13 problems

Let $G$) be an exponential solvable Lie group, and let ${\mathcal J}_1\subseteq{\mathcal J}_2$ be closed two-sided ideals of $C^*(G)$ such that ${\mathcal J}_2/{\mathcal J}_1$ is a…

Let $\Gamma$ be a torsion-free discrete group, and let $C^*_r(\Gamma)$ denote its reduced group $C^*$-algebra. An idempotent is nontrivial if it is neither the zero element nor the…

Let $G$ be a Lie group with maximal compact subgroup $K$, let $X=G/K$ with its proper left $G$-action, and for each highest weight $\mu$ of $K$ let $D_\mu$ be the Dirac operator as…

Real-rank-zero conjecture. Then $G$ is locally finite.

Baum–Connes' trace-range conjecture. The range of $\dim_\Gamma$ is contained in $Fin^{-1}(\Gamma)$. This was the original proposed trace-range statement; for torsion-free groups,…

Let $\Gamma$ be a torsion-free discrete group. The canonical trace on the reduced group $C^*$-algebra of $\Gamma$ induces a map on its $K$-theory, whose range is a subgroup of…

Cyclic cohomology conjecture. The cyclic cohomology of $A^{\infty}$ is given by

Let $A^{\infty}$ be the smooth subalgebra of the group $C^*$-algebra associated with the dihedral group, and let $HC^n(A^{\infty})$ denote its cyclic cohomology in degree $n$. Cycl…

Let $C^*(\mathbb{Z}[1/6])$ be regarded as a subalgebra of $C^*(\mathfrak{F}_{2,3})$, let $\operatorname{TSt}(C^*(\mathfrak{F}_{2,3}))$ denote the tracial states, call a state condi…

Simplicity–unique trace conjecture. The reduced group $C^*$-algebra $C^*_r(G)$ is simple if and only if $C^*_r(G)$ has the unique trace property.

Effros's conjecture. If $C_r^*(G)$ is an AF-algebra, then $G$ has to be locally finite.

Trace conjecture.

Let $M$ be a closed, connected, oriented and aspherical manifold with fundamental group $\pi$. Write $X$ for the universal cover of $M$ and let $C_r^*(\pi)$ denote the reduced grou…