Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userExponential rank and exponential length for Z-stable simple C*-algebras
219
0
0.0
(
0
)
Authors
Huaxin Lin
Ask ChatGPT about the research
No Arabic abstract
Let $A$ be a unital separable simple ${cal Z}$-stable C*-algebra which has rational tracial rank at most one and let $uin U_0(A),$ the connected component of the unitary group of $A.$ We show that, for any $epsilon>0,$ there exists a self-adjoint element $hin A$ such that $$ |u-exp(ih)|<epsilon. $$ The lower bound of $|h|$ could be as large as one wants. If $uin CU(A),$ the closure of the commutator subgroup of the unitary group, we prove that there exists a self-adjoint element $hin A$ such that $$ |u-exp(ih)| <epsilon and |h|le 2pi. $$ Examples are given that the bound $2pi$ for $|h|$ is the optimal in general. For the Jiang-Su algebra ${cal Z},$ we show that, if $uin U_0({cal Z})$ and $epsilon>0,$ there exists a real number $-pi<tle pi$ and a self-adjoint element $hin {cal Z}$ with $|h|le 2pi$ such that $$ |e^{it}u-exp(ih)|<epsilon. $$
rate research
Read More
Let $epsilon>0$ be a positive number. Is there a number $delta>0$ 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 ${v(t): tin [0,1]}subset A$ such that $$ v(0)=v, v(1)=1 and |uv(t)-v(t)u|<epsilon forall tin [0,1]. $$ 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 $phi: Cto A$ is a unital monomorphism and suppose that $vin A$ is a unitary with $[v]=0$ in $K_1(A)$ such that $v$ almost commutes with $phi.$ It is shown that there is a continuous path of unitaries ${v(t): tin [0,1]}$ in $A$ with $v(0)=v$ and $v(1)=1$ such that the entire path $v(t)$ almost commutes with $phi,$ provided that an induced Bott map vanishes. Oth
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 $phi, psi: Cto A$ are two unital monomorphisms. With some restriction on $C,$ we show that $phi$ and $psi$ are approximately unitarily equivalent if and only if [phi]=[psi] in KL(C,A) taucirc phi=taucirc psi for all tracial states of A and phi^{ddag}=psi^{ddag}, here phi^{ddag} and psi^{ddag} are homomorphisms from $U(C)/CU(C)to U(A)/CU(A) induced by phi and psi, respectively, and where CU(C) and CU(A) are closures of the subgroup generated by commutators of the unitary groups of C and B.
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.
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.
We study the structure and compute the stable rank of C*-algebras of finite higher-rank graphs. We completely determine the stable rank of the C*-algebra when the k-graph either contains no cycle with an entrance, or is cofinal. We also determine exactly which finite, locally convex k-graphs yield unital stably finite C*-algebras. We give several examples to illustrate our results.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
