A class of $C^*$-algebras, to be called those of generalized tracial rank one, is introduced, and... more 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.
We present a stable uniqueness theorem for non-unital C*-algebras. Generalized tracial rank one i... more We present a stable uniqueness theorem for non-unital C*-algebras. Generalized tracial rank one is defined for stably projectionless simple C*-algebras. Let $A$ and $B$ be two stably projectionless separable simple amenable C*-algebras with $gTR(A)\le 1$and $gTR(B)\le 1.$ Suppose also that $KK(A, D)=KK(B,D)=\{0\}$ for all C*-algebras $D.$ Then $A\cong B$ if and only if they have the same tracial cones with scales. We also show that every separable simple C*-algebra, $A$ with finite nuclear dimension which satisfies the UCT with non-zero traces must have $gTR(A)\le 1$ if $K_0(A)$ is torsion. In the next part of this research, we show similar results without the restriction on $K$-theory.
We present a classification theorem for amenable simple stably projectionless C*-algebras with ge... more We present a classification theorem for amenable simple stably projectionless C*-algebras with generalized tracial rank one whose $K_0$ vanish on traces which satisfy the Universal Coefficient Theorem. One of them is denoted by ${\cal Z}_0$ which has a unique tracial state and $K_0({\cal Z}_0)=\mathbb{Z}$ and $K_1({\cal Z}_0)=\{0\}.$ Let $A$ and $B$ be two separable simple $C^*$-algebras satisfying the UCT and have finite nuclear dimension. We show that $A\otimes {\cal Z}_0\cong B\otimes {\cal Z}_0$ if and only if ${\rm Ell}(B\otimes {\cal Z}_0)={\rm Ell}(B\otimes {\cal Z}_0).$ 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 $A\otimes {\cal Z}_0,$ where $A$ is any finite separable simple amenable $C^*$-algebras. Suppose that $A$ and $B$ are two finite separable simple $C^*$-algebras with finite nuclear dimension satisfying the UC...
The C* -algebra extensions of a topological space can be made into an abelian group which is natu... more The C* -algebra extensions of a topological space can be made into an abelian group which is naturally equivalent to the Khomology group of odd dimension [1] which has a close relation with index theory and is one of the starting points of KK theory [8]. The Cp -smoothness of an extension of a manifold was introduced in [3, 4], where Cp denotes the Schatten-von Neumann p-class [5]. We generalize the notion of Cp-smoothness to a finite CW complex and obtain necessary and sufficient conditions for an extension of a finite CW complex to be Cp -smooth modulo torsion.
A classification theorem is obtained for a class of unital simple separable amenable Z-stable C*-... more A classification theorem is obtained for a class of unital simple separable amenable Z-stable C*-algebras which exhausts all possible 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
ABSTRACT We present a class ${\cal N}_1$ of unital separable simple amenable \CA s whose Elliott ... more ABSTRACT We present a class ${\cal N}_1$ of unital separable simple amenable \CA s whose Elliott invariant exhausts all possible invariant sets for unital separable simple amenable ${\cal Z}$-stable \CA s. We show that if $A$ and $B$ are \CA s in ${\cal N}_1$ then $A\cong B$ if and only if they have the same Elliott invariant. This class includes all unital simple inductive limits of one dimensional non-commutative CW complexes and unital simple separable amenable \CA s of finite rational tracial rank in the UCT class.
Let $A$ be a simple separable unital locally approximately subhomogeneous C*-algebra (locally ASH... more Let $A$ be a simple separable unital locally approximately subhomogeneous C*-algebra (locally ASH algebra). It is shown that $A\otimes Q$ 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.
We prove that every unital stably finite simple amenable $C^*$-algebra $A$ with finite nuclear di... more 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 $A\otimes Q$ has generalized tracial rank at most one, where $Q$ is the universal UHF-algebra. Consequently, $A$ is classifiable in the sense of Elliott. In particular, if $A$ is a unital separable simple $C^*$-algebra with finite decomposition rank which satisfies the UCT, then $A$ is classifiable.
