Classification of direct limits of generalized Toeplitz algebras

Abstract

Let A be an algebra in the class. The invariant consists of the abelian semigroup V (A), the Murry-von Neumann equivalence classes of projections in matrices of A, an abelian semigroup k(A)+, some equivalence classes of homotopy classes of hyponormal partial isometries in matrices of A and a homomorphism d from k(A)+ into V (A). The main result of this paper states that the above invariant, together with the class of the identity, is complete for the class of C∗-algebras that we consider (cf. §5). For the subclass of the above class which consists of direct limits of finite direct sums of matrix algebras over only non-trivial extensions of the above type, the invariants are even simplier—just V (A). We also show that the algebras in the class exhaust all possible invariants (cf. 3.7). Our paper can be viewed as part of the program of classifying “amenable” C∗-algebras initiated by George A. Elliott. The classical model for the program is the classification of AF -algebras by their dimension groups [Ell1]. Elliott proved in [Ell2] that the class of real rank zero AT-algebras, direct limits of finite direct sums of matrix algebras over C(S) of real rank zero, could be classified by the graded group K0⊕K1 with its natural order. Since then a number of classification results appeared ([BEEK], [D1], [D2], [DL1], [DL2], [Ell3], [Ell4], [Ell5], [EE], [EG1], [EG2], [EGL], [EGLP],