arXiv subjects

Huaxin Lin

Publications and source records attributed to Huaxin Lin.

At least 55 records · Page 3Linked to original sources

arXiv2014

Minimal dynamical systems on connected odd dimensional spaces

Let $β: S^{2n+1}\to S^{2n+1}$ be a minimal homeomorphism ($n\ge 1$). We show that the crossed product $C(S^{2n+1})\rtimes_β \Z$ has rational tracial rank at most one. More generally, let $Ω$ be a connected compact metric space with finite covering dimension and with $H^1(Ω, \Z)=\{0\}.$ Suppose that $K_i(C(Ω))=\Z\oplus G_i$ for some finite abelian group $G_i,$ $i=0,1.$ Let $β: Ω\toΩ$ be a minimal homeomorphism. We also show that $A=C(Ω)\rtimes_β\Z$ has rational tracial rank at most one and is classifiable. In particular, this applies to the minimal dynamical systems on odd dimensional real projective spaces. This was done by studying the minimal homeomorphisms on $X\times Ω,$ where $X$ is the Cantor set.

math.OA↗

arXiv2014

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 Villadsen's simple AH--algebra of stable rank $k>1.$ We also prove that the map is an isomorphism for all Blackadar's 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.

math.OA↗

arXiv2014

Rotation algebras and Exel trace formula

We found that if $u$ and $v$ are any two unitaries in a unital $C^*$-algebra with $\|uv-vu\|<2$ such that $uvu^*v^*$ commutes with $u$ and $v,$ then the \SCA\, $A_{u,v}$ generated by $u$ and $v$ is isomorphic to a quotient of the rotation algebra $A_θ$ provided that $A_{u,v}$ has a unique tracial state. We also found that the Exel trace formula holds in any unital $C^*$-algebra. Let $θ\in (-1/2, 1/2)$ be a rational number. We prove the following: For any $\ep>0,$ there exists $\dt>0$ satisfying the following: if $u$ and $v$ are two unitary matrices such that $$ \|uv-e^{2πiθ}vu\|<\dt\andeqn {1\over{2πi}}τ(\log(uvu^*v^*))=θ, $$ then there exists a pair of unitary matrices $\tilde{u}$ and $\tilde{v}$ such that $$ \tilde{u}\tilde{v}=e^{2πiθ} \tilde{v}\tilde{u},\,\, \|u-\tilde{u}\|<\ep\andeqn \|v-\tilde{v}\|<\ep. $$ Furthermore, a generalization of this for all real $θ$ is obtained for unitaries in unital infinite dimensional simple $C^*$-algebras of tracial rank zero.

math.OA↗

arXiv2013

Kishimoto's Conjugacy Theorems in simple $C^*$-algebras of tracial rank one

Let $A$ be a unital separable simple amenable $C^*$-algebra with finite tracial rank which satisfies the Universal Coefficient Theorem (UCT). Suppose $\af$ and $\bt$ are two automorphisms with the Rokhlin property that {induce the same action on the $K$-theoretical data of $A$.} We show that $\af$ and $\bt$ are strongly cocycle conjugate and uniformly approximately conjugate, that is, there exists a sequence of unitaries $\{u_n\}\subset A$ and a sequence of strongly asymptotically inner automorphisms $σ_n$ such that $$ \af={\rm Ad}\, u_n\circ σ_n\circ \bt\circ σ_n^{-1}\andeqn \lim_{n\to\infty}\|u_n-1\|=0, $$ and that the converse holds. {We then give a $K$-theoretic description as to exactly when $\af$ and $\bt$ are cocycle conjugate, at least under a mild restriction. Moreover, we show that given any $K$-theoretical data, there exists an automorphism $\af$ with the Rokhlin property which has the same $K$-theoretical data.

math.OA↗

arXiv2013

Asymptotic unitary equivalence in $C^*$-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 $ϕ, ψ: C\to A$ are asymptotically unitarily equivalent, i.e., there exists a continuous path of unitaries $\{u_t: t\in [0,1)\}\subset A$ such that $$ \lim_{t\to 1} u_t^*ϕ(f)u_t=ψ(f) {\rm for all} \in C(X), $$ if and only if \beq [ϕ]&=&[ψ] {\rm in} KK(C, A), τ\circ ϕ&=&τ\circ ψ{\rm for all} τ\in T(A), and ϕ^†&=&ψ^†, \eneq where $T(A)$ is the simplex of tracial states of $A$ and $ϕ^†, ψ^†: U(M_{\infty}(C))/DU(M_{\infty}(C))\to$ $U(M_{\infty}(A))/DU(M_{\infty}(A))$ 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.

