scholarly journals On the Radon-Nikodym property in Jordan algebras

1983 ◽  
Vol 24 (2) ◽  
pp. 185-189
Author(s):  
Cho-Ho Chu
Keyword(s):  
Convex Hull ◽  
Banach Spaces ◽  
General Result ◽  
Hausdorff Space ◽  
Jordan Algebras ◽  
C Algebra ◽  
The Past ◽  
Projective Tensor ◽  
Nikodym Property ◽  
Pure States

Banach spaces whose duals possess the Radon-Nikodym property have been studied extensively in the past (cf. [5]). It has been shown recently in [4] that a C*-algebra is scattered if and only if its Banach dual possesses the Radon-Nikodym property. This result extends the well-known result of Pełczynski and Semandini [8] that a compact Hausdorff space Ωis dispersed if and only if C(Ω)* has the Radon-Nikodym property. The purpose of this note is to give a transparent proof of a more general result for Jordan algebras which unifies the aforementioned results. We prove that the dual of a JB-algebra A possesses the Radon-Nikodym property if and only if the state space of A is the cr-convex hull of its pure states. We also consider the projective tensor products of the duals of JB-algebras in this context.

Slicings, Selections and Their Applications

1992 ◽  
Vol 44 (3) ◽  
pp. 483-504 ◽  
Author(s):  
N. Ghoussoub ◽  
B. Maurey ◽  
W. Schachermayer
Keyword(s):  
Banach Spaces ◽  
List Type ◽  
Space Theory ◽  
Reflexive Banach Spaces ◽  
Open Problems ◽  
Infinite Dimensional ◽  
The Past ◽  
Asplund Spaces ◽  
Infinite Dimensional Banach Space ◽  
Nikodym Property

In the past few years, much progress have been made on several open problems in infinite dimensional Banach space theory. Here are some of the most recent results:1)The existence of boundedly complete basic sequences in a large class of Banach spaces including the ones with the so-called Radon-Nikodym property ([G-M2], [G-M4]).2)The embedding of separable reflexive Banach spaces into reflexive spaces with basis (fZl).3)The existence of long sequences of projections and hence of locally uniformly convex norms in the duals of Asplund spaces. ([F-G])

scholarly journals The dual of the space of bounded operators on a Banach space

2021 ◽  
Vol 8 (1) ◽  
pp. 48-59
Author(s):  
Fernanda Botelho ◽  
Richard J. Fleming
Keyword(s):  
Banach Space ◽  
Tensor Product ◽  
Banach Spaces ◽  
Dual Space ◽  
Tensor Products ◽  
Bounded Operators ◽  
Projective Tensor Product ◽  
Projective Tensor

Abstract Given Banach spaces X and Y, we ask about the dual space of the 𝒧(X, Y). This paper surveys results on tensor products of Banach spaces with the main objective of describing the dual of spaces of bounded operators. In several cases and under a variety of assumptions on X and Y, the answer can best be given as the projective tensor product of X ** and Y *.

scholarly journals SUBPROJECTIVITY OF PROJECTIVE TENSOR PRODUCTS OF BANACH SPACES OF CONTINUOUS FUNCTIONS

2021 ◽  
pp. 1-14
Author(s):  
R.M. CAUSEY
Keyword(s):  
Tensor Product ◽  
Banach Spaces ◽  
Tensor Products ◽  
Continuous Functions ◽  
Hausdorff Spaces ◽  
Projective Tensor Product ◽  
Projective Tensor ◽  
Spaces Of Continuous Functions

Abstract Galego and Samuel showed that if K, L are metrizable, compact, Hausdorff spaces, then $C(K)\widehat{\otimes}_\pi C(L)$ is c0-saturated if and only if it is subprojective if and only if K and L are both scattered. We remove the hypothesis of metrizability from their result and extend it from the case of the twofold projective tensor product to the general n-fold projective tensor product to show that for any $n\in\mathbb{N}$ and compact, Hausdorff spaces K1, …, K n , $\widehat{\otimes}_{\pi, i=1}^n C(K_i)$ is c0-saturated if and only if it is subprojective if and only if each K i is scattered.

scholarly journals Types of Radon-Nikodym properties for the projective tensor product of Banach spaces

