Research

I work in the broad area of Functional Analysis and Operator Algebras with a particular interest in groups, group actions, and "quantum" generalisations thereof. I also have some (decreasing) interests in Data Science, see the pages on computation.

Current activity

• Learning the ropes at Lancaster University

I am on the editorial board for the Bulletin of the Iranian Mathematical Society. To submit a paper please see the journal website.

I have acted as an editorial advisor for the Glasgow Mathematics Journal and the London Mathematical Society family of journals. My research has been supported by the EPSRC.

Publications

ORCID: 0000-0003-1707-4308 and arXiv.

1. "Averaging multipliers on locally compact quantum groups" with Jacek Krajczok and Christian Voigt. This paper is available on the ArXiV.
2. "The approximation property for locally compact quantum groups" with Jacek Krajczok and Christian Voigt. Published in Advances in Mathematics Volume 438 (2024) 109452. This paper is available on the ArXiV.
3. "Quantum graphs: different perspectives, homomorphisms and quantum automorphisms". Accepted by Communications of the American Mathematical Society. This paper is available on the ArXiV.
4. "A purely infinite Cuntz-like Banach ∗-algebra with no purely infinite ultrapowers" with Bence Horváth. Published in J. Funct. Anal. 283, No. 1, Article ID 109488, 36 p. (2022). This paper is available on the ArXiV.
5. "Completely bounded homomorphisms of the Fourier algebra revisited". Published in J. Group Theory 25, No. 3, 579-600 (2022). This paper is available on the ArXiV.
6. "Ring-theoretic (in)finiteness in reduced products of Banach algebras" with Bence Horváth. Published in Can. J. Math. 73 (2021), no. 5, 1423--1458. This paper is available on the ArXiV.
7. "One-parameter isometry groups and inclusions between operator algebras". Published in New York J. Math. 27 (2021) 164--204. This paper is available on the ArXiV.
8. "Admissibility Conjecture and Kazhdan's Property (T) for quantum groups" with Biswarup Das and Pekka Salmi. Published in J. Funct. Anal. 276 (2019), no. 11, 3484--3510. This paper is available on the ArXiV.
9. "On convoluters on $L^p$-spaces" with Nico Spronk. Published in Studia Math. 245 (2019), no. 1, 15--31. This paper is available on the ArXiV.
10. "Around Property (T) for quantum groups" with Adam Skalski and Ami Viselter. Published in Comm. Math. Phys. 353 (2017), no. 1, 69--118 This paper is available on the ArXiV.
11. "Categorical aspects of quantum groups: multipliers and intrinsic groups". Published in the Canadian Journal of Mathematics volume 68 (2016), 309--333. This paper is available on the ArXiV.
12. "Non-commutative separate continuity and weakly almost periodicity for Hopf von Neumann algebras". Published in J. Funct. Anal. Volume 269 (2015) 683--704. This paper is available on the ArXiV.
13. "Quantum Eberlein compactifications and invariant means" with Biswarup Das. Published in Indiana University Mathematics Journal Vol 65 No 1 (2016) 307--352. This paper is available on the ArXiV.
14. "Remarks on the Quantum Bohr Compactification". Published in Illinois J. Math. Volume 57, Issue 4, 1131--1171. This paper is available on the ArXiV.
15. "The Haagerup property for locally compact quantum groups" with Pierre Fima, Adam Skalski and Stuart White. Published in Crelle's Journal (Journal für die reine und angewandte Mathematik) Volume 2016 Issue 711, 189--229. This paper is available on the ArXiV.
16. "Equivalence of multi-norms" with H. Garth Dales, Hung Le Pham and Paul Ramsden. Published in Dissertationes Math. Volume 498 (2014) 53pp.
17. "Completely positive definite functions and Bochner's theorem for locally compact quantum groups" with Pekka Salmi. Published in J. Funct. Anal. Volume 264, Issue 7, 1525-1546. This paper is available on the ArXiV.
18. "Closed quantum subgroups of locally compact quantum groups" with Paweł Kasprzak, Adam Skalski and Piotr Sołtan. Published in Advances in Mathematics Volume 231 (2012) 3473-3501. This paper is available on the ArXiV.
19. "Completely positive multipliers of quantum groups". Published in International Journal Mathematics volume 23, no 12 (2012). This paper is available on the ArXiV.
20. "Completely bounded representations of convolution algebras of locally compact quantum groups" with Michael Brannan and Ebrahim Samei. Published in Münster Journal of Mathematics volume 6 (2013) 445-482. This paper is available on the ArXiV.
