Classifications Of A(?)-Algebras And Extension Algebras Of AT-Algebras

In this paper, the AT-algebras are classified without the restriction of being of real rank zero. and proved that such two algebras E_i(i = 1, 2) are isomorphic if and only if their invariants (V_*(E_i),T(E_i), [1_Ei],,r_Ei (i = 1, 2) are isomorphic. The so-called AT-algebras are inductive limits of finite direct sums of matrices over the extension algeras of circle algebra by K, where K is the C~* — algebra of all compact operators on a separable infinite dimensional Hilbert space. The invariant V_*(E) is a ternary Abelian semgroup, [1] is the Murray-von Neumann equivalence class of the unit, T(E) is the tracial state space, and r_E is the connecting map.The classification of extension algebras of simple AT-algebras by stable AT-algebras with real rank zero is given.We study the properties of AT-algebras in Chapter 7. We give the sufficient and necessary condition of the quotient algebras of AT-algebras being AT-algebras, and prove the uniqueness of the stable maximal AT ideals. The relations between quotient algebras, hereditary C*-subalgebras in AT-algebras and AT-algebras and extensions of AT-algebras are also given.In general, the algebras which are classified in this paper are not simple or finite, and their stable rank maybe not equal one and their real rank also maybe not equal zero.