Georg Lehner (FU Berlin)
The Rosenberg Conjecture through the lens of Solidification
The Rosenberg Conjecture states that for a given Banach Algebra A, the
comparison map between algebraic and topological K-theory becomes an
isomorphism on finite coefficients. The new framework of condensed
mathematics developed by Scholze et.al. allows one to frame this question
in a different way. Condensed sets are a replacement for topological
spaces that work well in their interplay with algebraic constructions. In
particular, if we view our Banach algebra A as a condensed ring, we can
equip the algebraic K-theory groups of A with the structure of condensed
abelian groups. There exists a natural notion of completion for condensed
abelian groups called solidification. The map from the K-theory of A into
its solidification recovers the comparion map from algebraic to
topological K-theory, allowing one to phrase the Rosenberg conjecture as a
topological statement about the condensed structure of the K-theory of A.
