Grant: Discovery Grants Program - Individual

This is an interdisciplinary proposal at the interface between operator algebras and logic. Most important operator algebras are algebras of bounded linear operators on a complex Hilbert space closed under the formation of adjoints and the norm topology (C*-algebras) or weak operator topology (von Neumann algebras, also assumed to contain the identity operator). The study of these algebras began with the work of mathematicians Murray and von Neumann in the 1930s and has since expanded to influence much of modern mathematics including number theory, geometry, ergodic theory, mathematical physics, and topology. Set theory and model theory are two areas of mathematical logic. The former is concerned with the foundational aspect of mathematics and to some extent (but by no means exclusively) with independence results. By Gödel's Incompleteness Theorem it is impossible to give a recursive set of axioms that extends the theory of natural numbers, that decides every statement expressible in its language, and which is consistent. This implies that some statements of number theory can be neither proved nor refuted on the basis of the standard Zermelo--Fraenkel axioms of set theory (ZFC). However, Gödel's statements code metamathematical statements and are unnatural from the point of view of number theory. Parts of core mathematics are generally considered immune to set-theoretic independence and such independent statements in the field of operator algebras (e.g. the existence of an outer automorphism of the Calkin algebra or the existence of nonisomorphic central sequence algebras of a fixed separable C*-algebra associated to free ultrafilters on the set of natural numbers) were found only recently. The answers to some prominent and long-standing open problems on operator algebras may be independent from ZFC, and this is the main theme of the present proposal. The most important questions that I will work on are the following. (1) The existence of a K-theory reversing automorphism of the Calkin algebra. This is a 50 year old question of Brown, Douglas, and Fillmore. I proved that such automorphisms do not exist if one assumes forcing axioms. The question remains whether some other additional set-theoretic axiom implies the opposite statement.  (2) Toms-Winter conjecture. One scenario for confirming the remaining part of this conjecture proceeds via properties of the central sequence algebra, i.e., the relative commutant of a C*-algebra in its ultrapower. I plan to study the structure theory of this algebra. This provides a multitude of interesting short-term objectives and problems suitable for HQP training. (3) The structure theory of von Neumann algebras with not necessarily separable predual, and the question of whether every von Neumann algebra can be disintegrated into factors. The case of tracial ultrapowers of type II1 von Neumann algebras is particularly interesting and important.