Archiv der Kategorie: Open Problems

Question: Let be a -factor. Is the unitary group contractible when equipped with the strong operator topology? Update (August 2025): A positive answer to the question was recently announced: Jekel. The unitary group of a II1 factor is SOT-contractible. preprint arXiv:2508.05834 Background/Motivation: By Kuiper’s theorem, if is an infinite-dimensional Hilbert space, then the unitary group… Read More: Contractibility of unitary groups of II-1 factors »

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… Read More: Scottish Book Problem 166 »

This is one of the few problems from the Scottish book that are still open. In modern terminology, the problem is: [Correction May 2026: Corrected sphere to closed ball] Let and be Banach spaces, and let be a bijective map with the following property: For every there exists such that for the closed ball ,… Read More: Scottish Book Problem 155 »

Given groups and , let us say that the pair has property (*) if any two injective homomorphisms are conjugate if (and only if) they are pointwise conjugate. Problem 1: Describe the class of groups such that has (*) for every group . Problem 2: Describe the class of groups such that has (*) for… Read More: Conjugacy of pointwise conjugate homomorphisms »

Question 1: If is a *-subalgebra of bounded linear operators on a separable Hilbert space such that is purely infinite as a ring, is the norm-closure purely infinite as a C*-algebra? This is Problem 8.4 in ​[1]​. As noted in Problem 8.6 in ​[1]​, this is even unclear if is a unital, simple, purely infinite… Read More: Purely infinite rings and C*-algebras »

Given a C*-algebra and a closed, two-sided ideal , is the image of the normal elements in under the quotient map a closed subset of ? Equivalently, if is a sequence of normal elements in that converge to , and if each admits a normal lift in , does admit a normal lift? This question… Read More: Liftable normal elements »

Do simple, -stable, stably projectionless C*-algebras have stable rank one? Definitions: A C*-algebra is said to be -stable if it tensorially absorbs the Jiang-Su algebra , that is, . Further, a simple C*-algebra is projectionless if it contains no nonzero projections, and it is stably projectionless if is projectionless. (In the nonsimple case, one should… Read More: Simple, Z-stable, projectionless C*-algebras »

Does the Calkin algebra admit an automorphism that induces the flip on ? Background/Motivation: Let be a separable, infinite-dimensional Hilbert space. The Calkin algebra is the quotient of the bounded, linear operators on by the closed, two-sided ideal of compact operators. The important problem of whether the Calkin algebra has outer automorphisms was eventually shown… Read More: Automorphisms of the Calkin algebra »