No Arabic abstract
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 Villadsens simple AH--algebra of stable rank $k>1.$ We also prove that the map is an isomorphism for all Blackadars 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.
We study purely atomic representations of C*-algebras associated to row-finite and source-free higher-rank graphs. We describe when purely atomic representations are unitarily equivalent and we give necessary and sufficient conditions for a purely atomic representation to be irreducible in terms of the associated projection valued measure. We also investigate the relationship between purely atomic representations, monic representations and permutative representations, and we describe when a purely atomic representation admits a decomposition consisting of permutative representations.
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.
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 $epsilon>0$ 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.