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 |
References:
| [1] | Blackadar, B., K-Theory for Operator Algebras (1986), Springer-Verlag: Springer-Verlag New York · Zbl 0597.46072 |
| [2] | Blackadar, B.; Kirchberg, E., Generalized inductive limits of finite-dimensional \(C^\ast \)-algebras, Math. Ann., 307, 343-380 (1997) · Zbl 0874.46036 |
| [3] | Brown, L. G.; Dadarlat, M., Extensions of \(C^\ast \)-algebras and quasidiagonality, J. Lond. Math. Soc. (2), 53, 582-600 (1996) · Zbl 0857.46045 |
| [4] | Lin, H., An Introduction to the Classification of Amenable \(C^\ast \)-Algebras (2001), World Scientific: World Scientific New Jersey/London/Singapore/Hong Kong · Zbl 1013.46055 |
| [5] | Murphy, G. J., Diagonality in \(C^\ast \)-algebras, Math. Z., 199, 279-284 (1988) · Zbl 0658.46051 |
| [6] | Spielberg, J. S., Embedding \(C^\ast \)-algebra extensions into AF-algebras, J. Funct. Anal., 81, 325-344 (1988) · Zbl 0678.46047 |
| [7] | Yao, H., On extensions of stably finite \(C^\ast \)-algebras, J. Oper. Theory, 67, 329-334 (2012) · Zbl 1259.46049 |
| [8] | Yao, H., On extensions of stably finite \(C^\ast \)-algebras, II, Can. Math. Bull., 52, 2, 435-439 (2016) · Zbl 1379.46044 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
