The Cuntz semigroup, a Riesz type interpolation property, comparison and the ideal property

We define a Riesz type interpolation property for the Cuntz semigroup of a $C^*$-algebra and prove it is satisfied by the Cuntz semigroup of every $C^*$-algebra with the ideal property. Related to this, we obtain two characterizations of the ideal property in terms of the Cuntz semigroup of the $C^*$-algebra. Some additional characterizations are proved in the special case of the stable, purely infinite $C^*$-algebras, and two of them are expressed in language of the Cuntz semigroup. We introduce a notion of comparison of positive elements for every unital $C^*$-algebra that has (normalized) quasitraces. We prove that large classes of $C^*$-algebras (including large classes of $AH$ algebras) with the ideal property have this comparison property.


Introduction
Elliott's classification program for separable, nuclear C * -algebras by discrete invariants including the K-theory is one of the most important and successful research areas in Operator Algebras ( [9], [10]; see also [24]). While it is clear that not all the separable, nuclear C * -algebras can be classified, very large classes of C * -algebras are known to be classifiable (see, e.g., [24]). Despite the success, counterexamples to Elliott's conjecture in the simple case were found by Rørdam ([25]) and by Toms ([28] and [29]). In [29] Toms used the Cuntz semigroup to distinguish simple, nuclear C * -algebras which cannot be distinguished by the conventional Elliott invariant, where the Cuntz semigroup of a C * -algebra A is a positively ordered, abelian semigroup whose elements are equivalence classes of positive elements in matrix algebras over A (see Section 2 for details). It then became clear that a further study of the Cuntz semigroup is needed, and that, in fact, it is very important in Elliott's classification program. On the other hand, as pointed out in [10], the understanding of the regularity properties of simple C * -algebras, including the comparison of positive elements, is essential in the new development of Elliott's program. Extending the notion of comparison of positive elements to classes of nonsimple C * -algebras-e.g., to the C * -algebras with the ideal property-and proving appropriate comparison results for these classes is clearly a necessary and nontrivial thing to do. In this paper we contribute to the study of the ideal property, to the understanding of the structure of the Cuntz semigroup of C * -algebras (with this property), and prove a certain type of comparison of positive elements for some classes of C * -algebras with the ideal property.
A C * -algebra is said to have the ideal property if each of its ideals is generated (as an ideal) by its projections (in this paper, by an ideal we mean a closed, two-sided ideal; the only exception here will be the Pedersen ideal, which is the smallest dense algebraic two-sided ideal in the C * -algebra). Note that every simple C * -algebra with an approximate unit of projections and every C * -algebra of real rank zero ( [4]) has the ideal property. The ideal property has been studied extensively by the first named author (alone or in collaboration), for example in [18], [19], [20], [21] and [13], [14]. The ideal property is important in Elliott's classification program. It appeared first in Ken Stevens' Ph.D. thesis in which he classified by a K-theoretical invariant a certain class of (non-simple) AI algebras with the ideal property. In [18] the first named author classified the AH algebras with the ideal property and with slow dimension growth up to a shape equivalence and gave several characterizations of when an arbitrary AH algebra has the ideal property. Recall that a C * -algebra A is said to be an AH algebra, if A is the inductive limit C * -algebra of: i] (C(X n,i ))P n,i , where the local spectra X n,i are finite, connected CW complexes, t n and [n, i] are strictly positive integers, and each P n,i is a projection of M [n,i] (C(X n,i )). In [13] and [14], jointly with Gong, Jiang and Li, the first named author proved a reduction theorem saying that every AH algebra with the ideal property and with the dimensions of the local spectra uniformly bounded (i.e., with no dimension growth) can be written as an AH algebra with the ideal property with (special) local spectra of dimensions ≤ 3. This result generalizes similar and strong reduction theorems for real rank zero AH algebras-proved by Dadarlat ( [8]) and Gong ([11])-and also for simple AH algebras-proved by Gong ([12])-which have been major steps in the classification of the corresponding classes of AH algebras. Also, in [18] and [19], the first named author proved several nonstable K-theoretical results for a large class of C * -algebras with the ideal property. Indeed, if A is an AH algebra with the ideal property and with slow dimension growth, it is proved in [18] that A has stable rank one (that means, in the unital case, that the set of the invertible elements in A is dense in A), that K 0 (A) is weakly unperforated in the sense of Elliott and is also a Riesz group ( [18] and [19]) and that the strict comparability of the projections in A is determined by the tracial states of A, when A is unital ( [18]). Also, jointly with Rørdam, the first named author proved in [20] that the ideal property is not preserved by taking minimal tensor products (even in the separable case). A characterization of the ideal property in the separable, purely infinite case is given by the first named author jointly with Rørdam in [21], in terms of the Jacobson topology of the primitive spectrum of the C * -algebra. The class of purely infinite C * -algebras have been introduced by Kirchberg and Rørdam in [17], extending the definition in the simple case given by Cuntz ([7]). A C * -algebra A is said to be purely infinite if A has no characters (or, equivalently, no non-zero abelian quotients) and if for every a, b ∈ A + such that a ∈ AbA (the ideal of A generated by b), it follows that there is a sequence {x n } of elements in A such that a = lim n→∞ x * n bx n ( [17]). The study of purely infinite C * -algebras was motivated by Kirchberg's classification of the separable, nuclear C * -algebras that tensorially absorb the Cuntz algebra O ∞ up to stable isomorphism by an ideal related KK-theory.
