Analysis Seminar

The analysis seminar is held on Tuesdays at 2:00-2:50 in 210 University Hall unless otherwise noted.

Winter Quarter, 2022

January 4, John Jasper (South Dakota State University)
Diagonals of compact operators

January 11, Chris Phillips (UO)
The tracial Rokhlin property for actions of infinite compact groups

Abstract: The tracial Rokhlin property for actions of finite groups is now well known, along with weakenings and versions for other classes of discrete groups. The Rokhlin property for actions of infinite infinite compact groups has also been studied. We define and investigate the tracial Rokhlin property for actions of second countable compact groups on simple unital C*-algebras. The naive generalization of the verrsion for finite groups does not appear to be good enough. We have a property which, first, allows one to prove the expected theorems, second, is “almost” implied by the version for finite groups when the group is finite, and, third, admits examples.

This is joint work with Javad Mohammadkarimi.

January 18, Marcin Bownik (UO)
Parseval wavelet frames on Riemaniann manifolds

Abstract: In this talk we discuss how to construct Parseval wavelet frames in $L^2(M)$ for a general Riemannian manifold $M$ . We also show the existence of wavelet unconditional frames in $L^p(M)$ for $1 < p <\infty$ . This construction is made possible thanks to smooth orthogonal projection decomposition of the identity operator on $L^2(M)$ , which is an operator version of a smooth partition of unity. We also show some applications such as a characterization of Triebel-Lizorkin $F_{p,q}^s(M)$ and Besov $B_{p,q}^s(M)$ spaces on compact manifolds in terms of magnitudes of coefficients of Parseval wavelet frames.

This talk is based on a joint work with Dziedziul and Kamont.

January 25, Chris Phillips (UO)
The dynamical radius of comparison

Abstract: The radius of comparison rc (A) of a unital C*-algebra A is one of a several quantities which can be interpreted as a measure of the “topological dimension” of a C*-algebra, that is, as a (rough) generalization to noncommutative C*-algebras of the function which assigns to a compact metric space X (and thus to the C*-algebra C (X)) the (covering) dimension dim (X). For example, it has been recently proved that rc (C (X)) is about half of dim (X) for any compact metric space X. It has been known for some time that for a simple unital AH algebra A, rc (A) can be any value in [0, \infty].

In this talk, we describe a radius of comparison for dynamical systems, that is, for actions of discrete groups on C*-algebras, based on the dynamical version of Cuntz comparison developed by Bosa, Perera, Wu, and Zacharias. It isn’t “equivariant” in the way that equivariant K-theory is equivariant. Rather, it potentially helps relate the radius of comparison of the crossed product to dynamical information about the action. In this talk, we describe the definition, some of what is known, and a collection of exotic examples involving finite groups which show that our dynamical radius of comparison can take many values.

This is joint work with M. Ali Asadi-Vasfi.

Fall Quarter, 2021

September 28, Organizational Meeting

October 5, Chris Phillips (UO)
The Radius of Comparison of a Commutative C*-algebra

Abstract: The radius of comparison of a C*-algebra is one measure of the generalization to C*-algebras of the dimension of a compact space. Part of the Toms-Winter conjecture says, informally, that a simple separable nuclear unital C*-algebra satisfying the UCT is classifiable if and only if its radius of comparison is zero. Nonzero radius of comparison played a key role in one of the main families of counterexamples to the original form of the Elliott classification program.

It has been known for some time that the radius of comparison of C (X) is, ignoring additive constants, at most half the covering dimension of X. (The factor 1/2 appears because of the use of complex scalars in C*-algebras.) In 2013, Elliott and Niu used Chern character arguments to show that the radius of comparison of C (X) is, again ignoring additive constants, at least half the rational cohomological dimension of X. This left open the question of which dimension the radius of comparison is really related to. The rational cohomological dimension can be strictly less than the integer cohomological dimension, and there are spaces with integer cohomological dimension 3 but infinite covering dimension.

