On order structure and operators in L ∞(μ)

2009 ◽  
Vol 7 (4) ◽  
Author(s):  
Irina Krasikova ◽  
Miguel Martín ◽  
Javier Merí ◽  
Vladimir Mykhaylyuk ◽  
Mikhail Popov
Keyword(s):  
Banach Space ◽  
Compact Operator ◽  
Linear Functional ◽  
Continuous Operator ◽  
The Other ◽  
Order Structure ◽  
Continuous Linear ◽  
Continuous Linear Functional ◽  
Other Hand ◽  
Continuous Operators

AbstractIt is known that there is a continuous linear functional on L ∞ which is not narrow. On the other hand, every order-to-norm continuous AM-compact operator from L ∞(μ) to a Banach space is narrow. We study order-to-norm continuous operators acting from L ∞(μ) with a finite atomless measure μ to a Banach space. One of our main results asserts that every order-to-norm continuous operator from L ∞(μ) to c 0(Γ) is narrow while not every such an operator is AM-compact.

scholarly journals On linear continuous operators on C_p-spaces

2021 ◽  
Vol 21 (1) ◽  
pp. 45-50
Author(s):  
A.P. Devyatkov ◽  
◽  
S.D. Shalaginov ◽  
Keyword(s):  
Linear Functional ◽  
Pointwise Convergence ◽  
Continuous Operator ◽  
Continuous Functions ◽  
Linear Continuous Operator ◽  
Continuous Linear ◽  
Space Of Continuous Functions ◽  
Continuous Linear Functional ◽  
Topology Of Pointwise Convergence ◽  
Continuous Operators

The paper describes the structure of a linear continuous operator on the space of continuous functions in the topology of pointwise convergence. The corresponding theorem is a generalization of A.V.Arkhangel'skii's theorem on the general form of a continuous linear functional on such spaces.

scholarly journals The states of a Banach algebra generate the dual

1971 ◽  
Vol 17 (4) ◽  
pp. 341-344 ◽  
Author(s):  
Allan M. Sinclair
Keyword(s):  
Banach Space ◽  
Banach Algebra ◽  
Linear Combination ◽  
Linear Functional ◽  
Measure Theory ◽  
Complex Field ◽  
Continuous Linear ◽  
Continuous Linear Functional ◽  
Unital Banach Algebra ◽  
Dual Banach Space

In this paper we prove that the states of a unital Banach algebra generate the dual Banach space as a linear space (Theorem 2). This is a result of R. T. Moore (4, Theorem 1(a)) who uses a decomposition of measures in his proof. In the proof given here the measure theory is replaced by a Hahn-Banach separation argument. We shall let A denote a unital Banach algebra over the complex field, and D(1) denote {f ∈ A′: ‖f‖ = f(1) = 1} where A′ is the dual of A. The motivation of Moore's results is the theorem that in a C*-algebra every continuous linear functional is a linear combination of four states (the states are the elements of D(1)) (see (2, 2.6.4, 2.1.9, 1.1.10)).

Properties (V) and (w V) on C(Ω X)

1995 ◽  
Vol 117 (3) ◽  
pp. 469-477 ◽  
Author(s):  
Elizabeth M. Bator ◽  
Paul W. Lewis
Keyword(s):  
Banach Space ◽  
Continuous Function ◽  
Linear Functional ◽  
Formal Series ◽  
Continuous Linear ◽  
Continuous Linear Functional ◽  
Weakly Compact ◽  
Image Position ◽  
Continuous Function Space ◽  
Fundamental Paper

A formal series Σxn in a Banach space X is said to be weakly unconditionally converging, or alternatively weakly unconditionally Cauchy (wuc) if Σ|x*(xn)| < ∞ for every continuous linear functional x* ∈ X*. A subset K of X* is called a V-subset of X* iffor each wuc series Σxn in X. Further, the Banach space X is said to have property (V) if the V-subsets of X* coincide with the relatively weakly compact subsets of X*. In a fundamental paper in 1962, Pelczynski [10] showed that the Banach space X has property (V) if and only if every unconditionally converging operator with domain X is weakly compact. In this same paper, Pelczynski also showed that all C(Ω) spaces have property (V), and asked if the abstract continuous function space C(Ω, X) has property (F) whenever X has property (F).

