Tracial limit of \(C^{\ast}\)-algebras
A \(C^*\)-algebra \(A\) is said to be the tracial limit of a sequence of unital \(C^*\)-algebras \(A_n\) with \(1_{A_n}=p_n\) if, for any \(\varepsilon>0\) and for any finite subset \(F\subset A\) there is an \(m\) such that for all \(n>m\) one has (i) for any \(x\in F\), \(\| [x,p_n]\| <\varepsilon\) and there is \(x'\in A_n\) such that \(\| p_nxp_n-x'\| <\varepsilon\); (ii) \(1-p_n\) is small in some sense (the authors call it `small in `measure''). This notion is related to the tracial rank. Let \(I^{(k)}\) be the class of all \(C^*\)-algebras of the form \(pM_n(C(X))p\), where \(p\in M_n(C(X))\) is a projection and \(X\) is a finite CW complex of dimension \(k\). It is shown that a separable \(C^*\)-algebra \(A\) is a tracial limit of \(C^*\)-algebras in \(I^{(k)}\) iff \(\text{TR}(A)\leq k\), where TR denotes the tracial rank.
scientific article; zbMATH DE number 2002326