# Kategorie-Archive: Open Problems

## Tensor products with pure C*-algebras

Von | August 4, 2024

Question: Is the minimal tensor product of two C*-algebras pure whenever one of them is? Background: Following Winter [1], a C*-algebra is said to be pure if it has very good comparison and divisibility properties, specifically its Cuntz semigroup is almost unperforated and almost divisible. This notion is closely related to that of -stability, where… Weiterlesen »

## Commutators and square-zero elements in C*-algebras

Von | April 14, 2024

Given elements and in a ring, the element is called an (additive) commutator. Given a C*-algebra , we use to denote the additive subgroup (equivalently, the linear subspace) generated by the set of additive commutators in . Note that is not necessarily a closed subspace. Given additive subgroups and , it is customary to use… 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 »

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 »

## Realizing Cuntz classes in commutative subalgebras

Von | Juli 18, 2021

Let be a unital, simple C*-algebra of stable rank one. Does there exist a commutative sub-C*-algebra such that for every lower-semicontinuous function there exists an open subset such that for ? Here, denotes the Choquet simplex of normalized -quasitraces on (if is exact, then this is just the Choquet simplex of tracial states on ),… 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 »

## C*-algebras complemented in their biduals

Von | Dezember 20, 2020

Question: Let be a C*-algebra that is complemented in its bidual by a *-homomorphism, that is, there exists a *-homomorphism such that for all . Is a von Neumann algebra? The converse is true: Let be a von Neumann algebra. Then has a (unique) isometric predual . Let be the natural inclusion of the Banach… 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 »