C.J.Mulvey MA (Cambridge) MSc (Sheffield)
DPhil (Sussex)
Reader Emeritus in Mathematics
Research
- A fairly comprehensive description of current
research interests may be found on my home page. The following
recent research articles may be viewed online, or downloaded and
printed. The journals in which they appear may be found by referring to
the list of research publications.
- It should be noted that the files in pdf format are both viewable
online, and far more efficiently downloadable for printing than those
in ps format. In particular, any fonts required are included within the
pdf file. The Adobe Acrobat software required to view pdf files is
available on most platforms without charge from www.adobe.com. Alternatively, please
email a request to c.j.mulvey@sussex.ac.uk
for a hardcopy version of the paper.
- A list of research
publications and information on forthcoming
papers is also available from this page, together with a brief curriculum vitae and my contact details. It is also possible
to download slides of recent
lectures and seminar talks.
A noncommutative theory of Penrose tilings
- This paper (with P.M.Resende) investigates the way in which the
noncommutative space of Penrose tilings of the plane can be presented
as an involutive unital quantale by considering a noncommutative theory
of Penrose tilings. Interestingly,
the theory is axiomatised in a way motivated by considering experiments
undertaken
on Penrose tilings in a way similar to that of a quantum situation, the
state
of the tiling being modified geometrically by interaction of the
experiments
with the tiling. The Lindenbaum quantale of this theory is shown to
have
relational points that correspond precisely with equivalence classes of
Penrose
sequences. The paper also shows explicitly the way in which the concept
of
an algebraically irreducible representation arises naturally in this
context.
- [PDF Version] [PS Version]
&
- This paper, presented at the Topology Meeting in Taormina, Sicily
in April, 1984, introduced the concept of quantale, outlining the
programme of
work in the spectral theory of C*-algebras and the constructive
foundations of quantum mechanics to which it was expected to
contribute. The paper is a slight development of that which appeared in
the Tagungsbericht of the Category
Meeting at Oberwolfach in September, 1983. It is included here since,
although
often quoted, it is more difficult to obtain in its published form in
the Rendiconti del Circulo Matematico di Palermo.
- [PDF Version] [PS
Version]
On the quantisation of points
- This paper (with J.W.Pelletier) investigates the extension of the
concept of point to non-commutative space, by examining the case of the
spectrum of
a C*-algebra. Axiomatising the spectrum by a (non-commutative)
propositional geometric theory motivates the consideration of
irreducible representations of the spectrum on orthocomplemented
sup-lattices. An intrinsic characterisation is obtained of those
representations that correspond to irreducible representations of the
C*-algebra on Hilbert space, yielding a concept of point which extends
that of a locale. In particular, this characterisation yields
independently that a C*-algebra is indeed determined by its spectrum.
- [PDF Version] [PS Version]
On the quantisation of spaces
- This paper (with J.W.Pelletier) applies the results of On the
quantisation of points to investigate the concept of spatiality for
involutive unital quantales and to identify the notion of a quantal
space. At the centre of the paper is the notion of a discrete von
Neumann quantale, providing the concept of a discrete quantal space,
and hence of the quantal power set of a non-commutative set. The work
is outlined in the article Quantales for the Encyclopaedia
of Mathematics. In particular, the paper shows that the spectrum of
a C*-algebra is a quantal space, providing a quantale context for the
work of Giles and Kummer and completing this aspect of the work
envisaged in the paper &.
- [PDF Version] [PS Version]
The Stone-Cech compactification of locales, III
- This paper (with B. Banaschewski) continues the investigation,
commenced in the papers The Stone-Cech compactification of locales and
The Stone-Cech compactification of locales, II (both with
B. Banaschewski), of constructive aspects of the Stone-Cech
compactification.
In the first of these papers, constructions of the compact completely
regular reflection and of the compact regular reflection of a locale
were obtained, these coinciding in the presence of the Axiom of
Countable
Dependent Choice. There was, however, a fundamental difference in the
approaches taken between these cases: the compact completely regular
reflection is obtainable directly as the locale of completely regular
ideals of the locale, while the compact regular reflection has a more
indirect description. In the second paper, the compact completely
regular
reflection was shown to be obtained as the Lindenbaum locale of the
propositional
theory of maximal ideals in the ring of bounded continuous real
functions
on the locale. In the present paper, it is shown that each of these
compactifications (and indeed any compactification) may be
obtained as the Lindenbaum locale of the theory of almost prime filters
of an appropriate kind.