The paper is divided into three sections. In Section 2 we remind the reader some relevant definitions and notation-including the definition of the Cuntz semigroupand define a Riesz type interpolation property for the Cuntz semigroup W (A) of a C * -algebra A (Definition 2.1). We prove that W (A) has this property, whenever A is a C * -algebra the ideal property (Theorem 2.2). Related to this, we obtain two characterizations of the ideal property in terms of the Cuntz semigroup of the C *algebra (Theorem 2.4). Some additional characterizations are proved in the special case of the stable, purely infinite C * -algebras, and two of them are expressed in language of the Cuntz semigroup of the algebra (Theorem 2.5). In Section 3, after reminding the reader some definitions, notation and results, we introduce a notion of comparison of positive elements for every unital C * -algebra that has (normalized) quasitraces (Definition 3.1). We prove that large classes of C * -algebras (including large classes of AH algebras) with the ideal property have this comparison property (Theorems 3.6, 3.7 and 3.8).
The symbol ⊗ will mean the minimal tensor product of C * -algebras.

A Riesz type interpolation property for the Cuntz semigroup and the ideal property
We start by recalling mainly some definitions and notation (see also [2]). If A is a C * -algebra and a, b ∈ A + , then we write a b if there is a sequence {x n } of elements of A such that a = lim x n bx * n where the x n are suitable rectangular matrices.) If both a b and b a, we will write a ∼ b and will call a and b Cuntz equivalent (see [6]). We shall denote W (A) = M ∞ (A) + / ∼, and a ∈ W (A) will denote the Cuntz equivalence class of an element a of M ∞ (A) + (so that W (A) = { a : a ∈ M ∞ (A) + }). Then W (A) is a positively ordered abelian semigroup when equipped with the relations: We shall refer to W (A) as the Cuntz semigroup of A. One shortcoming of this construction is that this semigroup fails to be continuous with respect to sequential inductive limits. This was remedied in [5] by constructing an ordered semigroup, termed Cu(A), in terms of countably generated Hilbert modules. This new object turned out to be intimately related to W (A), in that Cu(A) is order isomorphic to W (A ⊗ K), where K is the C * -algebra of the compact operators on ℓ 2 (N) ( [5]). The semigroup Cu(A) is, as opposed to W (A), closed under (order-theoretic) suprema of increasing sequences and, in significant cases, can be regarded as a completion of W (A) (see [5], [1]). Moreover, every element in Cu(A) is a supremum of a rapidly increasing sequence. More precisely, we write x ≪ y to mean that whenever y ≤ sup y n for an increasing sequence {y n }, there is m such that x ≤ y m . A sequence {x n } is rapidly increasing if x n ≪ x n+1 for all n. It is shown in [5] that, for any positive a, the sequence { (a − 1/n) + } is rapidly increasing (and with supremum a ).
If A is a (local) C * -algebra, then denote the set of projections in A by P( For each a ∈ A + and for each ε > 0, we shall write (a − ε) + for the positive element of A given by h ε (a), where h ε (t) = max{t − ε, 0}.
This property just defined is of course related to the property of Riesz interpolation by general positive elements. This can be achieved in the real rank zero setting ( [22]), as well as for simple stable algebras that absorb the Jiang-Su algebra ( [27]).
We want to prove the following result: A be a C * -algebra with the ideal property. Then W (A) has the weak Riesz interpolation by projections property.
The proof of the above theorem will use the following, of which part (ii) is known and essentially contained in [17]; we include it here just for convenience.
