C.J.Mulvey

C.J.Mulvey MA (Cambridge) MSc (Sheffield) DPhil (Sussex)
Reader Emeritus in Mathematics

This is my home page in the Department of Mathematics at the University of Sussex, with links to that of my home page in the Department of Pure Mathematics and Mathematical Statistics of the Centre for Mathematical Sciences of the University of Cambridge, at which I am now based, and to more detailed pages concerned with my research.

Areas of Research Interest

My research lies between ring theory and functional analysis, on the one hand, and category theory and logic, on the other. The unifying thread through this work has been the application of sheaf, topos and locale theoretic techniques to problems concerning representations of rings and C*-algebras, particularly in the non-commutative case. The applications of these ideas which have recently emerged have been to theoretical computer science, particularly in elucidating the links between quantisations of the calculus of relations and linear logic, and to foundations of quantum physics, particularly the algebraicisation of super-symmetry and the constructivisation of quantum mechanics and quantum gravity.

The motivating concern behind the research has been to develop the sheaf- and bundle-theoretic foundations to extend the Gelfand representation of C*-algebras from the commutative to the non-commutative case. Early work providing the algebraic foundation for this, led to an algebraic generalisation of the Gelfand representation to Gelfand rings, initially of importance in algebraic K-theory and more recently of significance in the algebraic foundations of super-symmetry theory.

The techniques developed in obtaining a generalisation of Swan's theorem to the case of Gelfand rings initiated a connection with constructive logic by locating these ideas within the broader context of the theory of toposes. In its turn, this allowed the development of a theory of Banach bundles unifying work stretching back to the fields of Hilbert spaces of Dixmier, while providing functional analytic techniques needed to work constructively with equivariant Banach bundles through syntactic and semantic aspects of the theory of locales.

More recently, the understanding of the geometric and the logical sides of spaces to which this led has allowed the introduction of a generalisation of the concept of locale to the non-commutative case in the paper, &. The concept of quantale introduced there allows a Gelfand representation for C*-algebras to be obtained, generalising that known in the commutative case, while providing insight into the connections between quantum mechanics and C*-algebras. In recent work, this has led to the concept of a point of a quantale, and thereby to the notion of a quantal space. In particular, the spectrum of any C*-algebra is a quantal space.

Selected Publications

&, Rend. Circ. Mat. Palermo 12 (1986), pp. 99-104.

A globalisation of the Hahn-Banach theorem (with J.W. Pelletier), Advances in Mathematics, 89 (1991), 1-59.

A quantisation of the calculus of relations (with J.W. Pelletier), Category Theory 1991, CMS Conference Proceedings 13, Amer. Math. Soc., 1992, 345-360.

Quantales: Quantal sets (with M. Nawaz), Non-Classical Logics and their Application to Fuzzy Subsets: A Handbook of the Mathematical Foundations of Fuzzy Set Theory, 159-217. Kluwer, 1995.

On the quantisation of points (with J.W. Pelletier). J. Pure Applied Algebra, 159 (2001), 231-295.

On the quantisation of spaces (with J.W. Pelletier). J. Pure Applied Algebra, 175 (2002), 289-325.

The Stone-Cech compactification of locales, III (with B. Banaschewski). J. Pure Applied Algebra, 185 (2003), 25-33.

A noncommutative theory of Penrose tilings (with P. Resende). Int. J. Theoret. Phys., To appear.

A globalisation of the Gelfand duality theorem (with B. Banaschewski). To appear.

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