[PDF Version] [PS Version]
Quantales
- This article, written as the entry on quantales for the Encyclopaedia
of Mathematics (published by Kluwer) briefly surveys some aspects
of the
theory of quantales, including a description in outline of recent work
(with
J.W.Pelletier) investigating spatiality of quantales. This work,
building on that contained in the earlier paper On the quantisation
of points, introduces the concept of quantal space, establishing
that the spectrum Max A of a C*-algebra A is indeed
spatial.
- [PDF Version] [PS Version]
A quantisation of the calculus of relations
- This paper (with J.W.Pelletier) introduces the concept of a
Hilbert quantale, generalising the quantale of relations on a set to
considering the
quantale of sup-preserving mappings on a sup-lattice, with the
motivating case being that of the sup-lattice of projections on a
Hilbert space. It is
shown that such a quantale has a natural involution exactly in the case
that
the sup-lattice is orthocomplemented, as is the case of that of
projections on a Hilbert space, with respect to which the Hilbert
quantale is a Gelfand quantale, of which the sup-lattice of right-sided
elements is canonically isomorphic to the given orthocomplemented
sup-lattice. Applying this, an intrinsic
characterisation of Hilbert quantales is obtained.
- [PDF Version] [PS Version]
On the geometry of choice
- This paper examines the way in which the application of the Axiom
of Choice may be avoided by carefully chosen constructive argument of a
geometric nature. Taking the case of the Hahn-Banach theorem as an
instance of this approach, the paper examines the way in which
successive intrusions of the Axiom of Choice in the classical proof may
be stripped away to yield a constructive treatment of the theorem,
rephrased within the context of locales rather than
topological space, obtained in joint work with J.J.C.Vermeulen. The
paper
is intended to be accessible to a broader audience within mathematics,
particularly
those working within non-classical logical contexts.
- [PDF Version] [PS Version]
Quantales: Quantal sets
- This paper (with M.Nawaz) investigates the axiomatisation and
equivalence of quantal sets and sheaves over an idempotent right-sided
quantale, providing an extension of the classical equivalence in the
case of a locale. The work is based in part on work contained in Nawaz'
1985 DPhil thesis, Quantales: Quantal Sets, at the University
of Sussex. It is shown that, with carefully appropriate axiomatisation,
the concepts of quantal set, complete quantal set, presheaf, and sheaf
may be defined in such a way that complete quantal sets over the
quantale correspond exactly to sheaves on the quantale. Moreover, the
category of quantal sets is naturally equivalent to that of sheaves,
through
a reflection corresponding to sheafification. The logical structure
with
which the category of sheaves on the quantale is endowed is explored in
a
forthcoming paper (with M.Nawaz) Quantales: Quantal logic.
- [PDF Version] [PS Version]
A constructive proof of the Stone-Weierstrass theorem
- This paper (with B.Banaschewski) establishes a constructive
version of the Stone-Weierstrass theorem. With the paper The
spectral theory of commutative C*-algebras (with B.Banaschewski)
and the forthcoming paper A globalisation of the Gelfand duality
theorem (also with B.Banaschewski), it forms part of a series of
paper establishing Gelfand duality for commutative C*-algebras in any
Grothendieck topos. The present paper establishes constructively that
any closed involutive subalgebra of the commutative C*-algebra of
continuous complex functions on a compact, completely regular locale
that separates the
locale is necessarily the C*-algebra itself. The method of proof is to
constructivise
the existence of finite partitions of unity over the locale to obtain
constructive
forms of results of one of the authors on Banach spaces over a compact
space.
Observing that the commutative C*-algebra concerned is that of global
sections
of the Dedekind complex numbers in the topos of sheaves on the compact,
completely
regular locale, the Stone-Weierstrass theorem is proved by observing
that
constructively the rational numbers are dense in the Dedekind real
numbers.
