Autor-Archive: hannesthiel

Von Neumann’s problem for II-1 factors

Von | Dezember 8, 2023

In the 1920s, von Neumann introduced the notion of amenability for groups, and he showed that a group is nonamenable whenever it contains the free group . The question of whether this characterizes (non)amenability became known as von Neumann’s problem, and it was finally answered negatively by Olshanskii in 1980: There exist nonamenable groups that… Weiterlesen »

Nonseparable, exact C*-algebras

Von | Dezember 8, 2023

It is known that every nuclear C*-algebra is exact, and that exactness passes to sub-C*-algebras. It follows that every sub-C*-algebra of a nuclear C*-algebra is exact. For separable C*-algebras, the converse holds. In fact, Kirchberg’s -embedding theorem shows that a separable C*-algebra is exact if and only if it embeds into the Cuntz algebra (which… Weiterlesen »

Traces on purely infinite C*-algebras

Von | Oktober 26, 2022

Does there exist a purely infinite C*-algebra that admits a tracial weight taking a finite, nonzero value? Here a C*-algebra is purely infinite if every element is properly infinite ( is Cuntz subequivalent to in ). This notion was introduced and studied by Kirchberg-Rørdam in ​[1]​ and ​[2]​. Further, a weight on a C*-algebra is… Weiterlesen »

Commutators in factors

Von | Oktober 8, 2022

Question: Is every element of trace zero in a factor a commutator? Background on commutators: It is a classical question to determine the elements in an algebra (over a field ) that are commutators, that is, of the form for some . A related (and often easier) question is to determine the commutator subspace ,… Weiterlesen »

The Blackadar-Handelman conjectures

Von | November 1, 2021

In ​[1]​, Blackadar and Handelman made two conjectures: Conjecture 1: (Below Theorem I.2.4 in ​[1]​) Let be a unital C*-algebra. Then the set of lower-semicontinuous dimension functions is dense in the set of dimension functions. Conjecture 2: (Below Theorem II.4.4 in ​[1]​) Let be a unital C*-algebra. Then the compact, convex set is a Choquet… Weiterlesen »

Inductive limits of semiprojective C*-algebras

Von | Dezember 20, 2020

Question: Is every separable C*-algebra an inductive limit of semiprojective C*-algebras? This question was first raised by Blackadar in ​[1]​. If we think of C*-algebras as noncommutative topological spaces, then semiprojective C*-algebras are noncommutative absolute neighborhood retracts (ANRs). It is a classical result from shape theory that every metrizable space is homeomorphic to an inverse… Weiterlesen »

Contractibility of unitary groups of II-1 factors

Von | Oktober 7, 2020

Question: Let be a -factor. Is the unitary group contractible when equipped with the strong operator topology? Background/Motivation: By Kuiper’s theorem, if is an infinite-dimensional Hilbert space, then the unitary group of the -factor is contractible in the norm topology. This was generalized by Breuer ​[1]​ to (certain) and von Neumann algebras, and eventually Brüning-Willgerodt… Weiterlesen »