Vorträge des SFB 1442

When do the spectra of operators vary continuously? How to show such a continuity if the operators depend on a underlying geometry and dynamics that changes? The theory of continuous fields of C*-algebras has a long history and goes back to the works of Kaplansky and Fell [5, 6, 3, 4]. Already Kaplansky [6] realized that this notion implies continuity of the spectra of self-adjoint elements with respect to the Hausdorff metric. During the talk, we review the connections to the spectra and discuss its applications. In the second part of the talk, we focus on more recent developments [2] controlling the rate of convergence where we will focus on dynamical defined operators. Tools to construct continuous fields of C*-algebras often involve some amenability condition. Also in this recent work the interplay with amenability leads to interesting observations.

Lecture in Memoriam W. Scharlau - In the early development stages of the algebraic theory of quadratic forms, the following property was discussed: two quaternion algebras over a field are said to be linked if they contain pure quaternions whose squares are equal. The celebrated Hasse-Minkowski theorem shows that any two quaternion algebras over a number field are linked. This talk will explore how far this property extends to more than two quaternion algebras over various fields. In particular, as a result of a joint work with Adam Chapman (Annals of K-theory, 2019), we will exhibit four quaternion algebras over the field of rational functions in two variables over the complex numbers that are not linked. The proof uses reduction of a system of quadratic equations modulo 2.

TBA