21. "Isometries between quantum convolution algebras" with Hung Le Pham. Published in The Quarterly Journal of Mathematics 64 (2013) 373-396. This paper is available on the ArXiV.
22. "Shift invariant preduals of $\ell_1(\Z)$" with Richard Haydon, Thomas Schlumprecht and Stuart White. Published in Israel J. Math. 192 (2012), 541–585. This paper is available on the ArXiV.
23. "Multi-norms and the injectivity of $L^p(G)$" with H. Garth Dales, Hung Le Pham and Paul Ramsden. Published in J. Lond. Math. Soc. Volume 86 (2012) 779-809. This paper is available on the ArXiV.
24. "Multipliers of locally compact quantum groups via Hilbert C$^*$-modules". Published in J. Lond. Math. Soc. Volume 84 (2011) 385-407. This paper is available on the ArXiV.
25. "Multipliers, Self-Induced and Dual Banach Algebras", published in Dissertationes Mathematicae volume 470 2010, 62pp. This paper is available on the ArXiV.
26. "A bicommutant theorem for dual Banach algebras". Published in the Mathematical Proceedings of the Royal Irish Academy 111 (2011) 21-28. This paper is available on the ArXiV.
27. "Characterising weakly almost periodic functionals on the measure algebra". This paper is available on the ArXiV. published in Studia Mathematica Volume 204, 2011, 213--234.
28. "Representing multipliers of the Fourier algebra on non-commutative $L^p$ spaces", published in Canadian Journal of Mathematics Volume 63, 2011, 798--825. This paper is available on the ArXiV.
29. "Operator Biprojectivity of Compact Quantum Groups", published in Proceedings of the American Mathematical Society Volume 138, 2010, 1349-1359. This paper is available on the ArXiV.
30. "Preduals of semigroup algebras" with Hung Le Pham and Stuart White, published in Semigroup Forum Volume 80, 2010, 61--78. This paper is available on the ArXiV.
31. "Reiter's properties (P_1) and (P_2) for locally compact quantum groups", with Volker Runde, published in Journal of Mathematical Analysis and Applications Volume 364, Issue 2, 2010, 352-365. This paper is available on the ArXiV. See Corrections!.
32. "Weakly almost periodic functionals on the measure algebra", published in Mathematische Zeitschrift Volume 265 (2010) 285-296. This paper is available on the ArXiV.
33. "Conditions implying the uniqueness of the weak$^*$-topology on certain group algebras" with Hung Le Pham and Stuart White. Published in Houston Journal of Mathematics Volume 35 (number 1) 2009, 253-276. This paper is available on the ArXiV.
34. "Can B(l^p) ever be amenable?" with Volker Runde, published in Studia Mathematica Volume 188 (number 2) 2008, 151-174. See also Erratum, Studia Mathematica, Volume 195 2009, 297-298. This paper is available on the ArXiV. See Corrections!.
35. "Ultrapowers of Banach algebras and modules", published in Glasgow Mathematics Journal, Volume 50, Issue 03, 2008, 539-555. DOI:10.1017/S0017089508004400. This paper is available on the ArXiV.
36. "Amenability of ultrapowers of Banach algebras", published in Proceedings of the Edinburgh Mathematical Society, Volume 52, 2009, 307-338. DOI:10.1017/S0013091507001083. This paper is available on the ArXiV. See Corrections!.
37. "p-Operator Spaces and Figa-Talamanca-Herz Algebras", published in The Journal of Operator Theory, Volume 63, 2010, 47-83. This paper is available on the ArXiV.
38. "Arens-regularity of Algebras arising from Tensor Norms", published in the New York Journal of Mathematics Volume 13 (2007) 215-270. This is a free to access online journal.
39. "Dual Banach algebras: representations and injectivity", published in Studia Mathematica Volume 178 (number 3) 2007, 231-275. This paper is available on the ArXiV.
40. "Connes-amenability of bidual and weighted semigroup algebras", published in Mathematica Scandinavica Volume 99, Issue 2, 2006, 217-246. This paper is available on the ArXiV (warning: old version).
41. "Closed ideals in the Banach algebra of operators on classical nonseparable spaces" published in Math. Proc. Camb. Phil. Soc. Volume 140, Issue 02, March 2006, pp 317-332. DOI:10.1017/S0305004105009102. Download preprint: Acrobat PDF.
42. "Semi-simplicity of B(E)**" with Charles Read, published in Journal of Functional Analysis, Volume 219, Part 1, (2005), pp 177-204. DOI:10.1016/j.jfa.2004.04.014. Download preprint: Acrobat PDF.
43. "Arens regularity of the algebra of operators on a Banach space" published in Bulletin of the London Mathematical Society, Volume 36, Part 4, (2004), pp 493-503. DOI:10.1112/S0024609303003072. Download preprint: Acrobat PDF.