scholarly journals A noncommutative theory of Bade functionals

1991 ◽  
Vol 33 (1) ◽  
pp. 73-81 ◽  
Author(s):  
Don Hadwin ◽  
Mehmet Orhon
Keyword(s):  
Banach Space ◽  
Linear Functional ◽  
Hausdorff Space ◽  
Boolean Algebras ◽  
Representation Space ◽  
Continuous Linear ◽  
Continuous Linear Functional ◽  
Banach Modules ◽  
Key Concepts ◽  
Stone Representation

Since the pioneering work of W. G. Bade [3, 4] a great deal of work has been done on bounded Boolean algebras of projections on a Banach space ([11, XVII.3.XVIII.3], [21, V.3], [16], [6], [12], [13], [14], ]17], [18], [23], [24]). Via the Stone representation space of the Boolean algebra, the theory can be studied through Banach modules over C(K), where K is a compact Hausdorff space. One of the key concepts in the theory is the notion of Bade functionals. If X is a Banach C(K)-module and x ε X, then a Bade functional of x with respect to C(K) is a continuous linear functional α on X such that, for each a in C(K) with a ≥ 0, we have(i) α (ax) ≥0,(ii) if α (ax) = 0, then ax = 0.

Representation of Linear Functionals on Köthe Spaces

1953 ◽  
Vol 5 ◽  
pp. 568-575 ◽  
Author(s):  
G. G. Lorentz ◽  
D. G. Wertheim
Keyword(s):  
Banach Space ◽  
Linear Functional ◽  
Continuous Linear ◽  
Linear Functionals ◽  
Continuous Linear Functional ◽  
Köthe Spaces ◽  
Integrable Functions ◽  
Köthe Space ◽  
Vector Valued

Kothe spaces, in the terminology of Diendonné [2], are certain spaces X of real valued integrable functions. In this paper we consider the problem of representation of continuous linear functional on vector valued Kothe spaces. The elements of a Kôthe space X(B) are functions with values in a Banach space B (see §2).

scholarly journals The Kreps-Yan theorem forL∞

2005 ◽  
Vol 2005 (17) ◽  
pp. 2749-2756 ◽  
Author(s):  
D. B. Rokhlin
Keyword(s):  
Banach Space ◽  
Convex Cone ◽  
Linear Functional ◽  
Closed Convex Cone ◽  
Continuous Linear ◽  
Continuous Linear Functional ◽  
Usual Approach

We prove the following version of the Kreps-Yan theorem. For any norm-closed convex coneC⊂L∞such thatC∩L+∞={0}andC⊃−L+∞, there exists a strictly positive continuous linear functional, whose restriction onCis nonpositive. The technique of the proof differs from the usual approach, applicable to a weakly Lindelöf Banach space.

scholarly journals TEOREMA REPRESENTASI RIESZ–FRECHET PADA RUANG HILBERT

2011 ◽  
Vol 5 (2) ◽  
pp. 1-8
Author(s):  
Mozart W. Talakua ◽  
Stenly J. Nanuru
Keyword(s):  
Banach Space ◽  
Hilbert Space ◽  
Hilbert Spaces ◽  
Representation Theorem ◽  
Linear Functional ◽  
Inner Product ◽  
Continuous Linear ◽  
Continuous Linear Functional ◽  
Important Idea

Hilbert space is a very important idea of the Davids Hilbert invention. In 1907, Riesz and Fréchet developed one of the theorem in Hilbert space called the Riesz-Fréchet representationtheorem. This research contains some supporting definitions Banach space, pre-Hilbert spaces, Hilbert spaces, the duality of Banach and Riesz-Fréchet representation theorem. On Riesz-Fréchet representation theorem will be shown that a continuous linear functional that exist in the Hilbert space is an inner product, in other words, there is no continuous linear functional on a Hilbert space except the inner product.

