Unsolved questions in $C^*$-algebras

Question

Can anyone list some unsolved questions in the context of $C^*$-algebras. With questions, I can force myself to learn a lot of the background knowledge concerning the questions, and I find it effectively change my aimless study habits.

Answer 1

One very difficult open problem with a lot of historical importance in the field which is nonetheless simple to state is the Kadison-Kaplansky conjecture.

Suppose that $\Gamma$ is an infinite discrete group and $H \subset \Gamma$ is a finite subgroup. Then, one may check that the element $p=\frac{1}{|H|} \sum_{h \in H} h$ is a projection (meaning $p=p^*=p^2$) in the group algebra $\mathbb{C}[\Gamma]$. In particular, if $\Gamma$ has a nontrivial torsion element, the reduced group C*-algebra $C^*_r(\Gamma)$ contains a nontrivial projection.

The Kadison-Kaplansky question concerns the converse: if $\Gamma$ is a discrete group with no torsion, is it possible that the reduced group C*-algebra $C^*_r(\Gamma)$ still contains some projection, other than the trivial projections $1$ and $0$?

Answer 2

Another very prominent open question is the free group factor isomorphism problem.

For a countable discrete group $\Gamma$ let $L(\Gamma)$ be the weak closure of the reduced group $C^\ast$-algebra from Mike F's answer. It is called the group von Neumann algebra and it is much bigger than $C_r^\ast(\Gamma)$ -- for example, it always contains many projections.

In fact, the group von Neumann algebra is so big that one may wonder how much of the group it remembers at all. There has been some progress in the last decades, but the most emblematic question, the free group isomorphism problem, is still widely open: Are the group algebras of all non-abelian free groups isomorphic?

Answer 3

There's the UCT problem in $KK$-theory: Given a pair of $C^*$-algebras $A,B$ with $A$ separable and $B$ $\sigma$-unital, one can define a map $KK^*(A,B)\to \operatorname{Hom}(K_*(A),K_*(B))$, and elements of the kernel correspond to elements of $\operatorname{Ext}_{\mathbb Z}^1(K_*(A),K_*(B))$. We say that the separable $C^*$-algebra $A$ is in the UCT class (UCT is short for universal coefficient theorem, due to its similarity to such results in algebraic topology) if for all separable $C^*$-algebras $B$, the following sequence $$0\to \operatorname{Ext}_{\mathbb Z}^1(K_*(A),K_*(B))\to KK^*(A,B)\to \operatorname{Hom}(K_*(A),K_*(B))\to 0$$ is exact.

The UCT problem asks whether every separable nuclear $C^*$-algebra is in the UCT class. There been some reductions made, and recently some approaches have been claimed, but the problem is still open.