An extension of a simply inequality between von Neumann–Jordan and James constants in Banach spaces

2017 ◽  
Vol 33 (9) ◽  
pp. 1287-1296 ◽  
Author(s):  
Chang Sen Yang ◽  
Feng Hui Wang
Keyword(s):  
Banach Spaces ◽  
Von Neumann

scholarly journals Some Properties Concerning the JL(X) and YJ(X) Which Related to Some Special Inscribed Triangles of Unit Ball

Symmetry ◽  
2021 ◽  
Vol 13 (7) ◽  
pp. 1285
Author(s):  
Asif Ahmad ◽  
Yuankang Fu ◽  
Yongjin Li
Keyword(s):  
Unit Ball ◽  
Banach Spaces ◽  
Uniformly Convex ◽  
Geometric Properties ◽  
James Constant ◽  
Von Neumann ◽  
Geometric Constants

In this paper, we will make some further discussions on the JL(X) and YJ(X) which are symmetric and related to the side lengths of some special inscribed triangles of the unit ball, and also introduce two new geometric constants L1(X,▵), L2(X,▵) which related to the perimeters of some special inscribed triangles of the unit ball. Firstly, we discuss the relations among JL(X), YJ(X) and some geometric properties of Banach spaces, including uniformly non-square and uniformly convex. It is worth noting that we point out that uniform non-square spaces can be characterized by the side lengths of some special inscribed triangles of unit ball. Secondly, we establish some inequalities for JL(X), YJ(X) and some significant geometric constants, including the James constant J(X) and the von Neumann-Jordan constant CNJ(X). Finally, we introduce the two new geometric constants L1(X,▵), L2(X,▵), and calculate the bounds of L1(X,▵) and L2(X,▵) as well as the values of L1(X,▵) and L2(X,▵) for two Banach spaces.

scholarly journals Completely bounded isomorphisms of injective von Neumann algebras

1989 ◽  
Vol 32 (2) ◽  
pp. 317-327 ◽  
Author(s):  
Erik Christensen ◽  
Allan M. Sinclair
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Von Neumann Algebras ◽  
Linear Structure ◽  
Type I ◽  
Hausdorff Spaces ◽  
Bounded Degree ◽  
Von Neumann ◽  
Separable Predual ◽  
Image Position

Milutin's Theorem states that if X and Y are uncountable metrizable compact Hausdorff spaces, then C(X) and C(Y) are isomorphic as Banach spaces [15, p. 379]. Thus there is only one isomorphism class of such Banach spaces. There is also an extensive theory of the Banach–Mazur distance between various classes of classical Banach spaces with the deepest results depending on probabilistic and asymptotic estimates [18]. Lindenstrauss, Haagerup and possibly others know that as Banach spaceswhere H is the infinite dimensional separable Hilbert space, R is the injective II 1-factor on H, and ≈ denotes Banach space isomorphism. Haagerup informed us of this result, and suggested considering completely bounded isomorphisms; it is a pleasure to acknowledge his suggestion. We replace Banach space isomorphisms by completely bounded isomorphisms that preserve the linear structure and involution, but not the product. One of the two theorems of this paper is a strengthened version of the above result: if N is an injective von Neumann algebra with separable predual and not finite type I of bounded degree, then N is completely boundedly isomorphic to B(H). The methods used are similar to those in Banach space theory with complete boundedness needing a little care at various points in the argument. Extensive use is made of the conditional expectation available for injective algebras, and the methods do not apply to the interesting problems of completely bounded isomorphisms of non-injective von Neumann algebras (see [4] for a study of the completely bounded approximation property).

Ergodic theorems for semifinite von Neumann algebras: II

1980 ◽  
Vol 88 (1) ◽  
pp. 135-147 ◽  
Author(s):  
F. J. Yeadon
Keyword(s):  
Banach Spaces ◽  
Ergodic Theorem ◽  
Von Neumann Algebra ◽  
Von Neumann Algebras ◽  
Operator Norm ◽  
Ergodic Theorems ◽  
Unbounded Operators ◽  
Von Neumann ◽  
The Mean ◽  
Image Position

In (7) we proved maximal and pointwise ergodic theorems for transformations a of a von Neumann algebra which are linear positive and norm-reducing for both the operator norm ‖ ‖∞ and the integral norm ‖ ‖1 associated with a normal trace ρ on . Here we introduce a class of Banach spaces of unbounded operators, including the Lp spaces defined in (6), in which the transformations α reduce the norm, and in which the mean ergodic theorem holds; that is the averagesconverge in norm.

