Waterloo, December 8 - 11, 2017
This is joint work with Taras Kolomatski and Daniel Spivak.
There are C*-subalgebras of $B(\ell_2)$ with the following properties:
1) A nonseparable AF algebra with no nonseparable abelian C*-subalgebra,
2) An extension of $K(\ell_2(2^\omega))$ by $K(\ell_2)$ which is not stable,
3) An inductive limit of an increasing system of separable stable AF algebras which is not stable,
There are C*-algebras with the following properties, and the existence of such operator algebras in $B(\ell_2)$ is undecidable:
4) A scattered C*-algebra which is not AF,
5) An extension of an AF algebra by AF algebra which is not AF,
6) An LF algebra which is not AF,
7) An algebra which does not have $\ll$-increasing approximate unit ($a\ll b$ iff $a=ab$),
All the above algebras are scattered.
The methods are inspired by commutative combinatorial set theory (trees, almost disjoint families, gaps,
and often mix with logic.