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THE CUNTZ SEMIGROUP OF THE TENSOR PRODUCT OF C*-ALGEBRAS 6 1 CRISTIAN IVANESCU AND DAN KUCˇEROVSKY´ 0 2 Abstract. We calculate the Cuntz semigroup of a tensor prod- t c uct C∗-algebra A⊗A. We restrict our attention to C∗-algebras O whichareunital,simple,separable,nuclear,stablyfinite,Z-stable, 2 satisfy the UCT, with finitely generated K0(A) and have trivial A] K1(A). O . h t a ∗ On calcule le semigroupe de Cuntz d’une C -algèbre produit m tensoriel A⊗A. On considère seulement les C*-algèbres simples, [ séparable,nucléaires,àélémentunité,stablementfinies,Z-stables, 9 v satisfaisantauUCT,dontlegroupeK0(A)estdetypefini,etdont 8 le groupe K1(A) est trivial. 3 0 1 0 Key words and phrases. C∗-algebra, Cuntz semigroup, K -group, 0 . 1 tensor product. 0 5 1 : v i X r a Date: Received December 8, 2014. 2000 Mathematics Subject Classification. Primary 46L35, 46L06. 1 2 CRISTIAN IVANESCUAND DANKUCˇEROVSKY´ 1. Introduction The Cuntz semigroup has been studied since the late seventies but onlyrecentlyhasitproventobeanimportantinvariantforC∗-algebras. First, in the early 2000s, M. Rordam and A. Toms constructed exam- ples of C∗-algebras that appeared to be counterexamples to the Elliott conjecture. Shortly afterwards, Toms realized that the Cuntz semigroup distinguishes some of the newly constructed algebras; hence, the Cuntz semigroup could be added to the Elliott invariant. Toms’s dis- covery obviously prompted major questions, such as: “What is the range of the Cuntz semigroup?” or “What is the relation between the Cuntz semigroup and the Elliott invariant?” or “What are the properties of the Cuntz semigroup?” In this paper we propose to study one property of the Cuntz semigroup, namely, how the Cuntz semigroup of the tensor product, A⊗A, of two identical copies of the C*-algebra A relates to the Cuntz semigroup of A. It is well known that the tensor product of two positive elements is still a positive element. This property allows us to define a naturaltensorproductmapfromA+⊗A+ to(A⊗A)+. Theusualinter- pretation of A+⊗A+ is as a subset of the usual tensor product of (nuclear) C∗-algebras. However, defining maps at the level of Cuntz semigroups requires defining tensor products of Cuntz semigroups, which are a priori tensor products of abelian semigroups. Hence we must consider semigroup tensor products, discussed below. Our approach to tensor products of Cuntz semigroups is to first take an algebraic tensor product of abelian semigroups and then to take a completion with respect to a suitable topology. See [19, para. 6.L] for more information on topological completions. The basic reason for introducing completions is that if we use only algebraic tensor products we can obtain surjectivity results only in very limited situations, such as the finite-dimensional case. In the first three sections of this paper we work with the algebraic tensor product, and we use the term “dense range” for results from which we later obtain surjectivity as a corollary, after THE TENSOR PRODUCT OF CUNTZ SEMIGROUPS 3 taking a completion. We consider completions in the last section of the paper. As defined by Grillet [15], the tensor product of two abelian semigroups is constructed by forming a free abelian semigroup and passing to the quotient by the relations (a+ a′) ⊗ b = (a ⊗ b) + (a′ ⊗ b) and (a⊗b′)+(a⊗b) = a⊗(b+b′). This definition is equivalent [15] to the definition by a universal property. Stating the universal product definition for a family (A ) of semigroups, we first say that a mapping i i∈I s of the Cartesian product of semigroups A into a semigroup C is Q i I-linear if the mapping is a semigroup homomorphism in each variable separately. Then, if an I-linear