arXiv subjects
Publications and source records attributed to Zhuang Niu.
We introduce Property (C) for a unital commutative sub-C*-algebra $D$ of a unital C*-algebra $A$, which is a version of the relative comparison property using almost normalizers. In the case that $D = \mathrm{C}(X)$ and $A = \mathrm{C}(X) \rtimesΓ$, where $(X, Γ)$ is a free and minimal dynamical system with uniform Rokhlin property, it turns out that this Property (C) is equivalent to the small boundary property of $(X, Γ)$.
It is shown that there is a simple unital separable AH algebra which is not singly generated.
Let $Γ$ be a countable discrete amenable group, and let $A=l^\infty(Γ) \rtimes Γ$. It is shown that if $a, b \in A \otimes \mathcal K$ are positive elements such that $$\mathrm{d}_τ(a) < \mathrm{d}_τ(b),\quad τ\in \mathrm{T}(A),$$ then $a$ is Cuntz subequivalent to $b$. Moreover, consider the universal minimal set $(M, Γ)$. The simple C*-algebra $\mathrm{C}(M)\rtimesΓ$ is shown to be AH in the strong sense that there is an increasing net of unital sub-C*-algebras $A_λ\subseteq A$, $λ\in Λ$, such that each $A_λ$ is a simple (separable) $\mathcal Z$-absorbing approximately homogeneous C*-algebra with real rank zero and $A = \bigcup_{λ\in Λ} A_λ$. In particular, $\mathrm{C}(M)\rtimesΓ$ is approximately divisible.
We show that Villadsen algebras, which are not Z-stable, are singly generated. More generally, we show that any simple unital AH algebra with diagonal maps is singly generated.
The authors' recent classification of Jesper Villadsen's remarkable generalization (based on a self-reproducing seed space) of Glimm's infinite tensor product (UHF) C*-algebras, by means of the Cuntz semigroup (in the case of a fixed, well-behaved, seed space), is extended to the analogous generalization of Bratteli's approximately finite-dimensional (AF) C*-algebras. Some progress is made in the direction of distinguishing between algebras based on different seed spaces.
Let $X$ be a compact metrizable space, and let $Δ$ be a closed set of Borel probability measures on $X$. We study the small boundary property of the pair $(X, Δ)$. In particular, it is shown that $(X, Δ)$ has the small boundary property if it has a restricted version of property Gamma. As an application, it is shown that, if $A$ is the crossed product C*-algebra $\mathrm{C}(X)\rtimes\mathbb Z^d$, where $(X, \mathbb Z^d)$ is a free minimal topological dynamical system, or if $A$ is an AH algebra with diagonal maps, then, $A$ is $\mathcal Z$-stable if the set of extreme tracial states is compact, regardless of its dimension.
We give a brief survey of the development of the Elliott program of classification of separable simple amenable $C^*$-algebras.
It is shown that, for an arbitrary free and minimal $\mathbb Z^n$-action on a compact Hausdorff space $X$, the crossed product C*-algebra $\mathrm{C}(X)\rtimes\mathbb Z^n$ always has stable rank one, i.e., invertible elements are dense. This generalizes a result of Alboiu and Lutley on $\mathbb Z$-actions. In fact, for any free and minimal topological dynamical system $(X, Γ)$, where $Γ$ is a countable discrete amenable group, if it has the uniform Rokhlin property and Cuntz comparison of open sets, then the crossed product C*-algebra $\mathrm{C}(X)\rtimesΓ$ has stable rank one. Moreover, in this case, the C*-algebra $\mathrm{C}(X)\rtimesΓ$ absorbs the Jiang-Su algebra tensorially if, and only if, it has strict comparison of positive elements.
It is shown that certain unital simple C*-algebras constructed by Villadsen are classified by the K0-group together with radius of comparison.
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.
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 $A\otimes \mathcal W$ 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.
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.
Let $(X, Γ)$ be a free minimal dynamical system, where $X$ is a compact separable Hausdorff space and $Γ$ is a discrete amenable group. It is shown that, if $(X, Γ)$ has a version of Rokhlin property (uniform Rokhlin property) and if $\mathrm{C}(X)\rtimesΓ$ has a Cuntz comparison on open sets, then the comparison radius of the crossed product C*-algebra $\mathrm{C}(X) \rtimes Γ$ is at most half of the mean topological dimension of $(X, Γ)$. These two conditions are shown to be satisfied if $Γ= \mathbb Z$ or if $(X, Γ)$ is an extension of a free Cantor system and $Γ$ has subexponential growth. The main tools being used are Cuntz comparison of diagonal elements of a subhomogeneous C*-algebra and small subgroupoids.
Let $(X, Γ)$ be a free and minimal topological dynamical system, where $X$ is a separable compact Hausdorff space and $Γ$ is a countable infinite discrete amenable group. It is shown that if $(X, Γ)$ has the Uniform Rokhlin Property and Cuntz comparison of open sets, then $\mathrm{mdim}(X, Γ)=0$ implies that $(\mathrm{C}(X) \rtimesΓ)\otimes\mathcal Z \cong \mathrm{C}(X) \rtimesΓ$, where $\mathrm{mdim}$ is the mean dimension and $\mathcal Z$ is the Jiang-Su algebra. In particular, in this case, $\mathrm{mdim}(X, Γ)=0$ implies that the C*-algebra $\mathrm{C}(X) \rtimesΓ$ is classified by the Elliott invariant.
Consider an arbitrary extension of a free $\mathbb Z^d$-action on the Cantor set. It is shown that it has dynamical asymptotic dimension at most $3^d - 1$.
It is shown that every Jiang-Su stable approximately subhomogeneous C*-algebra has finite decomposition rank. Previously, it was not even known that such algebras have finite nuclear dimension. A key step in the proof is that subhomogeneous C*-algebra are locally approximated by a certain class of more tractable subhomogeneous algebras, namely, a non-commutative generalization of the class of cell complexes. The result is applied to show that Jiang-Su stable minimal Z-crossed products have finite decomposition rank.
We show that there is a simple separable unital (non-nuclear) tracially AF algebra $A$ which does not absorb the Jiang-Su algebra $\mathcal Z$ tensorially, i.e., $A \ncong A\otimes\mathcal Z$.
We study homeomorphisms of a Cantor set with $k$ ($k < +\infty$) minimal invariant closed (but not open) subsets; we also study crossed product C*-algebras associated to these Cantor systems and their certain orbit-cut sub-C*-algebras. In the case that $k\geq 2$, the crossed product C*-algebra is stably finite, has stable rank 2, and has real rank zero if in addition $(X, σ)$ is aperiodic. The image of the index map is connected to certain directed graphs arising from the Bratteli-Vershik-Kakutani model of the Cantor system. Using this, it is shown that the ideal of the Bratteli diagram (of the Bratteli-Vershik-Kakutani model) must have at least $k$ vertices at each level, and the image of the index map must consist infinitesimals.