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Bimodule coefficients, Riesz transforms on Coxeter groups and strong solidity

Matthijs Borst, Martijn Caspers, Mateusz Wasilewski

Abstract

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 $\mathbb{C}[Γ]$, 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 $\mathcal{S}_p$ class $p \in [2, \infty)$ then the $n$-fold tensor power $H^{\otimes n}_Γ$ for $n \geq p/2$ 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 $\mathcal{S}_p$ for the gradient bimodule constructed from the word length function. We get equivalence of: (1) the gradient-$\mathcal{S}_p$ 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.

Bimodule coefficients, Riesz transforms on Coxeter groups and strong solidity

Abstract

In deformation-rigidity theory it is often important to know whether certain bimodules are weakly contained in the coarse bimodule. Consider a bimodule over the group algebra , with a discrete group. The starting point of this paper is that if a dense set of the so-called coefficients of is contained in the Schatten class then the -fold tensor power for 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 for the gradient bimodule constructed from the word length function. We get equivalence of: (1) the gradient- 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 -integrable representation.

Paper Structure

This paper contains 37 sections, 46 theorems, 110 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Von Neumann algebras
  4. Operator spaces
  5. Approximation properties
  6. Bimodules and containment
  7. Schatten classes
  8. Group algebras
  9. Hyperbolic groups
  10. Functions on groups
  11. Tensor products
  12. Coefficients of bimodules
  13. Coefficients and quasi-containment
  14. Tensoring bimodules
  15. The Akemann-Ostrand property AO$^+$ and strong solidity
  16. ...and 22 more sections

Key Result

Theorem 1.1

Let $H$ be a $\mathbb{C}[\Gamma]$ bimodule and let $V: \ell_2(\Gamma) \rightarrow H$ be bounded. Assume that $H$ is quasi-contained in the coarse bimodule of $\Gamma$, that $V$ is almost bimodular and that $V^\ast V$ is Fredholm. Assume that $C_r^\ast(\Gamma)$ is locally reflexive. Then $\mathcal{L}

Figures (1)

  • Figure 1: The graph $\mathsf{Graph}_{S}(W)$ is denoted for three different Coxeter systems $W = \langle S|M\rangle$ with $|S|=6$. In each of the graphs the label $m_{i,j}$ is shown on the edge $\{s_i,s_j\}$. We colored the edge orange when the label is even, we colored it blue when the label is odd, and we colored it black when the label is infinity. The relations we imposed on the generators are almost the same in the three cases. They only differ on the edges $\{s_4,s_5\}$ and $\{s_5,s_6\}$. The graph in (A) satisfies the assumptions of Proposition \ref{['proposition:characterization-parity-paths']} and hence does not contain a cyclic parity path. The graph in (B) does not satisfy the assumptions of the proposition as for the connected component $C=\{s_3,s_4\}$ of $(V,E_1)$ there are two distinct edges $\{s_2,s_3\}$ and $\{s_4,s_5\}$ with even label and with (at least) one endpoint in $C$. Therefore the graph contains a cyclic parity path. One is given by $P=(s_3,s_2,s_3,s_4,s_4,s_5,s_4,s_3)$ (another cyclic parity path uses the node $s_1$). The graph in (C) does also not satisfy the assumptions of the proposition as it contains a cycle with odd labels. Here a cyclic parity path is given by $P=(s_1,s_5,s_5,s_6,s_6,s_1)$ (another cyclic parity path is obtained by walking in reverse order).

Theorems & Definitions (100)

  • Theorem 1.1: Proposition 5.2 of caspersL2CohomologyDerivationsQuantum2021
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5: Application A
  • Theorem 1.6: Application B
  • Definition 3.1: Coefficients
  • Remark 3.2
  • Proposition 3.3: Quasi-containment
  • proof
  • ...and 90 more