Proof. (i). Since I is generated by its projections and the element (a − ε) + belongs to the Pedersen ideal of I, we can write (a − ε) from which the conclusion follows using [17, Lemma 2.5(ii) and Lemma 2.8(ii)].
(ii). Fix some 0 < ε < 1. Since q ∈ P(AaA), using [17, Proposition 2.7(v)], we get for this ε > 0 that there exists some n ∈ N such that: for i = 1, 2. Then, by [23], for our ε > 0, there exists δ > 0 such that: i.e. c ∈ I := Ab 1 A ∩ Ab 2 A. Note that since A has the ideal property and I is an ideal of A, it follows that I is generated (as an ideal) by P(I). Then, by Lemma 2.3, it follows that for our δ > 0 there exists p ∈ P(M ∞ (A)) such that p is a finite direct sum of projections of I and there exists m ∈ N such that: Finally, (1) and (2) imply that: which ends the proof.
We now prove a theorem inspired by the proof of Theorem 2.2 and that gives two characterizations of the ideal property in terms of the Cuntz semigroup of the C * -algebra. In particular, it implies Theorem 2.2.
Theorem 2.4. Let A be a C * -algebra. Then, the following are equivalent: (i) A has the ideal property; and m ∈ N such that (a − ε) + ≤ p ≤ m a and p is a finite direct sum of projections of A.
Proof. (i) ⇒ (ii). It follows from the proof of Theorem 2.2.
(ii) ⇒ (iii). Let a be an arbitrary element of A + . Then choose a i = b i = a, 1 ≤ i ≤ 2 and use (ii).
(iii) ⇒ (i). Let I be an ideal of A, a ∈ I ∩ A + and ε > 0. Then, by (iii), there exist n ∈ N, p 1 , p 2 , . . . , p n ∈ P(A) and there exists m ∈ N such that: Increasing m and then n, if necessary, (defining the new p ′ k s to be 0) we may suppose that m = n in (3). Working in M n (A), it is easy to see that the first inequality in (3) implies that: while the second inequality in (3) implies immediately that: Observe that (4) and (5) imply that (a − ε) + belongs to the ideal of A generated by P(AaA) ⊆ P(I). Therefore, a = lim ε→0 (a − ε) + belongs to the ideal of A generated by P(I) and hence, because each element of I is a linear combination of four positive elements of I, I is generated as an ideal of A by P(I). Since I is an arbitrary ideal of A, this proves that A has the ideal property.
The result below gives some additional characterizations of the ideal property in the case of stable, purely infinite C * -algebras. Two of these characterizations are in terms of the Cuntz semigroup of the C * -algebra. As already mentioned in the introduction, a C * -algebra A is termed purely infinite if A does not have nonzero abelian quotients and a b whenever a ∈ AbA (see [17]). Recall also that a positive, non-zero element a of a C * -algebra A is said to be properly infinite if a ⊕ a a ⊕ 0 in M 2 (A). It was shown in [17,Theorem 4.16] that a C * -algebra A is purely infinite if and only if all non-zero positive elements are properly infinite.
Theorem 2.5. Let A be a purely infinite, stable C * -algebra. Then, the following are equivalent: (i) A has the ideal property; (ii) For every a ∈ A + , there exists a sequence {p n } of projections in A such that a = sup n∈N p n (in W (A)); (iii) For every a ∈ A + , there exists a sequence {q n } of projections in A such that { q n } is increasing in W (A) and a = sup n∈N q n (in W (A)); (iv) For all a ∈ A + , we have that AaA = ∪ n≥1 I n , where {I n } is an increasing sequence of ideals of A and each I n is generated (as an ideal) by a single projection.
The proof of the above theorem will use the following: (a − ε n ) + ≤ p n ≤ a , for all n ∈ N (we also used the fact that m a = a , for every m ∈ N, since a is either properly infinite or zero). Let us prove now that sup n∈N p n = a . For this, observe first that by (6) we have p n ≤ a for every n ∈ N. Now, let x ∈ W (A) be such that p n ≤ x, for every n ∈ N. Then, (6) implies that (a − ε n ) + ≤ x, for every n ∈ N. Therefore, since also {ε n } is a strictly decreasing sequence of strictly positive numbers and lim n→∞ ε n = 0, we have: This ends the proof of the equality sup n∈N p n = a .
(iii) ⇒ (ii). The proof of this implication is trivial.
