Infinite number of non-isomorphic von Neumann algebras with property Gamma?

A II$_1$ factor $\mathcal M$ with trace $\tau$ has property Gamma if for every $\epsilon > 0$ and finite set $\{x_1,\cdots, x_n\} \subset \mathcal M$ there exists a trace 0 unitary element $y\in\mathcal M$ such that $\sum_{i=1}^n \|x_iy – yx_i\|_2 < \epsilon$, where the 2-norm is $\|a\|_2 = \tau(a^*a)^{1/2}$. Murray and von Neumann defined this … Read more

Connectivity of the group of invertible elements of C(S2)⊗AC(S^{2})\otimes A

For what type of C∗ algebras A, the group of invertible elements of C(S2)⊗A is a connected group? All finite dimensional A satisfy this property. Is it true to say that A satisfy the above property if and only if the group of invertible elements of A is connected? Answer AttributionSource : Link , Question … Read more

The weak-star closure of closed left ideals corresponding to pure states

I asked this question at math.stackexchange and received no comment. Let A be a C*-algebra and ϕ be a positive linear functional on A. Let ˜ϕ be its unique w∗-continuous extension on A∗∗. Let us put Nϕ:={a∈A:ϕ(a∗a)=0}   ,   N˜ϕ:={x∈A∗∗:ϕ(x∗x)=0} Nϕ forms a closed left ideal in A and N˜ϕ forms a w∗-closed left ideal in A∗∗. It … Read more

Quantization of $S^2$ as $C^*$-algebra?

The general context for the question – is belief that quantization of compact symplectic manifolds can be endowed with the structure of $C^*$-algebra (see MO230695). The particular question is about the most simple example – sphere $S^2$ with the standard symplectic form. I think that corresponding $C^*$-algebra can be defined explicitly by generators and relations. … Read more

A strongly open set which is not measurable in the weak operator topology

Let H be a non-separable Hilbert space and {ei}i∈I be an orthonormal basis for H. Let J be a uncountable proper subset in I. Let us put E={x∈B(H):‖ One may check that E is an open set in the strong operator topology but not in the weak operator topology. Question1: I feel E is not … Read more

A continuous functional calculus on/positive elements in a Fréchet algebra?

I am trying to understand what (minimal) conditions one would need in order to obtain a functional calculus on a Fréchet algebra, which we demand to be equipped with an involution that leaves all semi-norms invariant. The motivation behind this is that I need to see if forcing a Fréchet algebra to have positive elements … Read more

Baum Connes Conjecture [closed]

Closed. This question needs details or clarity. It is not currently accepting answers. Want to improve this question? Add details and clarify the problem by editing this post. Closed 5 years ago. Improve this question I have recently decided on a topic for my master thesis. I want to compare the Baum Connes conjecture as … Read more

Coming up with a represenation for sum of functions in the Fourier algebra

This is my first overflow question, so let me apologize in advance if this belongs on Let G be a discrete group. Let λ:G→B(ℓ2(G)) be the left regular representation of G and A(G) the Fourier algebra of G. Given f,g∈A(G), and writing f(s)=⟨λ(s)x,y⟩,g(s)=⟨λ(s)w,z⟩ for some choice x,y,w,z∈ℓ2(G), is there a way to choose (in … Read more

Bott-type projections in C∗C^*-algebras

Let A be a unital C∗-algebra and a∈A. If aa∗+1 is invertible in A then the element β(a)=(aa∗+I)−1(aa∗aa∗I) is an idempotent. The formula is similar to the one defining the Bott projection in the classical case. I wanted to know what does the above β mean for the K-theory of A? Does it have any … Read more

Equations in finite subgroups of unitary groups

Let $n$ be an integer. Andreas Thom mentioned that Camille Jordan showed that there exists some $m \in {\mathbb N}$ (depending on $n$), such that for any pair of $n \times n$-unitaries $u,v \in U(n)$ which generate a finite group, one has $[u^m,v^m] = 1_n$. In particular, there are two distinct words of length $\ell=2m … Read more