Corrections

Some of my early papers have corrections, which can be download: corrections [PDF]. It seems I became better at checking as time went on!

Seminars

As the covid crisis continues, one silver lining has been the ability to attend virtual seminars (so I am in fact watching way more Mathematics than I usual can!). I currently attend seminars from these series, sadly increasingly rarely:

• Wales MPPM Zoom Seminar : Physics, but also operator algebras; Tuesdays 4:30pm (approx) UK time (need to joining mailing list for Zoom coordinates)
• Quantum groups seminar : With a large proportion of talks about Operator Algebraic Quantum Groups.
• Banach spaces webinars : Fridays 3pm UK time.
• Arizona C*-seminar : Wednesdays 9pm UK time (usually recorded).
• UT Groups and Dynamics : Link to their YouTube site

Some older seminars which are no longer running or which I cannot attend:

• UK Virtual Operator Algebra seminar : I helped organise, but now in abeyance.
• GOBA Webinar : For early careers academics, but veterans are allowed in the back row, as it were. Seems to have paused.
• Virtual Noncommutative Analysis Weekly Seminar, now stopped.
• NCG Learning seminar, now stopped.
• UCLA Functional Analysis : Some great talks, but out of my time zone
• Oslo Cstar seminar : One can join the mailing list for Zoom access.
• Quantum group seminar : Ran over the summer by Teo Banica, with a wide remit
• Research Seminars : A large clearing house of math seminars (so many as to be overwhelming)

Supervision

I have had one PhD student work with me:

• Dr Steven Trotter whose thesis can be found here.

I was the Postdoc supervisor of Dr Biswarup Das, now a lecturer at Wroclaw University, Poland.

I have acted as the external PhD examiner for:

• Matthew Tarbard University of Oxford. Thesis is arXiv:1309.7469 [math.FA]
• Andrew Monk University of Glasgow.
• Bence Horváth University of Lancaster

MathSciNet

I have written about 50 MathSciNet reviews; if you have access, you can see the processed reviews here.

MathOverFlow

I spend a little time on MathOverflow, although the math.FA and math.OA community is rather sparse these days.

Thesis

My thesis was mined fairly successfully for papers (I think 6 in the end) but the original may still be of interest:

• My PhD Thesis [PDF]

Notes

• The group algebra derivation problem [PDF]: A self-contained proof of the group derivation problem, using the new proof by Bader et al.
• "Functorial properties of weakly almost periodic functionals on the measure algebra". This has been superseded by a later paper, as the referee noticed a much, much better way of proving the main result. However, perhaps the techniques used in this paper might be of interest: in particular, is there a non-commutative generalisation? This paper is available on the ArXiV, and will not otherwise be published (be aware that material in the first and final section is dealt with better in arXiv:0904.0436 (current version)).
• "Weakly almost periodic functionals, representations, and operator spaces": This paper shows that a completely contractive dual Banach algebra is isometric to a weak*-closed subalgebra of the algebra of completely bounded operators on a reflexive operator space. This result has been shown independently by F. Uygul (math.FA/0702200 on the arXiv), using rather similar arguments to my presentation, which will hence remain only on the arXiv.
• Bases in Banach spaces [PDF]: Introduction to Schauder bases, basic sequences, unconditional bases, and the Gowers dichotomy theorem. Prepared for a talk.
• Interpolation spaces [PDF]: Some notes on Beauzamy's book. Prepared for a talk.

Research notes

• Compact Quantum Groups some original notes on Compact Quantum Groups, starting with a summary (presenting the material from the perspective of locally compact quantum groups) and then developing the theory from scratch. Very much a work in progress (now stalled).
• Unitaires multiplicatifs et dualite pour les produits croises de C*-algebres a translation of the much cited paper by Baaj and Skandalis.

