On tracial approximation

Let C be a class of unital C*-algebras. The class TAC of C*-algebras which can be tracially approximated (in the Egorov-like sense first considered by Lin) by the C*-algebras in C is studied (Lin considered the case that C consists of finite-dimensional C*-algebras or the tensor products of such with C([0,1])). In particular, the question is considered whether, for any simple separable A∈TAC, there is a C*-algebra B which is a simple inductive limit of certain basic homogeneous C*-algebras together with C*-algebras in C, such that the Elliott invariant of A is isomorphic to the Elliott invariant of B. An interesting case of this question is answered. In the final part of the paper, the question is also considered which properties of C*-algebras are inherited by tracial approximation. (Results of this kind are obtained which are used in the proof of the main theorem of the paper, and also in the proof of the classification theorem of the second author given in [Z. Niu, A classification of tracially approximately splitting tree algebra, in preparation] and [Z. Niu, A classification of certain tracially approximately subhomogeneous C*-algebras, PhD thesis, University of Toronto, 2005]—which also uses the main result of the present paper.)