We show that, up to a slightly worse additive constant, the radius of comparison of C (X) is at least half the covering dimension of X. The proof is fairly short and uses little machinery.

October 12, Ilan Hirshberg (Ben Gurion University of the Negev, Beersheva, Israel)
Mean cohomological independence dimension and radius of comparison

Abstract: The concept of mean dimension for topological dynamics was developed by Lindenstrauss and Weiss, based on ideas of Gromov. Independently, and for different reasons entirely, Toms introduced the concept of radius of comparison for C*-algebras. It appears, however, that there is a connection between those two notions: to each topological dynamical system one can associate a C*-algebra (known as the crossed product or the transformation group C*-algebra), and there appears to be a connection between the mean dimension of the dynamical system and the radius of comparison of the associated C*-algebra.

I will explain those concepts and a related concept which we call mean cohomological independence dimension, and discuss what is known about the connection between them. I don’t expect to prove anything in the talk.

This is joint work with N. Christopher Phillips.

October 19, Alonso Delfin (UO)
A representation theorem for Hilbert bimodules and C*-like modules over $L^p$ algebra

Abstract: During this talk I will discuss a representation theorem for Hilbert bimodules due to Ruy Exel (1993). I will sketch a new proof and compare it with Exel’s one. An immediate consequence of this theorem yields a representation for right Hilbert modules, which in turn motivates the definition for modules over $L^p$ operator algebras. Time permitting, I’ll give some examples of these modules and some $L^p$ operator algebras associated to them.

October 26, Marcin Bownik (UO)
Simultaneous dilation and translation tilings of $\mathbb{R}^n$

Abstract: In this talk we present a solution of the wavelet set problem. That is, we characterize the full-rank lattices $\Gamma\subset \mathbb{R}^n$ and invertible $n \times n$ matrices A for which there exists a measurable set W such that $\{W + \gamma: \gamma \in \Gamma\}$ and $\{A^j(W): j\in \mathbb{Z}\}$ are tilings of $\mathbb{R}^n$ . The characterization is a non-obvious generalization of the one found by Ionascu and Wang, which solved the problem in the case $n = 2$ . As an application of our condition and a theorem of Margulis, we also strengthen a result of Dai, Larson, and the second author on the existence of wavelet sets by showing that wavelet sets exist for matrix dilations, all of whose eigenvalues $\lambda$ satisfy $|\lambda| \ge 1$ . As another application, we show that the Ionascu-Wang characterization characterizes those dilations whose product of two smallest eigenvalues in absolute value is $\ge 1$ .
Based on joint work with Darrin Speegle.

November 2, Chris Phillips (UO)
Incompressible Banach algebras

Abstract: Let A be a Banach algebra. We say that A is incompressible if every injective homomorphism from A to another Banach algebra is bounded below, uniformly incompressible if there is a nonzero lower bound depending only on the norm of the homomorphism, and isometrically incompressible if every contractive injective homomorphism from A to another Banach algebra is isometric. Incompressibility has been considered (under other names) in the past, although without any systematic theory. For example, the algebra of bounded operators on a Banach space, certain Calkin algebras, and all C*-algebras are uniformly and isometrically incompressible. The last result has led to the idea that a “C* like” L^p operator algebra should be incompressible. The disk algebra and the algebra of bounded infinite upper triangular matrices are not incompressible.

We describe some of the basic theory of incompressible Banach algebras and some of the examples. This talk will use very little machinery.

This is joint work with Bill Johnson and Gideon Schechtman.

November 9, Yuan Xu (UO)
Analysis on homogeneous space with a localizable kernel

Abstract: Analysis on the unit sphere is based on spherical harmonics, in which an essential tool is the addition formula that provides a localizable kernel on the sphere. Much of what is known on the sphere can be carried over to homogeneous spaces, if an “addition formula” is assumed to exist and is localizable.

Previous years:
2019-20
2018-19
2017-18
2016-17
2015-16
2014-15
2013-14

Math Department of Mathematics

Analysis Seminar

Winter Quarter, 2022

Fall Quarter, 2021