math.OA↗

arXiv2013

Exponential rank and exponential length for Z-stable simple C*-algebras

Let $A$ be a unital separable simple ${\cal Z}$-stable C*-algebra which has rational tracial rank at most one and let $u\in U_0(A),$ the connected component of the unitary group of $A.$ We show that, for any $ε>0,$ there exists a self-adjoint element $h\in A$ such that $$ |u-\exp(ih)|<ε. $$ The lower bound of $|h|$ could be as large as one wants. If $u\in CU(A),$ the closure of the commutator subgroup of the unitary group, we prove that there exists a self-adjoint element $h\in A$ such that $$ |u-\exp(ih)| <εand |h|\le 2π. $$ Examples are given that the bound $2π$ for $|h|$ is the optimal in general. For the Jiang-Su algebra ${\cal Z},$ we show that, if $u\in U_0({\cal Z})$ and $ε>0,$ there exists a real number $-π<t\le π$ and a self-adjoint element $h\in {\cal Z}$ with $|h|\le 2π$ such that $$ |e^{it}u-\exp(ih)|<ε. $$

math.OA↗

arXiv2012

On generalized universal irrational rotation algebras and the operator $u+v$

We introduce a class of generalized universal irrational rotation $C^*$-algebras $A_{θ,γ}=C^*(x,w)$ which is characterized by the relations $w^*w=ww^*=1$, $x^*x=γ(w)$, $xx^*=γ(e^{-2πiθ}w)$, and $xw=e^{-2πiθ}wx$, where $θ$ is an irrational number and $γ(z)\in C(\mathbb{T})$ is a positive function. We characterize tracial linear functionals, simplicity, and $K$-groups of $A_{θ,γ}$ in terms of zero points of $γ(z)$. We show that if $A_{θ,γ}$ is simple then $A_{θ,γ}$ is an $A{\mathbb T}$-algebra of real rank zero. We classify $A_{θ,γ}$ in terms of $θ$ and zero points of $γ(z)$. Let $A_θ=C^*(u,v)$ be the universal irrational rotation $C^*$-algebra with $vu=e^{2πiθ}uv$. Then $C^*(u+v)\cong A_{θ,|1+z|^2}$. As an application, we show that $C^*(u+v)$ is a proper simple $C^*$-subalgebra of $A_θ$ which has a unique trace, $K_1(C^*(u+v))\cong \mathbb{Z}$, and there is an order isomorphism of $K_0(C^*(u+v))$ onto $\mathbb{Z}+\mathbb{Z}θ$. {Moreover, $C^*(u+v)$ is a unital simple $A{\mathbb T}$-algebra of real rank zero.} We also calculate the spectrum and the Brown measure of $u+v$.

math.OA↗

arXiv2012

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 $ϕ, ψ: A\to B$ are unital {monomorphisms}. There exists a sequence of unitaries $\{u_n\}\subset B$ such that $$ \lim_{n\to\infty} u_n^*ϕ(a) u_n=ψ(a)\tforal a\in A, $$ if and only if $$ [ϕ]=[ψ]\,\,\,\text{in}\,\,\, KL(A,B), ϕ_{\sharp}=ψ_{\sharp}\andeqnϕ^‡=ψ^‡, $$ where $ϕ_{\sharp}, ψ_{\sharp}: \aff(T(A))\to \aff(T(B))$ and $ϕ^‡, ψ^‡: U(A)/CU(A)\to U(B)/CU(B)$ 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 $κ\in KL(A,B)$ preserves the order and the identity, $λ: \aff(\tr(A))\to \aff(\tr(B))$ is a continuous affine map and $γ: U(A)/CU(A)\to U(B)/CU(B)$ is a \hm\, which are compatible, we also show that there is a unital \hm\, $ϕ: A\to B$ so that $([ϕ],ϕ_{\sharp},ϕ^‡)=(κ, λ, γ),$ at least in the case that $K_1(A)$ is a free group,

math.OA↗

arXiv2012

On Local AH algebras

We show that every unital amenable separable simple $C^*$-algebra with finite tracial rank which satisfies the UCT has in fact tracial rank at most one. We also show that unital separable simple $C^*$-algebrass which are "tracially" locally AH with slow dimension growth are ${\cal Z}$-stable. As a consequence, unital separable simple $C^*$-algebras which are locally AH with no dimension growth are isomorphic to a unital simple AH-algebra with no dimension growth.

