arXiv · 2610.34000
We investigate the regularity properties of C*-algebras admitting a weakly selfless unital completely positive map.
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Miles Gould, Leonel Robert. 2026-10-25. Regularity and selfless cp maps. https://arxiv.org/abs/2610.34000
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We say that a countable discrete group action $α$ on a C$^*$-algebra $A$ is \textit{$\mathcal{W}$-absorbing} if there exist a C$^*$-algebra $B$ and an action $β$ on $B$ such that $α$ is cocycle conjugate to $β\otimes \mathrm{id}_{\mathcal{W}}$ on $B\otimes \mathcal{W}$ where $\mathcal{W}$ is the Razak-Jacelon algebra. In this paper, we completely classify outer $\mathcal{W}$-absorbing actions of finite groups on $\mathcal{W}$ up to conjugacy and cocycle conjugacy.
In this paper, we investigate the existence of maximal algebraic two-sided ideals in nonunital $C^*$-algebras. We formulate singular-distribution estimates in terms of lower semicontinuous $2$-quasitraces and introduce a novel criterion, which we call an admissible compact quasitracial triple set. These tools allow us to control membership in algebraic ideals and to prove the nonexistence of maximal ideals in nonunital $C^*$-algebras under a closed finite-scale face condition. This nonexistence result applies to a broad class of simple $C^*$-algebras including simple separable stable AF algebras, and nonunital, simple, separable, stably finite, $\mathcal Z$-stable $C^*$-algebras $A$ that admit an approximate identity consisting of increasing projections $(p_n)$ and for which the simplex $\operatorname{QT}_2^1(A,p_1)$ of normalized traces at $p_1$ has finitely many extreme points. Moreover, we construct a simple AF algebra whose finite-scale face is dense but not closed and which nevertheless has a maximal dense algebraic ideal. This example yields a negative resolution to a question raised by Ozawa, asking whether every maximal algebraic two-sided ideal in a $C^*$-algebra must be closed.
Let $B_\infty$ be the braid group on infinitely many strands and $P_\infty$ its pure braid group. We show that $C^*_r(P_\infty)\subseteq C^*_r(B_\infty)$ and all its intermediate inclusions are $\mathcal{Z}$-stable and completely selfless. This gives natural examples of simple nonnuclear $\mathcal{Z}$-stable $C^*$-algebras. Further examples come from $\mathrm{SL}_\infty(\mathbb{Z})$ and $\mathrm{GL}_\infty(\mathbb{Z})$. We also show that $C^*_r(\mathbb{F}_\infty)\subseteq C^*_r(B_\infty)$ is completely selfless for the free group $\mathbb{F}_\infty$ of braids in which only the first strand moves. All results hold for twisted group $C^*$-algebras. In contrast, $\mathcal{Z}$-stability fails for $B_n$ with $n$ finite and for groups that are not inner amenable.