Non-commutative Factor theorem for tensor products of lattices in product groups

TL;DR

This work proves a non-commutative intermediate factor theorem for crossed products arising from irreducible product lattices in higher-rank groups. It extends rigidity phenomena to the tensor-product setting by developing and combining $G_i$-continuous elements, a non-commutative Nevo–Zimmer principle, and IRA analysis to classify all intermediate von Neumann algebras as boundary-crossed products $(L^ fty(G/Q,\nu_Q)\overline{\otimes}\mathcal{N})\rtimes\Lambda$. The main result shows that any intermediate algebra between $\mathcal{N}\rtimes\Lambda$ and $(L^ fty(G/P,\nu_P)\overline{\otimes}\mathcal{N})\rtimes\Lambda$ must be of the form above for some intermediate boundary $P<Q<G$, with a distinction between the cases $d\ge 2$ and $d=1$ treated via field-based nc-IFT. This work broadens operator-algebraic rigidity to product lattices, connects boundary theory with crossed-product structure, and includes concrete instances such as a Bernoulli shift example to illustrate the constructions.

Abstract

We establish a non-commutative version of the Intermediate Factor Theorem for crossed products associated with product lattices. Given an irreducible lattice $Γ< G= G_1 \times \dots \times G_d$ in higher rank semisimple algebraic groups and a trace-preserving irreducible action $G \curvearrowright (\mathcal{N}, τ)$, we show that every intermediate von Neumann algebra between $\mathcal{N}\rtimesΓ$ and $(L^\infty(G/P,ν_P)\overline{\otimes}\mathcal{N})\rtimesΓ$ is again a crossed product of the form $(L^\infty(G/Q,ν_Q)\overline{\otimes}\mathcal{N})\rtimesΓ$.

Theorems & Definitions (26)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Definition 2.1
• Theorem 2.2
• proof
• Theorem 2.3
• proof
• Theorem 2.4
• proof