math.OA↗

arXiv2011

A short return to simple AH-algebras with real rank zero

Let $A$ be a unital simple AH-algebra with stable rank one and real rank zero such that $kx=0$ for all $x\in {\rm ker}ρ_A,$ the subgroup of infinitesmal elements in $K_0(A),$ and for the same integer $k\ge 1.$ We show that $A$ has tracial rank zero and is isomorphic to a unital simple AH-algebra with no dimension growth.

math.OA↗

arXiv2010

Approximate Unitary Equivalence in Simple C^*-algebras of Tracial Rank One

Let $C$ be a unital AH-algebra and let $A$ be a unital separable simple C*-algebra with tracial rank no more than one. Suppose that $ϕ, ψ: C\to A$ are two unital monomorphisms. With some restriction on $C,$ we show that $ϕ$ and $ψ$ are approximately unitarily equivalent if and only if [ϕ]=[ψ] in KL(C,A) τ\circ ϕ=τ\circ ψfor all tracial states of A and ϕ^‡=ψ^‡, here ϕ^‡ and ψ^‡ are homomorphisms from $U(C)/CU(C)\to U(A)/CU(A) induced by ϕand ψ, respectively, and where CU(C) and CU(A) are closures of the subgroup generated by commutators of the unitary groups of C and B.

math.OA↗

arXiv2010

Cuntz semigroups of C*-algebras of stable rank one and projective Hilbert modules

Let $A$ be a simple C*-algebra of stable rank one and let $p$ and $q$ be two $σ$-compact open projections. It is proved that there is a continuous path of unitaries in ${\tilde A}$ which connects open sub-projections of $p$ which is compactly contained in $p$ to those in $q.$ It is also shown that every Hilbert module is projective in the category whose morphisms are bounded module maps with adjoints. A discussion of projective Hilbert modules (whose morphisms are bounded module maps) is also given.

math.OA↗

arXiv2010

The Rohlin property for automorphisms on simple C*-algebras

We study a general Kishimoto's problem for automorphisms on simple C*-algebras with tracial rank zero. Let $A$ be a unital separable simple C*-algebra with tracial rank zero and let $α$ be an automorphism. Under the assumption that $α$ has certain Rokhlin property, we present a proof that $A\rtimes_α\Z$ has tracial rank zero. We also show that if the induced map $α_{*0}$ on $K_0(A)$ fixes a "dense" subgroup of $K_0(A)$ then the tracial Rokhlin property implies a stronger Rokhlin property. Consequently, the induced crossed product C*-algebras have tracial rank zero.

math.OA↗

arXiv2009

Approximately diagonalizing matrices over C(Y)

Let $X$ be a compact metric space which is locally absolutely retract and let $ϕ: C(X)\to C(Y, M_n)$ be a unital homomorphism, where $Y$ is a compact metric space with ${\rm dim}Y\le 2.$ It is proved that there exists a sequence of $n$ continuous maps $\alfa_{i,m}: Y\to X$ ($i=1,2,...,n$) and a sequence of sets of mutually orthogonal rank one projections $\{p_{1, m}, p_{2,m},...,p_{n,m}\}\subset C(Y, M_n)$ such that $$ \lim_{m\to\infty} \sum_{i=1}^n f(\alfa_{i,m})p_{i,m}=ϕ(f) for all f\in C(X). $$ This is closely related to the Kadison diagonal matrix question. It is also shown that this approximate diagonalization could not hold in general when ${\rm dim}Y\ge 3.$

