Peter Dixon's Home Page
E-mail: P.Dixon with the usual extension: @sheffield.ac.uk
School of Mathematics and Statistics
University of Sheffield
For brief biographical information, click here.
Mathematical interests (research)
As an undergraduate, I was chiefly interested in mathematical logic,
but as a postgraduate I moved into functional analysis, specialising in
general Banach algebra theory (MR classes 46H, 46J). The great attraction
of this field as a research area is that it involves both analysis and
algebra in proportions which vary from problem to problem. Many interesting
developments come from asking natural algebraic questions about
Banach algebras which then turn into problems in pure analysis.
I am particularly interested in the following specific topics.
- Automatic continuity theory, including the longstanding Michael
problem of whether characters on Fréchet algebras are automatically
continuous. I have been working on this problem for more than thirty years,
with significant progress about once a decade.
- Topologically irreducible representations of Banach algebras.
Big problem: when do the topologically irreducible representations of a
Banach algebra separate points?
- Radicals in Banach algebras: for example, the intersection of
the kernels of the topologically irreducible representations. What can
we say about this and related radicals? The close relation between topologically
irreducible representations and the Invariant Subspace Problem makes this
whole area very intractible.
- Varieties of Banach algebras and questions related to the von
Neumann inequality (MR class 47D25). Sample problem: if a unital Banach
algebra satisfies the von Neumann inequality for polynomials (with constant
term) in one variable, does it follow that the algebra is isomorphic to
a (not necessarily self adjoint) closed subalgebra of the algebra B(H)
of bounded operators on some Hilbert space H? Another classic problem:
It is known that the von Neumann inequality for commuting operators fails
for polynomials in more than two variables; does it hold up to a constant
for polynomials in three variables?
Specific research topics can be seen from my list of publications.
I retired in October 2009, so I am no longer supervising research
students. Prospective postgraduates in this area are referred to Pure
Maths prospective postgraduates page and to the School of Mathematics and
Mathematical interests (teaching)
In 2009-10 I gave two courses: PMA324 Chaos
and PMA443 Fractals.
In 2003-4 I gave a level 1 mathematics for computer scientists course
PMA1050 Discrete Foundations (which
overlapped with an M.Sc. course under the course code PMA6853),
and a level 1 course PMA116 Further Subsidiary Mathematics
In 2003-4 I gave a level 3 course: PMA445 Functional Analysis.
In 1998-9, I gave a postgraduate course on set theory; a PDF file of
the notes may be downloaded by clicking here.
Dept. of Pure Maths
School of Mathematics and Statistics (SoMaS)