More recently, I have been experimenting with putting LaTeX source on GitHub, where you can now find various notes: MatthewDaws/Mathematics.

Talks

Most of my research talks.

• Noncommutative graphs given at the Lancaster Open Problems in Mathematics seminar: this is aimed at undergraduate students, to give them a flavour of what Mathematicians do, when no teaching. Notice that I lie near the end, and don't talk about commutants.
• Approximation properties and averaging for Drinfeld Doubles given at the Noncommutative analysis on groups and quantum groups conference in Besancon, December 2023.
• Non-commutative graphs given at the Functional Analysis Seminar in Oxford. Somewhat familiar theme to the talk, but maybe some new angles shown.
• Around the Approximation Property for Quantum Groups given at 2023 IMAC Abstract Harmonic Analysis day. Of course bears more than a passing similarity to some talks below, and I am incapable of keeping to time on this topic, it seems.
• Quantum graphs and homomorphisms given at the Noncommutative Harmonic Analysis and Quantum Information conference, Sweden, June 2023. Similar to other talks I've given of late.
• Perspectives on Noncommutative Graphs given at the Seoul National University seminar, April 23. Rather similar to talks below, but with some minor typos corrected.
• Around the Approximation Property for (Quantum) Groups given at the Glasgow Analysis seminar April 2023. Rather similar to some talks below.
• Around the Approximation Property for (Quantum) Groups given at the Functional Analysis and Operator Algebras in Athens Seminar March 2023. A more gentle version of the talk below, but making the rookie error that adding detail will, of course, take up more time. Still, explained the non-quantum bit.
• Around the Approximation Property for Quantum Groups given at the Quantum Groups Seminar January 2023.
• Perpsectives on Noncommutative Graphs given at the Noncommutative harmonic analysis and quantum groups conference, Sep 2022.
• A whiteboard only talk to Lancaster staff and graduate students: "Introduction to Compact Quantum Groups", May 2022.
• Perspectives on Noncommutative Graphs given at the Ottawa (virtual) analysis seminar, April 2022.
• Quantum automorphisms of quantum graphs given at a Hybrid meeting between Lancaster and Besançon Nov 21
• Noncommutative Graphs given at Lancaster, Nov 21
• Purely infinite algebras and ultrapowers given at the satellite meeting of IWOTA 2021, in honour of the retirement of Garth Dales. Similar to the talk below.
• Purely infinite algebras and ultrapowers given at the Athens Functional Analysis seminar, May 2021.
• An introduction to (quantum) symmetries of graphs given at the BM(A)C 2021. You might also consult Schmidt's talk at the Workshop Website.
• Asymptotic sequence algebras, and ultrapowers, of Banach algebras given at the Belfast Analysis seminar, October 2019.
• Fields of algebras given at the Workshop on Harmonic Analysis and Operator Theory, Istanbul August 2019.
• Kaplansky Density for automorphism groups given at Banach Algebras 2019 Canada July 2019.
• Kaplansky Density for automorphism groups given at the Glasgow Analysis seminar, May 2019.
• Kaplansky Density for automorphism groups given at the Newcastle Analysis seminar, March 2019.
• Reproducible research: assessing spatial predictions of crime given at the Leeds Institute for Data Science seminar, 2017.
• A non-commutative notion of separate continuity given to the Analysis seminar of TCD and UCD. About the paper: arXiv 1409.7302.
• Fourier Algebras to Quantum Groups. Pure Maths colloquia November 2014.
• I gave four lectures at the Fields Institute as part of the Theme Period on Operator Spaces, Locally Compact Quantum Groups and Amenability. My talks were an introduction to operator algebraic quantum groups. Videos of my talks can be viewed on Fields Video Archive. Here are local copies of the slides: Talk 1, Talk 2, Talk 3, Talk 4. Here is a list of further reading which appears to have disappeared from the Fields website.
• Almost periodic functionals: Talk given at: Banach Algebras 2013. In no way resembles the next talk...
• Almost periodic functionals: Talk given at 2nd QoP meeting: C*-algebras and Banach algebras.
• Groups meet Analysis: The Fourier Algebra: Colloquium style talk given at York, May 2013. Similar to Kent talk, but hopefully even less technical, as now a mathematics (not pure) colloquia.
• The Haagerup approximation property for discrete quantum groups: Talk at ICMS meeting on C*-algebras.
