8,377 research outputs found
Almost orthogonality and Hausdorff interval topologies of atomic lattice effect algebras
summary:We prove that the interval topology of an Archimedean atomic lattice effect algebra is Hausdorff whenever the set of all atoms of is almost orthogonal. In such a case is order continuous. If moreover is complete then order convergence of nets of elements of is topological and hence it coincides with convergence in the order topology and this topology is compact Hausdorff compatible with a uniformity induced by a separating function family on corresponding to compact and cocompact elements. For block-finite Archimedean atomic lattice effect algebras the equivalence of almost orthogonality and s-compact generation is shown. As the main application we obtain a state smearing theorem for these effect algebras, as well as the continuity of -operation in the order and interval topologies on them
Atomicity of lattice effect algebras and their sub-lattice effect algebras
summary:We show some families of lattice effect algebras (a common generalization of orthomodular lattices and MV-effect algebras) each element E of which has atomic center C(E) or the subset S(E) of all sharp elements, resp. the center of compatibility B(E) or every block M of E. The atomicity of E or its sub-lattice effect algebras C(E), S(E), B(E) and blocks M of E is very useful equipment for the investigations of its algebraic and topological properties, the existence or smearing of states on E, questions about isomorphisms and so. Namely we touch the families of complete lattice effect algebras, or lattice effect algebras with finitely many blocks, or complete atomic lattice effect algebra E with Hausdorff interval topology
Topologies on Quantum Effects
Quantum effects play an important role in quantum measurement theory. The set
of all quantum effects can be organized into an algebraical structure called
effect algebra. In this paper, we study various topologies on the Hilbert space
effect algebra and the projection lattice effect algebra
The Expectation Monad in Quantum Foundations
The expectation monad is introduced abstractly via two composable
adjunctions, but concretely captures measures. It turns out to sit in between
known monads: on the one hand the distribution and ultrafilter monad, and on
the other hand the continuation monad. This expectation monad is used in two
probabilistic analogues of fundamental results of Manes and Gelfand for the
ultrafilter monad: algebras of the expectation monad are convex compact
Hausdorff spaces, and are dually equivalent to so-called Banach effect
algebras. These structures capture states and effects in quantum foundations,
and also the duality between them. Moreover, the approach leads to a new
re-formulation of Gleason's theorem, expressing that effects on a Hilbert space
are free effect modules on projections, obtained via tensoring with the unit
interval.Comment: In Proceedings QPL 2011, arXiv:1210.029
Duality for Convexity
This paper studies convex sets categorically, namely as algebras of a
distribution monad. It is shown that convex sets occur in two dual adjunctions,
namely one with preframes via the Boolean truth values {0,1} as dualising
object, and one with effect algebras via the (real) unit interval [0,1] as
dualising object. These effect algebras are of interest in the foundations of
quantum mechanics
Representation of States on Effect-Tribes and Effect Algebras by Integrals
We describe -additive states on effect-tribes by integrals.
Effect-tribes are monotone -complete effect algebras of functions where
operations are defined by points. Then we show that every state on an effect
algebra is an integral through a Borel regular probability measure. Finally, we
show that every -convex combination of extremal states on a monotone
-complete effect algebra is a Jauch-Piron state.Comment: 20 page
Arc Operads and Arc Algebras
Several topological and homological operads based on families of projectively
weighted arcs in bounded surfaces are introduced and studied. The spaces
underlying the basic operad are identified with open subsets of a
compactification due to Penner of a space closely related to Riemann's moduli
space. Algebras over these operads are shown to be Batalin-Vilkovisky algebras,
where the entire BV structure is realized simplicially. Furthermore, our basic
operad contains the cacti operad up to homotopy, and it similarly acts on the
loop space of any topological space. New operad structures on the circle are
classified and combined with the basic operad to produce geometrically natural
extensions of the algebraic structure of BV algebras, which are also computed.Comment: Published by Geometry and Topology at
http://www.maths.warwick.ac.uk/gt/GTVol7/paper15.abs.htm
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