Mean dimension and AH-algebras with diagonal maps

Mean dimension for AH-algebras is introduced. It is shown that if a simple unital AH-algebra with diagonal maps has mean dimension zero, then it has strict comparison on positive elements. In particular, the strict order on projections is determined by traces. Moreover, a lower bound of the mean dimension is given in term of comparison radius. Using classification results, if a simple unital AH-algebra with diagonal maps has mean dimension zero, it must be an AH-algebra without dimension growth. Two classes of AH-algebras are shown to have mean dimension zero: the class of AH-algebras with at most countably many extremal traces, and the class of AH-algebras with numbers of extreme traces which induce same state on the K0-group being uniformly bounded (in particular, this class includes AH-algebras with real rank zero)

Lifting KK-elements, asymptotical unitary equivalence and classification of simple C*-algebras

Let $A$ and $C$ be two unital simple C*-algebas with tracial rank zero. Suppose that $C$ is amenable and satisfies the Universal Coefficient Theorem. Denote by ${{KK}}_e(C,A)^{++}$ the set of those $\kappa$ for which $\kappa(K_0(C)_+\setminus\{0\})\subset K_0(A)_+\setminus\{0\}$ and $\kappa([1_C])=[1_A]$. Suppose that $\kappa\in {KK}_e(C,A)^{++}.$ We show that there is a unital monomorphism $\phi: C\to A$ such that $[\phi]=\kappa.$ Suppose that $C$ is a unital AH-algebra and $\lambda: \mathrm{T}(A)\to \mathrm{T}_{\mathtt{f}}(C)$ is a continuous affine map for which $\tau(\kappa([p]))=\lambda(\tau)(p)$ for all projections $p$ in all matrix algebras of $C$ and any $\tau\in \mathrm{T}(A),$ where $\mathrm{T}(A)$ is the simplex of tracial states of $A$ and $\mathrm{T}_{\mathtt{f}}(C)$ is the convex set of faithful tracial states of $C.$ We prove that there is a unital monomorphism $\phi: C\to A$ such that $\phi$ induces both $\kappa$ and $\lambda.$ Suppose that $h: C\to A$ is a unital monomorphism and \gamma \in \mathrm{Hom}(\Kone(C), \aff(A)). We show that there exists a unital monomorphism $\phi: C\to A$ such that $[\phi]=[h]$ in ${KK}(C,A),$ $\tau\circ \phi=\tau\circ h$ for all tracial states $\tau$ and the associated rotation map can be given by $\gamma.$ Applications to classification of simple C*-algebras are also given.Comment: The new version made a correction and removed a number of typo

Homomorphisms into a simple Z-stable C*-Algebras

Let $A$ and $B$ be unital separable simple amenable \CA s which satisfy the Universal Coefficient Theorem. Suppose {that} $A$ and $B$ are $\mathcal Z$-stable and are of rationally tracial rank no more than one. We prove the following: Suppose that $\phi, \psi: A\to B$ are unital {monomorphisms}. There exists a sequence of unitaries $\{u_n\}\subset B$ such that \lim_{n\to\infty} u_n^*\phi(a) u_n=\psi(a)\tforal a\in A, if and only if [\phi]=[\psi]\,\,\,\text{in}\,\,\, KL(A,B), \phi_{\sharp}=\psi_{\sharp}\andeqn\phi^{\ddag}=\psi^{\ddag}, where \phi_{\sharp}, \psi_{\sharp}: \aff(T(A))\to \aff(T(B)) and $\phi^{\ddag}, \psi^{\ddag}: U(A)/CU(A)\to U(B)/CU(B)$ are {the} induced maps and where $T(A)$ and $T(B)$ are tracial state spaces of $A$ and $B,$ and $CU(A)$ and $CU(B)$ are closure of {commutator} subgroups of unitary groups of $A$ and $B,$ respectively. We also show that this holds for some AH-algebras $A.$ {Moreover, if $\kappa\in KL(A,B)$ preserves the order and the identity, \lambda: \aff(\tr(A))\to \aff(\tr(B)) is a continuous affine map and $\gamma: U(A)/CU(A)\to U(B)/CU(B)$ is a \hm\, which are compatible, we also show that there is a unital \hm\, $\phi: A\to B$ so that $([\phi],\phi_{\sharp},\phi^{\ddag})=(\kappa, \lambda, \gamma),$ at least in the case that $K_1(A)$ is a free group,Comment: The revision improves the original result. It is now 47 page

