Abstract
This thesis deals with classification of nonsimple C-algebras of real rank zero, and whether filtered K-theory is a suitable invariant for this purpose.
As a consequence of the result of E. Kirchberg for purely infinite, nuclear
C-algebras with a finite primitive ideal space, it suffices to lift isomorphisms on filtered K-theory to ideal related KK-equivalences to achieve the desired classification result. Results by R. Meyer and R. Nest, and by R. Bentmann and M. Köhler, describe for exactly which finite primitive ideal spaces this
is possible for general C-algebras.
The main question throughout the thesis is the following: is it possible
to achieve the desired classification result for arbitrary finite primitive ideal spaces by restricting to C-algebras of real rank zero that possibly satisfy further restrictions on K-theory? The thesis consists of an account of the relevant theory and the relevant results, plus two articles.
The smallest primitive ideal spaces that do not admit classification of
general C-algebras, are six four-point spaces. In the first article (with G. Restorff and E. Ruiz), these six four-point spaces are examined, and it is shown that for four of these spaces, isomorphisms are liftable for C-algebras of real rank zero.
In the second article (with R. Bentmann and T. Katsura) it is shown that for real rank zero C-algebras whose subquotients have free K1-groups, isomorphisms are liftable also for a fifth of the spaces. In this article, the range of filtered K-theory is determined for real rank zero graph algebras over primitive ideal spaces that admit classification. As a consequence of completeness of filtered K-theory combined with this range result, one can conclude that real rank zero extensions of stabilized Cuntz-Krieger algebras are stabilized Cuntz-Krieger algebras, provided the primitive ideal space permits classification.
As a consequence of the result of E. Kirchberg for purely infinite, nuclear
C-algebras with a finite primitive ideal space, it suffices to lift isomorphisms on filtered K-theory to ideal related KK-equivalences to achieve the desired classification result. Results by R. Meyer and R. Nest, and by R. Bentmann and M. Köhler, describe for exactly which finite primitive ideal spaces this
is possible for general C-algebras.
The main question throughout the thesis is the following: is it possible
to achieve the desired classification result for arbitrary finite primitive ideal spaces by restricting to C-algebras of real rank zero that possibly satisfy further restrictions on K-theory? The thesis consists of an account of the relevant theory and the relevant results, plus two articles.
The smallest primitive ideal spaces that do not admit classification of
general C-algebras, are six four-point spaces. In the first article (with G. Restorff and E. Ruiz), these six four-point spaces are examined, and it is shown that for four of these spaces, isomorphisms are liftable for C-algebras of real rank zero.
In the second article (with R. Bentmann and T. Katsura) it is shown that for real rank zero C-algebras whose subquotients have free K1-groups, isomorphisms are liftable also for a fifth of the spaces. In this article, the range of filtered K-theory is determined for real rank zero graph algebras over primitive ideal spaces that admit classification. As a consequence of completeness of filtered K-theory combined with this range result, one can conclude that real rank zero extensions of stabilized Cuntz-Krieger algebras are stabilized Cuntz-Krieger algebras, provided the primitive ideal space permits classification.
| Original language | English |
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| Publisher | Faculty of Science, University of Copenhagen |
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| Number of pages | 98 |
| ISBN (Print) | 978-87-91927-68-3 |
| Publication status | Published - 2012 |