- [PDF Version] [PS Version]
The spectral theory of commutative C*-algebras: the constructive
spectrum
- This paper (with B.Banaschewski) investigates the constructive
theory of the spectra of commutative C*-algebras. With the paper A
constructive proof of the Stone-Weierstrass theorem (with
B.Banaschewski), the paper The spectral theory of commutative
C*-algebras: the constructive Gelfand-Mazur theorem, and the
forthcoming paper A globalisation of the Gelfand duality
theorem (also with B.Banaschewski), it forms part of a series of
papers
establishing Gelfand duality for commutative C*-algebras in any
Grothendieck
topos. The present paper introduces the propositional geometric theory
describing
the spectrum of a commutative C*-algebra in terms which correspond
classically
to that of multiplicative linear functionals in the weak* topology,
establishing
that the spectrum thereby constructed is indeed a compact, completely
regular
locale in the topos concerned, laying the ground for the proof in its
successor
paper of the constructive form of the Gelfand-Mazur theorem in the
context
of any Grothendieck topos.
- [PDF Version] [PS Version]
The spectral theory of commutative C*-algebras: the constructive
Gelfand-Mazur theorem
- This paper (with B.Banaschewski) investigates the constructive
theory of the spectra of commutative C*-algebras. With the paper A
constructive proof of the Stone-Weierstrass theorem (with
B.Banaschewski), the paper The spectral theory of commutative
C*-algebras: the constructive spectrum, and the forthcoming paper A
globalisation of the Gelfand duality theorem (also with
B.Banaschewski), it forms part of a series of papers establishing
Gelfand duality for commutative C*-algebras in any Grothendieck topos.
The present paper introduces the propositional geometric theory
describing the spectrum of a commutative C*-algebra in terms which
correspond classically to that of maximal ideals in the Zariski
topology, and establishes that the canonical interpretation of this
theory in that of multiplicative linear functionals
introduced in its predecessor paper establishes an equivalence of
theories,
hence an isomorphism of the spectra constructed. This equivalence,
relying
ultimately on the geometric equivalence of the topologies on the
complex
plane determined respectively by rational open codiscs and rational
open
rectangles, provides the constructive form of the Gelfand-Mazur
theorem.
- [PDF Version] [PS Version]
A globalisation of the Gelfand duality theorem
- This paper (with B.Banaschewski) completes the series of papers
establishing
Gelfand duality for commutative C*-algebras in any Grothendieck topos,
of
which the others (also with B.Banaschewski) are The spectral theory
of
commutative C*-algebras and A constructive proof of the
Stone-Weierstrass
theorem. The present paper introduces the Gelfand representation of
a
commutative C*-algebra, and proves it to be an isometric *-isomorphism
with
the commutative C*-algebra of continuous complex functions on the
spectrum,
from which the Gelfand duality theorem may then be obtained, before a
number
of instances and applications are examined. The paper contains enough
background from the preceding papers to be read independently.
- [PDF Version] [PS Version]
A geometric characterization concerning compact, convex sets
- This paper (with J.W.Pelletier) presents a geometric form of the
Hahn-Banach theorem in any Grothendieck topos, aimed at providing a
constructive approach to the Krein-Milman theorem. The characterisation
established shows that an element lies in a compact, convex subset of a
normed linear space exactly if it is bounded by all linear bounds of
the subset. The theorem is proved first in the particular case of the
space of real numbers, identifying the compact, convex subsets as the
intervals, in an appropriate sense. Applying the constructive form of
the Hahn-Banach theorem obtained in this context in A globalisation of the Hahn-Banach theorem
(also with J.W.Pelletier) allows the characterisation to be extended to
the general case, by taking the image of the compact, convex subset
along the generic bounded linear functional into the space of real
numbers over the unit ball of the dual of the normed space.
- [PDF Version] [PS Version]
The maximality of filters
- This paper introduces a constructive notion of maximality with
respect to which the classical theorems that establish that the prime
filters of a Boolean algebra are exactly the maximal filters, and that
the completely prime filters of a compact regular locale are exactly
the maximal regular filters may be proved constructively. Since these
results with the classical definition of maximality are known to be
equivalent to the presence of De Morgan's Law, it follows that the
constructive definition coincides with the classical one exactly in the
presence of De Morgan's Law.
- [PDF Version] [PS Version]
- [CJM Home Page]
© Christopher
Mulvey, 2004/05