mapping t of A into a semigroup T Q i has the property that, for any I-linear mapping s of A into some Q i semigroup C, there exists a unique homomorphism u of T into C such that s = u◦t, then we call the pair (t,T), and also the semigroup T, a tensor product of the family (A ) . i i∈I It is well known that not every positive element of a tensor product can be written as a tensor product of positive elements, even after allowing sums. Thus, the naive tensor product map from A+ ⊗A+ to (A⊗A)+ is in general not surjective. It seems interesting that, as we shall see, in some cases this map becomes surjective if we pass to Cuntz equivalence classes. In a recent paper [2], the question of determining surjectivity, at the level of Cuntz semigroups, of the natural tensor product map is posed; and left as an open problem. In that paper, the authors state that surjectivity does hold in the cases of AF algebras and O -stable ∞ algebras. This is not surjectivity at the level of algebraic tensor products, rather it is with respect to a particular choice of Cuntz semigroup tensorproductintroducedin[2], calledtheCuntzcategorytensorprod- uct. We will consider the case of simple, separable, unital, stably finite, nuclear, Z-stable C∗-algebras, with finitely generated K group, triv- 0 ial K -group and satisfying the UCT, and we show that the image of 1 the natural tensor product map on algebraic elements is dense in the 4 CRISTIAN IVANESCUAND DANKUCˇEROVSKY´ sense that it becomes surjective after passing to a completion. We consider completions with respect to several different possible topologies, the coarsest of these being given by pointwise suprema, as will be explained in the last section of the paper. We use [23] to deduce that algebras in the abovementioned class have stable rank one. The stable rank one property and its consequences are used several times in our proofs. Brown, Perera and Toms, [9], showed an important representation result for the original version of the Cuntz semigroup. This result was extendedtothenonunitalcase, usingthestabilizedversionoftheCuntz semigroup, by Elliott, Robert and Santiago, [14], and with more abstract hypotheses by Tikuisis and Toms, [28]. Their results (see Theo- rems2.1and2.1)implythatforcertainsimpleexactC∗-algebras,apart of the Cuntz semigroup is order isomorphic to an ordered semigroup of lower semicontinuous functions defined on a compact Hausdorff space. 2. The Cuntz semigroup Let A be a separable C*-algebra. For positive elements a,b ∈ A⊗K, we say that a is Cuntz subequivalent to b, and write a (cid:22) b, if v bv∗ → a n n in the norm topology, for some sequence (v ) in A⊗K. We say that a n is Cuntz equivalent to b and write a ∼ b if a (cid:22) b and b (cid:22) a. Denote by Cu(A) the set of Cuntz equivalence classes of the positive cone of A ⊗ K, i.e. Cu(A) = (A ⊗ K)+/ . The order relation a (cid:22) b defined ∼ for the positive elements of A⊗K induces an order relation on Cu(A): [a] ≤ [b] if a (cid:22) b, where [a] denotes the Cuntz equivalence class of the positive element a. Note (cf. [24, page 151]) that this order relation does not need to be the algebraic order with respect to the addition operation defined by setting [a] + [b] := [a′ + b′], where a′ and b′ are orthogonalpositive elements. It turnsout thatina stabilizationwe can always find such orthogonal representatives, i.e., in (A⊗K)+ we have a ∼ a′, b ∼ b′ with a′b′ = 0. Moreover, the choice of the orthogonal representatives does not affect the Cuntz class of their sum. So the ordered set Cu(A) becomes an abelian semigroup, under an addition THE TENSOR PRODUCT OF CUNTZ SEMIGROUPS 5 operation that is sometimes called Brown-Douglas-Fillmore addition [8]. If A is unital, we denote by T(A) the simplex of tracial states. By V(A) we denote the projection semigroup defined by the Murray von Neumann equivalence classes of projections in A⊗ K. The order structureonV(A)isdefinedthroughMurray-vonNeumanncomparison of projections. 