scholarly journals A decomposition theorem for submeasures

1985 ◽  
Vol 26 (1) ◽  
pp. 69-74
Author(s):  
A. R. Khan ◽  
K. Rowlands
Keyword(s):  
Decomposition Theorem ◽  
Lebesgue Decomposition ◽  
Decomposition Theorems

In recent years versions of the Lebesgue and the Hewitt-Yosida decomposition theorems have been proved for group-valued measures. For example, Traynor [4], [6] has established Lebesgue decomposition theorems for exhaustive groupvalued measures on a ring using (1) algebraic and (2) topological notions of continuity and singularity, and generalizations of the Hewitt-Yosida theorem have been given by Drewnowski [2], Traynor [5] and Khurana [3]. In this paper we consider group-valued submeasures and in particular we have established a decomposition theorem from which analogues of the Lebesgue and Hewitt-Yosida decomposition theorems for submeasures may be derived. Our methods are based on those used by Drewnowski in [2] and the main theorem established generalizes Theorem 4.1 of [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].

scholarly journals LEBESGUE DECOMPOSITION FOR REPRESENTABLE FUNCTIONALS ON *-ALGEBRAS

2015 ◽  
Vol 58 (2) ◽  
pp. 491-501 ◽  
Author(s):  
ZSIGMOND TARCSAY
Keyword(s):  
Decomposition Theorem ◽  
Lebesgue Decomposition ◽  
Absolutely Continuous ◽  
Hermitian Forms ◽  
Lebesgue Type Decomposition ◽  
Representable Functional ◽  
Type Decomposition

AbstractWe offer a Lebesgue-type decomposition of a representable functional on a *-algebra into absolutely continuous and singular parts with respect to another. Such a result was proved by Zs. Szűcs due to a general Lebesgue decomposition theorem of S. Hassi, H.S.V. de Snoo, and Z. Sebestyén concerning non-negative Hermitian forms. In this paper, we provide a self-contained proof of Szűcs' result and in addition we prove that the corresponding absolutely continuous parts are absolutely continuous with respect to each other.

scholarly journals A wide Perron integral

1986 ◽  
Vol 34 (2) ◽  
pp. 233-251
Author(s):  
D. N. Sarkhel
Keyword(s):  
Decomposition Theorem ◽  
Mean Value ◽  
Lebesgue Decomposition ◽  
Integration By Parts ◽  
Mean Value Theorems ◽  
Limit Process ◽  
Real Functions ◽  
Marcinkiewicz Theorem ◽  
Perron Integral ◽  
Interesting Generalization

In terms of an arbitrary limit process T, defined abstractly for real functions, we define in a novel way a T-continuous integral of Perron type, admitting mean value theorems, integration by parts and the analogue of the Marcinkiewicz theorem for the ordinary Perron integral. The integral is shown to include, as particular cases, the various known continuous, approximately continuous, cesàro-continuous, mean-continuous and proximally Cesàro-continuous integrals of Perron and Denjoy types. An interesting generalization of the classical Lebesgue decomposition theorem is also obtained.

scholarly journals Decompositions of vector measures in Riesz spaces and Banach lattices

1986 ◽  
Vol 29 (1) ◽  
pp. 23-39 ◽  
Author(s):  
Klaus D. Schmidt
Keyword(s):  
Bounded Variation ◽  
Banach Lattice ◽  
Decomposition Theorem ◽  
Riesz Space ◽  
Jordan Decomposition ◽  
Countable Additivity ◽  
Vector Measures ◽  
Central Result ◽  
Decomposition Theorems ◽  
Natural Way

The present paper is mainly concerned with decomposition theorems of the Jordan, Yosida-Hewitt, and Lebesgue type for vector measures of bounded variation in a Banach lattice having property (P). The central result is the Jordan decomposition theorem due to which these vector measures may alternately be regarded as order bounded vector measures in an order complete Riesz space or as vector measures of bounded variation in a Banach space. For both classes of vector measures, properties like countable additivity, purely finite additivity, absolute continuity, and singularity can be defined in a natural way and lead to decomposition theorems of the Yosida-Hewitt and Lebesgue type. In the Banach lattice case, these lattice theoretical and topological decomposition theorems can be compared and combined.

