scholarly journals Jensen's Trace Inequality in Several Variables

2003 ◽  
Vol 14 (06) ◽  
pp. 667-681 ◽  
Author(s):  
Frank Hansen ◽  
Gert K. Pedersen
Keyword(s):  
Symmetric Matrices ◽  
Trace Inequality ◽  
C Algebra ◽  
Infinite Dimensional ◽  
Infinite Dimensional Hilbert Space ◽  
Several Variables ◽  
Lower Semi ◽  
Continuous Trace ◽  
Abelian Fields ◽  
Densely Defined

For a convex, real function f, we present a simple proof of the formula [Formula: see text] valid for each tuple (x1,…, xm) of symmetric matrices in [Formula: see text] and every unital column (a1,…, am) of matrices, i.e. [Formula: see text]. This is the standard Jensen trace ine-quality. If f ≥ 0 it holds also for the unbounded trace on [Formula: see text], where [Formula: see text] is an infinite-dimensional Hilbert space. We then investigate the more general case where τ is a densely defined, lower semi-continuous trace on a C*-algebra [Formula: see text] and f is a convex, continuous function of n variables, and show that we have the inequality [Formula: see text] for every family of abeliann-tuples [Formula: see text], i.e. tuples of self-adjoint elements in [Formula: see text] such that [xik, xjk] = 0 for all i, j and k, where 1 ≤ k ≤ m, and every unital m-column (a1,…, am) in [Formula: see text], provided that the elements [Formula: see text] also form an abelian n-tuple. We even establish this result for weak* measurable, self-adjoint, abelian fields (xit)t∈T, 1 ≤ i ≤ n, i.e. [xit, xjt] = 0 for all i, j and t, and weak* measurable, unital column field (at)t∈T in [Formula: see text] paired with any trace or trace-like functional φ, i.e. one that contains the n-tuple (presumed abelian) with elements [Formula: see text] in its centralizer. This takes the form of the inequality [Formula: see text] We also study functions of n variables that are monotone increasing in each variable, and show in two important cases that [Formula: see text] whenever [Formula: see text] and [Formula: see text] are abelian n-tuples with xi ≤ yi for each i and φ is a trace or a trace-like functional.

scholarly journals CONVEX MULTIVARIABLE TRACE FUNCTIONS

2002 ◽  
Vol 14 (07n08) ◽  
pp. 631-648 ◽  
Author(s):  
ELLIOTT H. LIEB ◽  
GERT K. PEDERSEN
Keyword(s):  
Convex Function ◽  
Short Proof ◽  
Trace Function ◽  
C Algebra ◽  
Operator Mean ◽  
Lower Semi ◽  
Continuous Trace ◽  
Densely Defined

For any densely defined, lower semi-continuous trace τ on a C*-algebra A with mutually commuting C*-subalgebras A1, A2, … An, and a convex function f of n variables, we give a short proof of the fact that the function (x1, x2, …, xn)→ τ (f (x1, x2, …, xn)) is convex on the space [Formula: see text]. If furthermore the function f is log-convex or root-convex, so is the corresponding trace function. We also introduce a generalization of log-convexity and root-convexity called ℓ-convexity, show how it applies to traces, and give some examples. In particular we show that the Kadison–Fuglede determinant is concave and that the trace of an operator mean is always dominated by the corresponding mean of the trace values.

scholarly journals 2-LOCAL ISOMETRIES OF SOME OPERATOR ALGEBRAS

2002 ◽  
Vol 45 (2) ◽  
pp. 349-352 ◽  
Author(s):  
Lajos Molnár
Keyword(s):  
Operator Algebras ◽  
Linear Operators ◽  
Primary 47B49 ◽  
Similar Statement ◽  
Bounded Linear ◽  
C Algebra ◽  
Infinite Dimensional ◽  
C Algebras ◽  
Infinite Dimensional Hilbert Space ◽  
Local Isometry

AbstractAs a consequence of the main result of the paper we obtain that every 2-local isometry of the $C^*$-algebra $B(H)$ of all bounded linear operators on a separable infinite-dimensional Hilbert space $H$ is an isometry. We have a similar statement concerning the isometries of any extension of the algebra of all compact operators by a separable commutative $C^*$-algebra. Therefore, on those $C^*$-algebras the isometries are completely determined by their local actions on the two-point subsets of the underlying algebras.AMS 2000 Mathematics subject classification: Primary 47B49

Homotopy Classification of Projections in the Corona Algebra of a Non-simple C*-algebra

2012 ◽  
Vol 64 (4) ◽  
pp. 755-777 ◽  
Author(s):  
Lawrence G. Brown ◽  
Hyun Ho Lee
Keyword(s):  
Hilbert Space ◽  
Compact Operators ◽  
Homotopy Classification ◽  
Multiplier Algebra ◽  
Von Neumann ◽  
C Algebra ◽  
Infinite Dimensional ◽  
Infinite Dimensional Hilbert Space ◽  
Dimensional Hilbert Space

AbstractWe study projections in the corona algebra of C(X) ⊗ K, where K is the C*-algebra of compact operators on a separable infinite dimensional Hilbert space and X = [0, 1], [0,∞), (−∞,∞), or [0, 1]/﹛0, 1﹜. Using BDF's essential codimension, we determine conditions for a projection in the corona algebra to be liftable to a projection in the multiplier algebra. We also determine the conditions for two projections to be equal in K0, Murray-von Neumann equivalent, unitarily equivalent, or homotopic. In light of these characterizations, we construct examples showing that the equivalence notions above are all distinct.

