## Nebraska-Iowa Functional Analysis Seminar

There will be a one day conference on Operator Algebras on Saturday, November 3rd, 2018, associated with the 2018 Howard Rowlee Lecture will be given by Ruy Exel on Friday, November 2nd, 2018.To register, go the registration form. Early registration is very helpful.

**Speakers**

- Ionut Chifan of the University of Iowa
- Valentin Deaconu of the University of Nevada, Reno
- Ruy Exel of the University of Nebraska, Lincoln and Universidade Federal de Santa Catarina
- Ken Dykema of Texas A&M University
- Elizabeth Gillaspy of the University of Montana
- Isaac Goldbring of the University of California, Irvine
- Jesse Peterson of Vanderbilt University

#### Ionut Chifan,

In the mid thirties F. J. Murray and J. von Neumann found a natural way to associate a von Neumann algebra L(G) to every countable discrete group G. Classifying L(G) in terms of G emerged from the beginning as a natural yet quite challenging problem as these algebras tend to have very limited "memory" of the underlying group. This is perhaps best illustrated by Connes' famous result asserting that all icc amenable groups give rise to isomorphic von Neumann algebras; thus in this case, besides amenability, the algebra has no recollection of the usual group invariants like torsion, rank, or generators and relations. In the non-amenable case the situation is radically different; many examples where the von Neumann algebraic structure is sensitive to various algebraic group properties have been discovered via Popa's deformation/rigidity theory. In this talk I will present several new instances where the von Neumann algebra completely retains canonical algebraic constructions in group group theory such us (infinite) direct product, amalgamated free product, or wreath product. In addition, I will present several applications of these results to the study of rigidity in the C*-setting.

#### Valentin Deaconu,

I will recall the definition of a #### Ken Dykema,

The spectrum of an operator contains essential information. Even better is when we can find invariant subspaces that break the operator into pieces with conditions on the spectrum. This is the meaning of decomposability of an operator, in the sense of Foias. An arbitrary element of a finite von Neumann algebra has a sort of spectral distribution measure called its Brown measure. Haagerup and Schultz proved existence of invariant subspaces that break up such an element according to its Brown measure. These Haagerup-Schultz subspaces have been used to provide Schur-type upper triangular forms for such elements. In this talk, we review these constructions and describe how these Schur-type upper triangular forms relate to decomposability in the sense of Foias.

#### Ruy Exel,

We introduce a generalization of the notion of approximately
proper equivalence relations studied by Renault. With it we build an
etale groupoid and discuss the Radon-Nikodym problem relative to the
one-cocycle arising from a given potential. Should time allow, we
will also show applications to the study of Dobrushin-Landford-Ruelle
(DLR) measures in statistical mechanics when the space of spins is
infinite.

#### Elizabeth Gillaspy,

The infinite path space

All the words in the title will be defined during the talk; no prior familiarity with higher-rank graphs, KMS states, or Hausdorff dimension will be assumed.

#### Isaac Goldbring,

I will discuss how the notion of building models by games allows us to give new reformulations of many of the embedding problems in
operator algebras, e.g. the Connes embedding problem, the Kirchberg embedding problem, and the quasidiagonality problem.
We will also indicate how all of these problems can be reformulated in terms of the enforceability of tensor square roots.

#### Jesse Peterson,

We will introduce a class of groups, which we call properly proximal,
which includes all nonelementary hyperbolic groups, all nonelementary bi-exact groups,
all convergence groups, all lattices in semisimple Lie groups, and is closed under
commensurability and taking direct products, but excludes all amenable and even
all inner-amenable groups. We will then discuss rigidity results for von Neumann
algebras associated to measure-preserving actions of these groups. This is joint
work with Remi Boutonnet and Adrian Ioana.

**Lodging:**A block of rooms is available at the Graduate Hotel, 141 N 9th, at a conference rate of $99 plus taxes (including breakfast). To make your reservation please call the hotel at 402-475-4011. Be sure to mention room block UNLMATH for the reduced rate. Please book your room before October 15th.## Schedule of Talks:

Time | Speaker | Title |
---|---|---|

8:45--9:30 | Ruy Exel | Quasi-invariant measures for generalized approximately proper equivalence relations |

9:30--9:50 | BREAK | |

9:50--10:35 | Isaac Goldbring | Embedding problems, games, and square roots |

10:45--11:30 | Jesse Peterson | Properly proximal groups and their von Neumann algebras |

11:30--1:00 | LUNCH | |

1:00--1:45 | Elizabeth Gillaspy | Generalized gauge actions, KMS states, and Hausdorff dimension for higher-rank graphs |