(ii) ⇒ (iv). Assume (ii). Let a ∈ A + . Then, by (ii), there exists a sequence {p n } of projections of A such that: Since A is stable, replacing each p n by a projection in its Murray-von Neumann equivalence class we may suppose that p m p n = 0 for every m = n, m, n ∈ N. Then, for every n ∈ N we have that p 1 + p 2 + · · · + p n ∈ P(A) and p n ≤ p 1 + p 2 + · · · + p n and hence, using [17, Lemma 2.8(iii) and Lemma 2.9] we have: since p k ≤ a for all k ∈ N and a is properly infinite or zero (by [17,Theorem 4.16]). Define: q n := p 1 + p 2 + · · · + p n (∈ P(A)), for all n ∈ N Since p 1 + p 2 + · · · + p n ≤ p 1 + p 2 + · · · + p n+1 , for all n ∈ N, it follows that { q n } is an increasing sequence in W (A). Also, (8) and (9) imply that a = sup n∈N q n (10) We want to prove that: (note that q n ≤ q n+1 implies that Aq n A ⊆ Aq n+1 A, for all n ∈ N). Indeed, observe that (10) implies that q n ≤ a , for all n ∈ N, from which we get q n ∈ AaA for all n ∈ N, and hence: ∪ n≥1 Aq n A ⊆ AaA (12) Let ε > 0. Since (a − ε) + ≪ a in Cu(A) = W (A) (take into account the fact that A is stable and see [5]), from (2.14) we deduce that: (a − ε) + ≤ q m for some m ∈ N, which implies that (a − ε) + ∈ Aq m A ⊆ ∪ n≥1 Aq n A. Since ε > 0 is arbitrary, we have that a = lim ε→0 (a − ε) + ∈ ∪ n≥1 Aq n A, and therefore: Observe that (12) and (13) prove the equality (11). Hence, if we define I n := Aq n A for every n ∈ N, we have that AaA = ∪ n≥1 I n , where {I n } is an increasing sequence of ideals of A and each I n is generated (as an ideal) by a single projection (namely q n ).
(iv) ⇒ (i). Assume (iv). Let I be an ideal of A and let a ∈ I + . Then, by (iv) we have that AaA = ∪ n≥1 I n , where {I n } is an increasing sequence of ideals of A and each I n is generated (as an ideal) by a single projection. This implies that a belongs to the ideal of A generated by P(AaA) ⊆ P(I), and hence a belongs to the ideal of A generated by P(I). Since a ∈ I + was arbitrary and each element of I is a linear combination of four positive elements of I, it follows that I is generated (as an ideal) by P(I). But I was an arbitrary ideal of A, so we deduce that A has the ideal property.
(We have used here that every non-zero positive element is properly infinite.) (ii) For a C * -algebra A, denote by Then the same argument as in (i) shows that W pi (A) is a subsemigroup of W (A) with Riesz interpolation. With this language, [17,Theorem 4.16] can be rephrased by saying that A is purely infinite if and only if W (A) = W pi (A).

Comparison of positive elements and the ideal property
We will introduce a notion of comparison of positive elements for unital C *algebras that have (normalized) quasitraces. We will prove that large classes of C * -algebras with the ideal property have this comparison property.
We begin recalling some definitions, notation and results. The notion of dimension function was introduced by Cuntz in [6]. A dimension function on a C * -algebra A is an additive order preserving function d : Let A be a unital C * -algebra. A (normalized) quasitrace on A ia a function τ : A → C satisfying: , for all a, b ∈ A sa , (iv) τ is linear on abelian sub-C * -algebras of A, (v) τ extends to a function from M n (A) to C satisfying (i)-(iv).
The set of all (normalized) quasitraces on A will be denoted QT (A). This notion was introduced in  [3] that d τ is a lower semicontinuous dimension function on A. Note that for all p ∈ P(M ∞ (A)) we have that d τ (p) = τ (p).
Definition 3.2. A unital C * -algebra A such that QT (A) = ∅ is said to have strict comparison of projections if p q whenever p, q ∈ P(M ∞ (A)) satisfy the inequality τ (p) < τ (q) for every τ ∈ QT (A). Theorem 3.3. Let A be a unital C * -algebra with the ideal property. Assume moreover that A has strict comparison of projections and that it has finitely many extremal quasitraces. Let a, b ∈ M ∞ (A) + such that: Proof. We may assume, without loss of generality, that a, b ∈ A + . Let τ 1 , τ 2 , . . . , τ l be the extremal quasitraces of A, for some l ∈ N. (Hence, each quasitrace of A is a convex combination of , for every τ ∈ QT (A), the above inequalities imply that: On the other hand, since A has the ideal property, Lemma 2.3 implies that for ε 0 > 0 there exist projections p and q in P(M ∞ (A)) such that p is a finite direct sum of projections of AaA, q is a finite direct sum of projections of AbA and there is some m ∈ N such that: Using (14), (16) and (17), we get: which obviously implies that: Multiplying each of these l inequalities with appropriate positive numbers and then summing up all the inequalities, we have that: (of course, we used here the fact that τ 1 , τ 2 , . . . , τ l are the extremal quasitraces of A).
