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Abstract We classify all essential extensions of the form $$ \begin{align*} &0 \rightarrow {\mathcal{W}} \rightarrow {D} \rightarrow A \rightarrow 0,\end{align*}$$where $${\mathcal {W}}$$ is the unique separable simple C*-algebra with a unique tracial state, which is $KK$-contractible and has finite nuclear dimension, and $$A$$ is a separable amenable $${\mathcal {W}}$$-embeddable C*-algebra, which satisfies the Universal Coefficient Theorem (UCT). We actually prove more general results. We also classify a class of amenable $C^*$-algebras, which have only one proper closed ideal $${\mathcal {W}}.$$more » « less
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Let A be a separable simple exact Z-stable C∗-algebra. We show that the unitary group of \tilde{A} has the cancellation property. If A has continuous scale then the Cuntz semigroup of A has strict comparison property and a weak cancellation property. Let C be a 1-dimensional noncommutative CW complex with K1(C) = {0}. Suppose that λ : Cu∼(C) → Cu∼(A) is a morphism in the augmented Cuntz semigroups which is strictly positive. Then there exists a sequence of homomorphisms φn : C → A such that limn→∞ Cu∼(φn) = λ. This result leads to the proof that every separable amenable simple C∗-algebra in the UCT class has rationally generalized tracial rank at most one.more » « less
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We construct two types of unital separable simple 𝐶∗-algebras: 𝐴𝐶1 𝑧 and 𝐴𝐶2 𝑧 , one exact but not amenable, the other nonexact. Both have the same Elliott invariant as the Jiang–Su algebra – namely, 𝐴𝐶𝑖 𝑧 has a unique tracial state, 𝐾0 𝐴𝐶𝑖 𝑧 , 𝐾0 𝐴𝐶𝑖 𝑧 + , 1 𝐴𝐶𝑖 𝑧 = (Z, Z+, 1), and 𝐾1 𝐴𝐶𝑖 𝑧 = {0} (𝑖 = 1, 2). We show that 𝐴𝐶𝑖 𝑧 (𝑖 = 1, 2) is essentially tracially in the class of separable 𝒵-stable 𝐶∗-algebras of nuclear dimension 1. 𝐴𝐶𝑖 𝑧 has stable rank one, strict comparison for positive elements and no 2-quasitrace other than the unique tracial state. We also produce models of unital separable simple nonexact (exact but not nuclear) 𝐶∗-algebras which are essentially tracially in the class of simple separable nuclear𝒵-stable 𝐶∗-algebras, and the models exhaust all possible weakly unperforated Elliott invariants.We also discuss some basic properties of essential tracial approximation. 1.more » « less
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null (Ed.)We present a classification theorem for separable amenable simple stably projectionless C -algebras with finite nuclear dimension whose K0 vanish on traces which satisfy the Universal Coefficient Theorem. One of C -algebras in the class is denoted by Z0 which has a unique tracial state, K_0(Z_0) = Z and K1(Z_0) = {0}. Let A and B be two separable amenable simple C -algebras satisfying the UCT. We show that A ⊗ Z_0 = B ⊗ Z_0 if and only if Ell(A ⊗ Z_0 ) = Ell(B ⊗ Z_0 ). A class of simple separable C -algebras which are approximately sub-homogeneous whose spectra having bounded dimension is shown to exhaust all possible Elliott invariant for C -algebras of the form A ⊗ Z_0 , where A is any finite separable simple amenable C -algebras.more » « less
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Elliott, G.A. (Ed.)A classification theorem is obtained for a class of unital simple separable amenable Z-stable C*-algebras which exhausts all possi- ble values of the Elliott invariant for unital stably finite simple separable amenable Z-stable C*-algebras. Moreover, it contains all unital simple separable amenable C∗-algebras which satisfy the UCT and have finite rational tracial rank.more » « less
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Elliott, G.A. (Ed.)A class of C*-algebras, to be called those of generalized tracial rank one, is introduced. A second class of unital simple separable amenable C*-algebras, those whose tensor products with UHF-algebras of infinite type are in the first class, to be referred to as those of rational generalized tracial rank one, is proved to exhaust all possible values of the Elliott invariant for unital finite simple separable amenable Z-stable C*- algebras. A number of results toward the classification of the second class are presented including an isomorphism theorem for a special sub-class of the first class, leading to the general classification of all unital simple C*- algebras with rational generalized tracial rank one in Part II.more » « less
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