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Register a new userHomotopy of unitaries in simple C*-algebras with tracial rank one
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Huaxin Lin
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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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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 $A$ 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 $$ af={rm Ad}, u_ncirc sigma_ncirc btcirc sigma_n^{-1}andeqn lim_{ntoinfty}|u_n-1|=0, $$ 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.
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.
Let $G$ be a locally compact group. It is not always the case that its reduced C*-algebra $C^*_r(G)$ admits a tracial state. We exhibit closely related necessary and sufficient conditions for the existence of such. We gain a complete answer when $G$ compactly generated. In particular for $G$ almost connected, or more generally when $C^*_r(G)$ is nuclear, the existence of a trace is equivalent to amenability. We exhibit two examples of classes of totally disconnected groups for which $C^*_r(G)$ does not admit a tracial state.
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 $$ |e^{it}u-exp(ih)|<epsilon. $$
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