Asymptotic unitary equivalence in C∗C^*-algebras

Let $C=C(X)$ be the unital $C^*$-algebra of all continuous functions on a finite CW complex $X$ and let $A$ be a unital simple $C^*$-algebra with tracial rank at most one. We show that two unital monomorphisms $\phi, \psi: C\to A$ are asymptotically unitarily equivalent, i.e., there exists a continuous path of unitaries $\{u_t: t\in [0,1)\}\subset A$ such that $$\lim_{t\to 1} u_t^*\phi(f)u_t=\psi(f) {\rm for all} \in C(X),$$ if and only if \beq [\phi]&=&[\psi] {\rm in} KK(C, A), \tau\circ \phi&=&\tau\circ \psi {\rm for all} \tau\in T(A), and \phi^{\dag}&=&\psi^{\dag}, \eneq where $T(A)$ is the simplex of tracial states of $A$ and $\phi^{\dag}, \psi^{\dag}: U(M_{\infty}(C))/DU(M_{\infty}(C))\to$ $U(M_{\infty}(A))/DU(M_{\infty}(A))$ are induced homomorphisms and where $U(M_{\infty}(A))$ and $U(M_{\infty}(C))$ are groups of union of unitary groups of $M_k(A)$ and $M_k(C)$ for all integer $k\ge 1,$ $DU(M_{\infty}(A))$ and $DU(M_{\infty}(C))$ are commutator subgroups of $U(M_{\infty}(A))$ and $U(M_{\infty}(C)),$ respectively. We actually prove a more general result for the case that $C$ is any general unital AH-algebra

The C*-algebra of a minimal homeomorphism of zero mean dimension

Let $X$ be an infinite compact metrizable space, and let $\sigma: X\to X$ be a minimal homeomorphism. Suppose that $(X, \sigma)$ has zero mean topological dimension. The associated C*-algebra $A=\mathrm{C}(X)\rtimes_\sigma\mathbb Z$ is shown to absorb the Jiang-Su algebra $\mathcal Z$ tensorially, i.e., $A\cong A\otimes\mathcal Z$. This implies that $A$ is classifiable when $(X, \sigma)$ is uniquely ergodic. Moreover, without any assumption on the mean dimension, it is shown that $A\otimes A$ always absorbs the Jiang-Su algebra.Comment: Typos are correcte

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

Let $A$ be a unital simple separable C*-algebra satisfying the UCT. Assume that $\mathrm{dr}(A)<+\infty$, $A$ is Jiang-Su stable, and $\mathrm{K}_0(A)\otimes \mathbb{Q}\cong \mathbb{Q}$. Then $A$ is an ASH algebra (indeed, $A$ is a rationally AH algebra).Comment: 10 pages; Part I of arXiv:1507.03437; To appear in "Operator Algebras and their Applications: A Tribute to Richard V. Kadison", Contemporary Mathematics, Amer. Math. Soc., Providence, R. I., 201

Classification of finite simple amenable Z{\cal Z}-stable C∗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.Comment: 272 pages. This revision has 283 page

Z\mathcal Z-stability of C(X)⋊Γ\mathrm{C}(X)\rtimes\Gamma

Let $(X, \Gamma)$ be a free and minimal topological dynamical system, where $X$ is a separable compact Hausdorff space and $\Gamma$ is a countable infinite discrete amenable group. It is shown that if $(X, \Gamma)$ has the Uniform Rokhlin Property and Cuntz comparison of open sets, then $\mathrm{mdim}(X, \Gamma)=0$ implies that $(\mathrm{C}(X) \rtimes\Gamma)\otimes\mathcal Z \cong \mathrm{C}(X) \rtimes\Gamma$, where $\mathrm{mdim}$ is the mean dimension and $\mathcal Z$ is the Jiang-Su algebra. In particular, in this case, $\mathrm{mdim}(X, \Gamma)=0$ implies that the C*-algebra $\mathrm{C}(X) \rtimes\Gamma$ is classified by the Elliott invariant

Comparison radius and mean topological dimension: Rokhlin property, comparison of open sets, and subhomogeneous C*-algebras

Let $(X, \Gamma)$ be a free minimal dynamical system, where $X$ is a compact separable Hausdorff space and $\Gamma$ is a discrete amenable group. It is shown that, if $(X, \Gamma)$ has a version of Rokhlin property (uniform Rokhlin property) and if $\mathrm{C}(X)\rtimes\Gamma$ has a Cuntz comparison on open sets, then the comparison radius of the crossed product C*-algebra $\mathrm{C}(X) \rtimes \Gamma$ is at most half of the mean topological dimension of $(X, \Gamma)$. These two conditions are shown to be satisfied if $\Gamma = \mathbb Z$ or if $(X, \Gamma)$ is an extension of a free Cantor system and $\Gamma$ has subexponential growth. The main tools being used are Cuntz comparison of diagonal elements of a subhomogeneous C*-algebra and small subgroupoids

Comparison radius and mean topological dimension: Zd\mathbb{Z}^d-actions

Consider a minimal free topological dynamical system $(X, T, \mathbb{Z}^d)$. It is shown that the comparison radius of the crossed product C*-algebra $\mathrm{C}(X) \rtimes \mathbb{Z}^d$ is at most the half of the mean topological dimension of $(X, T, \mathbb{Z}^d)$. As a consequence, the C*-algebra $\mathrm{C}(X) \rtimes \mathbb{Z}^d$ is classifiable if $(X, T, \mathbb{Z}^d)$ has zero mean dimension