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On the decomposition into Discrete, Type II and Type III C*-algebras

  • CHI–KEUNG NG (a1) and NGAI–CHING WONG (a2)

We obtained a “decomposition scheme” of C*-algebras. We show that the classes of discrete C*-algebras (as defined by Peligard and Zsidó), type II C*-algebras and type III C*-algebras (both defined by Cuntz and Pedersen) form a good framework to “classify” C*-algebras. In particular, we found that these classes are closed under strong Morita equivalence, hereditary C*-subalgebras as well as taking “essential extension” and “normal quotient”. Furthermore, there exist the largest discrete finite ideal A d,1, the largest discrete essentially infinite ideal A d,∞, the largest type II finite ideal A II,1, the largest type II essentially infinite ideal A II,∞, and the largest type III ideal A III of any C*-algebra A such that A d,1 + A d,∞ + A II,1 + A II,∞ + A III is an essential ideal of A. This “decomposition” extends the corresponding one for W*-algebras.

We also give a closer look at C*-algebras with Hausdorff primitive ideal spaces, AW*-algebras as well as local multiplier algebras of C*-algebras. We find that these algebras can be decomposed into continuous fields of prime C*-algebras over a locally compact Hausdorff space, with each fiber being non-zero and of one of the five types mentioned above.

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Mathematical Proceedings of the Cambridge Philosophical Society
  • ISSN: 0305-0041
  • EISSN: 1469-8064
  • URL: /core/journals/mathematical-proceedings-of-the-cambridge-philosophical-society
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