2021/09/01 by Matthijs Borst, Martijn Caspers, Borst, Matthijs +3 · 3 citations
Mathematics · Physics and Astronomy · #Advanced Operator Algebra Research #Algebraic structures and combinatorial models #FOS: Mathematics #Functional Analysis (math.FA) #Group Theory (math.GR) #Operator Algebras (math.OA) #Quantum many-body systems
paper · pdf · doi:10.48550/arxiv.2109.00588
openalex publication_date 2021/09/01 · openalex created_date 2022/07/25 · openalex updated_date 2026/07/28
In deformation-rigidity theory it is often important to know whether certain bimodules are weakly contained in the coarse bimodule. Consider a bimodule H over the group algebra ℂ[Γ], with Γ a discrete group. The starting point of this paper is that if a dense set of the so-called coefficients of H is contained in the Schatten Sp class p ∈ [2, ∞) then the n-fold tensor power H⊗ nΓ for n ≥ p/2 is quasi-contained in the coarse bimodule. We apply this to gradient bimodules associated with the carré du champ of a symmetric quantum Markov semi-group. For Coxeter groups we give a number of characterizations of having coefficients in Sp for the gradient bimodule constructed from the word length function. We get equivalence of: (1) the gradient-Sp property introduced by the second named author, (2) smallness at infinity of a natural compactification of the Coxeter group, and for a large class of Coxeter groups: (3) walks in the Coxeter diagram called parity paths. We derive several strong solidity results. In particular, we extend current strong solidity results for right-angled Hecke von Neumann algebras beyond right-angled Coxeter groups that are small at infinity. Our general methods also yield a concise proof of a result by T. Sinclair for discrete groups admitting a proper cocycle into a p-integrable representation.