Factorization of Analytic Functions with Values in Non-Commutative L1-spaces and Applications

1989 ◽  
Vol 41 (5) ◽  
pp. 882-906 ◽  
Author(s):  
Uffe Haagerup ◽  
Gilles Pisier
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Analytic Functions ◽  
Von Neumann Algebra ◽  
Interpolation Method ◽  
Complex Interpolation ◽  
Von Neumann ◽  
Nikodym Property ◽  
Image Position ◽  
Algebraic Formulation

Let X be a Banach space such that X* is a von Neumann algebra. We prove that X has the analytic Radon-Nikodym property (in short: ARNP). More precisely we show that for any function ƒ in H1(X) we have This implies the ARNP for X as well as for all the Banach spaces which are finitely representable in X. The proof uses a C*-algebraic formulation of the classical factorization theorems for matrix valued H1-functions. As a corollary we prove (for instance) that if A ⊂ B is a C*-subalgebra of a C*-algebra B, then every operator from A into H∞ extends to an operator from B into H∞ with the same norm. We include some remarks on the ARNP in connection with the complex interpolation method.

Generalized duality and product of some noncommutative symmetric spaces

2016 ◽  
Vol 27 (10) ◽  
pp. 1650082 ◽  
Author(s):  
Yazhou Han
Keyword(s):  
Banach Spaces ◽  
Product Space ◽  
Symmetric Spaces ◽  
Von Neumann Algebra ◽  
Interpolation Method ◽  
Complex Interpolation ◽  
Von Neumann

Let [Formula: see text] and [Formula: see text] be two symmetric quasi-Banach spaces and let [Formula: see text] be a semifinite von Neumann algebra. The purpose of this paper is to study the product space [Formula: see text] and the space of multipliers from [Formula: see text] to [Formula: see text], i.e. [Formula: see text]. These spaces share many properties with their classical counterparts. Let [Formula: see text] It is shown that if [Formula: see text] is [Formula: see text]-convex fully symmetric and [Formula: see text] is [Formula: see text]-convex, then [Formula: see text], where [Formula: see text] and [Formula: see text] is the space of multipliers from [Formula: see text] to [Formula: see text] As an application, we give conditions on when [Formula: see text] Moreover, we show that the product space can be described with the help of complex interpolation method.

Analytic functions with values in lattices and symmetric spaces of measurable operators

1991 ◽  
Vol 109 (3) ◽  
pp. 541-563 ◽  
Author(s):  
Quanhua Xu
Keyword(s):  
Function Space ◽  
Banach Lattice ◽  
Analytic Functions ◽  
Symmetric Spaces ◽  
Von Neumann Algebra ◽  
Banach Function Space ◽  
Von Neumann ◽  
Nikodym Property ◽  
Measurable Operators ◽  
Image Position

AbstractLet 0 < p,pi ≤ ∞, 0 < q,qi < ∞ (i = 1, 2) such thatLet E be a quasi-Banach lattice which fails to contain c0 and whose α-convexity constant is equal to 1 for some 0 < α < ∞. Then for every f∈H(E(q)) there exist g∈Hp, 0(E(q0)), h∈Hp1(E(q1)) such thatConsequently, E is q-concave for some finite q if and only if E is uniformly H1-convexifiable in the sense of [24]. Analogous results are also obtained for symmetric spaces of measurable operators. Another result proved in the paper says that if E is a symmetric quasi-Banach function space on (0, ∞) having the analytic Radon–Nikodym property then LE(M, τ) also possesses this property for any semifinite von Neumann algebra (M, τ).

Constructions of Non-Commutative Lp-Spaces with a Complex Parameter Arising from Modular Actions

1997 ◽  
Vol 08 (08) ◽  
pp. 1029-1066 ◽  
Author(s):  
Hideaki Izumi
Keyword(s):  
Banach Spaces ◽  
Von Neumann Algebra ◽  
Interpolation Method ◽  
Parameter Family ◽  
Complex Parameter ◽  
Complex Interpolation ◽  
Von Neumann ◽  
Lp Spaces ◽  
Natural Maps ◽  
Complete Extension

For a von Neumann algebra ℳ and a weight φ on ℳ, we will construct a complex one-parameter family [Formula: see text] of non-commutative Lp-spaces by using Calderón's complex interpolation method. This is a simultaneous and complete extension of the construction of non-commutative Lp-spaces by H. Kosaki and M. Terp. Moreover, we will show that for each p, all the parametrized Lp-spaces are mutually isometrically isomorphic as Banach spaces via natural maps.

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.