2.1. Representations of the Cuntz semigroup. Brown, Perera and Toms’s representation result [9] for the Cuntz semigroup is as follows: Theorem 2.1. Let A be a simple, separable, unital, exact, stably finite Z-stable C∗-algebra. Then there is an order preserving isomorphism of ordered semigroups, ∼ W(A) = V(A)⊔Lsc(T(A),(0,∞)). In the statement of the above theorem, W(A) is the original defini- tionoftheCuntzsemigroup, i.e.,W(A) = M (A)+/ ,andLsc(T(A),(0,∞)) ∞ ∼ denotes the set of lower semicontinuous, affine, strictly positive functions on the tracial state space of the unital C*-algebra A. Addition withinLsc(T(A),(0,∞))isdonepointwiseandorderisdefinedthrough pointwise comparison, as is usual for functions. For [p] ∈ V(A) and f ∈ Lsc(T(A),(0,∞]), addition is defined by ˆ [p]+f := [p]+f ∈ Lsc(T(A),(0,∞)), where [aˆ](τ) = lim τ(a1/n),τ ∈ T(A), which reduces to τ(a) when a is n→∞ a projection. The order relation is given by: ˆ [p] ≤ f if [p](τ) < f(τ) for all τ ∈ T(A), ˆ f ≤ [p] if f(τ) ≤ [p](τ) for all τ ∈ T(A). Elliott, Robert and Santiago’s representation result [14] is very sim- ilar, and uses the stabilized Cuntz semigroup. In this result, the functions that appear may take infinite values and the algebras are not necessarily unital. Since we restrict our attention to the case of unital 6 CRISTIAN IVANESCUAND DANKUCˇEROVSKY´ algebras, the domain, T(A), can be taken to be a compact simplex, which in turn gives a simplified version of their result: Theorem 2.2. Let A be a simple, separable, unital, exact, stably finite Z-stable C*-algebra. Then there is an order preserving isomorphism of ordered semigroups, ∼ Cu(A) = V(A)⊔Lsc(T(A),(0,∞]), where Lsc(T(A),(0,∞]) will denote the set of lower semicontinuous, possibly infinite, affine, strictly positive functions on the tracial state space of a unital C*-algebra A. Within Lsc(T(A),(0,∞]) addition is pointwise and pointwise comparison is used. For [p] ∈ V(A) and f ∈ Lsc(T(A),(0,∞]), addition is given by ˆ [p]+f := [p]+f ∈ Lsc(T(A),(0,∞]), where [aˆ](τ) = lim τ(a1/n),τ ∈ T(A), which reduces to τ(a) when a is n→∞ a projection. The order relation is given by: ˆ [p] ≤ f if [p](τ) < f(τ) for all τ ∈ T(A), ˆ f ≤ [p] if f(τ) ≤ [p](τ) for all τ ∈ T(A). In the proof of the above theorems, a semigroup map i : Cu(A) −→ Lsc(T(A),(0,∞]) is defined, i([a])(τ) = d (a), with d to be explained τ τ later. These theorems show that the Cuntz semigroup, say Cu(A), is the disjoint union of the semigroup of positive elements coming from projections in (A⊗K)+, denoted V(A), and the set of lower semicontinuous, affine, strictly positive, functions on the tracial state space of A, denoted by Lsc(T(A),(0,∞]). In [9], the elements of the Cuntz semi- groupthat correspond to lower semicontinuous, affine, strictly positive, functions onthe tracial state space aretermed purely positive elements. In general, the set of purely positive elements does not form an object intheCuntzcategory. Toseethis, consideranelement xwithspectrum THE TENSOR PRODUCT OF CUNTZ SEMIGROUPS 7 [ǫ,1] in a C*-algebra of stable rank 1. The increasing sequence (x−1) n + is at first purely positive, but has a supremum that is projection-class. Theorem 2.2 implies that the subsemigroup of purely positive elements of the Cuntz semigroup, for certain C*-algebras A, is isomorphic to the semigroup Lsc(T(A),(0,∞]). The convex structure of the space of tracial states, T(A), makes it a Choquet simplex when A is unital [13], metrizable when A is separable. Wewillneedaresultaboutlowersemicontinuousfunctionsonmetriz- able