2003 ◽  
Vol 47 (4) ◽  
pp. 1303-1326 ◽  
Author(s):  
Qingying Bu ◽  
Joe Diestel ◽  
Patrick Dowling ◽  
Eve Oja
Keyword(s):  
Tensor Product ◽  
Banach Spaces ◽  
Projective Tensor Product ◽  
Projective Tensor

scholarly journals The weak Radon-Nikodym property in Banach spaces

1979 ◽  
Vol 64 (2) ◽  
pp. 151-174 ◽  
Author(s):  
Kazimierz Musiał
Keyword(s):  
Banach Spaces ◽  
Nikodym Property

Hewitt Realcompactifications of Products

1970 ◽  
Vol 22 (3) ◽  
pp. 645-656 ◽  
Author(s):  
William G. McArthur
Keyword(s):  
General Result ◽  
Hausdorff Space ◽  
Ad Hoc ◽  
Completely Regular ◽  
Analogous Question ◽  
Open Question ◽  
Hewitt Realcompactification

The Hewitt realcompactification vX of a completely regular Hausdorff space X has been widely investigated since its introduction by Hewitt [17]. An important open question in the theory concerns when the equality v(X × Y) = vX × vY is valid. Glicksberg [10] settled the analogous question in the parallel theory of Stone-Čech compactifications: for infinite spaces X and Y, β(X × Y) = βX × β Y if and only if the product X × Y is pseudocompact. Work of others, notably Comfort [3; 4] and Hager [13], makes it seem likely that Glicksberg's theorem has no equally specific analogue for v(X × Y) = vX × vY. In the absence of such a general result, particular instances may tend to be attacked by ad hoc techniques resulting in much duplication of effort.

On a Theorem of Beurling and Livingston

1965 ◽  
Vol 17 ◽  
pp. 367-372 ◽  
Author(s):  
Felix E. Browder
Keyword(s):  
Banach Space ◽  
Fourier Series ◽  
Banach Spaces ◽  
General Result ◽  
Functional Equations ◽  
Linear Functional ◽  
Monotone Mappings ◽  
Conjugate Space ◽  
Non Linear ◽  
Duality Mappings

In their paper (1), Beurling and Livingston established a generalization of the Riesz-Fischer theorem for Fourier series in Lp using a theorem on duality mappings of a Banach space B into its conjugate space B*. It is our purpose in the present paper to give another proof of this theorem by deriving it from a more general result concerning monotone mappings related to recent results on non-linear functional equations in Banach spaces obtained by the writer (2, 3, 4, 5) and G. J. Minty (6).

Completions of Boolean algebras of projections and weak-star closures of C*-algebras on dual Banach spaces

2011 ◽  
Vol 54 (2) ◽  
pp. 515-529
Author(s):  
Philip G. Spain
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Boolean Algebras ◽  
C Algebra ◽  
C Algebras ◽  
Weak Star ◽  
Weakly Closed ◽  
Operator Closure ◽  
Dual Banach Space ◽  
Completion Theorem

AbstractPalmer has shown that those hermitians in the weak-star operator closure of a commutative C*-algebra represented on a dual Banach space X that are known to commute with the initial C*-algebra form the real part of a weakly closed C*-algebra on X. Relying on a result of Murphy, it is shown in this paper that this last proviso may be dropped, and that the weak-star closure is even a W*-algebra.When the dual Banach space X is separable, one can prove a similar result for C*-equivalent algebras, via a ‘separable patch’ completion theorem for Boolean algebras of projections on such spaces.

scholarly journals Dual functors and the Radon-Nikodym property in the category of Banach spaces

1979 ◽  
Vol 27 (4) ◽  
pp. 479-494 ◽  
Author(s):  
John Wick Pelletier
Keyword(s):  
Banach Spaces ◽  
Direct Limit ◽  
Integral Operators ◽  
Compact Operators ◽  
Property A ◽  
Weakly Compact ◽  
Nikodym Property ◽  
Weakly Compact Operators

AbstractThe notion of duality of functors is used to study and characterize spaces satisfying the Radon-Nikodym property. A theorem of equivalences concerning the Radon-Nikodym property is proved by categorical means; the classical Dunford-Pettis theorem is then deduced using an adjointness argument. The functorial properties of integral operators, compact operators, and weakly compact operators are discussed. It is shown that as an instance of Kan extension the weakly compact operators can be expressed as a certain direct limit of ordinary hom functors. Characterizations of spaces satisfying the Radon-Nikodym property are then given in terms of the agreement of dual functors of the functors mentioned above.

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