Decomposition rank of approximately subhomogeneous C*-algebras
Aaron Tikuisis (University of Aberdeen)
Tuesday 2nd December, 2014 16:00-17:00 Maths 416
Approximately subhomogeneous C*-algebras include a great number of
interesting C*-algebras - conceivably all stably finite, separable,
nuclear C*-algebras. By definition, they are built from subhomogeneous
C*-algebras, i.e., subalgebras of $C(X,M_n)$.
Phillips made major contributions towards understanding subhomogeneous
C*-algebras via his realisation of them as "recursive subhomogeneous
algebras". This theory allowed many arguments about subhomogeneous
C*-algebras to reduce to C*-algebras of the form $C(X,M_n)$. In
particular, since Winter and I proved that $C(X,M_n) \otimes \mathcal Z$
has finite decomposition rank, it seemed natural to expect that $A
\otimes \mathcal Z$ does as well, whenever $A$ is subhomogeneous.
This problem turned out to require more substantial techniques than
originally expected. We have solved it, and our argument involves
demonstrating that subhomogeneous algebras are approximated by certain
tractable subhomogeneous C*-algebras defined with reference to a cell
decomposition. I will say some things about this argument.
This is joint work with G. Elliott, Z. Niu, L. Santiago, and W. Winter.