A Simple C*-Algebra Generated by Two Finite-Order Unitaries

1979 ◽  
Vol 31 (4) ◽  
pp. 867-880 ◽  
Author(s):  
Man-Duen Choi
Keyword(s):  
Hilbert Space ◽  
Finite Order ◽  
Explicit Construction ◽  
Unitary Operators ◽  
Unitary Matrices ◽  
C Algebra ◽  
Infinite Dimensional ◽  
Central Object ◽  
Infinite Dimensional Hilbert Space ◽  
Image Position

We present an example which illustrates several peculiar phenomena that may occur in the theory of C*-algebras. In particular, we show that a C*-subalgebra of a nuclear (amenable) C*-algebra need not be nuclear (amenable).The central object of this paper is a pair of abstract unitary matrices,acting on a common Hilbert space. For an explicit construction, we may decompose an infinite-dimensional Hilbert space H into H = H0 ⴲ H1 , H1 = Hα ⴲ Hβ with dim H0 = dim H1 = dim Hα = dim Hβ, letting u, v Є B(H) be any two unitary operators such thatand u2 = 1, v3 = 1. Whereas many choices of u, v are possible, it might be surprising to see that C*(u, v), the C*-algebra generated by u and v, is algebraically unique; namely, if (u1,V1) is another pair of such unitaries, then C*(u, v) is canonically *-isomorphic with C*(u1, v1) (Theorem 2.6).

scholarly journals Krylov Solvability of Unbounded Inverse Linear Problems

2020 ◽  
Vol 93 (1) ◽  
Author(s):  
Noè Angelo Caruso ◽  
Alessandro Michelangeli
Keyword(s):  
The Self ◽  
Linear Combinations ◽  
Infinite Dimensional ◽  
Infinite Dimensional Hilbert Space ◽  
Unbounded Linear Operator ◽  
Gradient Based ◽  
Hilbert Norm ◽  
Linear Problems ◽  
Finite Linear ◽  
Densely Defined

AbstractThe abstract issue of ‘Krylov solvability’ is extensively discussed for the inverse problem $$Af=g$$ A f = g where A is a (possibly unbounded) linear operator on an infinite-dimensional Hilbert space, and g is a datum in the range of A. The question consists of whether the solution f can be approximated in the Hilbert norm by finite linear combinations of $$g,Ag,A^2g,\dots $$ g , A g , A 2 g , ⋯ , and whether solutions of this sort exist and are unique. After revisiting the known picture when A is bounded, we study the general case of a densely defined and closed A. Intrinsic operator-theoretic mechanisms are identified that guarantee or prevent Krylov solvability, with new features arising due to the unboundedness. Such mechanisms are checked in the self-adjoint case, where Krylov solvability is also proved by conjugate-gradient-based techniques.

Infinitely entangled states

2003 ◽  
Vol 3 (4) ◽  
pp. 281-306
Author(s):  
M. Keyl ◽  
D. Schlingemann ◽  
R.F. Werner
Keyword(s):  
Hilbert Spaces ◽  
Degrees Of Freedom ◽  
Entangled States ◽  
Entangled State ◽  
Bounded Operators ◽  
Infinite Dimensional ◽  
Infinite Dimensional Hilbert Space ◽  
Density Operators ◽  
Fundamental Paper ◽  
Extended Notion

For states in infinite dimensional Hilbert spaces entanglement quantities like the entanglement of distillation can become infinite. This leads naturally to the question, whether one system in such an infinitely entangled state can serve as a resource for tasks like the teleportation of arbitrarily many qubits. We show that appropriate states cannot be obtained by density operators in an infinite dimensional Hilbert space. However, using techniques for the description of infinitely many degrees of freedom from field theory and statistical mechanics, such states can nevertheless be constructed rigorously. We explore two related possibilities, namely an extended notion of algebras of observables, and the use of singular states on the algebra of bounded operators. As applications we construct the essentially unique infinite analogue of maximally entangled states, and the singular state used heuristically in the fundamental paper of Einstein, Rosen and Podolsky.

scholarly journals Isometric Group Actions on Hilbert Spaces: Structure of Orbits

2008 ◽  
Vol 60 (5) ◽  
pp. 1001-1009 ◽  
Author(s):  
Yves de Cornulier ◽  
Romain Tessera ◽  
Alain Valette
Keyword(s):  
Hilbert Space ◽  
Nilpotent Group ◽  
Hilbert Spaces ◽  
Metabelian Group ◽  
Isometric Action ◽  
Finitely Generated ◽  
Infinite Dimensional ◽  
Infinite Dimensional Hilbert Space ◽  
Isometric Group Actions ◽  
Dense Orbits

AbstractOur main result is that a finitely generated nilpotent group has no isometric action on an infinite-dimensional Hilbert space with dense orbits. In contrast, we construct such an action with a finitely generated metabelian group.

scholarly journals The projection operator in a Hilbert space and its directional derivative. Consequences for the theory of projected dynamical systems

2004 ◽  
Vol 2 (1) ◽  
pp. 71-95 ◽  
Author(s):  
George Isac ◽  
Monica G. Cojocaru
Keyword(s):  
Hilbert Space ◽  
Dynamical Systems ◽  
Representation Theorem ◽  
Projection Operator ◽  
Directional Derivative ◽  
Metric Projection ◽  
Pseudomonotone Operators ◽  
Projected Dynamical Systems ◽  
Infinite Dimensional ◽  
Infinite Dimensional Hilbert Space

In the first part of this paper we present a representation theorem for the directional derivative of the metric projection operator in an arbitrary Hilbert space. As a consequence of the representation theorem, we present in the second part the development of the theory of projected dynamical systems in infinite dimensional Hilbert space. We show that this development is possible if we use the viable solutions of differential inclusions. We use also pseudomonotone operators.

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.

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