scholarly journals Category Algebras and States on Categories

Symmetry ◽  
2021 ◽  
Vol 13 (7) ◽  
pp. 1172
Author(s):  
Hayato Saigo
Keyword(s):  
Hilbert Spaces ◽  
Category Theory ◽  
Linear Functional ◽  
Quantum Probability ◽  
Probability Measures ◽  
Mathematical Framework ◽  
Hilbert Modules ◽  
Matrix Algebras ◽  
Generalized Matrix ◽  
Category Algebras

The purpose of this paper is to build a new bridge between category theory and a generalized probability theory known as noncommutative probability or quantum probability, which was originated as a mathematical framework for quantum theory, in terms of states as linear functional defined on category algebras. We clarify that category algebras can be considered to be generalized matrix algebras and that the notions of state on category as linear functional defined on category algebra turns out to be a conceptual generalization of probability measures on sets as discrete categories. Moreover, by establishing a generalization of famous GNS (Gelfand–Naimark–Segal) construction, we obtain a representation of category algebras of †-categories on certain generalized Hilbert spaces which we call semi-Hilbert modules over rigs. The concepts and results in the present paper will be useful for the studies of symmetry/asymmetry since categories are generalized groupoids, which themselves are generalized groups.

scholarly journals Geometrical properties of the space of idempotent probability measures

2021 ◽  
Vol 22 (2) ◽  
pp. 399
Author(s):  
Kholsaid Fayzullayevich Kholturayev
Keyword(s):  
Metric Space ◽  
Category Theory ◽  
Compact Hausdorff Space ◽  
Hausdorff Space ◽  
Probability Measures ◽  
Compact Metric Space ◽  
Geometrical Properties ◽  
Idempotent Mathematics

Although traditional and idempotent mathematics are "parallel'', by an application of the category theory we show that objects obtained the similar rules over traditional and idempotent mathematics must not be "parallel''. At first we establish for a compact metric space X the spaces P(X) of probability measures and I(X) idempotent probability measures are homeomorphic ("parallelism''). Then we construct an example which shows that the constructions P and I form distinguished functors from each other ("parallelism'' negation). Further for a compact Hausdorff space X we establish that the hereditary normality of I<sub>3</sub>(X)\ X implies the metrizability of X.

scholarly journals An algebraic approach to Wigner's unitary-antiunitary theorem

1998 ◽  
Vol 65 (3) ◽  
pp. 354-369 ◽  
Author(s):  
Lajos Molnár
Keyword(s):  
Algebraic Approach ◽  
Ring Theory ◽  
Hilbert Modules ◽  
Type Result ◽  
Matrix Algebras

AbstractWe present an operator algebraic approach to Wigner's unitary-antiunitary theorem using some classical results from ring theory. To show how effective this approach is, we prove a generalization of this celebrated theorem for Hilbert modules over matrix algebras. We also present a Wigner-type result for maps on prime C*-algebras.

scholarly journals Real Functions and the Extension of Generalized Probability Measures

2013 ◽  
Vol 55 (1) ◽  
pp. 85-94
Author(s):  
Jana Havlíčková
Keyword(s):  
Category Theory ◽  
Continuous Functions ◽  
Probability Measures ◽  
Fuzzy Probability ◽  
Classical Probability ◽  
Random Events ◽  
Elementary Methods ◽  
Generalized Probability ◽  
Fuzzy Probability Theory

Abstract In the classical probability, as well as in the fuzzy probability theory, random events and probability measures are modelled by functions into the closed unit interval [0,1]. Using elementary methods of category theory, we present a classification of the extensions of generalized probability measures (probability measures and integrals with respect to probability measures) from a suitable class of generalized random events to a larger class having some additional (algebraic and/or topological) properties. The classification puts into a perspective the classical and some recent constructions related to the extension of sequentially continuous functions.

scholarly journals σ-derivations on generalized matrix algebras

2020 ◽  
Vol 28 (2) ◽  
pp. 115-135
Author(s):  
Aisha Jabeen ◽  
Mohammad Ashraf ◽  
Musheer Ahmad
Keyword(s):  
Commutative Ring ◽  
Matrix Algebra ◽  
Morita Context ◽  
Matrix Algebras ◽  
Generalized Matrix ◽  
Generalized Matrix Algebra

AbstractLet 𝒭 be a commutative ring with unity, 𝒜, 𝒝 be 𝒭-algebras, 𝒨 be (𝒜, 𝒝)-bimodule and 𝒩 be (𝒝, 𝒜)-bimodule. The 𝒭-algebra 𝒢 = 𝒢(𝒜, 𝒨, 𝒩, 𝒝) is a generalized matrix algebra defined by the Morita context (𝒜, 𝒝, 𝒨, 𝒩, ξ𝒨𝒩, Ω𝒩𝒨). In this article, we study Jordan σ-derivations on generalized matrix algebras.

