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- W1588735341 abstract "We obtained a 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 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 $W^*$-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." @default.
- W1588735341 created "2016-06-24" @default.
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- W1588735341 date "2013-10-21" @default.
- W1588735341 modified "2023-09-27" @default.
- W1588735341 title "On the decomposition into Discrete, type II and type III $C^*$-algebras" @default.
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