On invariant subalgebras of noncommutative Poisson boundaries for higher rank lattices

Abstract: Let $G$ be a real connected semisimple Lie group with trivial center, no non-trivial compact factors, and all simple factors of real rank at least two. Let $Γ<G$ be an irreducible lattice, let $P<G$ be a minimal parabolic subgroup, and consider the crossed product $L^{\infty(G/P,ν_P)\rtimes} Γ$. We prove that every $Γ$-invariant von Neumann subalgebra of $L^{\infty(G/P,ν_P)\rtimes} Γ$ is of the form $L^{\infty(G/Q,ν_Q)\rtimes} Λ$, where $P\leq Q\leq G$ and $Λ\lhdΓ$. This confirms a conjecture of Amrutam--Hartman.

• The paper confirms that each invariant von Neumann subalgebra is given by a crossed product L∞(G/Q, ν_Q) ⋊ Λ, with Q parabolic and Λ normal in Γ.
• It employs double metric ergodicity and slice decomposition to rigorously classify the structure of noncommutative Poisson boundaries.
• The findings advance operator algebraic rigidity by connecting parabolic and normal subgroup data with boundary actions for higher rank lattices.

Invariant Subalgebras of Noncommutative Poisson Boundaries for Higher Rank Lattices

Introduction and Background

The classification of invariant subalgebras of operator algebras arising from group actions lies at the intersection of operator algebras, ergodic theory, and rigidity in group theory. In the context of higher rank semisimple Lie groups and their lattices, Poisson boundaries are fundamental spaces capturing harmonic and boundary phenomena for random walks. For a real connected semisimple Lie group $G$ (trivial center, no nontrivial compact factors, all simple factors of real rank $\geq 2$), $\Gamma < G$ an irreducible lattice, and $P < G$ a minimal parabolic, the noncommutative Poisson boundary is realized as the crossed product $L^\infty(G/P, \nu_P) \rtimes \Gamma$, where $\nu_P$ is the unique $K$-invariant probability measure on $G/P$ with $K$ a maximal compact subgroup of $G$.

Previous works established that in various settings, invariant subalgebras of such crossed product von Neumann algebras can be understood in terms of subgroups and factors arising from the structure of $\geq 2$0 and $\geq 2$1—for example, in [KP23], it was shown that $\geq 2$2-invariant von Neumann subalgebras of $\geq 2$3 are the group von Neumann algebras of normal subgroups of $\geq 2$4.

However, a key conjecture by Amrutam–Hartman [AH24] posited an explicit complete structure theorem for all $\geq 2$5-invariant subalgebras of $\geq 2$6 in the higher rank case, namely that every such subalgebra is itself a crossed product $\geq 2$7 with $\geq 2$8 parabolic, $\geq 2$9 normal. This paper (2607.02450) confirms the conjecture and provides new proofs of essential intermediate structure theorems, leveraging double metric ergodicity and finer dualities.

Main Results

Structure Theorem for Invariant Von Neumann Subalgebras

The central result of the paper proves:

Let $\Gamma < G$0, $\Gamma < G$1, $\Gamma < G$2 be as above, and $\Gamma < G$3 a $\Gamma < G$4-invariant von Neumann subalgebra. Then there exist a parabolic $\Gamma < G$5 with $\Gamma < G$6 and a normal subgroup $\Gamma < G$7 such that $\Gamma < G$8

This settles the conjecture of [AH24]. The proof proceeds by considering the image $\Gamma < G$9 of $P < G$0 under the canonical $P < G$1-equivariant conditional expectation $P < G$2 onto $P < G$3, splitting into two exclusive cases:

1. Trivial expectation ($P < G$4): $P < G$5 is contained in $P < G$6. Applying [KP23], one concludes $P < G$7 for some normal subgroup $P < G$8.
2. Non-trivial expectation ($P < G$9): Here, $L^\infty(G/P, \nu_P) \rtimes \Gamma$0 must be of the form $L^\infty(G/P, \nu_P) \rtimes \Gamma$1 for a normal subgroup $L^\infty(G/P, \nu_P) \rtimes \Gamma$2. The identification utilizes a sophisticated ergodic-theoretic analysis, exploiting double metric ergodicity ([Ka03]) and essential freeness of the action $L^\infty(G/P, \nu_P) \rtimes \Gamma$3.