QUANTUM INFORMATION GEOMETRY AND NONCOMMUTATIVE Lp-SPACES

2005 ◽  
Vol 08 (02) ◽  
pp. 215-233 ◽  
Author(s):  
ANNA JENČOVÁ
Keyword(s):  
Quantum Information ◽  
Banach Spaces ◽  
Von Neumann Algebra ◽  
Classical Case ◽  
Information Geometry ◽  
Positive Cone ◽  
Uniformly Convex ◽  
Von Neumann ◽  
Lp Spaces ◽  
Uniformly Convex Banach Spaces

Let M be a von Neumann algebra. We define the noncommutative extension of information geometry by embeddings of M into noncommutative Lp-spaces. Using the geometry of uniformly convex Banach spaces and duality of the Lp and Lq spaces for 1/p +1/q =1, we show that we can introduce the α-divergence, for α∈(-1, 1), in a similar manner as Amari in the classical case. If restricted to the positive cone, the α-divergence belongs to the class of quasi-entropies, defined by Petz.

scholarly journals On James and Jordan–von Neumann constants and the normal structure coefficient of Banach spaces

2001 ◽  
Vol 144 (3) ◽  
pp. 275-295 ◽  
Author(s):  
Mikio Kato ◽  
Lech Maligranda ◽  
Yasuji Takahashi
Keyword(s):  
Banach Spaces ◽  
Normal Structure ◽  
Von Neumann ◽  
Structure Coefficient

NoncommutativeLp-spaces with 0

1981 ◽  
Vol 89 (3) ◽  
pp. 405-411 ◽  
Author(s):  
Kichi-Suke Saito
Keyword(s):  
Banach Spaces ◽  
Von Neumann Algebra ◽  
Unbounded Operators ◽  
Von Neumann ◽  
Lp Spaces ◽  
Normal Semifinite Trace ◽  
Gauge Space ◽  
Nikodym Theorem

The noncommutative Lp-spaces (1 ≤p≤ ∞) of unbounded operators associated with a regular gauge space (a von Neumann algebra equipped with a faithful normal semifinite trace) are studied by many authors ((4), (5) and (7)). It is well-known that the noncommutativeLp-spaces (1 ≤P< ∞) are Banach spaces and the dual ofLpisLq(1 ≤p< ∞, 1/p+ 1/q= 1) by means of a Radon-Nikodym theorem.

scholarly journals On some modifications of n-th von Neumann–Jordan constant for Banach spaces

2020 ◽  
Vol 14 (3) ◽  
pp. 650-673
Author(s):  
Maciej Ciesielski ◽  
Ryszard Płuciennik
Keyword(s):  
Banach Spaces ◽  
Lebesgue Spaces ◽  
Von Neumann ◽  
Lower Upper

AbstractWe study, among others, upper, lower, upper modified and lower modified n-th von Neumann–Jordan constant and relationships between them. There are characterized uniformly non-$$l_{n}^{1}$$ l n 1 Banach spaces in terms of the upper modified n-th von Neumann–Jordan constant. Moreover, this constant is calculated explicitly for Lebesgue spaces $$L^{p}$$ L p and $$l^{p}$$ l p $$(1\le p\le \infty ).$$ ( 1 ≤ p ≤ ∞ ) . Finally, it is shown that the sequence of n-th upper and modified upper von Neumann–Jordan constants for the space $$L^p$$ L p as well as $$l^p$$ l p $$(2<p<\infty )$$ ( 2 < p < ∞ ) converges to $$B_p^2$$ B p 2 , where $$B_p$$ B p is the best type (2, p) constant in the Khinthine inequality for the case $$2\le p<\infty $$ 2 ≤ p < ∞ .

scholarly journals The generalized von Neumann--Jordan constant and normal structure in Banach spaces

2015 ◽  
Vol 6 (4) ◽  
pp. 206-214 ◽  
Author(s):  
Xi Wang ◽  
Yunan Cui ◽  
Chiping Zhang
Keyword(s):  
Banach Spaces ◽  
Normal Structure ◽  
Von Neumann
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