Abstract
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 ϕ be a ⁎-homomorphism from a C⁎-algebra A to a C⁎-algebra B. It is obvious that ϕ(I(A))⊂I(B). We denote the restriction ϕ to I(A) by I(ϕ). 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.
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