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 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 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.

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.

Further results on the relative entropy

1987 ◽  
Vol 101 (2) ◽  
pp. 363-373 ◽  
Author(s):  
Matthew J. Donald
Keyword(s):  
Relative Entropy ◽  
Von Neumann Algebra ◽  
Measurement Problem ◽  
Bounded Operators ◽  
Many Worlds ◽  
Von Neumann ◽  
Normal States ◽  
Injective Von Neumann Algebra ◽  
Made In

Given any subset ℬ, containing the identity (1), of ℬ (ℋ) (the bounded operators on some Hilbert space ℋ), and given two states σ and ρ on ℬ(ℋ), a definition was given in [3] of entℬ (σℬ|ρ|ℬ) - ‘the entropy of σ relative to ρ given the information in ℬ’. It was shown that, for ℬ an injective von Neumann algebra, the resulting relative entropy agreed with those of Umegaki, Araki, Pusz and Woronowicz, and Uhlmann. The purpose of this paper is to explore this definition further. After some technical preliminaries in Section 2, in Section 3 a new characterization of entℬ(ℋ) (σ|ρ) for σ and ρ normal states will be given. In Section 4 it will be shown that under fairly general circumstances the relative entropy on algebras can be used for statistical inference. This is important for applications of the relative entropy. I shall given the briefest sketches of how I see these applications being made in the measurement problem in quantum theory and in a ‘many worlds’ interpretation. The vigilant reader will notice that the scheme proposed in Section 4 for modelling measurements subject to given compatibility requirements differs slightly from that proposed in the introduction to [3]. The reason for this is outlined in Section 5, where an explicit computation is made of the relative entropy for the simplest non-trivial case in which ℬ is not an algebra; when ℬ = {1, P, Q} for P and Q projections subject to certain conditions.

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 Averaging operators in non commutative Lp spaces I

1983 ◽  
Vol 24 (1) ◽  
pp. 71-74 ◽  
Author(s):  
Christopher Barnett
Keyword(s):  
Von Neumann Algebra ◽  
Measure Theory ◽  
Continuous Functions ◽  
Unbounded Operators ◽  
Averaging Operators ◽  
Von Neumann ◽  
Lp Spaces ◽  
Finite Von Neumann Algebra ◽  
Spaces Of Continuous Functions ◽  
Finite Trace

The origin of the theory of averaging operators is explained in [1]. The theory has been developed on spaces of continuous functions that vanish at infinity by Kelley in [3] and on the Lp spaces of measure theory by Rota [5]. The motivation for this paper arose out of the latter paper. The aim of this paper is to prove a generalisation of Rota's main representation theorem (every average is a conditional expectation) in the context of a ‘non commutative integration’. This context is as follows. Let be a finite von Neumann algebra and ϕ a faithful normal finite trace on such that ϕ(I) = 1, where I is the identity of . We can construct the Banach spaces Lp (, ϕ), where 1 ≤ p < °, with norm ∥x∥p = ϕ(÷x÷p)1/p, of possibly unbounded operators affiliated with , as in [9]. We note that is dense in Lp(, ϕ). These spaces share many of the features of the Lp spaces of measure theory; indeed if is abelian then Lp(,ϕ) is isometrically isomorphic to Lp of some measure space.

Isometries of non-commutative Lp-spaces

1981 ◽  
Vol 90 (1) ◽  
pp. 41-50 ◽  
Author(s):  
F. J. Yeadon
Keyword(s):  
Von Neumann Algebra ◽  
Function Spaces ◽  
Main Step ◽  
Main Result Theorem ◽  
Von Neumann ◽  
Lp Spaces ◽  
Result Theorem

The spaces Lp(, φ) for 1 ≤ p ≤ ∞, where φ is a faithful semifinite normal trace on a von Neumann algebra , are defined in (10),(2),(14). The problem of determining the general form of an isometry of one such space into another has been studied in (i), (6), (9), (12), (5). Our main result, Theorem 2, is a characterization of such isometries for 1 ≤ p ≤ ∞, ≠ 2. The method of proof is based on that of (7), where isometries between Lp function spaces are characterized. The main step in the proof is Theorem 1, which gives the conditions under which equality holds in Clarkson's inequality.

Non Commutative Lp Spaces II

1982 ◽  
Vol 34 (5) ◽  
pp. 1208-1214 ◽  
Author(s):  
A. Katavolos
Keyword(s):  
Measure Space ◽  
Linear Functional ◽  
Von Neumann Algebra ◽  
Linear Mapping ◽  
Natural Question ◽  
Von Neumann ◽  
Lp Spaces ◽  
Image Position ◽  
Natural Way

Let M be a w*-algebra (Von Neumann algebra), τ a semifinite, faithful, normal trace on M. There exists a w*-dense (i.e., dense in the σ(M, M*)-topology, where M* is the predual of M) *-ideal J of M such that τ is a linear functional on J, and(where |x| = (x*x)1/2) is a norm on J. The completion of J in this norm is Lp(M, τ) (see [2], [8], [7], and [4]).If M is abelian, in which case there exists a measure space (X, μ) such that M = L∞(X, μ), then Lp(X, τ) is isometric, in a natural way, to Lp(X, μ). A natural question to ask is whether this situation persists if M is non-abelian. In a previous paper [5] it was shown that it is not possible to have a linear mapping

Dual of non-commutative Lp-spaces with 0 < p < 1

1988 ◽  
Vol 103 (3) ◽  
pp. 503-509 ◽  
Author(s):  
Keiichi Watanabe
Keyword(s):  
Von Neumann Algebra ◽  
Von Neumann ◽  
Modular Theory ◽  
Lp Spaces

After the development of the modular theory one can construct non-commutative Lp-spaces associated with a von Neumann algebra M which is not necessarily semifinite (see [4], [12], etc.).

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].

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