Generalized inductive limits and maximal stably finite quotients of C*-algebras

For any C*-algebra A, there is the smallest ideal I(A) of A such that the quotient A/I(A) is stably finite. Let phi be a *-homomorphism from a C*-algebra A to a C* -algebra B. It is obvious that phi(I(A)) subset of I(B). We denote the restriction phi to I(A) by I(phi). Then I is a functor between categories of C*-algebras. In this paper, we will consider exactness of the functor I. For this purpose, we give the definition of a generalized inductive system of a directed set of C*-algebras and some general results about such limits. (c) 2021 Elsevier Inc. All rights reserved.