scholarly journals Positive linear functional on topological ∗-algebras

1979 ◽  
Vol 68 (1) ◽  
pp. 17-24 ◽  
Author(s):  
Atsushi Inoue
Keyword(s):  
Linear Functional ◽  
Topological Algebras ◽  
Positive Linear Functional

Infrasequential Topological Algebras

1979 ◽  
Vol 22 (4) ◽  
pp. 413-418 ◽  
Author(s):  
T. Husain
Keyword(s):  
Linear Functional ◽  
Topological Algebra ◽  
Topological Algebras ◽  
Locally Convex ◽  
Locally Convex Algebra

The notion of sequential topological algebra was introduced by this author and Ng [3], Among a number of results concerning these algebras, we showed that each multiplicative linear functional on a sequentially complete, sequential, locally convex algebra is bounded ([3], Theorem 1). From this it follows that every multiplicative linear functional on a sequential F-algebra (complete metrizable) is continuous ([3], Corollary 2).

A Lebesgue Decomposition Theorem for C* Algebras

1972 ◽  
Vol 15 (1) ◽  
pp. 87-91 ◽  
Author(s):  
Michael Henle
Keyword(s):  
Absolute Continuity ◽  
Linear Functional ◽  
Decomposition Theorem ◽  
Lebesgue Decomposition ◽  
C Algebras ◽  
Positive Linear Functional ◽  
Decomposition Theorems

This paper, by generalizing von Neumann's proof of the Radon-Nikodym and Lebesgue decomposition theorems [3], obtains analogous results for positive linear functional on a C* algebra. The concept of "absolute continuity" used and the Radon-Nikodym portion of the resulting theorem are due to Dye [2].

Integration Theory

2021 ◽  
pp. 341-444
Author(s):  
Adel N. Boules
Keyword(s):  
Linear Functional ◽  
Integration Theory ◽  
General Measure ◽  
Radon Measures ◽  
Lp Spaces ◽  
Approximation Theorems ◽  
Positive Linear Functional ◽  
Product Measures ◽  
Compactly Supported Functions ◽  
Nikodym Theorem

The Lebesgue measure on ?n (presented in section 8.4) is a pivotal component of this chapter. The approach in the chapter is to extend the positive linear functional provided by the Riemann integral on the space of continuous, compactly supported functions on ?n (presented in section 8.1). An excursion on Radon measures is included at the end of section 8.4. The rest of the sections are largely independent of sections 8.1 and 8.4 and constitute a deep introduction to general measure and integration theories. Topics include measurable spaces and measurable functions, Carathéodory’s theorem, abstract integration and convergence theorems, complex measures and the Radon-Nikodym theorem, Lp spaces, product measures and Fubini’s theorem, and a good collection of approximation theorems. The closing section of the book provides a glimpse of Fourier analysis and gives a nice conclusion to the discussion of Fourier series and orthogonal polynomials started in section 4.10.

scholarly journals Polymer-Fourier quantization of the scalar field revisited

2016 ◽  
Vol 31 (32) ◽  
pp. 1650166 ◽  
Author(s):  
Angel Garcia-Chung ◽  
J. David Vergara
Keyword(s):  
Scalar Field ◽  
Irreducible Representation ◽  
Linear Functional ◽  
Complex Structure ◽  
Singular Limit ◽  
Fock Representation ◽  
Positive Linear Functional ◽  
Spatial Volume ◽  
Volume Preserving ◽  
Algebraic Scheme

The polymer quantization of the Fourier modes of the real scalar field is studied within algebraic scheme. We replace the positive linear functional of the standard Poincaré invariant quantization by a singular one. This singular positive linear functional is constructed as mimicking the singular limit of the complex structure of the Poincaré invariant Fock quantization. The resulting symmetry group of such polymer quantization is the subgroup [Formula: see text] which is a subgroup of [Formula: see text] formed by spatial volume preserving diffeomorphisms. In consequence, this yields an entirely different irreducible representation of the canonical commutation relations, nonunitary equivalent to the standard Fock representation. We also compared the Poincaré invariant Fock vacuum with the polymer Fourier vacuum.

Sign in / Sign up

Export Citation Format

Share Document