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Huaxin Lin

Publications and source records attributed to Huaxin Lin.

At least 73 records · Page 4Linked to original sources
arXiv2008

The Range of a Class of Classifiable Separable Simple Amenable C*-Algebras

We study the range of a classifiable class ${\cal A}$ of unital separable simple amenable $C^*$-algebras which satisfy the Universal Coefficient Theorem. The class ${\cal A}$ contains all unital simple AH-algebras. We show that all unital simple inductive limits of dimension drop circle $C^*$-algebras are also in the class. This unifies some of the previous known classification results for unital simple amenable $C^*$-algebras. We also show that there are many other $C^*$-algebras in the class. We prove that, for any partially ordered, simple weakly unperforated rationally Riesz group $G_0$ with order unit $u,$ any countable abelian group $G_1,$ any metrizable Choquet simplex $S,$ and any surjective affine continuous map $r: S\to S_u(G_0)$ (where $S_u(G_0)$ is the state space of $G_0$) which preserves extremal points, there exists one and only one (up to isomorphism) unital separable simple amenable $C^*$-algebra $A$ in the classifiable class ${\cal A}$ such that $$ ((K_0(A), K_0(A)_+, [1_A]), K_1(A), T(A), λ_A)=((G_0, (G_0)_+, u), G_1,S, r).

math.OA↗
arXiv2008

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 $κ$ for which $κ(K_0(C)_+\setminus\{0\})\subset K_0(A)_+\setminus\{0\}$ and $κ([1_C])=[1_A]$. Suppose that $κ\in {KK}_e(C,A)^{++}.$ We show that there is a unital monomorphism $ϕ: C\to A$ such that $[ϕ]=κ.$ Suppose that $C$ is a unital AH-algebra and $λ: \mathrm{T}(A)\to \mathrm{T}_{\mathtt{f}}(C)$ is a continuous affine map for which $τ(κ([p]))=λ(τ)(p)$ for all projections $p$ in all matrix algebras of $C$ and any $τ\in \mathrm{T}(A),$ 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 $ϕ: C\to A$ such that $ϕ$ induces both $κ$ and $λ.$ Suppose that $h: C\to A$ is a unital monomorphism and $γ\in \mathrm{Hom}(\Kone(C), \aff(A)).$ We show that there exists a unital monomorphism $ϕ: C\to A$ such that $[ϕ]=[h]$ in ${KK}(C,A),$ $τ\circ ϕ=τ\circ h$ for all tracial states $τ$ and the associated rotation map can be given by $γ.$ Applications to classification of simple C*-algebras are also given.

math.OA↗
arXiv2008

Localizing the Elliott Conjecture at Strongly Self-absorbing C*-algebras --An Appendix

This note provides some technical support to the proof of a result of W. Winter which shows that two unital separable simple amenable ${\cal Z}$-absorbing C*-algebras with locally finite decomposition property satisfying the UCT whose projections separate the traces are isomorphic if their $K$-theory is finitely generated and their Elliott invariants are the same.

math.OA↗
arXiv2008

Asymptotically unitary equivalence and asymptotically inner automorphisms

Let $C$ be a unital AH-algebra and let $A$ be a unital separable simple \CA with tracial rank zero. Suppose that $ϕ_1, ϕ_2: C\to A$ are two unital monomorphisms. We show that there is a continuous path of unitaries $\{u_t: t\in [0, \infty)\}$ of $A$ such that $$ \lim_{t\to\infty}u_t^*ϕ_1(a)u_t=ϕ_2(a)\tforal a\in C $$ if and only if $[ϕ_1]=[ϕ_2]$ in $KK(C,A),$ $τ\circ ϕ_1=τ\circ ϕ_2$ for all $τ\in T(A)$ and the rotation map ${\tildeη}_{ϕ_1,ϕ_2}$ associated with $ϕ_1$ and $ϕ_2$ is zero. In particular, an automorphism $\af$ on a unital separable simple \CA $A$ in ${\cal N}$ with tracial rank zero is asymptotically inner if and only if $$ [\af]=[{\rm id}_A] \text{in} KK(A,A) $$ and the rotation map ${\tildeη}_{ϕ_1, ϕ_2}$ is zero. Let $A$ be a unital AH-algebra (not necessarily simple) and let $\af\in Aut(A)$ be an automorphism. As an application, we show that the associated crossed product $A\rtimes_{\af}\Z$ can be embedded into a unital simple AF-algebra if and only if $A$ admits a strictly positive $\af$-invariant tracial state.

