Examples of subhomogeneous Banach and operator algebras
Lead Research Organisation:
Lancaster University
Department Name: Mathematics and Statistics
Abstract
PROJECT DESCRIPTION FOR BLAKE GREEN.
[Intended classifications: 70% Mathematical Analysis, 30% Algebra and Geometry]
Banach algebras are abstract models that can, in certain settings, provide a unified framework for a range of related problems or phenomena in mathematical analysis. Studying them allows the transfer of ideas and techniques between algebra and analysis.
A Banach algebra is said to be n-subhomogeneous if all its irreducible representations have degree at most n. When n=1 this implies the algebra is commutative, and hence many tools from classical commutative algebra and the analytical variants developed by Gelfand, Shilov and others in the 1940s-1950s can be applied. In contrast, for n=2 these tools are inadequate, since we enter the noncommutative realm.
The best-understood class of noncommutative Banach algebras is the class of C*-algebras, and n-subhomogeneous C*-algebras admit a very satisfactory theory of their own. However, the foundational results for such objects rely heavily on features that seem to be unique to the C*-setting, and so there has been relatively little attention paid to more general n-subhomogeneous Banach algebras. The need for a better understanding of these more general objects is higlighted by recent work of Choi-Farah-Ozawa (2014), in which the main example is a 2-subhomogeneous operator algebra with certain "exotic" or "pathological" properties.
This PhD project lies in the intersection of mathematical analysis with noncommutative ring theory. It has two main objectives, which will be pursued as parallel strands.
1) Review and extend some of the existing structural results for subhomogeneous C*-algebras in the more general setting of subhomogeneous operator algebras, thereby putting the Choi-Farah-Ozawa example in proper context. There is currently very little in the existing literature, so the existing methodology of the C*-algebraic setting will need to be refined or augmented with new techniques developed during the PhD. This strand aims to organize some existing folklore but also to map out new territory by means of examples and counterexamples that do not occur in the C*-algebraic setting.
2) Explore particular families of subhomogeneous Banach algebras that arise naturally in mathematical analysis, specifically function theory and abstract harmonic analysis: e.g. Banach algebras of matrix-valued differentiable functions, or Banach convolution algebras associated to crystallographic groups. In particular, to what extent can the techniques developed by Gelfand et al. for the n=1 setting be adapted to analyse the structure of these examples? Here, the plan is to start developing new methodology that is a hybrid of Gelfand's functional-analytical techniques with ideas from the world of pure algebra (so-called "P.I. rings").
Taken together, the aim of the two strands is to obtain an improved understanding of a natural class of mathematical objects, with both concrete examples and general theory that could be built upon by subsequent researchers.
[Intended classifications: 70% Mathematical Analysis, 30% Algebra and Geometry]
Banach algebras are abstract models that can, in certain settings, provide a unified framework for a range of related problems or phenomena in mathematical analysis. Studying them allows the transfer of ideas and techniques between algebra and analysis.
A Banach algebra is said to be n-subhomogeneous if all its irreducible representations have degree at most n. When n=1 this implies the algebra is commutative, and hence many tools from classical commutative algebra and the analytical variants developed by Gelfand, Shilov and others in the 1940s-1950s can be applied. In contrast, for n=2 these tools are inadequate, since we enter the noncommutative realm.
The best-understood class of noncommutative Banach algebras is the class of C*-algebras, and n-subhomogeneous C*-algebras admit a very satisfactory theory of their own. However, the foundational results for such objects rely heavily on features that seem to be unique to the C*-setting, and so there has been relatively little attention paid to more general n-subhomogeneous Banach algebras. The need for a better understanding of these more general objects is higlighted by recent work of Choi-Farah-Ozawa (2014), in which the main example is a 2-subhomogeneous operator algebra with certain "exotic" or "pathological" properties.
This PhD project lies in the intersection of mathematical analysis with noncommutative ring theory. It has two main objectives, which will be pursued as parallel strands.
1) Review and extend some of the existing structural results for subhomogeneous C*-algebras in the more general setting of subhomogeneous operator algebras, thereby putting the Choi-Farah-Ozawa example in proper context. There is currently very little in the existing literature, so the existing methodology of the C*-algebraic setting will need to be refined or augmented with new techniques developed during the PhD. This strand aims to organize some existing folklore but also to map out new territory by means of examples and counterexamples that do not occur in the C*-algebraic setting.
2) Explore particular families of subhomogeneous Banach algebras that arise naturally in mathematical analysis, specifically function theory and abstract harmonic analysis: e.g. Banach algebras of matrix-valued differentiable functions, or Banach convolution algebras associated to crystallographic groups. In particular, to what extent can the techniques developed by Gelfand et al. for the n=1 setting be adapted to analyse the structure of these examples? Here, the plan is to start developing new methodology that is a hybrid of Gelfand's functional-analytical techniques with ideas from the world of pure algebra (so-called "P.I. rings").
Taken together, the aim of the two strands is to obtain an improved understanding of a natural class of mathematical objects, with both concrete examples and general theory that could be built upon by subsequent researchers.
Organisations
People |
ORCID iD |
| Blake Green (Student) |
Studentship Projects
| Project Reference | Relationship | Related To | Start | End | Student Name |
|---|---|---|---|---|---|
| EP/N509504/1 | 01/10/2016 | 30/09/2021 | |||
| 1943819 | Studentship | EP/N509504/1 | 01/10/2017 | 30/09/2021 | Blake Green |
| Description | Thus far, work was done to expand on results from a recent work of Choi-Farah-Ozawa (2014) by taking the example amenable operator algebra and doing further investigation on its irreducible representations (including that it is 2-subhomogeneous) and went on to show what the Fell topology looked like on its dual space (this topology was defined by Fell in 1965). The Fell topology for the algebra defined by Choi-Farah-Ozawa was shown to "look like" the natural numbers with some "interesting" points at infinity. All the results proven about this have been made ready to become a chapter in the final thesis. New work has also begun on an offshoot of the problem from the Choi-Farah-Ozawa paper. In the 2014 paper it was shown that there was an example of an amenable operator algebra that was not isomorphic to a C*-algebra. However, the example found was a subalgebra of bounded sequences of 2 x 2 matrices. The question currently is whether all amenable operator subalgebras of convergent sequences of 2 x 2 matrices are isomorphic to a C*-algebra. The belief is yes, and we seek to prove this. |
| Exploitation Route | Methods used in the current problem of proving that all amenable subalgebras of convergent sequences of 2 x 2 matrices expand on ideas used by Gifford (2006) and will hopefully evolve into a kind of inductive argument which can be used in other cases. |
| Sectors | Other |