Set theory and coronas of C*-algebras

30.06.2022 15:00 - 16:30

A. Vignati (U de Paris, FR)

As abelian C*-algebras correspond functorially to locally compact Hausdorff space, studying C*-algebras is often viewed as noncommutative topology. A locally compact Hausdorff space \(X\) can be embedded densely in its Čech-Stone compactification \(\beta X\), the ‘largest compact space in which \(X\) sits densely’. Similarly, to every nonunital C*-algebras \(A\) one can associate ‘the largest unital C*-algebra in which \(A\) sit densely’, the multiplier algebra \(M(A)\). Corona C*-algebras, quotients of the form \(M(A)/A\), correspond to Čech-Stone remainders (space of the form \(\beta X \setminus X\)).

Čech-Stone remainders have been studied with set theoretical methods since the '80s. The work of Rudin, Shelah, Steprans, Velickovic, and Farah among others, showed that the structure of the space \(\beta X \setminus X\), and its autohomeomorphisms, often depend on the set theoretic axioms in play. Similar phenomenons appear when studying corona C*-algebras, as Farah's work on the Calkin algebra (the corona algebra of the compact operators) witnesses. This talk is dedicated to overview how different axioms in set theory impact the structure of automorphisms of corona C*-algebras.




HS 13, 2. OG., OMP 1