math.OA↗
arXiv2008

Approximate Homotopy of Homomorphisms from $C(X)$ into a Simple $C^*$-algebra

Let $X$ be a finite CW complex and let $h_1, h_2: C(X)\to A$ be two unital \hm s, where $A$ is a unital C*-algebra. We study the problem when $h_1$ and $h_2$ are approximately homotopic. We present a $K$-theoretical necessary and sufficient condition for them to be approximately homotopic under the assumption that $A$ is a unital separable simple C*-algebra of tracial rank zero, or $A$ is a unital purely infinite simple C*-algebra. When they are approximately homotopic, we also give a bound for the length of the homotopy. Suppose that $h: C(X)\to A$ is a monomorphism and $u\in A$ is a unitary (with $[u]=\{0\}$ in $K_1(A)$). We prove that, for any $\ep>0,$ and any compact subset ${\cal F}\subset C(X),$ there exists $\dt>0$ and a finite subset ${\cal G}\subset C(X)$ satisfying the following: if $\|[h(f), u]\|<\dt$ and $\text{Bott}(h,u)=\{0\},$ then there exists a continuous rectifiable path $\{u_t: t\in [0,1]\}$ such that $$ \|[h(g),u_t]\|<\ep, \rforal g\in {\cal F},u_0=u\andeqn u_1=1_A. $$ Moreover, $$ \text{Length}(\{u_t\})\le 2π+\ep. $$ We show that if $\text{dim}X\le 1,$ or $A$ is purely infinite simple, then $\dt$ and ${\cal G}$ are universal (independent of $A$ or $h$). In the case that ${\rm dim} X=1,$ this provides an improvement of the so-called the Basic Homotopy Lemma of Bratteli, Elliott, Evans and Kishimoto for the case that $A$ is mentioned above. Moreover, we show that $\dt$ and ${\cal G}$ can not be universal whenever $\text{dim} X\ge 2.$ Nevertheless, we also found that $\dt$ can be chosen to be dependent on a measure distribution but independent of $A$ and $h.$

math.OA↗
arXiv2008

The Range of Approximate Unitary Equivalence Classes of Homomorphisms from AH-algebras

Let $C$ be a unital AH-algebra and $A$ be a unital simple C*-algebra with tracial rank zero. It has been shown that two unital monomorphisms $ϕ, ψ: C\to A$ are approximately unitarily equivalent if and only if $$ [ϕ]=[ψ] {\rm in} KL(C,A) and τ\circ ϕ=τ\circ ψ\tforal τ\in T(A), $$ where $T(A)$ is the tracial state space of $A.$ In this paper we prove the following: Given $κ\in KL(C,A)$ with $κ(K_0(C)_+\setminus \{0\})\subset K_0(A)_+\setminus \{0\}$ and with $κ([1_C])=[1_A]$ and a continuous affine map $λ: T(A)\to T_{\mathtt{f}}(C)$ which is compatible with $κ,$ where $T_{\mathtt{f}}(C)$ is the convex set of all faithful tracial states, there exists a unital monomorphism $ϕ: C\to A$ such that $$ [ϕ]=κ\andeqn τ\circ ϕ(c)=λ(τ)(c) $$ for all $c\in C_{s.a.}$ and $τ\in T(A).$ Denote by ${\rm Mon}_{au}^e(C,A)$ the set of approximate unitary equivalence classes of unital monomorphisms. We provide a bijective map $$ Λ: {\rm Mon}_{au}^e (C,A)\to KLT(C,A)^{++}, $$ where $KLT(C,A)^{++}$ is the set of compatible pairs of elements in $KL(C,A)^{++}$ and continuous affine maps from $T(A)$ to $T_{\mathtt{f}}(C).$ Moreover, we realized that there are compact metric spaces $X$, unital simple AF-algebras $A$ and $κ\in KL(C(X), A)$ with $κ(K_0(C(X))_+\setminus\{0\})\subset K_0(A)_+\setminus \{0\}$ for which there is no \hm $h: C(X)\to A$ so that $[h]=κ.$

math.OA↗
arXiv2007

AF-embedding of the crossed products of AH-algebras by finitely generated abelian groups

Let $X$ be a compact metric space and let $Λ$ be a $\Z^k$ ($k\ge 1$) action on $X.$ We give a solution to a version of Voiculescu's problem of AF-embedding: The crossed product $C(X)\rtimes_Λ\Z^k$ can be embedded into a unital simple AF-algebra if and only if $X$ admits a strictly positive $Λ$-invariant Borel probability measure. Let $C$ be a unital AH-algebra, let $G$ be a finitely generated abelian group and let $Λ: G\to Aut(C)$ be a monomorphism. We show that $C\rtimes_Λ G$ can be embedded into a unital simple AF-algebra if and only if $C$ admits a faithful $Λ$-invariant tracial state.