Technical Innovations and Methods

Double $L^\infty(G/P, \nu_P) \rtimes \Gamma$4-ergodicity

A central technical tool is the use of double $L^\infty(G/P, \nu_P) \rtimes \Gamma$5-ergodicity: for every separable Banach $L^\infty(G/P, \nu_P) \rtimes \Gamma$6-module $L^\infty(G/P, \nu_P) \rtimes \Gamma$7, every essentially bounded weak*-measurable, $L^\infty(G/P, \nu_P) \rtimes \Gamma$8-equivariant map from $L^\infty(G/P, \nu_P) \rtimes \Gamma$9 into $\nu_P$0 is essentially constant and valued in the fixed-point space $\nu_P$1. For Poisson boundaries of higher rank lattices, Kaimanovich’s result [Ka03] guarantees this property.

Slice Decomposition and Fiber Rigidity

For non-trivial $\nu_P$2, a careful slice analysis decomposes $\nu_P$3 into “$\nu_P$4-Fourier coefficients" indexed by $\nu_P$5, leading to $\nu_P$6 subspaces. Using double ergodicity, one proves that each $\nu_P$7 is either trivial or coincides with $\nu_P$8, and that the set of $\nu_P$9 where $K$0 is maximal forms a normal subgroup $K$1, proving the crossed product decomposition. Fubini-type and measure-theoretic methods on fiber products of boundary spaces are essential for establishing the constancy of certain kernel functions, a key step in the rigidity argument.

Novelty and Relation to Prior Work

Whereas previous proofs for intermediate subalgebras [AH24] relied on the noncommutative Nevo–Zimmer theorem [BH19], this work’s approach for the full structure theorem—especially the handling of the conditional expectation—offers a new, arguably more transparent perspective that sharply leverages boundary ergodicity. The argument is inextricably linked to the higher rank hypothesis; similar results in lower ranks or for other types of groups remain open.

Implications and Theoretical Consequences

The classification of $K$2-invariant von Neumann subalgebras as explicit crossed products enhances our understanding of the rigidity and structural simplicity of noncommutative Poisson boundaries for higher rank lattices. This has several notable theoretical implications:

• Operator Algebraic Rigidity: The result ties the possible invariant subalgebras directly with the algebraic structure of parabolic subgroups and normal subgroups, a strong form of rigidity echoing Margulis factor theorems and superrigidity principles.
• Connes' Rigidity Conjecture Connection: Understanding the subalgebra lattice feeds directly into operator algebraic rigidity problems for group von Neumann algebras, as flagged by Houdayer [Hou24].
• Extension to Quantum Symmetries: The ergodic-theoretic and representation-theoretic methods introduced may generalize to the analysis of quantum symmetries and boundary actions in broader noncommutative contexts.

Numerical and Structural Results

While the results are chiefly structural rather than numerical, it is a strong claim that the only $K$3-invariant von Neumann subalgebras one can ever obtain in $K$4 arise precisely as $K$5 for parabolics $K$6 and normal $K$7. This provides a complete and explicit classification.

Future Directions

Outstanding directions emerging from this work include:

• Lattices in Lower Rank and Non-semisimple Cases: The methods crucially use higher rank assumptions. Extending these classification results to groups of rank one, general semisimple, or non-algebraic settings remains open.
• Quantum Group Extensions: Further exploring analogues of these results for quantum groups and their boundaries may yield insights into quantum rigidity phenomena.
• Connections to Measurable Group Theory: The structural rigidity established here may have implications for orbit equivalence and cocycle superrigidity questions in dynamics and measurable group theory.

Conclusion

This paper confirms and proves the classification of $K$8-invariant von Neumann subalgebras of noncommutative Poisson boundaries for higher rank lattices, linking the structure tightly to the algebraic data of parabolic subgroups and normal subgroups of the lattice. Employing double metric ergodicity and an intricate decomposition of the crossed product, it provides a new proof approach distinct from previous reliance on advanced theorems like noncommutative Nevo–Zimmer, thereby advancing understanding of operator algebraic rigidity in higher rank settings.

Reference: "On invariant subalgebras of noncommutative Poisson boundaries for higher rank lattices" (2607.02450)