• Positive definite functions on locally compact quantum groups: Talk at Cergy-Pontoise (Slightly unsatisfactory website).
• "Algebraic quantum groups to compact quantum groups": A white-board talk at NAG 5.
• The Fourier Algebra: Colloquium style talk given at Kent, January 2013. I didn't use all the slides!
• Shift invariant preduals: This was a whiteboard talk at YFAG, so here are my hand-written notes. The paper is available on the arxiv.
• Compactifications of quantum groups: Talk given at Queen's University Belfast, October 2012.
• Completely Positive Multipliers: Talk given at the CEMPI Inaugural Conference, Lille, France, September 2012.
• Analysis in Non-Commutative Mathematics: Talk given at the Young Functional Analysts' Workshop 2012, Oxford, March 2012. Really an introduction to Compact Quantum Groups.
• Shift invariant preduals of group algebras: Talk given at the Banach Algebras 2011 conference in Waterloo, Canada, August 2011. A differing take on the talk below!
• Shift-invariant preduals of $\ell^1(\mathbb Z)$: Talk given to the Analysis seminar at UCD, Dublin, February 2011.
• The Fourier algebra and homomorphisms: Talk given to the Analysis and Algebra seminar at Glagow, December 2010.
• Some operator algebra problems from Quantum Information: Talk given to the algebra seminar at Leeds, November 2010.
• Multipliers and Abstract Harmonic Analysis: Talk given at CAHAS 2010 August 2010. Printable version.
• Quantum groups by way of Operator algebras: Talk given At Belfast university, April 2010.
• Multipliers and the Fourier algebras, and non-commutative L^p spaces: Talk given at Birmingham University, March 2010. A third attempt on a theme!
• Multipliers of quantum groups from Hilbert C*-modules: Talk given at Lancaster University, March 2010.
• Multipliers and the Fourier algebras: Talk given at Nottingham University, January 2010. Somewhat similar to the talk beow.
• Multipliers and the Fourier algebras: Talk given at Banach Algebras 2009, Stefan Banach International Mathematical Center, Bedlewo, Poland. A handout version of the slides can be downloaded here.
• Biprojectivity and compact quantum groups: Colloquium talk given to the Pure Mathematics department as Leeds, April 2009. A handout version of the slides can be downloaded here.
• Weakly almost periodic functionals on the measure algebra [PDF]: Talks given at the CIRM, November 2008. This is very similar to the talk below, so the PDF file is just a printable version of my slides (as given to conference participants).
• Weakly almost periodic functionals on the measure algebra [PDF]: Talk given in Newcastle, October 2008.
• Compactifications and the Fourier algebra [PDF]: Talk given in Glasgow, February 2008.
• Quantum compactifications of the Fourier algebra [PDF]: Talk given at the Fields Institute, Toronto, December 2007. To my horror, they have recorded my talk, so the audio is available for download here.
• Compact Quantum Groups: Series of three talks in Leeds, on the whiteboard, November 2007.
• Uniqueness of preduals [PDF]: Talk given at Banach Algebras 07, in Laval, July 2007. This bears some similarly to the two talks below.
• The predual of $l^1(G)$ [PDF]: Talk given in Leeds, May 2007. This is an extension of the BMC below immediately below.
• How to make the bilateral shift weak$^*$-continuous [PDF]: Talk given at the BMC in Swansea, April 2007.
• p-operator spaces and Figa-Talamanca-Herz algebras [PDF]: Talk given in February 2007, in Oxford and at Texas A+M.
• Operator spaces and the Sz-Nagy Similarity Problem [PDF]: Talk given to the junior seminar, November 2006, in Oxford.
• Involutions on algebras of operators [PDF]: Talk given in Glasgow, May 2006.
• Dual Banach algebras [PDF]: Talk given at the British Mathematics Conference, April 2006, in Newcastle.

Conferences

I was (weakly) an organiser of the Quantum groups and interactions in Glasgow, May 2023.

For a time, I was the treasurer, and de facto secretary, of the NBFAS network. I also ran the Leeds node of the Quantum groups, operators and non-commutative probability network.

In 2010 I helped to organise a "Semester" of activities in Leeds. The website for this has been lost, but as it contained slides of talks (and I like to archive such things) I have reproduced it here:

• Banach algebra and operator space techniques in topological group theory, Leeds, 2010

A website for the April 2024 Mini-meeting in memory of Garth Dales.