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Register a new userDistance between unitary orbits of normal elements in simple C*-algebras of real rank zero
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Let $x, y$ be two normal elements in a unital simple C*-algebra $A.$ We introduce a function $D_c(x, y)$ and show that in a unital simple AF-algebra there is a constant $1>C>0$ such that $$ Ccdot D_c(x, y)le {rm dist}({cal U}(x),{cal U}(y))le D_c(x,y), $$ where ${cal U}(x)$ and ${cal U}(y)$ are the closures of the unitary orbits of $x$ and of $y,$ respectively. We also generalize this to unital simple C*-algebras with real rank zero, stable rank one and weakly unperforated $K_0$-group. More complicated estimates are given in the presence of non-trivial $K_1$-information.
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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.
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 $kappain {KK}_e(C,A)^{++}.$ We show that there is a unital monomorphism $phi: Cto A$ 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 $tauin mathrm{T}(A),$ 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 $phi: Cto A$ such that $phi$ induces both $kappa$ and $lambda.$ Suppose that $h: Cto A$ is a unital monomorphism and $gamma in mathrm{Hom}(Kone(C), aff(A)).$ We show that there exists a unital monomorphism $phi: Cto A$ such that $[phi]=[h]$ in ${KK}(C,A),$ $taucirc phi=taucirc h$ 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.
Let $A$ be a $C^*$-algebra. Let $E$ and $F$ be Hilbert $A$-modules with $E$ being full. Suppose that $theta : Eto F$ is a linear map preserving orthogonality, i.e., $<theta(x), theta(y) > = 0$ whenever $<x, y > = 0$. We show in this article that if, in addition, $A$ has real rank zero, and $theta$ is an $A$-module map (not assumed to be bounded), then there exists a central positive multiplier $uin M(A)$ such that $<theta(x), theta(y) > = u < x, y>$ ($x,yin E$). In the case when $A$ is a standard $C^*$-algebra, or when $A$ is a $W^*$-algebra containing no finite type II direct summand, we also obtain the same conclusion with the assumption of $theta$ being an $A$-module map weakened to being a local map.
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 $phi_1, phi_2: Cto A$ are two unital monomorphisms. We show that there is a continuous path of unitaries ${u_t: tin [0, infty)}$ of $A$ such that $$ lim_{ttoinfty}u_t^*phi_1(c)u_t=phi_2(c)tforal cin C $$ if and only if $[phi_1]=[phi_2]$ in $KK(C,A),$ $phi_1^{ddag}=phi_2^{ddag},$ $(phi_1)_T=(phi_2)_T$ and a rotation related map $bar{R}_{phi_1,phi_2}$ associated with $phi_1$ and $phi_2$ 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.
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
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