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Lexicographic Direct Sums of Elementary C*-Algebras

Published online by Cambridge University Press:  20 November 2018

Horst Behncke  and
George A. Elliott
Horst Behncke
Affiliation:
University of Osnabrück, Osnabrück, West Germany
George A. Elliott
Affiliation:
University of Toronto, Toronto, Ontario
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Besides the simple ones, there are several other kinds of C*-algebras which it has proved interesting to try to classify. For instance, a large body of results relates to the extensions of one given C*-algebra, possibly simple, by another. Extrapolating in this direction, we have considered the class of C*-algebras which can be decomposed in the strongest possible nontrivial sense in terms of their simple subquotients, and such that these simple subquotients in turn are as uncomplicated as possible.

We have found that the classification of these C*-algebras, namely, the lexicographic direct sums of elementary C*-algebras, is to a large degree tractable, and yet involves an interesting new invariant in the antiliminary case, which is the case of no minimal ideals. Even the postliminary case, which is the case that the ordered set of simple subquotients satisfies the decreasing chain condition, is not without interest as an extension of the case of finitely many simple subquotients, analysed in the earlier papers [1] and [2].

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 39 , Issue 2 , 01 April 1987 , pp. 257 - 296
Copyright
Copyright © Canadian Mathematical Society 1987

References

1. Behncke, H. and Leptin, H., C*-algebras with a two-point dual, J. Functional Analysis 10 (1972), 330–335.Google Scholar
2. Behncke, H. and Leptin, H., Classification of C*-algebras with a finite dual, J. Functional Analysis 76 (1974). 241–257.Google Scholar
3. Behncke, H., Krauss, F. and Leptin, H., C*-algebren mit geordneten Ideal Folgen, J. Functional Analysis 10 (1972), 204–211.Google Scholar
4. Bratteli, O., Inductive limits of finite dimensional C*-algebras, Trans. Amer. Math. Soc. 171 (1972), 195–234.Google Scholar
5. Bratteli, O. and Elliott, G. A., Structure spaces oj approximately finite-dimensional C*-algebras, II, J. Functional Analysis 30 (1978), 74–82.Google Scholar
6. Cuntz, J., Simple C*-algebras generated by isometrics, Comm. Math. Phys. 57 (1977), 173–185.Google Scholar
7. Dixmier, J., Les C*-algèbres et leurs représentations, 2e édition (Gauthier-Villars, Paris, 1969).Google Scholar
8. Dixmier, J., On some C*-algebras considered by Glimm, J. Functional Analysis 7 (1967), 182–203.Google Scholar
9. Elliott, G. A., On the classification of inductive limits of sequences of semisimple finite-dimensional algebras, J. Algebra 38 (1976), 29–44.Google Scholar
10. Elliott, G. A., On totally ordered groups, and K0 , Ring Theory Waterloo (1978), 1–49; Lecture Notes in Mathematics 734 (Springer-Verlag, New York, 1979).Google Scholar
11. Elliott, G. A., Some simple C*-algebras constructed as crossed products with discrete outer automorphism groups, Publ. Res. Inst. Math. Sci. 16 (1980), 299–311.Google Scholar
12. Euchs, L., Riesz groups, Ann. Scuola Norm. Pisa 19 (1965), 1–34.Google Scholar
13. Glimm, J., On a certain class of operator algebras, Trans. Amer. Math. Soc. 95 (1960), 318–340.Google Scholar
14. Goodearl, K. R. and Handelman, D. E., Stenosis of dimension groups and A F C*-algebras, J. Reine Angew. Math. 332 (1982), 1–99.Google Scholar
15. Handelman, D. E., Extensions for A F C*-algebras and dimension groups. Trans. Amer. Math. Soc. 271 (1982), 537–573.Google Scholar
16. Jensen, H. E., Scattered C*-algebras with almost finite spectrum, J. Functional Analysis 50 (1983), 127–132.Google Scholar
17. Pedersen, G. K., Measure theory for C*-algebras II, Math. Scand. 22 (1968), 63–74.Google Scholar