SDU Operator Algebras Seminar

Title: Selfless Inclusions of C*-Algebras and Quantum Groups

Abstract: Recently, strong asymptotic freeness, or selflessness, in C*-algebras has emerged as a powerful technique to prove important regularity properties including simplicity, unique trace, stable rank 1, and strict comparison. In particular, in Fall 2024, Amrutam, Gao, Kunnwalkam Elayavalli, and I showed that the reduced group C*-algebras of all hyperbolic groups with trivial finite radical are selfless, which resolved the open problem of strict comparison for the reduced group C*-algebra of the free group on two generators. Since then, our result has been expanded to include a much larger class of groups. Work has also begun on isolating selflessness for C*-algebras not arising from groups, including the result of Hayes, Kunnawalkam Elayavalli, and Robert on selflessness of the reduced free product of a large class of C*-algebras (see also Flores-Klisse-Ó Cobhthaigh-Pagliero). I will introduce the general notion of a selfless inclusion of C*-algebras, with which we will see the selflessness of the reduced unitary compact matrix quantum groups. This work is joint with Ben Hayes, Srivatsav Kunnawalkam Elayavalli, and Leonel Robert.

Title: On Relative Invariant Subalgebra Rigidity Property

Abstract: A countable discrete group Γ is said to have the relative ISR-property if, for every non-trivial normal subgroup N ◁ Γ and every von Neumann subalgebra M ⊆ L(Γ) that is invariant under conjugation by N, one has M = L(K) for some subgroup K ≤ Γ. Similarly, Γ has the relative C*-ISR-property if every N-invariant unital C*-subalgebra A ⊆ C*_r(Γ) is of the form C*_r(K). We show that every torsion-free acylindrically hyperbolic group with trivial amenable radical satisfies the relative ISR-property. Moreover, we prove that all torsion-free hyperbolic groups have the relative C*-ISR-property. Furthermore, we establish an analogous relative ISR-property for irreducible lattices in higher-rank semisimple Lie groups, such as SL_d(ℤ) (d ≥ 3), with trivial center.

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