# Regularity in affiliated von Neumann algebras and applications to partial differential equations

Lead Research Organisation:
Imperial College London

### Abstract

The proposed research will concentrate on the development of the regularity theory in affiliated von Neumann algebras and its subsequent applications to several areas of analysis and the theory of partial differential equations.

The subject of the regularity of spectral and Fourier multipliers has been now a topic of intensive continuous research over many decades due to its importance to many areas. Indeed, solutions to main equations of mathematical physics such as Schrödinger, wave, Klein-Gordon, relativistic Klein-Gordon, and many other equations can be written as spectral multipliers, i.e. functions of the operator governing the equation (e.g. the Laplacian). Multiplier theorems and their further dependence (decay) for large times has been a building block of the so-called dispersive estimates, implying further Strichartz estimates, nowadays being the main tool for investigating the global in time well-posedness of nonlinear equations. This scheme has many variants motivated by a variety of settings of the mathematical physics, with different operators replacing the Laplacian, different types of potentials, and different types of nonlinearities.

The present project aims at bringing the modern techniques of von Neumann algebras into these investigations. Indeed, several results known in the simplest Euclidean setting allow for their interpretation in terms of the functional subspaces of affiliated von Neumann algebras, or rather of spaces of (densely defined) operators affiliated to the von Neumann algebra of the space. This can be the group von Neumann algebra if the underlying space has a group structure, or von Neumann algebras generated by given operators on the space, such as the Dirac operator of noncommutative geometry or the one in the setting of quantum groups.

In this approach we can think of multipliers as those operators that are affiliated to the given von Neumann algebra (the affiliation is an extension of the inclusion, setting up a rigorous framework, after John von Neumann, for doing spectral analysis or functional calculus of unbounded operators with complicated spectral structure). We are interested in developing a new approach to proving multiplier theorems for operators on different function spaces by looking at their regularity in the relevant scales of regularity in the affiliated von Neumann algebras. The aim of the project is two-fold: to make advances in a general theory, but keeping in mind all the particular important motivating examples of settings (groups, manifolds, fractals, and many others that are included in this framework) and of evolution PDEs, with applications to the global in time well-posedness for their initial and initial-boundary problems. As such, it will provide a new approach to establishing dispersive estimates for their solutions, the problem that is long-standing and notoriously difficult in the area of partial differential equations with variable coefficients or in complicated geometry.

This is important, challenging and timely research with deep implications in theories of noncommutative operator analysis and partial differential equations, as well as their relation to other areas and applications.

The subject of the regularity of spectral and Fourier multipliers has been now a topic of intensive continuous research over many decades due to its importance to many areas. Indeed, solutions to main equations of mathematical physics such as Schrödinger, wave, Klein-Gordon, relativistic Klein-Gordon, and many other equations can be written as spectral multipliers, i.e. functions of the operator governing the equation (e.g. the Laplacian). Multiplier theorems and their further dependence (decay) for large times has been a building block of the so-called dispersive estimates, implying further Strichartz estimates, nowadays being the main tool for investigating the global in time well-posedness of nonlinear equations. This scheme has many variants motivated by a variety of settings of the mathematical physics, with different operators replacing the Laplacian, different types of potentials, and different types of nonlinearities.

The present project aims at bringing the modern techniques of von Neumann algebras into these investigations. Indeed, several results known in the simplest Euclidean setting allow for their interpretation in terms of the functional subspaces of affiliated von Neumann algebras, or rather of spaces of (densely defined) operators affiliated to the von Neumann algebra of the space. This can be the group von Neumann algebra if the underlying space has a group structure, or von Neumann algebras generated by given operators on the space, such as the Dirac operator of noncommutative geometry or the one in the setting of quantum groups.

In this approach we can think of multipliers as those operators that are affiliated to the given von Neumann algebra (the affiliation is an extension of the inclusion, setting up a rigorous framework, after John von Neumann, for doing spectral analysis or functional calculus of unbounded operators with complicated spectral structure). We are interested in developing a new approach to proving multiplier theorems for operators on different function spaces by looking at their regularity in the relevant scales of regularity in the affiliated von Neumann algebras. The aim of the project is two-fold: to make advances in a general theory, but keeping in mind all the particular important motivating examples of settings (groups, manifolds, fractals, and many others that are included in this framework) and of evolution PDEs, with applications to the global in time well-posedness for their initial and initial-boundary problems. As such, it will provide a new approach to establishing dispersive estimates for their solutions, the problem that is long-standing and notoriously difficult in the area of partial differential equations with variable coefficients or in complicated geometry.

This is important, challenging and timely research with deep implications in theories of noncommutative operator analysis and partial differential equations, as well as their relation to other areas and applications.

