The classical Gaussian functor associates to every orthogonal representation of a locally compact group G a probability measure preserving action of G called a Gaussian action. In this paper, we generalize this construction by associating to every affine isometric action of G on a Hilbert space, a one-parameter family of nonsingular Gaussian actions whose ergodic properties are related in a very subtle way to the geometry of the original action. We show that these nonsingular Gaussian actions exhibit a phase transition phenomenon and we relate it to new quantitative invariants for affine isometric actions. We use the Patterson-Sullivan theory as well as Lyons-Pemantle work on tree-indexed random walks in order to give a precise description of this phase transition for affine isometric actions of groups acting on trees. Finally, we use Gaussian actions to show that every nonamenable locally compact group without property (T) admits a free nonamenable weakly mixing action of stable type III1.

The classical Gaussian functor associates to every orthogonal representation of a locally compact group G a probability measure preserving action of G called a Gaussian action. In this paper, we generalize this construction by associating to every affine isometric action of G on a Hilbert space, a one-parameter family of nonsingular Gaussian actions whose ergodic properties are related in a very subtle way to the geometry of the original action. We show that these nonsingular Gaussian actions exhibit a phase transition phenomenon and we relate it to new quantitative invariants for affine isometric actions. We use the Patterson-Sullivan theory as well as Lyons-Pemantle work on tree-indexed random walks in order to give a precise description of this phase transition for affine isometric actions of groups acting on trees. We also show that every locally compact group without property (T) admits a nonsingular Gaussian that is free, weakly mixing and of stable type III 1 .

We investigate the structure of the relative bicentralizer algebra B(N ⊂ M, ϕ) for inclusions of von Neumann algebras with normal expectation where N is a type III1 subfactor and ϕ ∈ N * is a faithful state. We first construct a canonical flow β ϕ : R * + B(N ⊂ M, ϕ) on the relative bicentralizer algebra and we show that the W * -dynamical system (B(N ⊂ M, ϕ), β ϕ ) is independent of the choice of ϕ up to a canonical isomorphism. In the case when N = M , we deduce new results on the structure of the automorphism group of B(M, ϕ) and we relate the period of the flow β ϕ to the tensorial absorption of Powers factors. For general irreducible inclusions N ⊂ M , we relate the ergodicity of the flow β ϕ to the existence of irreducible hyperfinite subfactors in M that sit with normal expectation in N . When the inclusion N ⊂ M is discrete, we prove a relative bicentralizer theorem and we use it to solve Kadison's problem when N is amenable.2010 Mathematics Subject Classification. 46L10, 46L30, 46L36, 46L37, 46L55.

We give a spectral gap characterization of fullness for type III factors which is the analog of a theorem of Connes in the tracial case. Using this criterion, we generalize a theorem of Jones by proving that if M is a full factor and σ : G → Aut(M ) is an outer action of a discrete group G whose image in Out(M ) is discrete then the crossed product von Neumann algebra M ⋊σ G is also a full factor. We apply this result to prove the following conjecture of Tomatsu-Ueda: the continuous core of a type III1 factor M is full if and only if M is full and its τ invariant is the usual topology on R.2010 Mathematics Subject Classification. 46L10.