On the decomposition into Discrete, type II and type III $C^{\ast }$-algebras

Abstract

We obtained a "decomposition scheme" of C*-algebras. We show that the classes of discrete C*-algebras (as defined by Peligard and Zsido), 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 $A_{d,\infty }$, the largest type II finite ideal $A_{II,1}$, the largest type II essentially infinite ideal $A_{II,\infty }$, and the largest type III ideal $A_{III}$ of any C*-algebra $A$ such that $A_{d,1}+A_{d,\infty }+A_{II,1}+A_{II,\infty }+A_{III}$ is an essential ideal of $A$. This "decomposition" extends the corresponding one for $W^{\ast }$-algebras. We also give a closer look at C*-algebras with Hausdorff primitive spectrum, 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.

41 pages; we added a lot of details and some new results

Cited by in corpus (1)

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