Noncommutative Geometry Seminar

Date: March 11, 2022

Time: 2:00PM - 3:00PM

Location: ZOOM

Speaker: Rufus Willett, University of Hawai’i

Title: Decomposable C*-algebras and the UCT

Abstract: 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 -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 -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 -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.

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