We know that a $C^*$-algebra $A$ has real rank zero iff every self-adjoint element of $A$ can be approximated in norm by self-adjoint element with finite spectra. My question is: …

Here's a research problem, which I think interesting. Suppose that $t$ is an invertible element in the Calkin algebra $\mathcal{Q} = \mathcal{B}(\ell_2)/\mathcal{K}(\ell_2)$ which …

This problem may be hard, so I don't expect an off-the-cuff solution. But can anyone suggest possible proof strategies? I have vectors $v_1, \ldots, v_k$ in ${\bf R}^n$. Each of t …

If I have two operators or finite-dimensional matrices $A$ and $B$, how can I quantify the amount to which they commute or don't commute? (I would consider it a big plus if it is c …

Elsewhere I asked about ultrapowers of the C*-algebra $A$ of compact operators on separable infinite-dimensional Hilbert space. My question was whether the process of taking ultrap …

Suppose $G$ is a finitely generated group, with given generating set $S={g_1, \dots, g_n}$. (Assume that if $g\in S$, then $g^{-1}\notin S$. (EDIT: Also assume that $S$ is minima …

A type $II_{1}$ factor $M$ with trace $\tau$ has Property $\Gamma$ if for every finite subset ${ x_{1}, x_{2},..., x_{n} } \subseteq M$ and each $\epsilon >0$, there is a unitary e …

Sorry if this is a naive question-- I'm trying to learn this stuff (cross-posted from http://math.stackexchange.com/questions/89248/induced-representations-of-topological-groups) …

Let $A$ be a $C^{*}$-algebra, represented on a Hilbert space $H$, and $D$ a selfadjoint unbounded operator on $H$ (note that we do not impose that $D$ have compact resolvent). Let …

A finitely presented, countable discrete group $G$ is amenable if there is a finitely additive measure $m$ on the subsets of $G \backslash${$e$} with total mass 1 and satisfying $m …

Who introduced the Strong Atiyah Conjecture? Recall that the conjecture says the following. Let $G$ be a group, $A$ a $n\times n$-matrix over ${\mathbb Z}G$. We view $A$ as a bo …

I am learning group C* algebras for my graduate research. I've known that C*(G)=C(\hat{G}), if G is abelian. What can we say if the group is not abelian? Do we have explicit descr …

I am looking for a reference concerning operator theoretical Models of $K(\mathbb{Z},3)$. Stolz-Teichner briefly say in "what is an elliptic object" that a cert …

Let $A$ be a $C^*$-algebra and $I$ be a two side closed (essential) ideal of $A$. Suppose that $p \in A\backslash I$ is a non trivial projection. Let $B=pIp$. My questions are: (1 …

Let $(M,\tau)$ be a finite von Neumann factor (in my case $M=R^\omega$, but I don't think this additional hypothesis might be useful for this particular problem) and fix a projecti …