Intersection properties of ball sequences and uniqueness of Hahn–Banach extensions

1999 ◽  
Vol 129 (6) ◽  
pp. 1251-1262 ◽  
Author(s):  
E. Oja ◽  
M. Põldvere
Keyword(s):  
Banach Space ◽  
Linear Functional ◽  
Geometric Property ◽  
Closed Subspace ◽  
Continuous Linear ◽  
Uniqueness Property ◽  
Continuous Linear Functional

Let X be a Banach space and Y a closed subspace. We introduce an intrinsic geometric property of Y—the k-ball sequence property—which is a weakening of the famous k-ball property due to Alfsen & Effros. We prove that Y satisfies the 2-ball sequence property if and only if Y has the Phelps uniqueness property U (i.e. every continuous linear functional g ∈Y* has a unique norm-preserving extension f ∈X*). We prove that Y is an ideal having property U if and only if Y satisfies the 3-ball sequence property, and in this case, Y satisfies the k-ball sequence property for all k.

scholarly journals Extreme points and support points of conformal mappings

2019 ◽  
Vol 17 (1) ◽  
pp. 23-31
Author(s):  
Ronen Peretz
Keyword(s):  
Remainder Term ◽  
Linear Functional ◽  
Convex Combination ◽  
Extreme Points ◽  
Holomorphic Functions ◽  
Support Point ◽  
Continuous Linear ◽  
Continuous Linear Functional ◽  
The Third ◽  
Support Points

Abstract There are three types of results in this paper. The first, extending a representation theorem on a conformal mapping that omits two values of equal modulus. This was due to Brickman and Wilken. They constructed a representation as a convex combination with two terms. Our representation constructs convex combinations with unlimited number of terms. In the limit one can think of it as an integration over a probability space with the uniform distribution. The second result determines the sign of ℜ L(z0(f(z))2) up to a remainder term which is expressed using a certain integral that involves the Löwner chain induced by f(z), for a support point f(z) which maximizes ℜ L. Here L is a continuous linear functional on H(U), the topological vector space of the holomorphic functions in the unit disk U = {z ∈ ℂ | |z| < 1}. Such a support point is known to be a slit mapping and f(z0) is the tip of the slit ℂ − f(U). The third demonstrates some properties of support points of the subspace Sn of S. Sn contains all the polynomials in S of degree n or less. For instance such a support point p(z) has a zero of its derivative p′(z) on ∂U.

scholarly journals AN OPERATOR SUMMABILITY OF SEQUENCES IN BANACH SPACES

2013 ◽  
Vol 56 (2) ◽  
pp. 427-437 ◽  
Author(s):  
ANIL KUMAR KARN ◽  
DEBA PRASAD SINHA
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Borel Measure ◽  
Operator Ideal ◽  
The Other ◽  
Summable Sequence ◽  
Other Hand ◽  
Limited Operator

AbstractLet 1 ≤ p < ∞. A sequence 〈 xn 〉 in a Banach space X is defined to be p-operator summable if for each 〈 fn 〉 ∈ lw*p(X*) we have 〈〈 fn(xk)〉k〉n ∈ lsp(lp). Every norm p-summable sequence in a Banach space is operator p-summable whereas in its turn every operator p-summable sequence is weakly p-summable. An operator T ∈ B(X, Y) is said to be p-limited if for every 〈 xn 〉 ∈ lpw(X), 〈 Txn 〉 is operator p-summable. The set of all p-limited operators forms a normed operator ideal. It is shown that every weakly p-summable sequence in X is operator p-summable if and only if every operator T ∈ B(X, lp) is p-absolutely summing. On the other hand, every operator p-summable sequence in X is norm p-summable if and only if every p-limited operator in B(lp', X) is absolutely p-summing. Moreover, this is the case if and only if X is a subspace of Lp(μ) for some Borel measure μ.

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