1:55--2:40 | Ken Dykema | Schur-type upper triangular forms and decomposability in finite von Neumann algebras |

2:40--3:00 | BREAK | |

3:00--3:45 | Valentin Deaconu | Symmetries of Cuntz-Pimsner algebras |

3:55--4:40 | Ionut Chifan | Rigidity in group von Neumann algebras |

## Abstracts:

#### Ionut Chifan, *Rigidity in group von Neumann algebras*

In the mid thirties F. J. Murray and J. von Neumann found a natural way to associate a von Neumann algebra L(G) to every countable discrete group G. Classifying L(G) in terms of G emerged from the beginning as a natural yet quite challenging problem as these algebras tend to have very limited "memory" of the underlying group. This is perhaps best illustrated by Connes' famous result asserting that all icc amenable groups give rise to isomorphic von Neumann algebras; thus in this case, besides amenability, the algebra has no recollection of the usual group invariants like torsion, rank, or generators and relations. In the non-amenable case the situation is radically different; many examples where the von Neumann algebraic structure is sensitive to various algebraic group properties have been discovered via Popa's deformation/rigidity theory. In this talk I will present several new instances where the von Neumann algebra completely retains canonical algebraic constructions in group group theory such us (infinite) direct product, amalgamated free product, or wreath product. In addition, I will present several applications of these results to the study of rigidity in the C*-setting.#### Valentin Deaconu, *Symmetries of Cuntz-Pimsner algebras*

I will recall the definition of a *C*^{*}-correspondence and of the Cuntz-Pimsner algebra. I will discuss group and groupoid actions on*C*^{*}-correspondences and the associated crossed products. I will illustrate with examples related to discrete graphs and to Hermitian vector bundles.#### Ken Dykema, *Schur-type upper triangular forms and decomposability in finite von Neumann algebras*

The spectrum of an operator contains essential information. Even better is when we can find invariant subspaces that break the operator into pieces with conditions on the spectrum. This is the meaning of decomposability of an operator, in the sense of Foias. An arbitrary element of a finite von Neumann algebra has a sort of spectral distribution measure called its Brown measure. Haagerup and Schultz proved existence of invariant subspaces that break up such an element according to its Brown measure. These Haagerup-Schultz subspaces have been used to provide Schur-type upper triangular forms for such elements. In this talk, we review these constructions and describe how these Schur-type upper triangular forms relate to decomposability in the sense of Foias.#### Ruy Exel, *Quasi-invariant measures for generalized approximately proper equivalence relations*

We introduce a generalization of the notion of approximately
proper equivalence relations studied by Renault. With it we build an
etale groupoid and discuss the Radon-Nikodym problem relative to the
one-cocycle arising from a given potential. Should time allow, we
will also show applications to the study of Dobrushin-Landford-Ruelle
(DLR) measures in statistical mechanics when the space of spins is
infinite.#### Elizabeth Gillaspy, *Generalized gauge actions, KMS states, and Hausdorff dimension for higher-rank graphs*

The infinite path space *L*of a higher-rank graph^{∞}*L*is (often) a Cantor set -- compact, perfect, totally disconnected. Together with Carla Farsi, Nadia Larsen, and Judith Packer, we have found several ways to put a metric on this Cantor set, and computed the associated Hausdorff dimension and measure. It turns out that the same data we needed to metrize*L*also gives us a generalized gauge action on^{∞}*C*-- and the KMS states associated to this action are intimately tied to the Hausdorff measure on^{*}(L)*L*. To us, this was an unexpected link between the dynamical information exhibited by a higher-rank graph (as exhibited in its KMS states) and its fractal structure.^{∞}All the words in the title will be defined during the talk; no prior familiarity with higher-rank graphs, KMS states, or Hausdorff dimension will be assumed.

#### Isaac Goldbring, *Embedding problems, games, and square roots*

I will discuss how the notion of building models by games allows us to give new reformulations of many of the embedding problems in
operator algebras, e.g. the Connes embedding problem, the Kirchberg embedding problem, and the quasidiagonality problem.
We will also indicate how all of these problems can be reformulated in terms of the enforceability of tensor square roots.#### Jesse Peterson, *Properly proximal groups and their von Neumann algebras*

We will introduce a class of groups, which we call properly proximal,
which includes all nonelementary hyperbolic groups, all nonelementary bi-exact groups,
all convergence groups, all lattices in semisimple Lie groups, and is closed under
commensurability and taking direct products, but excludes all amenable and even
all inner-amenable groups. We will then discuss rigidity results for von Neumann
algebras associated to measure-preserving actions of these groups. This is joint
work with Remi Boutonnet and Adrian Ioana.