Since A has strict comparison of projections, (18) implies that: Using again condition (ii) in Lemma 2.3 and the fact that q is a finite direct sum of projections of AbA, it follows that: for some n ∈ N.
Remark 3.4. For all a and b ∈ A + , the conclusion of the above result that, for every ε > 0, there is m such that (a − ε) + b ⊗ 1 m can be rephrased by saying that a ∈ AbA, as mentioned in [26, comment before Lemma 4.1]. We shall be using this below.
If A is a unital C * -algebra, we will denote by T (A) the set of tracial states of A.
Proof. Let U be an arbitrary U HF algebra, that is, U is the inductive limit of a sequence {M n(k) } k of matrix algebras via unital * -homomorphisms M n(k) → M n(k+1) . Define B := A ⊗ U . Then, clearly, B is a unital AH algebra with the ideal property and with slow dimension growth (in the sense of Gong ([11])). Then, since B is a unital exact C * -algebra, by a theorem of Haagerup ([15]) we have that QT (B) = T (B). Hence, by [18,Theorem 5.1(b)] it follows that B has strict comparison of projections. Let T (U ) = {σ}. Then, we have that T (B) = T (A ⊗ U ) = {τ ⊗ σ : τ ∈ T (A)}. Therefore, clearly, B = A ⊗ U has finitely many extremal tracial states, since A has finitely many extremal tracial states. We may assume, without any loss of generality, that a, b ∈ A + . Define a := a ⊗ 1, Then, for all ρ = τ ⊗ σ ∈ T (B)(τ ∈ T (A))), we have by hypothesis: Using now Theorem 3.3 and Remark 3.4 we deduce that a ⊗ 1 = a ∈ B bB = (A ⊗ U )(b ⊗ 1)(A ⊗ U ), from which we easily conclude (using, e.g., Fubini maps) that a ∈ AbA, which implies the conclusion, by [17,Proposition 2.7(v)].
The next two theorems are the main results of this section. Recall first that a positive ordered abelian semigroup W (in particular, the Cuntz semigroup of a C * -algebra) is said to be almost unperforated if for all x, y ∈ W and all m, n ∈ N, with nx ≤ my and n > m, one has x ≤ y (see, e.g. [26]). Proof. Observe first that since both A and B have the ideal property and A (or B) is exact, it follows that A ⊗ B has the ideal property (use, e.g., [20,Corollary 1.3]). On the other hand, by a result in [30], B is Z-stable, that is B ∼ = B ⊗ Z, where Z is the Jiang-Su algebra ( [16]). Hence the unital AH algebra with the ideal property A ⊗ B is Z-stable, i.e. A ⊗ B ∼ = (A ⊗ B) ⊗ Z, and then [26, Theorem 4.5] implies that W (A ⊗ B) is almost unperforated. Note that if T (B) = {σ}, then T (A ⊗ B) = {τ ⊗ σ : τ ∈ T (A)} and since A has finitely many extremal tracial states, it is obvious that A ⊗ B has also finitely many extremal tracial states. Now, the fact that A ⊗ B has weak strict comparison follows from Theorem 3.7. In the proof of Theorem 3.3 we showed, in particular, that in the case when a unital C * -algebra A has finitely many extremal quasitraces, then ( * ) =⇒ ( * * ). Therefore, in this case, if A has almost weak strict comparison, it follows that A has weak strict comparison. Note that if we drop the condition that the C * -algebra A has finitely many extremal quasitraces (tracial states), the conclusions of Theorem 3.3 and of Corollary 3.5 remain true if we replace in their hypotheses condition ( * ) by condition ( * * ) as above. Also, it is easy to see that, if in Theorems 3.6, 3.7, 3.8 we drop the condition that A has finitely many extremal quasitraces (tracial states) and the condition that B has a unique tracial state (in Theorem 3.8), then they remain true if we replace in their conclusions "weak strict comparison" by "almost weak strict comparison" (to show these results we use the same proofs).
In conclusion, we thus obtain generalizations of all the results proved in Section 3.