A classification of finite simple amenable ${cal Z}$-stable $C^*$-algebras, I: $C^*$-algebras with generalized tracial rank one

A class of $C^*$-algebras, to be called those of generalized tracial rank one, is introduced, and classified by the Elliott invariant. 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 ${cal Z}$-stable $C^*$-algebras. An isomorphism theorem for a special sub-class of those $C^*$-algebras are presented. This provides the basis for the classification of $C^*$-algebras with rational generalized tracial rank one in Part II.

The classification of simple separable KK-contractible C*-algebras with finite nuclear dimension

The class of simple separable KK-contractible (KK-equivalent to ${0}$) C*-algebras which have finite nuclear dimension is shown to be classified by the Elliott invariant. In particular, the class of C*-algebras $Aotimes mathcal W$ is classifiable, where $A$ is a simple separable C*-algebra with finite nuclear dimension and $mathcal W$ is the simple inductive limit of Razak algebras with unique trace, which is bounded.

On the classification of simple amenable $C*$-algebras with finite decomposition rank, II

We prove that every unital stably finite simple amenable $C^*$-algebra $A$ with finite nuclear dimension and with UCT such that every trace is quasi-diagonal has the property that $Aotimes Q$ has generalized tracial rank at most one, where $Q$ is the universal UHF-algebra. Consequently, $A$ is classifiable in the sense of Elliott.

The classification of simple separable unital locally ASH algebras

Let $A$ be a simple separable unital locally approximately subhomogeneous C*-algebra (locally ASH algebra). It is shown that $Aotimes Q$ can be tracially approximated by unital Elliott-Thomsen algebras with trivial $textrm{K}_1$-group, where $Q$ is the universal UHF algebra. In particular, it follows that $A$ is classifiable by the Elliott invariant if $A$ is Jiang-Su stable.

Classification of finite simple amenable ${cal Z}$-stable $C^*$-algebras

We present a classification theorem for a class of unital simple separable amenable ${cal Z}$-stable $C^*$-algebras by the Elliott invariant. This class of simple $C^*$-algebras exhausts all possible Elliott invariant for unital stably finite simple separable amenable ${cal Z}$-stable $C^*$-algebras. Moreover, it contains all unital simple separable amenable $C^*$-alegbras which satisfy the UCT and have finite rational tracial rank.

Determinant Rank of C*-algebras

Let $A$ be a unital $C^*$-algebra and let $U_0(A)$ be the group of unitaries of $A$ which are path connected to the identity. Denote by $CU(A)$ the closure of the commutator subgroup of $U_0(A).$ Let $i_A^{(1, n)}colon U_0(A)/CU(A)rightarrow U_0(mathrm M_n(A))/CU(mathrm M_n(A))$ be the hm, defined by sending $u$ to ${rm diag}(u,1_n).$ We study the problem when the map $i_A^{(1,n)}$ is an isomorphism for all $n.$ We show that it is always surjective and is injective when $A$ has stable rank one. It is also injective when $A$ is a unital $C^*$-algebra of real rank zero, or $A$ has no tracial state. We prove that the map is an isomorphism when $A$ is the Villadsens simple AH--algebra of stable rank $k>1.$ We also prove that the map is an isomorphism for all Blackadars unital projectionless separable simple $C^*$-algebras. Let $A=mathrm M_n(C(X)),$ where $X$ is any compact metric space. It is noted that the map $i_A^{(1, n)}$ is an isomorphism for all $n.$ As a consequence, the map $i_A^{(1, n)}$ is always an isomorphism for any unital $C^*$-algebra $A$ that is an inductive limit of finite direct sum of $C^*$-algebras of the form $mathrm M_n(C(X))$ as above. Nevertheless we show that there are unital $C^*$-algebras $A$ such that $i_A^{(1,2)}$ is not an isomorphism.