Noncommutative Analysis Ical Atom

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.

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.

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.

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.

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.

The Pick interpolation theorem states that the existence of a function on the complex unit disc that is analytic, bounded by 1, and satisfies some interpolation data is equivalent to the positivity of a matrix that depends on the interpolation data. In 1979 Abrahamse generalized this result from the disk to any g-holed multiply connected domain. However, in the result of Aabrahamse, a family of matrices parametrized by the g-dimensional torus was needed. In 2010, A variation of the Pick interpolation problem was studied by Davidson, Paulsen, Raghupathi, and Singh, who discovered that if the constraint of zero derivative at a point is applied to the interpolating function, then there is a family of matrices parametrized by the unit sphere that need to be positive. In my thesis, I have combined these results to solve a constrained interpolation problem on a multiply connected domain. I will present the ideas that prove these kinds of interpolation theorems, that were first applied to that cause by Sarason, and will show how I used them for the constraint-multiply connected problem. If time allows it, I will also say a few words about matrix-valued interpolation.

This lecture extends some standard results familiar from undergraduate calculus to the setting of higher order noncommutative functions. These extensions are accomplished using a difference-differential operator combining algebraic and topological properties. The results include the Taylor formula and an antiderivative. The Taylor formula is applied to obtain results about power series on nilpotent matrices and convergence in different topologies.