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].

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.

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.

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 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.

scholarly journals Kadec–Pełczyński decomposition for Haagerup Lp-spaces

2002 ◽  
Vol 132 (1) ◽  
pp. 137-154 ◽  
Author(s):  
NARCISSE RANDRIANANTOANINA
Keyword(s):  
Von Neumann Algebra ◽  
Von Neumann Algebras ◽  
Bounded Sequence ◽  
Point Property ◽  
Weakly Compact ◽  
Von Neumann ◽  
Lp Spaces ◽  
Reflexive Subspace ◽  
Finite Von Neumann Algebras ◽  
Bounded Sequences

Let [Mscr ] be a von Neumann algebra (not necessarily semi-finite). We provide a generalization of the classical Kadec–Pełczyński subsequence decomposition of bounded sequences in Lp[0, 1] to the case of the Haagerup Lp-spaces (1 [les ] p < 1 ). In particular, we prove that if { φn}∞n=1 is a bounded sequence in the predual [Mscr ]∗ of [Mscr ], then there exist a subsequence {φnk}∞k=1 of {φn}∞n=1, a decomposition φnk = yk+zk such that {yk, k [ges ] 1} is relatively weakly compact and the support projections supp(zk) ↓k 0 (or similarly mutually disjoint). As an application, we prove that every non-reflexive subspace of the dual of any given C*-algebra (or Jordan triples) contains asymptotically isometric copies of [lscr ]1 and therefore fails the fixed point property for non-expansive mappings. These generalize earlier results for the case of preduals of semi-finite von Neumann algebras.

Correlations in a General Theory of Quantum Measurement

2007 ◽  
Vol 14 (04) ◽  
pp. 445-458 ◽  
Author(s):  
Hanna Podsędkowska
Keyword(s):  
Hilbert Space ◽  
Quantum Theory ◽  
General Theory ◽  
Quantum Measurement ◽  
Von Neumann Algebra ◽  
Classical Case ◽  
Von Neumann ◽  
Full Algebra ◽  
Algebraic Formalism ◽  
Algebra Of Operators

The paper investigates correlations in a general theory of quantum measurement based on the notion of instrument. The analysis is performed in the algebraic formalism of quantum theory in which the observables of a physical system are described by a von Neumann algebra, and the states — by normal positive normalized functionals on this algebra. The results extend and generalise those obtained for the classical case where one deals with the full algebra of operators on a Hilbert space.

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.

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