∞ , which implies that f = 0. Thus, ( V C , V + C ) is an ordered vector space.In [18], Maloney and Tikuisis obtained conditions guaranteeing that the Rieszinterpolation property holds in a finite dimensional ordered vector space. The prop-erties of the sets P w obtained in Lemma 7.6 (i) and (ii) are precisely those propertiesin [18, Corollary 5.1] shown to guarantee that the Riesz interpolation property holdsin ( V C , V + C ). (cid:3) Let us define a pairing ( · , · ) : C × V + C → [0 , ∞ ] as follows: for each y ∈ C and f ∈ V + C , write y = P ni =1 α i x i + w , the representation of y described in Proposition7.4, and then set ( y, f ) = n P i =1 α i f ( x i ) if w ≤ supp( f ) , ∞ otherwise . Theorem 7.9.
The pairing defined above is bilinear. Moreover, the map x ( x, · ) ,from C to Hom( V + C , [0 , ∞ ]) , is an isomorphism of extended Choquet cones.Proof. Let x, y ∈ C and f ∈ V + C . Write x = m X i =1 α i x i + v,y = n X j =1 β j y j + w, with v, w ∈ Idem( C ) and x i , y j ∈ X as in Proposition 7.4. Then x + y = m X i =1 α i x i + n X j =1 β j y j + v + w. Observe that ǫ ( x i ) , ǫ ( y j ) ≤ v + w and that α i , β j ∈ (0 , ∞ ) for all i, j . Thus, the sumon the right side is the representation of x + y described in Proposition 7.4, exceptfor the possible repetition of elements of X appearing both among the x i s and the y j s. If v + w ≤ supp( f ), then v ≤ supp( f ) and w ≤ supp( f ), and so( x, f ) + ( y, f ) = m X i =1 α i f ( x i ) + n X j =1 β j f ( y j ) = ( x + y, f ) . If, on the other hand, v + w (cid:2) supp( f ), then either v (cid:2) supp( f ) or w (cid:2) supp( f ),and in either case ( x, f ) + ( y, f ) = ∞ = ( x + y, f ). This proves additivity on the first coordinate. Homogeneity with respect to scalar multiplication follows automaticallyfrom additivity.Let f, g ∈ V + C and w ∈ Idem( C ). Then w ≤ supp( f + g ) if and only if w ≤ supp( f ) and w ≤ supp( g ) (Lemma 7.7). This readily shows linearity on the secondcoordinate.For each x ∈ C , let Λ x ∈ Hom( V + C , [0 , ∞ ]) be defined by the pairing above:Λ x ( f ) = ( x, f ) for all f ∈ V + C . Let Λ : C → Hom( V + C , [0 , ∞ ]) be the map given by y Λ y for all y ∈ C . To prove that Λ is injective, suppose that y, z ∈ C are suchthat Λ y = Λ z . Choose any f ∈ V + C such that supp( f ) = ǫ ( y ). If ǫ ( y ) ǫ ( z ) thenΛ y ( f ) is finite, while Λ z ( f ) = ∞ . This contradicts that Λ y = Λ z . Hence ǫ ( y ) ≤ ǫ ( z ).By a similar argument ǫ ( z ) ≤ ǫ ( y ), and so we get equality. Set w = ǫ ( y ) = ǫ ( z ).Then we can write y = m X i =1 α i y i + w,z = n X i =1 β i z i + w with y i , z i ∈ P w for all i . Let f ∈ V C be such that supp( f ) = w . Then f ( y i ) , f ( z i ) > m X i =1 α i f ( y i ) = Λ y ( f ) = Λ z ( f ) = n X i =1 β i f ( z i ) . Let V + w = { f ∈ V + C : supp( f ) = w } , i.e., f ∈ V + w if f is positive on P w and zerooutside O w . It is clear that V + w − V + w consists of all the functions on X that vanishoutside O w . It then follows from (3) that n = m and that, up to relabelling, y i = z i for all 1 ≤ i ≤ n . Consequently y = z .Let us show that Λ is surjective. Let λ ∈ Hom( V + C , [0 , ∞ ]). By Lemma 7.7, theset { w ∈ Idem( C ) : w = supp( f ) for some f ∈ V + C such that λ ( f ) < ∞} is closed under infima. Since this set is also finite, it has a minimum element w . Weclaim that for each f ∈ V + C we have λ ( f ) < ∞ ⇔ w ≤ supp( f ) . Indeed, from the definition of w it is clear that if λ ( f ) < ∞ then w ≤ supp( f ).Suppose on the other hand that f ∈ V + C is such that w ≤ supp( f ). Let f ∈ V + w besuch that λ ( f ) < ∞ . Then αf − f is positive (with support w ) for a sufficientlylarge scalar α ∈ (0 , ∞ ). Thus, λ ( f ) ≤ αλ ( f ) < ∞ .Let us extend λ by linearity to the vector subspace V w := V + w − V + w . As remarkedabove, V w consists of all the functions f : X → R vanishing on the complementof O w . That is, V w = span( { x : x ∈ O w } ), where x denotes the characteristicfunction of { x } .If x ∈ P w , then x + ǫ P w ∈ V + w for all ǫ >
0; here P w denotes the characteristicfunction of P w . It follows that λ ( x + ǫ P w ) ≥
0, and letting ǫ →
0, that λ ( x ) ≥ ONES OF TRACES ARISING FROM AF C*-ALGEBRAS 37 for all x ∈ P w . If x ∈ O w \ P w , then λ ( P w − α x ) ≥ α ∈ R . It follows that λ ( x ) = 0 for all x ∈ O w \ P w . Thus λ ( f ) = X x ∈ P w λ ( x ) f ( x )for all f ∈ V w . Since V + v ⊆ V w for any idempotent v such that w ≤ v , the formulaabove holds for all f ∈ V + C such that w ≤ supp( f ).Define y = X x ∈ P w λ ( x ) x + w. By the previous arguments, λ ( f ) = Λ y ( f ) for all f such that w ≤ supp( f ). On theother hand, λ ( f ) = ∞ = Λ y ( f ) for all f such that w (cid:2) supp( f ). Hence, λ = Λ y . (cid:3) References [1] Erik M. Alfsen.
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