Extensions by simple C∗C^*-algebras -- Quasidiagonal extensions

Let $A$ be an amenable separable \CA and $B$ be a non-unital but $\sigma$-unital simple \CA with continuous scale. We show that two essential extensions $\tau_1$ and $\tau_2$ of $A$ by $B$ are approximately unitarily equivalent if and only if $$[\tau_1]=[\tau_2] {\rm in} KL(A, M(B)/B).$$ If $A$ is assumed to satisfy the Universal Coefficient Theorem, there is a bijection from approximate unitary equivalence classes of the above mentioned extensions to $KL(A, M(B)/B).$ Using $KL(A, M(B)/B),$ we compute exactly when an essential extension is quasidiagonal. We show that quasidiagonal extensions may not be approximately trivial. We also study the approximately trivial extensions.Comment: to appear Canad J. Mat

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

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 $\phi, \psi: C\to A$ are approximately unitarily equivalent if and only if [\phi]=[\psi] {\rm in} KL(C,A) and \tau\circ \phi=\tau\circ \psi \tforal \tau\in T(A), where $T(A)$ is the tracial state space of $A.$ In this paper we prove the following: Given $\kappa\in KL(C,A)$ with $\kappa(K_0(C)_+\setminus \{0\})\subset K_0(A)_+\setminus \{0\}$ and with $\kappa([1_C])=[1_A]$ and a continuous affine map $\lambda: T(A)\to T_{\mathtt{f}}(C)$ which is compatible with $\kappa,$ where $T_{\mathtt{f}}(C)$ is the convex set of all faithful tracial states, there exists a unital monomorphism $\phi: C\to A$ such that [\phi]=\kappa\andeqn \tau\circ \phi(c)=\lambda(\tau)(c) for all $c\in C_{s.a.}$ and $\tau\in T(A).$ Denote by ${\rm Mon}_{au}^e(C,A)$ the set of approximate unitary equivalence classes of unital monomorphisms. We provide a bijective map $$\Lambda: {\rm Mon}_{au}^e (C,A)\to KLT(C,A)^{++},$$ 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 $\kappa\in KL(C(X), A)$ with $\kappa(K_0(C(X))_+\setminus\{0\})\subset K_0(A)_+\setminus \{0\}$ for which there is no \hm $h: C(X)\to A$ so that $[h]=\kappa.

Full extensions and approximate unitary equivalences

Let $A$ be a unital separable amenable \CA and $C$ be a unital \CA with certain infinite property. We show that two full monomorphisms $h_1, h_2: A\to C$ are approximately unitarily equivalent if and only if $[h_1]=[h_2]$ in $KL(A,C).$ Let $B$ be a non-unital but $\sigma$-unital \CA for which $M(B)/B$ has the certain infinite property. We prove that two full essential extensions are approximately unitarily equivalent if and only if they induce the same element in $KL(A, M(B)/B).$ The set of approximately unitarily equivalence classes of full essential extensions forms a group. If $A$ satisfies the Universal Coefficient Theorem, it is can be identified with $KL(A, M(B)/B).

Classification of simple C*-algebras and higher dimensional noncommutative tori

We show that unital simple C*-algebras with tracial topological rank zero which are locally approximated by subhomogeneous C^-algebras can be classified by their ordered $K$-theory. We apply this classification result to show that certain simple crossed products are isomorphic if they have the same ordered $K$-theory. In particular, irrational higher dimensional non-commutative tori of the form $C({\mathbb T}^k)\times_{\theta}{\mathbb Z}$ are in fact inductive limits of circle algebras.Comment: 24 pages published versio

Furstenberg Transformations and Approximate Conjugacy

Let $\alpha$ and $\beta$ be two Furstenberg transformations on 2-torus associated with irrational numbers $\theta_1,$ $\theta_2,$ integers $d_1, d_2$ and Lipschitz functions $f_1$ and $f_2.$ We show that $\alpha$ and $\beta$ are approximately conjugate in a measure theoretical sense if (and only if) $\bar{\theta_1\pm \theta_2}=0$ in $\R/\Z.$ Closely related to the classification of simple amenable $C^*$-algebras, we show that $\alpha$ and $\beta$ are approximately $K$-conjugate if (and only if) $\bar{\theta_1\pm \theta_2}=0$ 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

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

Let $X$ be a compact metric space and let $\Lambda$ 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 $C(X)\rtimes_{\Lambda}\Z^k$ can be embedded into a unital simple AF-algebra if and only if $X$ admits a strictly positive $\Lambda$-invariant Borel probability measure. Let $C$ be a unital AH-algebra, let $G$ be a finitely generated abelian group and let $\Lambda: G\to Aut(C)$ be a monomorphism. We show that $C\rtimes_{\Lambda} G$ can be embedded into a unital simple AF-algebra if and only if $C$ admits a faithful $\Lambda$-invariant tracial state.Comment: 46 page

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 $\widehat{a}(\tau)=\tau(a).$ We show that $u$ can be approximated by unitaries in $A$ with finite spectrum if and only if $u\in CU(A)$ and $\widehat{u^n+(u^n)^*},i(\widehat{u^n-(u^n)^*})\in \overline{\rho_A(K_0(A)}$ 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

Kishimoto's Conjugacy Theorems in simple C∗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 $\sigma_n$ such that \af={\rm Ad}\, u_n\circ \sigma_n\circ \bt\circ \sigma_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

Simple C*-algebras with a unique tracial state

The oreginal paper has been withdrawn