C.J.Mulvey

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

Forthcoming Papers

A fairly comprehensive description of current research interests may be found on my home page. Information on recent research articles is also available from this page. The following are outlines of papers that are presently not in final form.

If you wish to be notified when a particular paper is available for viewing online, or downloading for printing, or to be notified of the posting of papers in one of my fields of interest, please email a request to c.j.mulvey@sussex.ac.uk.

A list of research publications is also available from this page, together with a brief curriculum vitae and information on my contact details.

A constructive proof of the Hahn-Banach theorem

This paper (with the late J.J.C.Vermeulen) completes the constructivisation of the Hahn-Banach initiated in the paper A globalisation of the Hahn-Banach theorem (with J.W.Pelletier), by applying techniques developed in Vermeulen's 1986 DPhil thesis, Constructive Methods in Functional Analysis, at the University of Sussex to develop a geometrically constructive approach to the theorem that stays extremely close to the classical proof. An outline of the approach taken may be found in the paper On the geometry of choice. The paper gives a self-contained introduction to these techniques and their applications.

Gelfand quantales

This paper examines the basic concepts of Gelfand quantales, introduced to place the insights of Rosicky in his paper Multiplicative lattices and C*-algebras in the context of involutive quantales, allowing a more elegant approach to the spectrum of C*-algebras proposed in my paper &. In particular, the paper extends the concept of a compact, completely regular quantale to this context, and introduces the concept of a von Neumann quantale. Considering the work of Pelletier and Rosicky on simple involutive quantales in the context of Gelfand quantales allows their characterisations to be relativised to this case, providing rather straightforward conditions for a Gelfand quantale to be simple. In turn, this yields insights into the concept of spatiality for Gelfand quantales, in the context of the characterisations of points obtained in the paper On the quantisation of points and its sequel On the quantisation of spaces (each with J.W.Pelletier).

**The spectrum of a C*-algebra**

This paper introduces the spectrum of a C*-algebra in terms of the Gelfand quantale of closed linear subspaces of the C*-algebra. The spectrum is shown to be a compact, completely regular quantale, on which the Gelfand representation of the C*-algebra in the C*-algebra of continuous complex functions on the spectrum may be defined. A Gelfand representation theorem establishes that this is an isometric *-isomorphism of C*-algebras, in a way that generalises precisely the commutative case, placing into the context of quantales the insights of Giles and Kummer concerning the spectral representation of C*-algebras, as proposed in the paper &.

Quantales: Quantal logic

This paper (with M.Nawaz) continues the investigation, commenced in the paper Quantales: Quantal sets (with M.Nawaz), of quantal sets and sheaves over an idempotent right-sided quantale. It is shown that the topos of quantal sets over such a quantale admits a natural logical structure on the quantal subsets of each quantal set, together with existential and universal quantification, in a way that extends non-commutatively the concept of formal topos introduced by J.Benabou. In particular, it is shown that the topos is endowed with a quantal subset classifier, as well as its subobject classifier, which may be considered to determine the quantised structure of the topos, and from which the quantale itself may be recovered.

Foulis quantales

This paper places the Foulis duality between orthomodular lattices and Foulis semigroups in the context of quantales, by specialising the natural duality between orthocomplemented sup-lattices and Hilbert quantales to the case of Foulis quantales. In doing so, the paper examines the structure of Foulis quantales, in particular the links between the Sasaki projections in the quantale and its right-sided elements which allow Foulis duality to be established within this context.

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