A note on ideals in synaptic algebras

2012 ◽  
Vol 62 (6) ◽  
Author(s):  
Sylvia Pulmannová
Keyword(s):  
Jordan Algebra ◽  
Normed Space ◽  
Orthomodular Lattice ◽  
Effect Algebra ◽  
Algebra Structure ◽  
Order Unit ◽  
Synaptic Algebra ◽  
Special Jordan Algebra

AbstractThe notion of a synaptic algebra was introduced by David Foulis. Synaptic algebras unite the notions of an order-unit normed space, a special Jordan algebra, a convex effect algebra and an orthomodular lattice. In this note we study quadratic ideals in synaptic algebras which reflect its Jordan algebra structure. We show that projections contained in a quadratic ideal from a p-ideal in the orthomodular lattice of projections in the synaptic algebra and we find a characterization of those quadratic ideals which are generated by their projections.

Synaptic algebras

2010 ◽  
Vol 60 (5) ◽  
Author(s):  
David Foulis
Keyword(s):  
Hilbert Space ◽  
Jordan Algebra ◽  
Normed Space ◽  
Orthomodular Lattice ◽  
Effect Algebra ◽  
Order Unit ◽  
Natural Conditions ◽  
Spectral Order ◽  
Synaptic Algebra ◽  
Special Jordan Algebra

AbstractA synaptic algebra is both a special Jordan algebra and a spectral order-unit normed space satisfying certain natural conditions suggested by the partially ordered Jordan algebra of bounded Hermitian operators on a Hilbert space. The adjective “synaptic”, borrowed from biology, is meant to suggest that such an algebra coherently “ties together” the notions of a Jordan algebra, a spectral order-unit normed space, a convex effect algebra, and an orthomodular lattice.

scholarly journals Symmetries in synaptic algebras

2014 ◽  
Vol 64 (3) ◽  
Author(s):  
David Foulis ◽  
Sylvia Pulmannová
Keyword(s):  
Jordan Algebra ◽  
Equivalence Relation ◽  
Orthomodular Lattice ◽  
Von Neumann Algebra ◽  
Unit Element ◽  
Von Neumann ◽  
Synaptic Algebra ◽  
Projection Lattice ◽  
Finite Sequences

AbstractA synaptic algebra is a generalization of the Jordan algebra of self-adjoint elements of a von Neumann algebra. We study symmetries in synaptic algebras, i.e., elements whose square is the unit element, and we investigate the equivalence relation on the projection lattice of the algebra induced by finite sequences of symmetries. In case the projection lattice is complete, or even centrally orthocomplete, this equivalence relation is shown to possess many of the properties of a dimension equivalence relation on an orthomodular lattice.

scholarly journals Eisenstein Series Arising from Jordan Algebras

2019 ◽  
Vol 72 (1) ◽  
pp. 183-201 ◽  
Author(s):  
Marcela Hanzer ◽  
Gordan Savin
Keyword(s):  
Eisenstein Series ◽  
Jordan Algebra ◽  
Jordan Algebras ◽  
Algebra Structure ◽  
Parabolic Subgroups ◽  
Automorphic Representations ◽  
Unipotent Radicals ◽  
Maximal Parabolic Subgroups

AbstractWe describe poles and the corresponding residual automorphic representations of Eisenstein series attached to maximal parabolic subgroups whose unipotent radicals admit Jordan algebra structure.

Equivalent norms on Banach Jordan algebras

1979 ◽  
Vol 86 (2) ◽  
pp. 261-270 ◽  
Author(s):  
M. A. Youngson
Keyword(s):  
Jordan Algebra ◽  
Sufficient Conditions ◽  
Equivalent Norm ◽  
Jordan Algebras ◽  
The Self ◽  
Equivalent Norms ◽  
Positive Elements ◽  
Definition Of ◽  
Necessary And Sufficient

1. Introduction. Recently Kaplansky suggested the definition of a suitable Jordan analogue of B*-algebras, which we call J B*-algebras (see (10) and (11)). In this article, we give a characterization of those complex unital Banach Jordan algebras which are J B*-algebras in an equivalent norm. This is done by generalizing results of Bonsall ((3) and (4)) to give necessary and sufficient conditions on a real unital Banach Jordan algebra under which it is the self-adjoint part of a J B*-algebra in an equivalent norm. As a corollary we also obtain a characterization of the cones in a Banach Jordan algebra which are the set of positive elements of a J B*-algebra.

