A subgroup is normal exactly when the group C*-algebra or von Neumann algebra it generates is invariant under conjugation by the ambient group, so invariant subalgebras of the reduced group C*-algebra and of L(Gamma) — and the equivariant conditional expectations onto them — can be read as noncommutative generalisations of normal subgroups. For an irreducible lattice in a centre-free semisimple Lie group whose simple factors all have real rank at least two, no genuinely noncommutative examples exist: every invariant von Neumann subalgebra of L(Gamma) equals L(Lambda) for an ordinary normal subgroup, and every equivariant conditional expectation restricts on the representation to the indicator of a normal subgroup. The mechanism is a co-finiteness property (the commutant of the subalgebra admits an invariant ultraweakly continuous state), proved by transposing the two halves of Margulis' Normal Subgroup Theorem into operator algebras: Furstenberg boundaries and the noncommutative Poisson boundary supply the amenability half, property (T) the other. Co-finiteness lifts amenability and the Haagerup property to the ambient algebra, and the resulting noncommutative just-infiniteness is strictly stronger than the group-theoretic notion.
Kalantar and Panagopoulos study what they call the "noncommutative normal subgroups" of a discrete group: the invariant von Neumann subalgebras of the group von Neumann algebra L(Gamma), the invarian…