2019年度RIMS共同研究(グループ型)

作用素環の分類理論における新展開

1月20日--22日@RIMS 111

  • Schedule

    1月20日(月) 1月21日(火) 1月22日(水)
    9:30--10:30
    佐藤僚亮(R. Sato) 武石拓也(T. Takeishi)
    10:45--11:45
    磯野優介(Y. Isono) C. Houdayer
    13:30--14:30 磯野優介(Y. Isono) 磯野優介(Y. Isono)
    14:45--15:45 C. Houdayer 窪田陽介(Y. Kubota) Program in pdf
    16:00--17:00 曽我部太郎(T. Sogabe) C. Houdayer

  • Titles and Abstracts
    C. Houdayer Stationary actions of higher rank lattices on von Neumann algebras

    In this lecture series, I will talk about a recent joint work with Remi Boutonnet in which we show that for higher rank lattices (e.g. SL(3, Z)), the left regular representation is weakly contained in any weakly mixing unitary representation. This strengthens Margulis’ normal subgroup theorem (1978), Stuck-Zimmer’s stabilizer rigidity result (1992) as well as Peterson’s character rigidity result (2014). We also show that Uniformly Recurrent Subgroups (URS) of higher rank lattices are finite, answering a question of Glasner-Weiss (2014). I will explain the main novelty of our work that consists in proving a structure theorem for stationary actions of higher rank lattices on von Neumann algebras.
    Y. Isono Ergodic theory of affine isometric actions on Hilbert spaces

    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. 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 III$_1$. This is joint work with Y. Arano and A. Marrakchi.
    Y. Kubota Reconstruction of profinite abelian groups from the K-theory of crossed product C*-algebras

    For a free abelian group and a decreasing sequence of its finite index subgroups, we associate a compact group, the profinite completion, including the original group as a dense subgroup. In this talk we show that the K-theory of the crossed product C*-algebra, together with the homomorphism induced from the inclusion, has rich information enough to reconstruct the profinite completion. This is a central part of our reconstruction of the Bost-Connes semigroup action from the associated C*-algebra. This is a joint work with Takuya Takeishi.
    R. Sato q-Schur generating functions and tensor product representations

    We give a formulation of inductive limits of compact quantum groups and investigate the unitary representation theory of the infinite-dimensional quantum unitary group, which is the inductive limit of finite rank quantum unitary groups, based on our formulation. In particular, we describe tensor products of representations associated with quantized characters using q-Schur generating functions, which is an important tool in integrable probability.
    T. Sogabe A topological invariant of the continuous fields of the Cuntz algebras

    We would like to discuss about a topological invariant of the continuous fields of the Cuntz algebras. We will explain construction of the invariant and that the invariant is trivial if the continuous field coming from a vector bundle as a Cuntz-Pimsner algebra.
    T. Takeishi Partition functions as C*-dynamical invariants and actions of congruence monoids

    C*-algebras of ax+b-semigroups of congruence monoids $C_\lambda^*(R\rtimes R_{\mathfrak{m},\Gamma})$ are introduced by C. Bruce, which behaves similarly to the C*-algebras examined by Cuntz--Deninger--Laca. Both kinds of algebras have canonical time evolutions, and have similar phase transition phenomena. In this talk, we determine the partition functions and associated representations of $(C_\lambda^*(R\rtimes R_{\mathfrak{m},\Gamma}), \sigma)$, inspired by the construction of the representations of Cuntz--Deninger--Laca. As a consequence, we recover several number theoretic invariants from those C*-dynamical sysmtems. In the case of $(C_\lambda^*(R\rtimes R^\times), \sigma)$, we in fact obtain slightly different partition functions from those suggested in the work of Cuntz--Deninger--Laca. This is a joint work with C. Bruce and M. Laca.