Rufus Willett
(University of Hawai’i)
March 11, 2022
Organized by Americas GNCG Seminar.

Decomposable C*-algebras and the UCT

A $C^*$-algebra satisfies the UCT if it is K-theoretically the same as a commutative $C^*$-algebra, in some sense.  Whether or not all (separable) nuclear $C^*$-algebras satisfy the UCT is an important open problem; in particular, it is the last remaining ingredient needed to prove the ‘best possible’ classification result for simple nuclear $C^*$-algebras in the sense of the Elliott classification program.

We introduce a notion of a ‘decomposition’ of a $C^*$-algebra over a class of $C^*$-algebras. Roughly, this means that there are almost central elements of the $C^*$-algebra that cut it into two pieces from the class, with well-behaved intersection. Our main result shows that the class of nuclear $C^*$-algebras that satisfy the UCT is closed under decomposability.   

Decomposability introduces a natural ‘complexity hierarchy’ on the class of $C^*$-algebras: one starts with finite-dimensional $C^*$-algebras, and the ‘complexity rank’ of a $C^*$-algebra is roughly the number of decompositions one needs to get to down to the finite-dimensional level. There are interesting examples: we show that all UCT Kirchberg (i.e. purely infinite, separable, simple, unital, nuclear) $C^*$-algebras have complexity rank one or two, and characterize when each of these cases occur.  The UCT for all nuclear $C^*$-algebras thus becomes equivalent to the statement that all Kirchberg algebras have finite complexity rank.

This is based on joint work with Arturo Jaime, and with Guoliang Yu.

Share on email
Share on facebook
Share on google
Share on twitter
Share on linkedin