scholarly journals Characterizations of the Automorphisms¶of Hilbert Space Effect Algebras

2001 ◽  
Vol 223 (2) ◽  
pp. 437-450 ◽  
Author(s):  
Lajos Molnár
Keyword(s):  
Hilbert Space ◽  
Effect Algebras ◽  
Space Effect

scholarly journals Coexistency on Hilbert Space Effect Algebras and a Characterisation of Its Symmetry Transformations

2020 ◽  
Vol 379 (3) ◽  
pp. 1077-1112 ◽  
Author(s):  
György Pál Gehér ◽  
Peter Šemrl
Keyword(s):  
Hilbert Space ◽  
Effect Algebra ◽  
Mathematical Structure ◽  
Quantum Measurements ◽  
The Other ◽  
Effect Algebras ◽  
Natural Question ◽  
Fuzzy Event ◽  
Space Effect ◽  
Symmetry Transformations

Abstract The Hilbert space effect algebra is a fundamental mathematical structure which is used to describe unsharp quantum measurements in Ludwig’s formulation of quantum mechanics. Each effect represents a quantum (fuzzy) event. The relation of coexistence plays an important role in this theory, as it expresses when two quantum events can be measured together by applying a suitable apparatus. This paper’s first goal is to answer a very natural question about this relation, namely, when two effects are coexistent with exactly the same effects? The other main aim is to describe all automorphisms of the effect algebra with respect to the relation of coexistence. In particular, we will see that they can differ quite a lot from usual standard automorphisms, which appear for instance in Ludwig’s theorem. As a byproduct of our methods we also strengthen a theorem of Molnár.

scholarly journals Extensions of ordering sets of states from effect algebras onto their MacNeille completions

10.29007/lkdv ◽  
2018 ◽  
Author(s):  
Jiří Janda ◽  
Zdenka Riečanová
Keyword(s):  
Hilbert Space ◽  
Effect Algebra ◽  
Effect Algebras ◽  
Macneille Completion ◽  
Atomic Lattice ◽  
Space Effect ◽  
Lattice Effect

In [Riečanová Z, Zajac M.: Hilbert Space Effect-Representations of Effect Algebras] it was shown that an effect algebra E with an ordering set M of states can by embedded into a Hilbert space effect algebra E(l<sub>2</sub>(M)). We consider the problem when its effect algebraic MacNeille completion Ê can be also embedded into the same Hilbert space effect algebra E(l<sub>2</sub>(M)). That is when the ordering set M of states on E can be be extended to an ordering set of states on Ê. We give an answer for all Archimedean MV-effect algebras and Archimedean atomic lattice effect algebras.

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