math.OA↗

arXiv2009

Homotopy of unitaries in simple C*-algebras with tracial rank one

Let $ε>0$ be a positive number. Is there a number $δ>0$ satisfying the following? Given any pair of unitaries $u$ and $v$ in a unital simple $C^*$-algebra $A$ with $[v]=0$ in $K_1(A)$ for which $$ \|uv-vu\|<\dt, $$ there is a continuous path of unitaries $\{v(t): t\in [0,1]\}\subset A$ such that $$ v(0)=v, v(1)=1 \and \|uv(t)-v(t)u\|<ε\forall t\in [0,1]. $$ An answer is given to this question when $A$ is assumed to be a unital simple $C^*$-algebra with tracial rank no more than one. Let $C$ be a unital separable amenable simple $C^*$-algebra with tracial rank no more than one which also satisfies the UCT. Suppose that $ϕ: C\to A$ is a unital monomorphism and suppose that $v\in A$ is a unitary with $[v]=0$ in $K_1(A)$ such that $v$ almost commutes with $ϕ.$ It is shown that there is a continuous path of unitaries $\{v(t): t\in [0,1]\}$ in $A$ with $v(0)=v$ and $v(1)=1$ such that the entire path $v(t)$ almost commutes with $ϕ,$ provided that an induced Bott map vanishes. Other versions of the so-called Basic Homotopy Lemma are also presented.

math.OA↗

arXiv2009

Unitaries in a Simple C*-algebra of Tracial Rank One

Let $A$ be a unital separable simple infinite dimensional \CA with tracial rank no more than one and with the tracial state space $T(A)$ and let $U(A)$ be the unitary group of $A.$ Suppose that $u\in U_0(A),$ the connected component of $U(A)$ containing the identity. We show that, for any $\ep>0,$ there exists a selfadjoint element $h\in A_{s.a}$ such that $$ \|u-\exp(ih)\|<\ep. $$ We also study the problem when $u$ can be approximated by unitaries in $A$ with finite spectrum. Denote by $CU(A)$ the closure of the subgroup of unitary group of $U(A)$ generated by its commutators. It is known that $CU(A)\subset U_0(A).$ Denote by $\widehat{a}$ the affine function on $T(A)$ defined by $\widehat{a}(τ)=τ(a).$ We show that $u$ can be approximated by unitaries in $A$ with finite spectrum if and only if $u\in CU(A)$ and $\widehat{u^n+(u^n)^*},i(\widehat{u^n-(u^n)^*})\in \overline{ρ_A(K_0(A)}$ for all $n\ge 1.$ Examples are given that there are unitaries in $CU(A)$ which can not be approximated by unitaries with finite spectrum. Significantly these results are obtained in the absence of amenability.

math.OA↗

arXiv2009

Asymptotically Unitary Equivalence and Classification of Simple Amenable C*-algebras

Let $C$ and $A$ be two unital separable amenable simple C*-algebras with tracial rank no more than one. Suppose that $C$ satisfies the Universal Coefficient Theorem and 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 $$ \lim_{t\to\infty}u_t^*ϕ_1(c)u_t=ϕ_2(c)\tforal c\in C $$ if and only if $[ϕ_1]=[ϕ_2]$ in $KK(C,A),$ $ϕ_1^‡=ϕ_2^‡,$ $(ϕ_1)_T=(ϕ_2)_T$ and a rotation related map $\bar{R}_{ϕ_1,ϕ_2}$ associated with $ϕ_1$ and $ϕ_2$ is zero. Applying this result together with a result of W. Winter, we give a classification theorem for a class ${\cal A}$ of unital separable simple amenable \CA s which is strictly larger than the class of separable \CA s whose tracial rank are zero or one. The class contains all unital simple ASH-algebras whose state spaces of $K_0$ are the same as the tracial state spaces as well as the simple inductive limits of dimension drop circle algebras. Moreover it contains some unital simple ASH-algebras whose $K_0$-groups are not Riesz. One consequence of the main result is that all unital simple AH-algebras which are ${\cal Z}$-stable are isomorphic to ones with no dimension growth.

math.OA↗

arXiv2008

Inductive Limits of Subhomogeneous $C^*$-algebras with Hausdorff Spectrum

We consider unital simple inductive limits of generalized dimension drop C*-algebras They are so-called ASH-algebras and include all unital simple AH-algebras and all dimension drop $C^*$-algebras. Suppose that $A$ is one of these C*-algebras. We show that $A\otimes Q$ has tracial rank no more than one, where $Q$ is the rational UHF-algebra. As a consequence, we obtain the following classification result: Let $A$ and $B$ be two unital simple inductive limits of generalized dimension drop algebras with no dimension growth. Then $A\cong B$ if and only if they have the same Elliott invariant.

math.OA↗