Decomposition of totally transcendental modules

1980 ◽  
Vol 45 (1) ◽  
pp. 155-164 ◽  
Author(s):  
Steven Garavaglia
Keyword(s):  
Quantifier Elimination ◽  
Decomposition Theorem ◽  
Noetherian Rings ◽  
Projective Modules ◽  
Artinian Modules ◽  
Injective Modules ◽  
Unique Decomposition ◽  
Special Cases ◽  
Decomposition Theorems ◽  
Coherent Rings

The main theorem of this paper states that if R is a ring and is a totally transcendental R-module, then has a unique decomposition as a direct sum of indecomposable R-modules. Natural examples of totally transcendental modules are injective modules over noetherian rings, artinian modules over commutative rings, projective modules over left-perfect, right-coherent rings, and arbitrary modules over Σ – α-gens rings. Therefore, our decomposition theorem yields as special cases the purely algebraic unique decomposition theorems for these four classes of modules due to Matlis; Warfield; Mueller, Eklof, and Sabbagh; and Shelah and Fisher. These results and a number of other corollaries about totally transcendental modules are covered in §1. In §2, I show how the results of § 1 can be used to give an improvement of Baur's classification of ω-categorical modules over countable rings. In §3, the decomposition theorem is used to study modules with quantifier elimination over noetherian rings.The goals of this section are to prove the decomposition theorem and to derive some of its immediate corollaries. I will begin with some notational conventions. R will denote a ring with an identity element. LR is the language of left R-modules described in [4, p. 251] and TR is the theory of left R-modules. “R-module” will mean “unital left R-module”. A formula will mean an LR-formula.

scholarly journals Purity and decomposition theorems for staggered sheaves

2012 ◽  
Vol 11 (4) ◽  
pp. 695-745
Author(s):  
Pramod N. Achar ◽  
David Treumann
Keyword(s):  
Decomposition Theorem ◽  
Derived Category ◽  
Perverse Sheaves ◽  
Perverse Sheaf ◽  
Coherent Sheaves ◽  
Direct Sum ◽  
Constructible Sheaves ◽  
Decomposition Theorems ◽  
Pure Complex

AbstractTwo major results in the theory of ℓ-adic mixed constructible sheaves are the purity theorem (every simple perverse sheaf is pure) and the decomposition theorem (every pure object in the derived category is a direct sum of shifts of simple perverse sheaves). In this paper, we prove analogues of these results for coherent sheaves. Specifically, we work with staggered sheaves, which form the heart of a certain t-structure on the derived category of equivariant coherent sheaves. We prove, under some reasonable hypotheses, that every simple staggered sheaf is pure, and that every pure complex of coherent sheaves is a direct sum of shifts of simple staggered sheaves.

scholarly journals A Unified Approach to the Decomposition Theorems in Baer $$*$$-Rings

2021 ◽  
Vol 76 (3) ◽  
Author(s):  
Zbigniew Burdak ◽  
Marek Kosiek ◽  
Patryk Pagacz ◽  
Marek Słociński
Keyword(s):  
Hilbert Space ◽  
Linear Algebra ◽  
Decomposition Theorem ◽  
Unified Approach ◽  
Hereditary Property ◽  
Hilbert Space Operators ◽  
Baer Rings ◽  
Decomposition Theorems

AbstractThe aim of the paper is to generalize decomposition theorems showed in Bagheri-Bardi et al. (Linear Algebra Appl 583:102–118, 2019; Linear Algebra Appl 539:117–133, 2018) by a unified approach. We show a general decomposition theorem with respect to a hereditary property. Then the vast majority of decompositions known in the algebra of Hilbert space operators is generalized to elements of Baer $$*$$ ∗ -rings by this theorem. The theorem yields also results which are new in the algebra of bounded Hilbert space operators. Additionally, the model of summands in Wold–Słociński decomposition is given in Baer $$*$$ ∗ -rings.

Lebesgue decomposition theorems

2013 ◽  
Vol 79 (1-2) ◽  
pp. 219-233
Author(s):  
Zoltan Sebestyén ◽  
Zsigmond Tarcsay ◽  
Tamas Titkos
Keyword(s):  
Lebesgue Decomposition ◽  
Decomposition Theorems
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