We give a classification theorem for unital separable simple nuclear C * -algebras with tracial topological rank zero which satisfy the Universal Coefficient Theorem. We prove that if A and B are two such C * -algebras and

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Let C and A be two unital separable amenable simple C * -algebras with tracial rank no more than one. Suppose that C satisfies the Universal Coefficient Theorem and suppose that ϕ 1 , ϕ 2 : C → A are two unital monomorphisms. We show that there is a continuous path ofT and a rotation related map R ϕ1,ϕ2 associated with ϕ 1 and ϕ 2 is zero.Applying this result together with a result of W. Winter, we give a classification theorem for a class A of unital separable simple amenable C * -algebras which is strictly larger than the class of separable C * -algebras whose tracial rank are zero or one. Tensor products of two C * -algebras in A are again in A. Moreover, this class is closed under inductive limits and contains all unital simple ASH-algebras whose state spaces of K 0 is the same as the tracial state spaces as well as some unital simple ASH-algebras whose K 0 -group is Z and tracial state spaces are any metrizable Choquet simplex. One consequence of the main result is that all unital simple AH-algebras which are Z-stable are isomorphic to ones with no dimension growth.

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...

We introduce the notion of tracial topological rank for C*‐algebras. In the commutative case, this notion coincides with the covering dimension. Inductive limits of C*‐algebrasof the form PMn(C(X))P, where X is a compact metric space with dim X ⩽ k, and P is a projection in Mn(C(X)), have tracial topological rank no more than k. Non‐nuclear C*‐algebras can have small tracial topological rank. It is shown that if A is a simple unital C*‐algebra with tracial topological rank k (< ∞), then A is quasidiagonal, A has stable rank 1, A has weakly unperforated K0(A), A has the following Fundamental Comparability of Blackadar: if p, q ∈ A are two projections with τ(p) < τ(q) for all tracial states τ on A, then p ≼ q . 2000 Mathematics Subject Classification: 46L05, 46L35.