Inspired largely by the fact that commutative C*-algebras correspond to
(locally compact Hausdorff) topological spaces, C*-algebras are often
viewed as noncommutative topological spaces. In particular, this
perspective has inspired notions of noncommutative dimension: numerical
isomorphism invariants for C*-algebras, whose value at C(X) is equal to
the dimension of X. This talk will focus on certain recent notions of
dimension, especially decomposition rank as defined by Kirchberg and Winter.
A particularly interesting part of the dimension theory of C*-algebras
is occurrences of dimension reduction, where the act of tensoring
certain canonical C*-algebras (e.g. UHF algebras, Cuntz' algebras O_2
and O_infinity) can have the effect of (drastically) lowering the
dimension. This is in sharp contrast to the commutative case, where
taking a tensor product always increases 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 explain a relationship between dimension reduction in C(X,A) and
the well-known topological fact that S^n is not a retract of D^{n+1}.