Mondays 11:00-12:00

A place to learn about all things noncommutative analysis. From operator algebras, systems, and spaces to noncommutative functions and their commutative shadows.

The seminar meets on Mondays, 11:00-12:00, in 32/114

2021–22–B meetings

Apr 4 The Radius of Comparison of a Commutative C*-algebra Chris Phillips (University of Oregon)

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.

Apr 11 A new universal AF-algebra Wieslaw Kubis (Institute of Mathematics, Prague)

We introduce and study a new class of separable approximately finite-dimensional (AF) C* -algebras, namely, AF-algebras with “Cantor property”. We show the existence of a separable AF-algebra A that is universal in the sense of quotients, i.e. every separable AF-algebra is a quotient of A. Moreover, a natural extension property involving left-invertible embeddings describes it uniquely up to isomorphism.

This is a joint work with Saeed Ghasemi. The paper is Universal AF-algebras. J. Funct. Anal. 279 (2020), no. 5, 108590, 32 pp.

May 23, In Building 32, room 114 Structure of crossed product $C^*$-algebras Zhuang Niu (University of Wyoming)

Consider a dynamical system, and let us study the structure of the corresponding crossed product $C^*$-algebra, in particular on the classifiability, comparison, and stable rank. More precisely, let us introduce a uniform Rokhlin property and a relative comparison property (these two properties hold for all free and minimal $Z^d$ actions). With these two properties, the crossed product $C^*$-algebra is shown to always have stable rank one, to satisfy the Toms-Winter conjecture, and that the comparison radius is dominated by half of the mean dimension of the dynamical system.

Jun 6, In Building 32, Room 114 The tracial Rokhlin property for actions of infinite compact groups N. Christopher Phillips (University of Oregon)

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.

Jun 13, In seminar room, minus 101 Gap labelling for electron motion in quasicrystals and C*-algebra of minimal actions of Z^d on the Cantor set N. Christopher Phillips (University of Oregon)

This talk will be a survey of the mathematics of the gap labelling problem for quasicrystals, but will assume no knowledge of physics.

In one standard approximation, the possible energy levels of an electron moving in a crystal form a collection of bands. These energy levels constitute the spectrum of a suitable Schr$\"{o}$dinger operator, and the gaps between the bands are gaps in the spectrum.

Quasicrystals are not periodic, but exhibit long range order. The structure of the spectrum of the Schr$\"{o}$dinger operator for quasicrystals is addressed by the ``Gap Labelling Conjecture’’. This conjecture was made in 1989, and some results are known.

An infinite quasicrystal has an associated action of ${\mathbb{Z}}^d$ on the Cantor set $X$, and thus a transformation group C*-algebra $A$. The physics is supposed to give an invariant measure on $X$, and hence a tracial state on $A$. The gaps in the spectrum of Schr"{o}dinger operator correspond to the values of this tracial state on projections in $A$, and the Gap Labelling Theorem states that these values all already occur as values of the measure on compact open subsets of $X$.

In this talk, I will give a more careful description of the situation, including sketches of how the objects above are constructed and how they are related to each other. Then I will say something about the results that have been proved, and outline what goes into their proofs.