53 research outputs found
Divisible effect algebras and interval effect algebras
summary:It is shown that divisible effect algebras are in one-to-one correspondence with unit intervals in partially ordered rational vector spaces
Spectral resolutions in effect algebras
Effect algebras were introduced as an abstract algebraic model for Hilbert
space effects representing quantum mechanical measurements. We study additional
structures on an effect algebra that enable us to define spectrality and
spectral resolutions for elements of akin to those of self-adjoint
operators. These structures, called compression bases, are special families of
maps on , analogous to the set of compressions on operator algebras, order
unit spaces or unital abelian groups. Elements of a compression base are in
one-to-one correspondence with certain elements of , called projections. An
effect algebra is called spectral if it has a distinguished compression base
with two special properties: the projection cover property (i.e., for every
element in there is a smallest projection majorizing ), and the
so-called b-comparability property, which is an analogue of general
comparability in operator algebras or unital abelian groups. It is shown that
in a spectral archimedean effect algebra , every admits a unique
rational spectral resolution and its properties are studied. If in addition
possesses a separating set of states, then every element is determined
by its spectral resolution. It is also proved that for some types of interval
effect algebras (with RDP, archimedean divisible), spectrality of is
equivalent to spectrality of its universal group and the corresponding rational
spectral resolutions are the same. In particular, for convex archimedean effect
algebras, spectral resolutions in are in agreement with spectral
resolutions in the corresponding order unit space.Comment: 31 pages, accepted in Quantum, comments welcom
Orthomodular lattices with almost orthogonal sets of atoms
summary:The set of all atoms of an atomic orthomodular lattice is said to be almost ortho\-go\-nal if the set is finite for every . It is said to be strongly almost ortho\-go\-nal if, for every , any sequence of atoms such that contains at most finitely many distinct elements. We study the relation and consequences of these notions. We show among others that a complete atomic orthomodular lattice is a compact topological one if and only if the set of all its atoms is almost ortho\-go\-nal
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