Operators which Factor through Convex Banach Lattices

1980 ◽  
Vol 32 (6) ◽  
pp. 1482-1500 ◽  
Author(s):  
Shlomo Reisner
Keyword(s):  
Banach Spaces ◽  
Banach Lattice ◽  
Interpolation Method ◽  
Banach Lattices ◽  
Uniform Convexity ◽  
Complex Interpolation ◽  
Power Type ◽  
Convex Operator ◽  
Image Position ◽  
The Ideal

We investigate here classes of operators T between Banach spaces E and F, which have factorization of the formwhere L is a Banach lattice, V is a p-convex operator, U is a q-concave operator (definitions below) and jF is the cannonical embedding of F in F”. We show that for fixed p, q this class forms a perfect normed ideal of operators Mp, q, generalizing the ideal Ip,q of [5]. We prove (Proposition 5) that Mp, q may be characterized by factorization through p-convex and q-concave Banach lattices. We use this fact together with a variant of the complex interpolation method introduced in [1], to show that an operator which belongs to Mp, q may be factored through a Banach lattice with modulus of uniform convexity (uniform smoothness) of power type arbitrarily close to q (to p). This last result yields similar geometric properties in subspaces of spaces having G.L. – l.u.st.

NATURAL BILINEAR FORMS, NATURAL SESQUILINEAR FORMS AND THE ASSOCIATED DUALITY ON NON-COMMUTATIVE Lp-SPACES

1998 ◽  
Vol 09 (08) ◽  
pp. 975-1039 ◽  
Author(s):  
HIDEAKI IZUMI
Keyword(s):  
General Theory ◽  
Von Neumann Algebra ◽  
Interpolation Method ◽  
Parameter Family ◽  
Complex Interpolation ◽  
Bilinear Forms ◽  
Sesquilinear Forms ◽  
Von Neumann ◽  
Lp Spaces

In the author's previous paper, he constructed a complex one-parameter family of non-commutative Lp-spaces [Formula: see text], [Formula: see text], 1 < p < ∞, for a von Neumann algebra ℳ with respect to a fixed faithful normal semi-finite weight φ on ℳ by using Calderón's complex interpolation method. In this paper, we will construct bounded non-degenerate bilinear forms < , >p,(α) on [Formula: see text], [Formula: see text], 1 < p <∞, 1/p + 1/q = 1, and bounded non-degenerate sesquilinear forms ( | )p,(α) on [Formula: see text], [Formula: see text], 1 < p < ∞, 1/p + 1/q = 1, and by using general theory of the complex interpolation method we show the reflexivity of [Formula: see text] and the duality between [Formula: see text] and [Formula: see text] via < , >p,(α) (or the duality between [Formula: see text] and [Formula: see text] via ( | )p,(α)). Moreover, we discuss bimodule properties of [Formula: see text].

scholarly journals An Analogue of the Radon-Nikodym Property for Non-Locally Convex Quasi-Banach Spaces

1979 ◽  
Vol 22 (1) ◽  
pp. 49-60 ◽  
Author(s):  
N. J. Kalton
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Measurable Function ◽  
Nikodym Property ◽  
Image Position ◽  
Locally Convex

In recent years there has been considerable interest in Banach spaces with the Radon-Nikodym Property; see (1) for a summary of the main known results on this class of spaces.We may define this property as follows: a Banach space X has the Radon-Nikodym Property if whenever T ∈ ℒ (L1, X)(where L1 = L1(0, 1)) then T is differentiable i.e.where g:(0, 1)→X is an essentially bounded strongly measurable function. In this paper we examine analogues of the Radon-Nikodym Property for quasi-Banach spaces. If 0>p > 1, there are several possible ways of defining “differentiable” operators on Lp, but they inevitably lead to the conclusion that the only differentiable operator is zero.

scholarly journals Entire Analytic Functions of Unbounded Type on Banach Spaces and Their Lineability

Axioms ◽  
2021 ◽  
Vol 10 (3) ◽  
pp. 150
Author(s):  
Andriy Zagorodnyuk ◽  
Anna Hihliuk
Keyword(s):  
Banach Space ◽  
Banach Spaces ◽  
Analytic Functions ◽  
Entire Functions ◽  
Dimensional Algebra ◽  
Dimensional Banach Space ◽  
Infinite Dimensional ◽  
Linear Subspaces ◽  
Infinite Dimensional Banach Space

In the paper we establish some conditions under which a given sequence of polynomials on a Banach space X supports entire functions of unbounded type, and construct some counter examples. We show that if X is an infinite dimensional Banach space, then the set of entire functions of unbounded type can be represented as a union of infinite dimensional linear subspaces (without the origin). Moreover, we show that for some cases, the set of entire functions of unbounded type generated by a given sequence of polynomials contains an infinite dimensional algebra (without the origin). Some applications for symmetric analytic functions on Banach spaces are obtained.

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