Mathematics Department

# Operator Algebras and Dynamics Seminar

## Spring 2022

All talks are from 3:45-4:45 p.m. in the Seminar room, unless otherwise specified.

• Apr
11
• Low Complexity Subshifts admitting Mixing Measures
Darren Creutz
Time: 03:45 PM

#### View Abstract

Word complexity is a fine-grained quantification of determinism for subshifts: closed, shift-invariant subsets of A^\mathbb{Z} for some finite alphabet A. A natural question is the extent to which measure-theoretic mixing, a very strong form of dynamical randomness, implies complexity in the symbolic sense. Despite initially conjecturing that mixing subshifts have superpolynomial complexity, Ferenczi in 1996 established that the staircase transformation, which is mixing, has quadratic complexity. However, he could only prove that mixing implies a superlinear lower bound. We introduce a class of modified staircases with complexity much lower than quadratic: for any $f : \mathbb{N} \to \mathbb{N}$ with $f(n)/n$ increasing and $\Sum 1/f(n) < \infty$, we prove there exists an extremely elevated staircase transformation with word complexity $p$ satisfying $p(q) = o(f(q))$. This is joint work with Ronnie Pavlov and Shaun Rodock.
• Mar
28
• Amenability of C*-dynamical systems
Alex Bearden
University of Texas at Tyler
Time: 03:45 PM

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The notion of amenability of a locally compact group was generalized to certain operator algebraic settings by Anantharaman-Delaroche in the late 70s. We will review the history and triumphs of amenability in W*- and C*-dynamical systems, and then describe recent progress in the understanding of the various competing notions of amenability from the work of Buss/Echterhoff/Willett, Ozawa/Suzuki, and our joint work with Jason Crann.
• Mar
07
• A quantization of coarse structures and uniform Roe algebras
David Sherman
University of Virginia
Time: 03:45 PM

#### View Abstract

A coarse structure is a way of talking about "large-scale" properties. It is encoded in a family of relations that often, but not always, come from a metric. A coarse structure naturally gives rise to Hilbert space operators that in turn generate a so-called uniform Roe algebra. In ongoing work with Bruno Braga and Joe Eisner, we use ideas of Weaver to construct "quantum" coarse structures and uniform Roe algebras in which the underlying set is replaced with an arbitrary represented von Neumann algebra. The general theory immediately applies to quantum metrics (suitably defined), but it is much richer. We explain another source of examples based on measure instead of metric, leading to a large and easy-to-understand class of new C*-algebras. I will present the big picture: where uniform Roe algebras come from, how Weaver's framework facilitates our definitions. I will focus on a few illustrative examples and will not assume any familiarity with coarse structures or von Neumann algebras.