math.OA↗
arXiv2007

AF-embedding of crossed products of AH-algebras by $\Z$ and asymptotic AF-embedding

Let $A$ be a unital AH-algebra and let $α\in Aut(A)$ be an automorphism. A necessary condition for $A\rtimes_α\Z$ being embedded into a unital simple AF-algebra is the existence of a faithful tracial state. If in addition, there is an automorphism $κ$ with $κ_{*1}=-{\rm id}_{K_1(A)}$ such that $α\circ κ$ and $κ\circ \af$ are asymptotically unitarily equivalent, then $A\rtimes_{\af}\Z$ can be embedded into a unital simple AF-algebra. Consequently, in the case that $A$ is a unital AH-algebra (not necessarily simple) with torsion $K_1(A),$ $A\rtimes_α\Z$ can be embedded into a unital simple AF-algebra if and only if $A$ admits a faithful $α$-invariant tracial state. We also show that if $A$ is a unital A$\T$-algebra then $A\rtimes_α\Z$ can be embedded into a unital simple AF-algebra if and only if $A$ admits a faithful $\af$-invariant tracial state. If $X$ is a compact metric space and $Λ: \Z^2\to Aut(C(X))$ is a \hm then $C(X)\rtimes_Λ\Z^2$ can be asymptotically embedded into a unital simple AF-algebra provided that $X$ admits a strictly positive $Λ$-invariant probability measure. Consequently $C(X)\rtimes_Λ\Z^2$ is quasidiagonal if $X$ admits a strictly positive $Λ$-invariant Borel probability measure.

math.OA↗
arXiv2006

Embedding Crossed Products into a Unital Simple AF-algebra

Let $X$ be a compact metric space and let $\af$ be a homeomorphism on $X.$ Related to a theorem of Pimsner, we show that $C(X)\rtimes_{\af}\Z$ can be embedded into a unital simple AF-algebra if and only if there is a strictly positive $\af$-invariant Borel probability measure. Suppose that $Λ$ is a $\Z^d$ action on $X.$ If $C(X)\rtimes_Λ\Z$ can be embedded into a unital simple AF-algebra, then there must exist a strictly positive $Λ$-invariant Borel probability measure. We show that, if in addition, there is a generator $\af_1$ of $Λ$ such that $(X, \af_1)$ is minimal and unique ergodic, then $C(X)\rtimes_Λ\Z^d$ can be embedded into a unital simple AF-algebra with a unique tracial state. Let $A$ be a unital separable amenable simple \CA with tracial rank zero and with a unique tracial state which satisfies the Universal Coefficient Theorem and let $G$ be a finitely generated discrete abelian group. Suppose $Λ: G\to Aut(A)$ is a \hm. Then $A\rtimes_Λ G$ can always be embedded into a unital simple AF-algebra.

math.OA↗
arXiv2006

Tracial Rokhlin property for automorphisms on simple $A{\mathbb T}$-algebras

Let $A$ be a unital simple $A\T$-algebra of real rank zero. Given an isomorphism $γ_1: K_1(A)\to K_1(A),$ we show that there is an automorphism $\af: A\to A$ such that $\af_{*1}=γ_1$ which has the tracial Rokhlin property. Consequently, the crossed product $A\rtimes_{\af}\Z$ is a simple unital AH-algebra with real rank zero. We also show that automorphism with Rokhlin property can be constructed from minimal homeomorphisms on a connected compact metric space.

math.OA↗
arXiv2005

Furstenberg Transformations and Approximate Conjugacy

Let $α$ and $β$ be two Furstenberg transformations on 2-torus associated with irrational numbers $θ_1,$ $θ_2,$ integers $d_1, d_2$ and Lipschitz functions $f_1$ and $f_2.$ We show that $α$ and $β$ are approximately conjugate in a measure theoretical sense if (and only if) $\bar{θ_1\pm θ_2}=0$ in $\R/\Z.$ Closely related to the classification of simple amenable $C^*$-algebras, we show that $α$ and $β$ are approximately $K$-conjugate if (and only if) $\bar{θ_1\pm θ_2}=0$ in $\R/\Z$ and $|d_1|=|d_2|.$ This is also shown to be equivalent to that the associated crossed product $C^*$-algebras are isomorphic.

