On the decomposition into discrete, type II and type III C*-algebras

Chi-Keung Ng

14:00 pm to 15:00 pm, October 29th, 2014   Science Building A1510


We finish a classifying scheme of $C^*$-algebras. We show that the classes of discrete $C^*$-algebras (as defined by Peligard and Zsid\'{o}), type ${\rm I\!I}$, and type ${\rm I\!I\!I}$ $C^*$-algebras (as defined by Cuntz and Pedersen) are closed under strong Morita equivalence and taking ``essential extension''. Furthermore, there exist the largest discrete finite ideal $A_{{\rm d},1}$, the largest discrete anti-finite ideal $A_{{\rm d},\infty}$, the largest type ${\rm I\!I}$ finite ideal $A_{{\rm I\!I},1}$, the largest type ${\rm I\!I}$ anti-finite ideal $A_{{\rm I\!I},\infty}$, and the largest type ${\rm I\!I\!I}$ ideal $A_{\rm I\!I\!I}$ of a $C^*$-algebra $A$ with $A_{{\rm d},1} + A_{{\rm d},\infty} + A_{{\rm I\!I},1} + A_{{\rm I\!I},\infty} + A_{\rm I\!I\!I}$ being an essential ideal of $A$. When $A$ is a $W^*$-algebra, these ideals coincide with the largest type ${\rm I}$ finite part, type ${\rm I}$ infinite part, type ${\rm I\!I}$ finite part, type ${\rm I\!I}$ infinite part and type ${\rm I\!I\!I}$ part, respectively. Moreover, this classification scheme observes many good rules. We find that any prime $C^*$-algebra is of one of the five types: finite discrete, anti-finite discrete, finite type ${\rm I\!I}$, anti-finite type ${\rm I\!I}$ or type ${\rm I\!I\!I}$. If $A$ has a Hausdorff primitive spectrum, or $A$ is an $AW^*$-algebra, or $A$ is the local multiplier algebra of another $C^*$-algebra, then $A$ is a continuous field of prime $C^*$-algebras over a locally compact Hausdorff space, with each fiber being non-zero and of one of the five types. If, in addition, $A$ is discrete (respectively, anti-finite), there is an open dense subset of $\Omega$ on which each fiber is discrete (respectively, anti-finite).

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