Choquet simplices. Proposition 2.3. Let S be a compact metrizable Choquet simplex. Then every positive element of Lsc(S,(0,∞]), bounded or not, is the pointwise supremum of some pointwise nondecreasing sequence of continuous positive functions on S. Proof. Let S be a compact metrizable Choquet simplex. Using Ed- ward’s separation theorem inductively shows, see Lemma 6.1 in [1], thatevery lower semicontinuous positive affinefunctiononthesimplex, possibly with infinite values, is the pointwise supremum of a strictly increasing sequence of affine continuous functions without infinite values. Compactness lets us arrange that the functions are everywhere positive, for example, we may replace each f by the pointwise supremum n of {f ,ǫ1} for a suitably small ǫ. (cid:3) n The dual of the Cuntz semigroup is denoted by D(A), and is referred to as the set of all dimension functions. It is the set of all additive, suprema-preserving, and order preserving maps d : Cu(A) → (0,∞] such that, in the unital case, d([1]) = 1. If the map on A+ given by x 7→ d([x]) is lower semicontinuous, we say that the dimension function is lower semicontinuous. The lower semicontinuous dimension functions correspond to the 2-quasitraces, by Proposition 2.24 of [6]; for more information, see [14]. In the general case, once Theorem 2.2 is no longer applicable, the dual space of the Cuntz semigroup is strictly larger than the set of traces, T(A). See [5, page 307] for an example of a dimension function that is not lower semicontinuous and thus does 8 CRISTIAN IVANESCUAND DANKUCˇEROVSKY´ not come from either a trace or quasitrace. This example arises from a nonsimple and nonunital C*-algebra. We don’t know if there is an example coming from a simple and nuclear C*-algebra. We also note that nonsimple purely infinite algebras may have a nontrivial Cuntz semigroup, but do not have traces. Thus, their dimension functions do not come from traces. Any (necessarily not exact; see [18]) C*-algebra for which the quasitraces are not all traces will be an example where the states on the Cuntz semigroup do not correspond to the traces. Reviewing the liter- ature dealing with the Cuntz semigroup, the algebraic structure of the Cuntz semigroup is generally the main topic of interest, and the topological structure is hardly ever explicitly mentioned. We mention here some minor but apparently new observations about the topology of the Cuntz semigroup. We have Hausdorff metrics: D(A) is metrizable when A is separable, the metric being given by |d (x )−d (x )|2−k, P 1 k 2 k where x is a dense subsequence of the positive part of the unit ball k of A. Similarly, the Cuntz semigroup itself has at least a pseudomet- ric, in the presence of separability, of the form |d (x )−d (x )|2−k, P k 1 k 2 where d is a dense subsequence of D(A). We note that in general there k mayexist projection-classelements thatareequal, underthedimension functions, to purely positive elements. In the stable rank 1 case, it is possible to discriminate between projection-class elements and purely positive elements on the basis of a spectral criterion. 2.2. Dimension functions and a conjecture of Blackadar and Handelman. We have seen, as in Theorem 2.2, that the map i is useful in describing the order on the Cuntz semigroup. The map i is i(a) = d (a), where we define d (a) to be an extended version of the τ τ rankofa: d (a) = lim τ(a1/n), whereτ isatracialstate. Thismap, d , τ τ n→∞ also calleda dimension function, islower semicontinuous as amap from A+ to [0,∞], possibly taking infinite values, and defines a state on the Cuntz semigroup. In 1982, Blackadar and Handelman conjectured, see [5], that the set of lower semicontinuous dimension functions that come from traces