Cartan sub-C*-algebras: existence, variety, and rigidity

Cartan subalgebras in operator algebras are objects of dynamical nature that have a long history, both in von Neumann algebras and C*-algebras. A II_1 factor can behave in many different ways, from admitting no Cartan subalgebra, to having a unique one, to having unclassifiably many (up to suitable notions of equivalence).

 

Much less is known for C*-algebras; while many C*-algebras have canonical Cartan subalgebras, these are usually far from unique even if one prescribes certain topological features, as has been established by now mainly via applications of classification theory. In this talk, we will discuss some situations showcasing the variety of Cartans that a C*-algebra may exhibit, some relevant open questions, and we shall discuss some examples, namely essential extensions of C(S^1) by the compacts, where a form of rigidity occurs, in the sense that all their Cartan subalgebras with spectrum the one point compactification of the naturals can be described.

 

The talk is based on joint work with Wilhelm Winter, and joint work (in progress) with Philipp Sibbel.