Interactions Between Dynamical Systems and C* Algebras

Lead Research Organisation: University of Glasgow
Department Name: School of Mathematics & Statistics

Abstract

I am interested in Dynamical Systems, Ergodic Theory, Bifurcation Theory, Chaos Theory, Functional Analysis and systems with symmetry or rigidity. In particular I have experience in research involving dynamical systems with random elements (studying the decay rates of randomly iterated Liverani-Saussol-Vaienti maps) and would therefore also be interested in pursuing research in this area.

Publications

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Studentship Projects

Project Reference Relationship Related To Start End Student Name
EP/N509668/1 01/10/2016 30/09/2021
1653952 Studentship EP/N509668/1 01/10/2015 31/03/2019 Luke Ito
 
Description The main result obtained so far in the award concerns C*-algebras associated to tilings. Mathematically, the concept of a tiling encapsulates the natural intuition one would expect - it is a covering of a space (possibly higher-dimensional) by prescribed shapes (tiles) that overlap only on their edges. We enforce additional assumptions on our tilings amounting to the existence of only finitely many patterns in any ball of a given radius, an aperiodicity condition, and a condition called repetitivity which roughly means that if a pattern appears somewhere in the tiling, it keeps reappearing with uniform regularity throughout the tiling. These assumptions ensure that the tilings we consider make good models for physical quasicrystals, which share these features. A C*-algebra is (isomorphic to) a subalgebra of bounded operators on a Hilbert space which is closed under taking adjoints. Such objects arose in the study of quantum mechanics as a mathematical model for algebras of physical observables. We can associate a C*-algebra to a tiling by considering translational equivalence on the closure of the set of translates of the given tiling in the "big ball" metric, in which tilings are close if they agree on a large ball up to a small translation. There is a prescribed way to form a C*-algebra from the translational equivalence relation (for the interested reader, this is an example of a groupoid C*-algebra, where we think of the equivalence relation as describing a groupoid). Since our tilings model quasicrystals, these tiling C*-algebras are closely related to the physics of such objects. More precisely, the image of the K-theory of the C*-algebra under the canonical trace provides a labelling of the gaps in the spectrum of a Schrödinger operator, which gives us a sense of what the spectrum looks like even if we do not know its precise form. Over the last 50 years or so, there has been much activity towards classifying C*-algebras, that is creating and computing an invariant which distinguishes isomorphism classes of C*-algebras (at least in the presence of certain properties of the algebra). We have proven that the C*-algebra of any tiling satisfying the assumptions described above is quasidiagonal. Quasidiagonality was an important ingredient in the early classification results, along with the regularity property of finite nuclear dimension, but has since been shown to be automatic with the recent completion of the classification program for separable, simple, unital, nuclear C*-algebras in the presence of the UCT (see work of Elliott-Gong-Lin-Niu and Tikuisis-White-Winter). Nevertheless, this recent route to classification relies heavily on complex abstract machinery which our proof of quasidiagonality allows us to avoid in the case that there is a unique trace on our C*-algebra. In this setting, work of Matui-Sato, Lin-Niu and Winter will provide a more direct proof of classification when our algebra has finite nuclear dimension.

With further work on the project, we have now successfully proved that C*-algebras associated to tilings are classifiable by their Elliott invariant by showing that the algebras tensorially absorb the Jiang-Su algebra Z. In order to do this, we extend existing notions of a property known as "almost finiteness" to the groupoid actions from which these algebras can be thought of as arising. This property allows us to approximate the action arbitrarily well in a sense by permuting a finite number of "blocks". This in turn allows us to access Z-stability of our algebra using a matrix embeddability condition due to Hishberg and Orovitz.
Exploitation Route Nuclear dimension estimates always appear as upper bounds on the true value. In our case, we know that our upper bound must be greater than or equal to one (because for separable C*-algebras, nuclear dimension zero is equivalent to being AF, which some tiling algebras are not). The general sense in the community appears to be that the "correct" value of nuclear dimension is either zero or one. Our methods certainly do not preclude obtaining such a bound, but any finite bound will do for our purposes. Therefore, if we do end up with a finite nuclear dimension bound that is greater than one, there is scope for this bound to be improved in future work.
From an applications standpoint, the classifiability of algebras which are associated to quasicrystals may provide additional motivation to actually compute the associated invariants. Doing so may have applications for the construction and analysis of quasicrystal structures, leading to developments in materials science. Quasicrystalline materials have potential uses in a wide range of areas due to their low friction and heat conductivity and resistance to corrosion.
Sectors Aerospace, Defence and Marine,Chemicals,Construction,Digital/Communication/Information Technologies (including Software),Electronics,Energy,Environment,Healthcare,Manufacturing, including Industrial Biotechology,Transport
URL https://arxiv.org/abs/1908.00770