On the statistical estimation of the logarithmic derivative of a measure in a Hilbert space

2010 ◽  
Vol 17 (4) ◽  
pp. 741-747
Author(s):  
Elizbar Nadaraya ◽  
Grigol Sokhadze
Keyword(s):  
Hilbert Space ◽  
Statistical Estimation ◽  
Logarithmic Derivative ◽  
Observation Data ◽  
Independent Observation ◽  
Infinite Dimensional ◽  
Infinite Dimensional Hilbert Space ◽  
Finite Dimensional ◽  
Dimensional Hilbert Space ◽  
Approximating Sequence

Abstract We consider the problem of statistical estimation of the logarithmic derivative of a measure in an infinite-dimensional Hilbert space. It is shown that an approximating sequence of finite-dimensional estimates can be constructed for the unknown logarithmic derivative using independent observation data.

Non-Isomorphic Non-Hyperfinite Factors

1969 ◽  
Vol 21 ◽  
pp. 1293-1308 ◽  
Author(s):  
Wai-Mee Ching
Keyword(s):  
Hilbert Space ◽  
Von Neumann Algebra ◽  
Crossed Product ◽  
Type Ii ◽  
Von Neumann ◽  
Infinite Dimensional ◽  
Type Iii ◽  
Infinite Dimensional Hilbert Space ◽  
Finite Dimensional ◽  
Dimensional Hilbert Space

A von Neumann algebra is called hyperfinite if it is the weak closure of an increasing sequence of finite-dimensional von Neumann subalgebras. For a separable infinite-dimensional Hilbert space the following is known: there exist hyperfinite and non-hyperfinite factors of type II1 (4, Theorem 16’), and of type III (8, Theorem 1); all hyperfinite factors of type Hi are isomorphic (4, Theorem 14); there exist uncountably many non-isomorphic hyperfinite factors of type III (7, Theorem 4.8); there exist two nonisomorphic non-hyperfinite factors of type II1 (10), and of type III (11). In this paper we will show that on a separable infinite-dimensional Hilbert space there exist three non-isomorphic non-hyperfinite factors of type II1 (Theorem 2), and of type III (Theorem 3).Section 1 contains an exposition of crossed product, which is developed mainly for the construction of factors of type III in § 3.

scholarly journals EXTENSION ALGEBRAS OF CUNTZ ALGEBRA, II

2009 ◽  
Vol 80 (1) ◽  
pp. 83-90 ◽  
Author(s):  
SHUDONG LIU ◽  
XIAOCHUN FANG
Keyword(s):  
Hilbert Space ◽  
Compact Operators ◽  
Cuntz Algebra ◽  
Infinite Dimensional ◽  
Algebra Ii ◽  
Infinite Dimensional Hilbert Space ◽  
Dimensional Hilbert Space ◽  
K Theory ◽  
Extension Algebra

AbstractIn this paper, we construct the unique (up to isomorphism) extension algebra, denoted by E∞, of the Cuntz algebra 𝒪∞ by the C*-algebra of compact operators on a separable infinite-dimensional Hilbert space. We prove that two unital monomorphisms from E∞ to a unital purely infinite simple C*-algebra are approximately unitarily equivalent if and only if they induce the same homomorphisms in K-theory.

scholarly journals Partial automorphisms of stable C*-algebras and Hilbert C*-bimodules

2005 ◽  
Vol 79 (3) ◽  
pp. 391-398
Author(s):  
Kazunori Kodaka
Keyword(s):  
Hilbert Space ◽  
Compact Operators ◽  
Equivalence Classes ◽  
Infinite Dimensional ◽  
C Algebras ◽  
Infinite Dimensional Hilbert Space ◽  
Dimensional Hilbert Space ◽  
Countably Infinite

AbstractLet A be a C*-algebra and K the C*-algebra of all compact operators on a countably infinite dimensional Hilbert space. In this note, we shall show that there is an isomorphism of a semigroup of equivalence classes of certain partial automorphisms of A ⊗ K onto a semigroup of equivalence classes of certain countably generated A-A-Hilbert bimodules.

The Set of Finite Operators is Nowhere Dense

1989 ◽  
Vol 32 (3) ◽  
pp. 320-326 ◽  
Author(s):  
Domingo A. Herrero
Keyword(s):  
Hilbert Space ◽  
Linear Operator ◽  
Bounded Linear Operator ◽  
Dense Subset ◽  
Bounded Linear ◽  
Infinite Dimensional ◽  
Infinite Dimensional Hilbert Space ◽  
Dimensional Hilbert Space

AbstractA bounded linear operator A on a complex, separable, infinite dimensional Hilbert space is called finite if for each . It is shown that the class of all finite operators is a closed nowhere dense subset of

scholarly journals A SIMPLE PROOF OF THE FUNDAMENTAL THEOREM ABOUT ARVESON SYSTEMS

2006 ◽  
Vol 09 (02) ◽  
pp. 305-314 ◽  
Author(s):  
MICHAEL SKEIDE
Keyword(s):  
Hilbert Space ◽  
Simple Proof ◽  
Fundamental Theorem ◽  
Hilbert Module ◽  
Bounded Operators ◽  
Infinite Dimensional ◽  
Infinite Dimensional Hilbert Space ◽  
Product Systems ◽  
Dimensional Hilbert Space ◽  
The One

With every E0-semigroup (acting on the algebra of of bounded operators on a separable infinite-dimensional Hilbert space) there is an associated Arveson system. One of the most important results about Arveson systems is that every Arveson system is the one associated with an E0-semigroup. In these notes we give a new proof of this result that is considerably simpler than the existing ones and allows for a generalization to product systems of Hilbert module (to be published elsewhere).

scholarly journals Tomography in Abstract Hilbert Spaces

2006 ◽  
Vol 13 (03) ◽  
pp. 239-253 ◽  
Author(s):  
V. I. Man'ko ◽  
G. Marmo ◽  
A. Simoni ◽  
F. Ventriglia
Keyword(s):  
Hilbert Space ◽  
Quantum State ◽  
Trace Formula ◽  
Hilbert Spaces ◽  
Scalar Product ◽  
Linear Operators ◽  
Infinite Dimensional ◽  
Infinite Dimensional Hilbert Space ◽  
Dimensional Hilbert Space ◽  
Rank One

The tomographic description of a quantum state is formulated in an abstract infinite-dimensional Hilbert space framework, the space of the Hilbert-Schmidt linear operators, with trace formula as scalar product. Resolutions of the unity, written in terms of over-complete sets of rank-one projectors and of associated Gram-Schmidt operators taking into account their non-orthogonality, are then used to reconstruct a quantum state from its tomograms. Examples of well known tomographic descriptions illustrate the exposed theory.

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