Simple stably projectionless C*-algebras with generalized tracial rank one

Abstract

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...

Tracial oscillation zero and stable rank one

“…The following is known. The proof of this is exactly the same as that of the case T (A) = QT (A) (see 5.1, 5.2, 5.3 and 5.4 of [13] for details, and also see the remark after Definition 6.3 of [17]).…”

“…Theorem 2.19 (cf. Theorem 5.3 and Proposition 5.4 of [13], also [23]). Let A be a σ-unital simple C * -algebra with a strict positive element e A and with continuous scale and with QT (A) = ∅.…”

“…Proof. By Lemma 4.4 of [13], there is e 1 ∈ M n (A) with 0 ≤ e 1 ≤ 1 and x ∈ M n (A) such that e 1 x * x = x * xe 1 = x * x and a 0 = xx * is a strictly positive element of A (for some n ∈ N). Put S = {τ ∈ QT [0,1] (A) : τ (e 1 ) ≥ 1}.…”

“…Definition 2.8. Recall (Theorem 4.7 of [13]) that a σ-unital C * -algebra A is compact if and The following is a quasitrace version of Lemma 4.5 of [13].…”

“…. By (e 2.26), {c 0,n }, {c → 0 (see, 5.1 of [13] and 2.5 of [23], for example). It follows that { bn } ∈ N cu .…”