Rigidity and arithmetic of lattice von Neumann algebras

Let \Gamma\in\mathscr K, let \Lambda be any countably infinite ICC group, and let \mu and \omega be arbitrary normalized scalar cocycles on these groups.

1. (i)

   Every nonzero bifinite L_\mu(\Gamma)–L_\omega(\Lambda) correspondence is a closed bimodule summand of a finite direct sum of the models in Equation (1.3).

2. (ii)

   Every such correspondence has a finite orthogonal decomposition into full-germ orbit bundles. After choosing distinct germ orbits and allowing zero fibres, all bounded bimodule maps are

   \Hom_{M\text{--}N}\!\left(\bigoplus_d H(d,\sigma_d), \bigoplus_d H(d,\tau_d)\right) =\bigoplus_d\Hom_{C_d,\alpha}(\sigma_d,\tau_d).

   Norm, adjoint, and composition are fibrewise. Projection summands are exactly projective-representation summands in the fibres. A correspondence K=\bigoplus_d H(d,\sigma_d) has dimensions

   \dim_M K=\sum_d\dim_{\mathbb C}(\sigma_d)[\Lambda:B_d],\qquad \dim K_N=\sum_d\dim_{\mathbb C}(\sigma_d)[\Gamma:A_d].

   Both are positive integers when K\ne0.

3. (iii)

   Conjugation and Connes fusion correspond to conjugation and balanced tensor product of these bundles. They are equivalently computed by finite double-coset induction. The scalar transports are given explicitly in Equations (3.4) and (3.17) and Section 3.