Operator Algebras and Dynamics Seminars

Fall 2021

Dates subject to change depending on the colloquia schedule.

• Oct

  14
• Continuous Actions of Lattices (in Higher-Rank Semisimple Groups) are Free

  Darren Creutz

  USNA

  Time: 03:45 PM

  View Abstract

  Let \Gamma be an irreducible lattice in a higher-rank semisimple group G. Generalizing Margulis' Normal Subgroup theorem in the connected case, Stuck and Zimmer (1994) showed that every nonatomic probability-preserving action of \Gamma in a connected higher-rank Lie group G is essentially free. The general case was proven by the speaker and J. Peterson (2017). However, being (very) nonamenable, not every action of such a lattice on a compact metric space admits an invariant measure, leaving open the question of whether every minimal action of such a lattice on an infinite compact metric space is (topologically) free. I will present the complete result: every action of such a lattice is indeed (topologically) free. The main idea is to replace invariant measure by stationary measure (which always exist) and show that every stationary action is essentially free. This answers the general form of a question of Glasner and Weiss on URS's for such lattices.

• Sep

  29
• Complexity of Elevated Staircase Transformations

  ENS Shaun Rodock

  USNA

  Time: 03:45 PM

  View Abstract

  We introduce elevated staircase transformations, a type of rank-one transformation. The classical staircase transformation was shown to attain quadratic complexity and it was conjectured in 1995 that this was minimal among all mixing rank one transformations. We show that elevated staircase transformations are mixing and attain complexity q(log q)^(1+eps) sharply disproving the conjecture.

• Sep

  22
• Compressions and Operator Systems

  Ben Passer

  USNA

  Time: 03:45 PM

  View Abstract

  The matrix range of two tuples of operators can be identical even if the tuples themselves have very different structure. Using a notion of extreme point for operator systems, we show that the spatial structure of a tuple T can be determined from the matrix range under minimality conditions. This generalizes work in the literature on free spectrahedra, matrix convex sets, and compact operators.

• Sep

  13
• Dilation Scales and the Structure of Operator Systems

  Ben Passer

  USNA

  Time: 03:45 PM

  View Abstract

  The structure of a compact convex set K is determined by affine functions, and the extreme points of K are detectable in the embedding of affine functions on K into the continuous functions on K. The noncommutative analogue of this is to examine an operator system inside a unital C*-algebra. I will discuss some basics of operator systems and show how this framework leads into dilation problems for noncommuting operators and problems about the spatial structure of operator systems.