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Zbl 1137.46035
Elliott, George A.; Niu, Zhuang
On tracial approximation.
(English)
[J] J. Funct. Anal. 254, No. 2, 396-440 (2008). ISSN 0022-1236

Let $T$ be a tree with finitely many vertices $\{v_i\}_{i=1}^n$ and let $(\bar{k}_i=\{k_{i1},\dots,k_{ij_i}\})_{i=1}^n$ be $n$ partitions of an integer $k$, with all numbers non-zero. The splitting tree algebra $S(\bar{k}_1,\dots,\bar{k}_n;T)$ is the algebra of all continuous $M_k(\Bbb C)$-valued functions $f$ on $T$ such that $f(v_i)\in M_{k_{i1}}(\Bbb C)\oplus\dots\oplus M_{k_{ij_i}}(\Bbb C)$ for all $i$. Let $\Cal S$ be a class of splitting tree algebras and let $TA\Cal S$ be the class of $C^*$-algebras that can be tracially approximated by the $C^*$-algebras in $\Cal S$. Let $A$ be a simple separable $C^*$-algebra in $TA\Cal S$. It is shown that there exists a simple inductive limit $C^*$-algebra $B$ of $C^*$-algebras in the class $\Cal S'$ consisting of $\Cal S$ together with the Gong standard homogeneous $C^*$-algebras such that the Elliott invariant of $A$ is isomorphic to the Elliott invariant of $B$.
[Vladimir M. Manuilov (Moskva)]
MSC 2000:
*46L35 Classifications and factors of C*-algebras
46L05 General theory of C*-algebras

Keywords: Elliott invariant; tracial approximation

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