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

Published online by Cambridge University Press:  22 August 2017

CHI–KEUNG NG  and
NGAI–CHING WONG
CHI–KEUNG NG
Affiliation:
Chern Institute of Mathematics and LPMC, Nankai University, Tianjin 300071, China. e-mail: ckng@nankai.edu.cn
NGAI–CHING WONG
Affiliation:
Department of Applied Mathematics, National Sun Yat-sen University, Kaohsiung, 80424, Taiwan. e-mail: wong@math.nsysu.edu.tw
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Abstract

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 Ad,1, the largest discrete essentially infinite ideal Ad,∞, the largest type II finite ideal AII,1, the largest type II essentially infinite ideal AII,∞, and the largest type III ideal AIII of any C*-algebra A such that Ad,1 + Ad,∞ + AII,1 + AII,∞ + AIII 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.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 165 , Issue 3 , November 2018 , pp. 475 - 509
Copyright
Copyright © Cambridge Philosophical Society 2017 

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