Inductive limits of K-theoretic complexes with torsion coefficients

具有平凡 $K_1$-群的一维 NCCW 复形归纳极限的 $C^*$-对角

$C^*$-diagonal of inductive limit of $1$-dimensional NCCW complexes

本文证明了一维 NCCW 复形在 $K_1$-群平凡时其归纳极限存在 $C^*$-对角，并分析了逼近此类复形间 $^*$-同态的局限与推论。

A decomposition theorem for real rank zero inductive limits of 1-dimensional non-commutative CW complexes

In this paper, we consider the real rank zero $\mathrm{C}^*$-algebras which can be written as an inductive limit of the Elliott-Thomsen building blocks and prove a decomposition result for the connecting homomorphisms; this technique will be used in the classification theorem.

Classification of inductive limits of 1-dimensional NCCW complexes

A classification result is obtained for the C*-algebras that are (stably isomorphic to) inductive limits of 1-dimensional noncommutative CW complexes with trivial $K_1$-group. The classifying functor Cu is defined in terms of the Cuntz semigroup of the unitization of the algebra. For the simple C*-algebras covered by the classification, Cu reduces to the ordered $K_0$-group, the cone of traces, and the pairing between them. As an application of the classification, it is shown that the crossed products by a quasi-free action $O_2\rtimes_λ\mathbb{R}$ are all isomorphic for a dense set of positiv…

Inductive limits of semiprojective C*-algebras

We prove closure properties for the class of C*-algebras that are inductive limits of semiprojective C*-algebras. Most importantly, we show that this class is closed under shape domination, and so in particular under shape and homotopy equivalence. It follows that the considered class is quite large. It contains for instance the stable suspension of any nuclear C*-algebra satisfying the UCT and with torsion-free $K_0$-group. In particular, the stabilized C*-algebra of continuous functions on the pointed sphere is isomorphic to an inductive limit of semiprojectives.

Classification of certain simple $C^*$-algebras with torsion in $K_1$

We show that the Elliott invariant is a classifying invariant for the class of $C^*$-algebras that are simple unital infinite dimensional inductive limits of sequences of finite direct sums of building blocks of the form $$ \{f\in C(\T)\otimes M_n: f(x_i)\in M_{d_i}, i=1,2,...,N\}, $$ where $x_1,x_2,...,x_N\in\T$, $d_1,d_2,...,d_N$ are integers dividing $n$, and $M_{d_i}$ is embedded unitally into $M_n$. Furthermore we prove existence and uniqueness theorems for *-homomorphisms between such algebras and we identify the range of the invariant.

On the inductive limit of direct sums of simple TAI algebras

An ATAI (or ATAF, respectively) algebra, introduced in [Jiang1] (or in [Fa] respectively) is an inductive limit $\lim\limits_{n\rightarrow\infty}(A_{n}=\bigoplus\limits_{i=1}A_{n}^{i},ϕ_{nm})$, where each $A_{n}^{i}$ is a simple separable nuclear TAI (or TAF) C*-algebra with UCT property. In [Jiang1], the second author classified all ATAI algebras by an invariant consisting orderd total K-theory and tracial state spaces of cut down algebras under an extra restriction that all element in $K_{1}(A)$ are torsion. In this paper, we remove this restriction, and obtained the classification for all A…