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1 answer
204 views

A certain class of representations

Let $g$ be a non-identity element in a torsion-free amenable group, does there exist a finite-dimensional unitary representation $\pi$ with $\pi(g)\neq 1$? (The word "finite-dimensional" was ...
MSMalekan's user avatar
  • 2,118
2 votes
0 answers
96 views

Could we assume without loss of generality that all coefficients are positive?

Let $\alpha$ be an element in the group algebra $\mathbb CG$ of a torsion-free group $G$. Assume that, as an operator acting on $\ell^2(G)$, $\alpha$ is positive. Does there exist $\beta\in\mathbb CG$ ...
MSMalekan's user avatar
  • 2,118
3 votes
1 answer
388 views

Fundamental group and group measure space construction

Let $N$ be a type ${\rm II}$ factor, with trace $\tau$. Consider its fundamental group$$ \mathcal{F}(N)= \{ \tau(p)/\tau(q) \ | \ p,q \text{ non-zero finite projections in } N \text{ and } pNp \simeq ...
Sebastien Palcoux's user avatar
5 votes
1 answer
306 views

Cartan subalgebra and group measure space construction

Let $N$ be a ${\rm II}_1$ factor. A maximal abelian self-adjoint subalgebra (MASA) is a $*$-subalgebra $A \subset N$ such that $A' \cap N = A$. It is called a Cartan subalgebra if moreover $\mathcal{N}...
Sebastien Palcoux's user avatar
3 votes
0 answers
206 views

Do these limits exist?

Let $G$ be a (discrete) torsion free group with identity $e$. Recall that for an element $\alpha\in\mathbb C[G]$, the set of complex functions on $G$ with finite support, $\alpha^*\in\mathbb C[G]$ is ...
MSMalekan's user avatar
  • 2,118
4 votes
1 answer
106 views

Does the inclusion of the discretized group into itself lift to the group von Neumann algebras?

Let $G$ be a locally compact, Hausdorff and $2^{nd}$-countable group and let $G_{disc}$ be the same group with the discrete topology. We have a continuous (and bijective) homomorphism given by $$ ...
Adrián González Pérez's user avatar
0 votes
2 answers
225 views

Isomorphism theorem for subfactors?

It's about the existence of a generalization of the first isomorphism theorem for groups, for subfactors : Let $(N \subset M)$ and $(N' \subset M')$ be irreducible inclusions of hyperfinite $II_1$ ...
Sebastien Palcoux's user avatar
5 votes
0 answers
241 views

Find a lower bound for a pre-invariant $Fol(L(F_m), X_m)$

In the paper of Bannon and Ravichandran, A Folner invariant for type $\rm{II}_1$ factors, they defined an invariant $Fol(M)$ for a separable type $\rm{II}_1$ factor $M$, especially for the free group ...
Jiang's user avatar
  • 1,528
23 votes
4 answers
2k views

Are almost commuting hermitian matrices close to commuting matrices (in the 2-norm)?

I consider on $M_n(\mathbb C)$ the normalized $2$-norm, i.e. the norm given by $\|A\|_2 = \sqrt{\mathrm{Tr}(A^* A)/n}$. My question is whether a $k$-uple of hermitian matrices that are almost ...
Mikael de la Salle's user avatar