Invariant von Neumann subalgebras of crossed products and tensor products
Abstract
Let Γ be a countable discrete group acting on a tracial von Neumann algebra (Q,τQ) in a trace-preserving way. We study invariant von Neumann subalgebras of QrtimesΓ and of L(Γ)⊗̅ Q, where the latter is equipped with the diagonal action. We show that, under the pointwise weak mixing assumption, an invariant subalgebra whose image under the canonical conditional expectation onto Q is scalar is necessarily contained in L(Γ). Further, under relative weak mixing and central freeness assumptions, we determine the structure of invariant subalgebras of the crossed product with non-trivial intersection with Q. For p.m.p. actions of groups with trivial amenable radical and the invariant subalgebras rigidity property, pointwise relative weak mixing gives a complete classification of invariant subalgebras in the tensor product case. We construct essentially free, prime, pointwise weakly mixing p.m.p. actions of 𝔽n for every 2≤ n≤∞, as well as non-prime examples with explicitly described factors.
Related
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BibTeX
@misc{amrutam2026invariant,
title = {{Invariant von Neumann subalgebras of crossed products and tensor products}},
author = {Tattwamasi Amrutam and Artem Dudko and Yongle Jiang and Adam Skalski},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.19023},
eprint = {2610.19023},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.19023},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Invariant von Neumann subalgebras of crossed products and tensor products
AU - Tattwamasi Amrutam
AU - Artem Dudko
AU - Yongle Jiang
AU - Adam Skalski
PY - 2026
DA - 2026/10/13
PB - arXiv
DO - 10.48550/arxiv.2610.19023
UR - https://arxiv.org/abs/2610.19023
AB - Let Γ be a countable discrete group acting on a tracial von Neumann algebra (Q,τ_Q) in a trace-preserving way. We study invariant von Neumann subalgebras of QrtimesΓ and of L(Γ)⊗̅ Q, where the latter is equipped with the diagonal action. We show that, under the pointwise weak mixing assumption, an invariant subalgebra whose image under the canonical conditional expectation onto Q is scalar is necessarily contained in L(Γ). Further, under relative weak mixing and central freeness assumptions, we determine the structure of invariant subalgebras of the crossed product with non-trivial intersection with Q. For p.m.p. actions of groups with trivial amenable radical and the invariant subalgebras rigidity property, pointwise relative weak mixing gives a complete classification of invariant subalgebras in the tensor product case. We construct essentially free, prime, pointwise weakly mixing p.m.p. actions of 𝔽_n for every 2≤ n≤∞, as well as non-prime examples with explicitly described factors.
ER -