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A category is described to which the Cuntz semigroup belongs and as a functor into which it preserves inductive limits.1. Recently, Toms in [26] used the refinement of the invariant K 0 introduced by Cuntz almost thirty years ago in [4] to show that certain C*-algebras are not isomorphic. Anticipating the possible use of this invariant to establish isomorphism, we take the liberty of reporting some observations concerning it. (In particular, we present what might be viewed as an embryonic isomorphism theorem.)One of the first things that might be noted in connection with this invariant, which considers, instead of the finitely generated projective modules over a given C*-algebra, the larger class of modules consisting of the countably generated Hilbert C*-modules (see [12], [15], and [21]; see also [10]) is that, whereas the equivalence relation between finitely generated projective modules would appear to be inevitable, namely, just isomorphism, in the wider setting of Hilbert C*-modules it is no longer quite so clear what the equivalence relation should be. While it is tempting just to choose isomorphism again, one should note that, even in the stably finite case (which is perhaps the case that this invariant is of most interest), whereas the isomorphism classes of algebraically finitely generated Hilbert C*-modules (which are of course also algebraically projective, and up to isomorphism exhaust the finitely generated projective modules) form an ordered set with respect to inclusion (in other words, if each of two such modules is isomorphic to a submodule of the other, then they must be isomorphic-indeed, any two such isomorphisms, from each of

The cone of lower semicontinuous traces is studied with a view to its use as an invariant. Its properties include compactness, Hausdorffness, and continuity with respect to inductive limits. A suitable notion of dual cone is given. The cone of lower semicontinuous 2-quasitraces on a (non-exact) C*-algebra is considered as well. These results are applied to the study of the Cuntz semigroup. It is shown that if a C*-algebra absorbs the Jiang-Su algebra, then the subsemigroup of its Cuntz semigroup consisting of the purely non-compact elements is isomorphic to the dual cone of the cone of lower semicontinuous 2-quasitraces. This yields a computation of the Cuntz semigroup for the following two classes of C*-algebras: C*-algebras that absorb the Jiang-Su algebra and have no non-zero simple subquotients, and simple C*-algebras that absorb the Jiang-Su algebra.

We study a class of stably projectionless simple C*-algebras which may be viewed as having generalized tracial rank one in analogy with the unital case. A number of structural questions concerning these simple C*-algebras are studied, pertinent to the classification of separable stably projectionless simple amenable Jiang-Su stable C*-algebras. NotationDefinition 2.1. Let A be a C*-algebra. Denote by Ped(A) the Pedersen ideal (see Section 5.6 of [38]). Definition 2.4. Let A be a C*-algebra and let a, b ∈ A + . We write a b if there exists a sequence (x n ) in A such that x * n bx n → a in norm. If a b and b a, we write a ∼ b and say that a and b are Cuntz equivalent. It is known that ∼ is an equivalence relation. Denote by Cu(A) the set of Cuntz equivalence classes of positive elements of A ⊗ K. It is an ordered abelian semigroup ([7]). Denote by Cu(A) + the subset of those elements which cannot be represented by projections. We shall write a for the equivalence class represented by a. Thus, a b will be also written as a ≤ b . Recall that we write a ≪ b if the following holds: for any increasing sequence ( y n ), if b ≤ sup{ y n } then there exists n 0 ≥ 1 such that a ≤ y n 0 In what follows we will also use the objects Cu ∼ (A) and Cu ∼ (ϕ) introduced in [43].Definition 2.5. If B is a C*-algebra, we will use QT(B) for the set of quasitraces τ with τ = 1 (see [2]). Let A be a σ-unital C*-algebra. Suppose that every quasitrace of every hereditary sub-C*-algebra B of A is a trace.If τ ∈ T(A), we will extend it to (A ⊗ K) + by the rule τ (a ⊗ b) = τ (a)Tr(b), for all a ∈ A and b ∈ K, where Tr is the canonical densely defined trace on K.Recall that A has the (Blackadar) property of strict comparison for positive elements, if for any two elements a, b ∈ (A ⊗ K) + with the property that d τ (a) < d τ (b) < +∞ for all τ ∈ T(A) \ {0}, necessarily a b. In general (without knowing that quasitraces are traces), this property will be called strict comparison for positive elements using traces.Let S be a topological convex set. Denote by Aff(S) the set of all real continuous affine functions, and by Aff + (S) the set of all real continuous affine functions f with f (s) > 0 for all s, together with zero function.Recall T(A) w denotes the closure of T(A) in T(A) with respect to pointwise convergence on Ped(A) (see the end of 2.1). Suppose that 0 ∈ T(A) w and that T(A) generates T(A), in particular. (By 4.5 below, these properties hold, in the case that A = Ped(A).) Then A has strict comparison for positive elements using traces if and only if d τ (a) < d τ (b) for all τ ∈ T(A) w implies a b, for any a, b ∈ (A ⊗ K) + .Definition 2.6. Let A be a C*-algebra such that 0 ∈ T(A) w . There is a linear map r aff : A s.a. → Aff(T(A)w ), from A s.a. to the set of all real affine continuous functions on T(A) w , defined byw and for all a ∈ A s.a. . Denote by A q s.a. the space r aff (A s.a. ) and by A q + the cone r aff (A + ) (see [9]). Denote by Aff 0 (T 1 (A)) the set of all real continuous affine functions which vanish at zero, and de...