Kategorie-Archive: Open Problems

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 »

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 »

Scottish Book Problem 166

Von | Oktober 7, 2020

This is one of the few problems from the Scottish book that are still open. In slightly modernized form, and correcting the typo (in the book, and should be switched in the last sentence) the problem is: Let be a topological manifold, and let be a continuous function. Let denote the subgroup of homeomorphisms that… Weiterlesen »

Scottish Book Problem 155

Von | Oktober 7, 2020

This is one of the few problems from the Scottish book that are still open. In modern terminology, the problem is: Let and be Banach spaces, and let be a bijective map with the following property: For every there exists such that for the sphere , the restriction is isometric. Does it follow that is… Weiterlesen »