is weakly dense in the set of dimension functions (or states THE TENSOR PRODUCT OF CUNTZ SEMIGROUPS 9 on the Cuntz semigroup). The conjecture is known to be true for a large class of C*-algebras, see [29, page 426], that includes the algebras that we propose to study in this paper, namely: simple, unital, stably finite, nuclear, Z-stable C*-algebras, with stable rank one, trivial K - 1 group and with the UCT property. We note that by [23], the stable rank one property follows from the other properties. From now on, we thus assume that the Blackadar-Handelman conjecture holds. Consider the map t : A+ ×A+ → Cu(A⊗A) defined by t(a,b) = [a⊗b]. Let us check that the above map t respects Cuntz equivalence. Lemma 2.4. Let A be a σ-unital C*-algebra. Given positive elements a,a′,b in A such that a′ (cid:22) a we have a⊗b (cid:22) a′ ⊗b. Proof. Let e be a countable approximate unit. Since a′ (cid:22) a, choose n an e such that e ae∗ → a′. We have n n n ∗ ′ ∗ ∗ ′ (e ⊗e )(a⊗b)(e ⊗e ) −a ⊗b = e ae ⊗e be −a ⊗b, n n n n n n n n and so ∗ ′ ||(e ⊗e )(a⊗b)(e ⊗e ) −a ⊗b|| → 0 n n n n (cid:3) If a ∼ a′ then a⊗b ∼ a′ ⊗b by applying the lemma twice, and thus we obtain the Corollary: Corollary 2.5. Consider the map t : A+ ×A+ → Cu(A⊗A) defined by t(a,b) = [a⊗b]. If a and a′ are positive elements of A that are Cuntz equivalent, then t(a,b) = t(a′,b). 3. Main result We begin with a technical lemma that is used in proving our main results. Lemma 3.1. Let S and S be compact metrizable Choquet simplices. 1 2 Let F be a positive, (bi)affine, continuous finite-valued function on S ×S . The function F can be approximated uniformly from below by 1 2 10 CRISTIAN IVANESCUAND DANKUCˇEROVSKY´ (1) (2) (k) a finite sum a f (x)f (y) where the f are continuous affine Pi,j ij i j i positive functions on S and the a are positive scalars. k ij Proof. Suppose first that S = S = S. The affine continuous functions 1 2 A(S) on the compact Choquet simplex S happen to be a Banach space whose dual is an L -space, see, e.g., [21, pg.181]. The space A(S) is 1 separable because S is metrizable. We can therefore apply Theorem 3.2 in [21] to obtain an inductive limit decomposition of A(S) in the form A(S) = [En, where the E are finite-dimensional l -spaces andthe connecting maps n ∞ are inclusion maps. We denote the dimension of E by m . Each n n subspace E = ℓmi has a basis, {f }mi , of elements of A(S )+ with i ∞ j j=1 mi f = 1 in that subspace. Since the connecting maps in the above Pj j inductive limit are inclusion maps, we can inductively choose the bases in such a way that the set of basis functions for ℓmi is contained in the ∞ set of basis functions for ℓ∞mi+1. The union of these sets of basis functions gives an infinite sequence (f ) ∈ A(S)+. Choose a dual sequence i (x ) ∈ S; thus, f (x ) is 1 if i = j, and is zero otherwise. Passing to i i j duals we also obtain a projective limit decomposition of S in the form S ✛ S ✛ S ✛ ··· m1 m2 m3 where the S are m -dimensional simplexes and the maps are affine mi i surjective maps. Considerthe(bi)affinefunctionF (x,y) := mn a f (x)f (y)where n Pj,k jk j k a is the given function F(x,y) evaluated at (x ,x ). The F are an jk j k n increasing sequence of positive functions that converges pointwise, on a compact space, and Dini’s theorem implies uniform convergence. De- noting the (bi)affine function obtained in the limit by G, we note that G is equal to the given function F at points of the form (x ,x ). Be- i j cause both G and F are already known to be continuous and positive, to prove equality it suffices to show that the (x ) are (affine) linearly i dense in a suitable sense. We recall that the affine span of a subset of anaffine space is the set of all finite linear combinations of the points of