A class of $C^*$-algebras, to be called those of generalized tracial rank one, is introduced, and... more 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.
We present a stable uniqueness theorem for non-unital C*-algebras. Generalized tracial rank one i... more We present a stable uniqueness theorem for non-unital C*-algebras. Generalized tracial rank one is defined for stably projectionless simple C*-algebras. Let $A$ and $B$ be two stably projectionless separable simple amenable C*-algebras with $gTR(A)\le 1$and $gTR(B)\le 1.$ Suppose also that $KK(A, D)=KK(B,D)=\{0\}$ for all C*-algebras $D.$ Then $A\cong B$ if and only if they have the same tracial cones with scales. We also show that every separable simple C*-algebra, $A$ with finite nuclear dimension which satisfies the UCT with non-zero traces must have $gTR(A)\le 1$ if $K_0(A)$ is torsion. In the next part of this research, we show similar results without the restriction on $K$-theory.
We present a classification theorem for amenable simple stably projectionless C*-algebras with ge... more We present a classification theorem for amenable simple stably projectionless C*-algebras with generalized tracial rank one whose $K_0$ vanish on traces which satisfy the Universal Coefficient Theorem. One of them is denoted by ${\cal Z}_0$ which has a unique tracial state and $K_0({\cal Z}_0)=\mathbb{Z}$ and $K_1({\cal Z}_0)=\{0\}.$ Let $A$ and $B$ be two separable simple $C^*$-algebras satisfying the UCT and have finite nuclear dimension. We show that $A\otimes {\cal Z}_0\cong B\otimes {\cal Z}_0$ if and only if ${\rm Ell}(B\otimes {\cal Z}_0)={\rm Ell}(B\otimes {\cal Z}_0).$ 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 $A\otimes {\cal Z}_0,$ where $A$ is any finite separable simple amenable $C^*$-algebras. Suppose that $A$ and $B$ are two finite separable simple $C^*$-algebras with finite nuclear dimension satisfying the UC...
The C* -algebra extensions of a topological space can be made into an abelian group which is natu... more The C* -algebra extensions of a topological space can be made into an abelian group which is naturally equivalent to the Khomology group of odd dimension [1] which has a close relation with index theory and is one of the starting points of KK theory [8]. The Cp -smoothness of an extension of a manifold was introduced in [3, 4], where Cp denotes the Schatten-von Neumann p-class [5]. We generalize the notion of Cp-smoothness to a finite CW complex and obtain necessary and sufficient conditions for an extension of a finite CW complex to be Cp -smooth modulo torsion.
A classification theorem is obtained for a class of unital simple separable amenable Z-stable C*-... more A classification theorem is obtained for a class of unital simple separable amenable Z-stable C*-algebras which exhausts all possible 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
ABSTRACT We present a class ${\cal N}_1$ of unital separable simple amenable \CA s whose Elliott ... more ABSTRACT We present a class ${\cal N}_1$ of unital separable simple amenable \CA s whose Elliott invariant exhausts all possible invariant sets for unital separable simple amenable ${\cal Z}$-stable \CA s. We show that if $A$ and $B$ are \CA s in ${\cal N}_1$ then $A\cong B$ if and only if they have the same Elliott invariant. This class includes all unital simple inductive limits of one dimensional non-commutative CW complexes and unital simple separable amenable \CA s of finite rational tracial rank in the UCT class.
Let $A$ be a simple separable unital locally approximately subhomogeneous C*-algebra (locally ASH... more Let $A$ be a simple separable unital locally approximately subhomogeneous C*-algebra (locally ASH algebra). It is shown that $A\otimes Q$ 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.
We prove that every unital stably finite simple amenable $C^*$-algebra $A$ with finite nuclear di... more 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 $A\otimes Q$ has generalized tracial rank at most one, where $Q$ is the universal UHF-algebra. Consequently, $A$ is classifiable in the sense of Elliott. In particular, if $A$ is a unital separable simple $C^*$-algebra with finite decomposition rank which satisfies the UCT, then $A$ is classifiable.
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Papers by Guihua Gong