Characterizations of Lie triple derivations on generalized matrix algebras

2020 ◽  
Vol 48 (9) ◽  
pp. 3651-3660
Author(s):  
Mohammad Ashraf ◽  
Mohd Shuaib Akhtar
Keyword(s):  
Matrix Algebras ◽  
Generalized Matrix

scholarly journals ADJOINT FUNCTORS BETWEEN CATEGORIES OF HILBERT -MODULES

2016 ◽  
Vol 17 (2) ◽  
pp. 453-488 ◽  
Author(s):  
Pierre Clare ◽  
Tyrone Crisp ◽  
Nigel Higson
Keyword(s):  
Category Theory ◽  
Reductive Groups ◽  
Hilbert Modules ◽  
Adjoint Functors ◽  
Left Action ◽  
Parabolic Induction ◽  
Left And Right ◽  
Restriction Functor ◽  
Tempered Representation ◽  
Standard Concept

Let$E$be a (right) Hilbert module over a$C^{\ast }$-algebra$A$. If$E$is equipped with a left action of a second$C^{\ast }$-algebra$B$, then tensor product with$E$gives rise to a functor from the category of Hilbert$B$-modules to the category of Hilbert$A$-modules. The purpose of this paper is to study adjunctions between functors of this sort. We shall introduce a new kind of adjunction relation, called a local adjunction, that is weaker than the standard concept from category theory. We shall give several examples, the most important of which is the functor of parabolic induction in the tempered representation theory of real reductive groups. Each local adjunction gives rise to an ordinary adjunction of functors between categories of Hilbert space representations. In this way we shall show that the parabolic induction functor has a simultaneous left and right adjoint, namely the parabolic restriction functor constructed in Clareet al.[Parabolic induction and restriction via$C^{\ast }$-algebras and Hilbert$C^{\ast }$-modules,Compos. Math.FirstView(2016), 1–33, 2].

scholarly journals THE INDEX OF (WHITE) NOISES AND THEIR PRODUCT SYSTEMS

2006 ◽  
Vol 09 (04) ◽  
pp. 617-655 ◽  
Author(s):  
MICHAEL SKEIDE
Keyword(s):  
Tensor Product ◽  
Hilbert Spaces ◽  
Hilbert Modules ◽  
Product System ◽  
Invariant Vector ◽  
Product Systems ◽  
System A ◽  
Definition Of ◽  
General Product ◽  
White Noises

Almost every paper about Arveson systems (i.e. product systems of Hilbert spaces) starts by recalling their basic classification assigning to every Arveson system a type and an index. So it is natural to ask in how far an analogue classification can also be proposed for product systems of Hilbert modules. However, while the definition of type is plain, there are obstacles for the definition of index. But all obstacles can be removed when restricting to the category which we introduce here as spatial product systems and that matches the usual definition of spatial in the case of Arveson systems. This is not really a loss because the definition of index for nonspatial Arveson systems is rather formal and does not reflect the information the index carries for spatial Arveson systems.E0-semigroups give rise to product systems. Our definition of spatial product system, namely, existence of a unital unit that is central, matches Powers' definition of spatial in the sense that the E0-semigroup from which the product system is derived admits a semigroup of intertwining isometries. We show that every spatial product system contains a unique maximal completely spatial subsystem (generated by all units) that is isomorphic to a product system of time ordered Fock modules. (There exist nonspatial product systems that are generated by their units. Consequently, these cannot be Fock modules.) The index of a spatial product system we define as the (unique) Hilbert bimodule that determines the Fock module. In order to show that the index merits the name index we provide a product of product systems under which the index is additive (direct sum). While for Arveson systems there is the tensor product, for general product systems the tensor product does not make sense as a product system. Even for Arveson systems our product is, in general, only a subsystem of the tensor product. Moreover, its construction depends explicitly on the choice of the central reference units of its factors.Spatiality of a product system means that it may be derived from an E0-semigroup with an invariant vector expectation, i.e. from a noise. We extend our product of spatial product systems to a product of noises and study its properties.Finally, we apply our techniques to show the module analogue of Fowler's result that free flows are comletely spatial, and we compute their indices.

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