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.

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].

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.

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.

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.

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 Local rigidity for actions of Kazhdan groups on noncommutative Lp-spaces

2015 ◽  
Vol 26 (08) ◽  
pp. 1550064
Author(s):  
Bachir Bekka
Keyword(s):  
Von Neumann Algebra ◽  
Regular Representation ◽  
Von Neumann ◽  
Lp Spaces ◽  
Local Rigidity ◽  
Left Regular Representation ◽  
Left Regular ◽  
Property T ◽  
Finite Factor ◽  
Finite Index Subgroup

Let Γ be a discrete group and 𝒩 a finite factor, and assume that both have Kazhdan's Property (T). For p ∈ [1, +∞), p ≠ 2, let π : Γ →O(Lp(𝒩)) be a homomorphism to the group O(Lp(𝒩)) of linear bijective isometries of the Lp-space of 𝒩. There are two actions πl and πr of a finite index subgroup Γ+ of Γ by automorphisms of 𝒩 associated to π and given by πl(g)x = (π(g) 1)*π(g)(x) and πr(g)x = π(g)(x)(π(g) 1)* for g ∈ Γ+ and x ∈ 𝒩. Assume that πl and πr are ergodic. We prove that π is locally rigid, that is, the orbit of π under O(Lp(𝒩)) is open in Hom (Γ, O(Lp(𝒩))). As a corollary, we obtain that, if moreover Γ is an ICC group, then the embedding g ↦ Ad (λ(g)) is locally rigid in O(Lp(𝒩(Γ))), where 𝒩(Γ) is the von Neumann algebra generated by the left regular representation λ of Γ.

scholarly journals ON THE STABILITY OF THE DIFFERENTIAL PROCESS GENERATED BY COMPLEX INTERPOLATION

2020 ◽  
pp. 1-32
Author(s):  
Jesús M. F. Castillo ◽  
Willian H. G. Corrêa ◽  
Valentin Ferenczi ◽  
Manuel González
Keyword(s):  
Unit Circle ◽  
Banach Spaces ◽  
Local Stability ◽  
Interpolation Method ◽  
Complex Interpolation ◽  
Analytic Family ◽  
Köthe Spaces ◽  
The Stability ◽  
Stability Results ◽  
Differential Process

We study the stability of the differential process of Rochberg and Weiss associated with an analytic family of Banach spaces obtained using the complex interpolation method for families. In the context of Köthe function spaces, we complete earlier results of Kalton (who showed that there is global bounded stability for pairs of Köthe spaces) by showing that there is global (bounded) stability for families of up to three Köthe spaces distributed in arcs on the unit circle while there is no (bounded) stability for families of four or more Köthe spaces. In the context of arbitrary pairs of Banach spaces, we present some local stability results and some global isometric stability results.

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.

scholarly journals CONNECTIONS ON STATISTICAL MANIFOLDS OF DENSITY OPERATORS BY GEOMETRY OF NONCOMMUTATIVE Lp-SPACES

1999 ◽  
Vol 02 (01) ◽  
pp. 169-178 ◽  
Author(s):  
PAOLO GIBILISCO ◽  
TOMMASO ISOLA
Keyword(s):  
Unit Sphere ◽  
Von Neumann Algebra ◽  
Noncommutative Space ◽  
Statistical Manifold ◽  
Commutative Case ◽  
Von Neumann ◽  
Lp Spaces ◽  
Density Operators ◽  
Statistical Manifolds ◽  
Made In

Let [Formula: see text] be a statistical manifold of density operators, with respect to an n.s.f. trace τ on a semifinite von Neumann algebra M. If Sp is the unit sphere of the noncommutative space Lp(M, τ), using the noncommutative Amari embedding [Formula: see text], we define a noncommutative α-bundle-connection pair (ℱα, ∇α), by the pullback technique. In the commutative case we show that it coincides with the construction of nonparametric Amari–Čentsov α-connection made in Ref. 8 by Gibilisco and Pistone.

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