math.OA↗
arXiv2005

Minimal Homeomorphisms and Approximate Conjugacy in Measure

Let X be an infinite compact metric space with finite covering dimension. Let $\afhpa,\bt: X\to X$ be two minimal homeomorphisms. Suppose that the range of $K_0$-groups of both crossed product C*-algebras s are dense in the space of real affine continuous functions. We show that $\af$ and $\bt$ are approximately conjugate uniformly in measure if and only if they have affine homeomorphic invariant probability measure spaces.

math.OA↗
arXiv2004

Minimal dynamical systems on the product of the Cantor set and the circle II

Let $X$ be the Cantor set and $ϕ$ be a minimal homeomorphism on $X\times\T$. We show that the crossed product $C^*$-algebra $C^*(X\times\T,ϕ)$ is a simple $A\T$-algebra provided that the associated cocycle takes its values in rotations on $\T$. Given two minimal systems $(X\times\T,ϕ)$ and $(Y\times\T,ψ)$ such that $ϕ$ and $ψ$ arise from cocycles with values in isometric homeomorphisms on $\T$, we show that two systems are approximately $K$-conjugate when they have the same $K$-theoretical information.

math.OA↗
arXiv2004

Classification of homomorphisms and dynamical systems

Let $A$ be a unital simple C*-algebra with tracial rank zero and $X$ be a compact metric space. Suppose that $h_1, h_2: C(X)\to A$ are two unital monomorphisms. We show that $h_1$ and $h_2$ are approximately unitarily equivalent if and only if $$ [h_1]=[h_2] {\rm in} KL(C(X),A) {\rm and} τ\circ h_1(f)=τ\circ h_2(f) $$ for every $f\in C(X)$ and every trace $τ$ of $A.$ Adopting a theorem of Tomiyama, we introduce a notion of approximate conjugacy for minimal dynamical systems. Let $X$ be a compact metric space and $α, β: X\to X$ be two minimal homeomorphisms. Using the above mentioned result, we show that two dynamical systems are approximately conjugate in that sense if and only if a $K$-theoretical condition is satisfied. In the case that $X$ is the Cantor set, this notion coincides with strong orbit equivalence of Giordano, Putnam and Skau and the $K$-theoretical condition is equivalent to saying that the associate crossed product C*-algebras are isomorphic. Another application of the above mentioned result is given for $C^*$-dynamical systems related to a problem of Kishimoto. Let $A$ be a unital simple AH-algebra with no dimension growth and with real rank zero, and let $α\in Aut(A).$ We prove that if $α^r$ fixes a large subgroup of $K_0(A)$ and has the tracial Rokhlin property then $A\rtimes_α\Z$ is again a unital simple AH-algebra with no dimension growth and with real rank zero.

math.OA↗
arXiv2004

Simple nuclear $C^*$-algebras of tracial topological rank one

We give a classification theorem for unital separable nuclear simple \CA s with tracial rank no more than one. Let $A$ and $B$ be two unital separable simple nuclear \CA s with $TR(A), TR(B)\le 1$ which satisfy the universal coefficient theorem. We show that $A\cong B$ if and only if $$ (K_0(A), K_0(A)_+, [1_A], K_1(A), T(A)) \cong (K_0(B), K_0(B)_+, [1_B], K_1(B), T(B)). $$

math.OA↗
arXiv2004

Crossed products by minimal homeomorphisms

Let X be an infinite compact metric space with finite covering dimension and let h be a minimal homeomorphism of X. Let A be the associated crossed product C*-algebra. We show that A has tracial rank zero whenever the image of K_0 (A) in the affine functions on the tracial state space of A is dense. As a consequence, we show that these crossed product C*-algebras are in fact simple AH algebras with real rank zero. When X is connected and h is further assumed to be uniquely ergodic, then the above happens if and only if the rotation number associated to h has irrational values. By applying the classification theorem for nuclear simple C*-algebras with tracial rank zero, we show that two such dynamical systems have isomorphic crossed products if and only if they have isomorphic scaled ordered K-theory.

math.OA↗
arXiv2004

Minimal dynamical systems on the product of the Cantor set and the circle

We prove that a crossed product algebra arising from a minimal dynamical system on the product of the Cantor set and the circle has real rank zero if and only if that system is rigid. In the case that cocycles take values in the rotation group, it is also shown that rigidity implies tracial rank zero, and in particular, the crossed product algebra is isomorphic to a unital simple AT-algebra of real rank zero. Under the same assumption, we show that two systems are approximately $K$-conjugate if and only if there exists a sequence of isomorphisms between two associated crossed products which approximately maps $C(X\times \T)$ onto $C(X\times \T)$.

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