scholarly journals Some Properties oflp(A,X)Spaces

2009 ◽  
Vol 2009 ◽  
pp. 1-8 ◽  
Author(s):  
Xiaohong Fu ◽  
Songxiao Li
Keyword(s):  
Simple Proof ◽  
Normed Space ◽  
Convex Space ◽  
Unconditional Basis ◽  
Locally Convex Space ◽  
Locally Convex

We provide a representation of elements of the spacelp(A,X)for a locally convex spaceXand1≤p<∞and determine its continuous dual for normed spaceXand1<p<∞. In particular, we study the extension and characterization of isometries onlp(N,X)space, whenXis a normed space with an unconditional basis and with a symmetric norm. In addition, we give a simple proof of the main result of G. Ding (2002).

scholarly journals Aplikasi Pemetaan terkait Semi-Inner Produk pada Ruang Bernorma Real

2021 ◽  
Vol 1 (1) ◽  
pp. 7-14
Author(s):  
Aulia Khifah Futhona ◽  
Supama
Keyword(s):  
Normed Space ◽  
Inner Product ◽  
Birkhoff Orthogonality ◽  
Real Normed Space ◽  
Lower Semi

In this article, we give the properties of mappings associated with the upper semi-inner product , lower semi-inner product  and Lumer semi-inner product  which generate the norm on a real normed space. Furthermore, we establish applications to the Birkhoff orthogonality and characterization of best approximants.

scholarly journals Kadison’s antilattice theorem for a synaptic algebra

2018 ◽  
Vol 51 (1) ◽  
pp. 1-7 ◽  
Author(s):  
David J. Foulis ◽  
Sylvia Pulmannová
Keyword(s):  
Orthomodular Lattice ◽  
Operator Algebras ◽  
Synaptic Algebra

Abstract We prove that if A is a synaptic algebra and the orthomodular lattice P of projections in A is complete, then A is a factor if and only if A is an antilattice.We also generalize several other results of R. Kadison pertaining to infima and suprema in operator algebras.

scholarly journals On Carlsson type orthogonality and characterization of inner product spaces

Filomat ◽  
2012 ◽  
Vol 26 (4) ◽  
pp. 859-870 ◽  
Author(s):  
Eder Kikianty ◽  
Sever Dragomir
Keyword(s):  
Normed Space ◽  
Product Space ◽  
Inner Product Space ◽  
Inner Product ◽  
Product Spaces ◽  
Isosceles Orthogonality ◽  
Inner Product Spaces ◽  
Pythagorean Orthogonality

In an inner product space, two vectors are orthogonal if their inner product is zero. In a normed space, numerous notions of orthogonality have been introduced via equivalent propositions to the usual orthogonality, e.g. orthogonal vectors satisfy the Pythagorean law. In 2010, Kikianty and Dragomir [9] introduced the p-HH-norms (1 ? p < ?) on the Cartesian square of a normed space. Some notions of orthogonality have been introduced by utilizing the 2-HH-norm [10]. These notions of orthogonality are closely related to the classical Pythagorean orthogonality and Isosceles orthogonality. In this paper, a Carlsson type orthogonality in terms of the 2-HH-norm is considered, which generalizes the previous definitions. The main properties of this orthogonality are studied and some useful consequences are obtained. These consequences include characterizations of inner product space.

scholarly journals Conditional Probability, Three-Slit Experiments, and the Jordan Algebra Structure of Quantum Mechanics

2012 ◽  
Vol 2012 ◽  
pp. 1-20 ◽  
Author(s):  
Gerd Niestegge
Keyword(s):  
Quantum Mechanics ◽  
Conditional Probability ◽  
Mathematical Structure ◽  
Higher Order ◽  
Complete Characterization ◽  
Jordan Algebras ◽  
Algebra Structure ◽  
Quantum Logics ◽  
Close Link

Most quantum logics do not allow for a reasonable calculus of conditional probability. However, those ones which do so provide a very general and rich mathematical structure, including classical probabilities, quantum mechanics, and Jordan algebras. This structure exhibits some similarities with Alfsen and Shultz's noncommutative spectral theory, but these two mathematical approaches are not identical. Barnum, Emerson, and Ududec adapted the concept of higher-order interference, introduced by Sorkin in 1994, into a general probabilistic framework. Their adaption is used here to reveal a close link between the existence of the Jordan product and the nonexistence of interference of third or higher order in those quantum logics which entail a reasonable calculus of conditional probability. The complete characterization of the Jordan algebraic structure requires the following three further postulates: a Hahn-Jordan decomposition property for the states, a polynomial functional calculus for the observables, and the positivity of the square of an observable. While classical probabilities are characterized by the absence of any kind of interference, the absence of interference of third (and higher) order thus characterizes a probability calculus which comes close to quantum mechanics but still includes the exceptional Jordan algebras.

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