C*-selflessness and $\mathcal{Z}$-stability for Thompson's group $F$

We give a characterization of linear isometries with finite corank between two arbitrary unital C$^*$-algebras in terms of Jordan $^*$-homomorphisms and ``finite-dimensional remainders''. Also, the corresponding results are given for linear isometries and linear order embeddings between self-adjoint parts of C$^*$-algebras. In the finite-dimensional case, we give new examples of linear isometries between matrix algebras.

math.OA↗

arXiv2026

A boundary approach to rigidity of $q$-Araki--Woods factors

In this paper, we study structural and rigidity properties of $q$-Araki--Woods factors arising from strongly continuous orthogonal representations of $\R$. We first introduce an AO-type boundary property that is formulated in terms of length projections on the $q$-Fock spaces, and prove that all $q$-Araki--Woods factors satisfy this condition. We next characterize uniform convergence of the associated malleable deformations. We then use these two ingredients as a substitute for weak containment in the coarse bimodule and deduce several rigidity theorems in the framework of Popa's deformation/rigidity theory. As a main consequence, we prove that every $q$-Araki--Woods factor is stably strongly solid. We also prove a normalizer rigidity theorem for inclusions arising from direct sum representations. Finally, we compute Connes' $τ$-invariant, and construct a continuum of pairwise non-isomorphic type $\mathrm{III}_1$ $q$-Araki--Woods factors with the same $τ$-invariant.

math.OA↗

arXiv2026

Random $C^*$-subalgebras

Given a discrete countable group $Γ$, we introduce the concept of random subalgebras of the reduced group $C^*$-algebra $C_r^*(Γ)$. This extends the existing theory of invariant random subgroups and the one of invariant random subalgebras of the group von Neumann algebra ${\rm L}(Γ)$. As applications, we provide a characterisation of $C^*$-simplicity, a condition for amenability of the group and show the existence and uniqueness of a maximal nuclear $Γ$-invariant unital $C^*$-subalgebra of $C^*_r(Γ)$. Moreover, we give a topological characterisation of exactness of any separable unital $C^*$-algebra. Finally, we study exotic random subalgebras of $C^*_r(Γ)$ and super-exotic random subalgebras of ${\rm L}(Γ)$, that is, random subalgebras which do not arise purely from random subgroups of $Γ$ and not from random subalgebras of $C^*_r(Γ)$, respectively.

math.OA↗