39 research outputs found
Constructive pointfree topology eliminates non-constructive representation theorems from Riesz space theory
In Riesz space theory it is good practice to avoid representation theorems
which depend on the axiom of choice. Here we present a general methodology to
do this using pointfree topology. To illustrate the technique we show that
almost f-algebras are commutative. The proof is obtained relatively
straightforward from the proof by Buskes and van Rooij by using the pointfree
Stone-Yosida representation theorem by Coquand and Spitters
The Gelfand spectrum of a noncommutative C*-algebra: a topos-theoretic approach
We compare two influential ways of defining a generalized notion of space.
The first, inspired by Gelfand duality, states that the category of
'noncommutative spaces' is the opposite of the category of C*-algebras. The
second, loosely generalizing Stone duality, maintains that the category of
'pointfree spaces' is the opposite of the category of frames (i.e., complete
lattices in which the meet distributes over arbitrary joins). One possible
relationship between these two notions of space was unearthed by Banaschewski
and Mulvey, who proved a constructive version of Gelfand duality in which the
Gelfand spectrum of a commutative C*-algebra comes out as a pointfree space.
Being constructive, this result applies in arbitrary toposes (with natural
numbers objects, so that internal C*-algebras can be defined). Earlier work by
the first three authors, shows how a noncommutative C*-algebra gives rise to a
commutative one internal to a certain sheaf topos. The latter, then, has a
constructive Gelfand spectrum, also internal to the topos in question. After a
brief review of this work, we compute the so-called external description of
this internal spectrum, which in principle is a fibered pointfree space in the
familiar topos Sets of sets and functions. However, we obtain the external
spectrum as a fibered topological space in the usual sense. This leads to an
explicit Gelfand transform, as well as to a topological reinterpretation of the
Kochen-Specker Theorem of quantum mechanics, which supplements the remarkable
topos-theoretic version of this theorem due to Butterfield and Isham.Comment: 12 page
A localic theory of lower and upper integrals
An account of lower and upper integration is given. It is constructive in the sense of geometric logic. If the integrand takes its values in the non-negative lower reals, then its lower integral with respect to a valuation is a lower real. If the integrand takes its values in the non-negative upper reals,then its upper integral with respect to a covaluation and with domain of
integration bounded by a compact subspace is an upper real. Spaces of valuations and of covaluations are defined.
Riemann and Choquet integrals can be calculated in terms of these lower and upper integrals
Integrals and Valuations
We construct a homeomorphism between the compact regular locale of integrals
on a Riesz space and the locale of (valuations) on its spectrum. In fact, we
construct two geometric theories and show that they are biinterpretable. The
constructions are elementary and tightly connected to the Riesz space
structure.Comment: Submitted for publication 15/05/0