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On tracial approximation. (English) Zbl 1137.46035
Let $T$ be a tree with finitely many vertices ${\left\{{v}_{i}\right\}}_{i=1}^{n}$ and let ${\left({\overline{k}}_{i}=\left\{{k}_{i1},\cdots ,{k}_{i{j}_{i}}\right\}\right)}_{i=1}^{n}$ be $n$ partitions of an integer $k$, with all numbers non-zero. The splitting tree algebra $S\left({\overline{k}}_{1},\cdots ,{\overline{k}}_{n};T\right)$ is the algebra of all continuous ${M}_{k}\left(ℂ\right)$-valued functions $f$ on $T$ such that $f\left({v}_{i}\right)\in {M}_{{k}_{i1}}\left(ℂ\right)\oplus \cdots \oplus {M}_{{k}_{i{j}_{i}}}\left(ℂ\right)$ for all $i$. Let $𝒮$ be a class of splitting tree algebras and let $TA𝒮$ be the class of ${C}^{*}$-algebras that can be tracially approximated by the ${C}^{*}$-algebras in $𝒮$. Let $A$ be a simple separable ${C}^{*}$-algebra in $TA𝒮$. It is shown that there exists a simple inductive limit ${C}^{*}$-algebra $B$ of ${C}^{*}$-algebras in the class ${𝒮}^{\text{'}}$ consisting of $𝒮$ together with the Gong standard homogeneous ${C}^{*}$-algebras such that the Elliott invariant of $A$ is isomorphic to the Elliott invariant of $B$.

##### MSC:
 46L35 Classifications of ${C}^{*}$-algebras 46L05 General theory of ${C}^{*}$-algebras
##### Keywords:
Elliott invariant; tracial approximation