### Planned Impact

As it is often the case with pure mathematics, the main impact will be academic. However, the range of the academic beneficiaries will be potentially very wide as the area of the operator analysis and its applications to partial differential equations influences advances in a variety of subjects. As it is written more specifically in the "Academic Beneficiaries" section, the expected impact on mathematics (and possibly on theoretical physics) is expected to be substantial. Besides these, there is a link to a range of applications of nonlinear PDEs through the planned work in the direction of dispersive and Strichartz estimates, and thus part of our research will be applicable there. The impact to this end is specified in more detail in the Pathways to Impact supplement to this application.

In addition to the academic aspects, in order to maximise the impact and exploitation of the EPSRC investment in this research, and to increase the knowledge transfer, we plan to organise an intensive workshop/conference devoted to the topic of the grant. A high-profile meeting would be extremely useful, to communicate the obtained results to the leading experts in the field of noncommutative analysis, the main topic of the EPSRC grant, to colleagues working in its applications, to discuss the achievements and future developments, thus also increasing the long-term influence of the conducted research. Inviting the leading mathematicians from a variety of countries working in the field will certainly significantly contribute to the worldwide academic advancement of the area highlighting in a unique way the results obtained during our project. Communicating the research findings in an especially designed meeting to an internationally wide-spread selection of world leaders in the field would be an ideal way to facilitate and to maximise the knowledge transfer related to this research. A minicourse given by the PI/RA planned in the framework of the meeting will contribute to the training of highly skilled researchers and the participation of PhD students and young postdocs will be very useful for improving teaching and learning. The meeting will also serve as an excellent way of identifying further research areas that would be influenced by the conducted research in a longer run. Consequently, we will consider editing and producing a volume of research papers originating from the meeting to increase its visibility and impact.

In addition to the academic aspects, in order to maximise the impact and exploitation of the EPSRC investment in this research, and to increase the knowledge transfer, we plan to organise an intensive workshop/conference devoted to the topic of the grant. A high-profile meeting would be extremely useful, to communicate the obtained results to the leading experts in the field of noncommutative analysis, the main topic of the EPSRC grant, to colleagues working in its applications, to discuss the achievements and future developments, thus also increasing the long-term influence of the conducted research. Inviting the leading mathematicians from a variety of countries working in the field will certainly significantly contribute to the worldwide academic advancement of the area highlighting in a unique way the results obtained during our project. Communicating the research findings in an especially designed meeting to an internationally wide-spread selection of world leaders in the field would be an ideal way to facilitate and to maximise the knowledge transfer related to this research. A minicourse given by the PI/RA planned in the framework of the meeting will contribute to the training of highly skilled researchers and the participation of PhD students and young postdocs will be very useful for improving teaching and learning. The meeting will also serve as an excellent way of identifying further research areas that would be influenced by the conducted research in a longer run. Consequently, we will consider editing and producing a volume of research papers originating from the meeting to increase its visibility and impact.

### Organisations

## People |
## ORCID iD |

Michael Ruzhansky (Principal Investigator) |

### Publications

Akylzhanov R
(2019)

*Hardy-Littlewood, Hausdorff-Young-Paley inequalities, and L-L Fourier multipliers on compact homogeneous manifolds*in Journal of Mathematical Analysis and Applications
Akylzhanov R
(2018)

*Smooth Dense Subalgebras and Fourier Multipliers on Compact Quantum Groups*in Communications in Mathematical Physics
Akylzhanov R
(2020)

*L-L multipliers on locally compact groups*in Journal of Functional Analysis
Akylzhanov R
(2020)

*Re-expansions on compact Lie groups*in Analysis and Mathematical Physics
Altybay A
(2021)

*The heat equation with strongly singular potentials*in Applied Mathematics and Computation
Altybay A
(2020)

*A parallel hybrid implementation of the 2D acoustic wave equation*in International Journal of Nonlinear Sciences and Numerical Simulation
Botchway L
(2020)

*Difference equations and pseudo-differential operators on Z n*in Journal of Functional Analysis
Cardona D
(2023)

*Boundedness of pseudo-differential operators in subelliptic Sobolev and Besov spaces on compact Lie groups*in Complex Variables and Elliptic Equations
Daher R
(2019)

*Titchmarsh theorems for Fourier transforms of Hölder-Lipschitz functions on compact homogeneous manifolds*in Monatshefte für Mathematik
Dasgupta A
(2017)

*Erratum to "The Gohberg Lemma, compactness, and essential spectrum of operators on compact Lie groups"*in Journal d'Analyse MathématiqueDescription | The general procedure has been developed for deriving Lp-Lq estimates for Fourier and spectral multipliers. In the setting of locally compact groups and compact quantum groups the technique relies heavily on the von Neumann algebras theory that becomes instrumental in handling the spectral properties of the appearing operators. The research continues into the direction of simplification of the proofs as well as in its extensions to other settings. The main research is accompanied by the related sub Riemannian research allowing one to obtain additional properties of the sub elliptic operators. |

Exploitation Route | The developed techniques are now actively applied in a variety of related settings (non-harmonic analysis, symmetric spaces, Jacobi operators, anharmonic oscillators). |

Sectors | Other |

URL | https://analysis-pde.org/von-neumann/ |