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# Classification of Certain Simple $C^*$-Algebras with Torsion in $K_1$

Published:2001-12-01
Printed: Dec 2001
• Jesper Mygind
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## Abstract

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 finite direct sums of building blocks of the form $$\{f \in C(\T) \otimes M_n : f(x_i) \in M_{d_i}, i = 1,2,\dots,N\},$$ where $x_1,x_2,\dots,x_N \in \T$, $d_1,d_2,\dots,d_N$ 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.
 MSC Classifications: 46L80 - $K$-theory and operator algebras (including cyclic theory) [See also 18F25, 19Kxx, 46M20, 55Rxx, 58J22] 19K14 - $K_0$ as an ordered group, traces 46L05 - General theory of $C^*$-algebras