Dimension reduction in $C(X, \mbox{UHF})$

Aaron Peter Tikuisis  (University of Aberdeen)

10:00 am to 11:00 am, Apr 17th, 2014   Science Building A1510

Abstract:

Viewing $C^*$-algebras as noncommutative topological spaces, it makes sense to study generalizations of dimension to this context. Stable rank, real rank, tracial rank, decomposition rank, and nuclear dimension are some of the more prominent dimension-type invariants that have been developed. A particularly interesting feature of the dimension theory of $C^*$-algebras is occurrences of dimension reduction, where the act of tensoring with certain canonical $C^*$-algebras (e.g. Cuntz' $O_2$ or $O_{\infty}$, the Jiang-Su algebra, or a UHF algebra) can have the effect of lowering the dimension.

I will discuss some results of this nature, in particular comparing the dimension of $C(X,A)$ to the dimension of $X$, for various $C^*$-algebras $A$. I will discuss a relationship between dimension reduction in $C(X,A)$ and the well-known topological fact that there is no retraction from $D^{n+1}$ to $S^n$.