I find myself needing to construct some noncommutative geometries. I want to take various (algeba-) geometric objects and look at their noncommutative analogs. Is there a constructive way to do this? ...
While analyzing convergence speed of stochastic-gradient methods for convex optimization problems, Recht et al (2011) posed a tantalizing conjecture. It seems quite tricky, so after having struggled a ...
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 computable easily for ...
What are the simplest examples of rings that are not isomorphic to their opposite rings? Is there a science to constructing them? The only simple example known to me: In Jacobson's Basic Algebra ...
The unitary reduction of normal matrices is a well-known fact: if $A\in M_n(\mathbb C)$ commutes with its Hermitian adjoint $A^*$, then there exists a unitary $U\in\mathbb U_n$ and a diagonal matrix ...
Definition 1: Suppose $B$ is a $C^* $-algebra. $A$ is massive $C^* $-subalgebra of $B$ iff 1. $A$ is a subalgebra of $B$; 2. for each irreducible representation $\pi$ of $B$ representation $\pi|_A$ is ...
Suppose we have the Hilbert space L2(Rn) and we have n operators Qi and n operators Pi defined in the usual way by: Qi ψ(q1,q2,...,qn) = qi ψ(q1,q2,...,qn) Pi ψ(q1,q2,...,qn) = -i ...
Theo's question made me wonder if there are other "noncommutative analogs" outside of operator algebras. Some noncommutative analogs from operator algebras include: A $C^\ast$-algebra is a ...
One of the reasons I study von Neumann algebras is that they always have plenty of projections. There are many projectionless $C^\ast$-algebras ($0$ and possibly $1$ are the only projections), but the ...
Suppose M is a von Neumann algebra. Denote by L its maximal noncommutative localization, i.e., the Ore localization with respect to the set of all left and right regular elements, i.e., elements whose ...
Hochschild homology gives invariants of (unital) $k$-algebras for $k$ a unital, commutative ring. If we let our algebra $A$ be the group ring $k[G]$ for $G$ a finite group, we get group homology. ...
One can easily prove that factors have no nontrivial ultraweakly closed 2-sided ideals as these are equivalent to nontrivial central projections. One can also show type $I_n$, type $II_1$, and type ...