Generalized inductive limits and maximal stably finite quotients of \(C^\ast \)-algebras. (English) Zbl 1485.46057
Summary: For any \(C^\ast \)-algebra \(A\), there is the smallest ideal \(I(A)\) of \(A\) such that the quotient \(A / I(A)\) is stably finite. Let \(\varphi\) be a \(^\ast \)-homomorphism from a \(C^\ast \)-algebra \(A\) to a \(C^\ast \)-algebra \(B\). It is obvious that \(\phi(I(A)) \subset I(B)\). We denote the restriction \(\varphi\) to \(I(A)\) by \(I(\phi)\). Then \(I\) is a functor between categories of \(C^\ast \)-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 \(C^\ast \)-algebras and some general results about such limits.
MSC:
46L05 | General theory of \(C^*\)-algebras |
46M40 | Inductive and projective limits in functional analysis |
46L80 | \(K\)-theory and operator algebras (including cyclic theory) |
46M15 | Categories, functors in functional analysis |
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