Every primitive ideal of a C*-algebra is closed and prime. A long-standing problem of Dixmier asked whether the converse holds, and this was settled by Weaver in 2003, when he gave the first example [1] of a prime C*-algebra that is not primitive. Between primitive ideals and closed prime ideals lies the class of factor…
Question: Is pure if (and only if) is non-amenable? Here, denotes the reduced group C*-algebra of a discrete group . A C*-algebra is said to be \emph{pure} if it is Jiang-Su stable at the level of Cuntz semigroups, in the sense that Equivalently, the Cuntz semigroup is almost unperforated and almost divisible (see…
Introduction: The study of automatic continuity for homomorphisms (multiplicative, linear maps) between Banach algebras has a long history. In 1960, Bade and Curtis [1] proved the existence of discontinuous homomorphisms between commutative Banach algebras in ZFC, showing that automatic continuity results can only be achieved by imposing additional hypotheses on the algebras , or on…
Question: Is the minimal tensor product of two C*-algebras pure whenever one of them is? Update (February 2026): A partial positive answer has been given by Seth and Vilalta [6]. In particular, is pure whenever is simple and pure and is an ASH algebra. Background: Following Winter [1], a C*-algebra is said to be pure…
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…
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…
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…
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…
Question: Is every element of trace zero in a factor a commutator? Update (August 2024): A positive answer to the questions was recently given In the following article: S. Wen, J. Fang, Z. Yao. A stronger version of Dixmier’s averaging theorem and some applications. J. Funct. Anal. 287 (2024). Background on commutators: It is a…
Surprisingly, the following question is open: Question: Are maximal ideals in C*-algebras closed? Let be a C*-algebra. By an ideal in we mean a two-sided ideal that is not necessarily closed. We say that is a maximal ideal if the only ideals satisfying are and . An ideal is proper if . If is unital,…
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