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…
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 ),…
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…
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…
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…
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…
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 ,…
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…
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…
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…
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