Noncommutative dimension and tensor products

13 May 2014
Aaron Tikuisis
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}.
  • Functional Analysis Seminar