Nonzero Projections Research Articles

On finite sums of projections and Dixmier's averaging theorem for type II1 factors

Let M be a type II1 factor and let τ be the faithful normal tracial state on M. In this paper, we prove that given an X∈M, X=X⁎, then there is a decomposition of the identity into N∈N mutually orthogonal nonzero projections Ej∈M, I=∑j=1NEj, such that EjXEj=τ(X)Ej for all j=1,…,N. Equivalently, there is a unitary operator U∈M with UN=I and 1N∑j=0N−1U⁎jXUj=τ(X)I. As the first application, we prove that a positive operator A∈M can be written as a finite sum of projections in M if and only if τ(A)≥τ(RA), where RA is the range projection of A. This result answers affirmatively Question 6.7 of [9]. As the second application, we show that if X∈M, X=X⁎ and τ(X)=0, then there exists a nilpotent element Z∈M such that X is the real part of Z. This result answers affirmatively Question 1.1 of [4]. As the third application, we show that for X1,…,Xn∈M, there exist unitary operators U1,…,Uk∈M such that 1k∑i=1kUi⁎XjUi=τ(Xj)I,∀1≤j≤n. This result is a stronger version of Dixmier's averaging theorem for type II1 factors.

A stronger version of Dixmier's averaging theorem and some applications

Let M be a type II1 factor and let τ be the faithful normal tracial state on M. In this paper, we prove that given finite elements X1,⋯,Xn∈M, there is a finite decomposition of the identity into integer N∈N mutually orthogonal nonzero projections Ej∈M, I=∑j=1NEj, such that EjXiEj=τ(Xi)Ej for all j=1,⋯,N and i=1,⋯,n. Equivalently, there is a unitary operator U∈M such that 1N∑j=0N−1U⁎jXiUj=τ(Xi)I for i=1,⋯,n. This result is a stronger version of Dixmier's averaging theorem for type II1 factors. As the first application, we show that all elements of trace zero in a type II1 factor are single commutators and any self-adjoint elements of trace zero are single self-commutators. This result answers affirmatively Question 1.1 in [6]. As the second application, we prove that any self-adjoint element in a type II1 factor can be written a linear combination of 4 projections. This result answers affirmatively Question 6(2) in [12]. As the third application, we show that if (M,τ) is a finite factor, X∈M, then there exists a normal operator N∈M and a nilpotent operator K such that X=N+K.

Optimal measures, nonzero projections, and irreducible representations

This brief note considers the problem of maximizing the probability of detection for a finite number of mixed quantum states. It is shown that for linearly independent density matrices, an optimal measure that is simple and unique contains nonzero operators that are orthogonal projections on subspaces associated to irreducible representations (irreps) of a finite-dimensional Hilbert space.

Smale space $C^*$-algebras have nonzero projections

The main result of the present paper is that the stable and unstable C*-algebras associated to a mixing Smale space always contain nonzero projections. This gives a positive answer to a question of the first listed author and Karen Strung and has implications for the structure of these algebras in light of the Elliott program for simple C*-algebras. Using our main result, we also show that the homoclinic, stable, and unstable algebras each have real rank zero.

Multiplicative ∗-Jordan type higher derivations on von Neumann algebras

Let be a von Neumann algebra without nonzero central abelian projections on a complex Hilbert space . Let pn (X 1 , X 2 , · · ·, Xn ) be the polynomial defined by n indeterminates X 1, · · ·, Xn and their Jordan multiple ∗-products. In this paper it is shown that a family 𝒟 = {dm } m ∈ℕ of mappings such that , the identity map on satisfies the condition for all U 1 , U 2 , · · ·, Un ∈ and for each m ∈ ℕ if and only if 𝒟 = {dm } m ∈ℕ is an additive ∗-higher derivation.

Primitive Ideal Space of Ultragraph $C^*$-algebras

In this paper, we describe the primitive ideal space of the $C^*$-algebra $C^*(mathcal G)$ associated to the ultragraph $mathcal{G}$. We investigate the structure of the closed ideals of the quotient ultragraph $ C^* $-algebra $C^*left(mathcal G/(H,S)right)$ which contain no nonzero set projections and then we characterize all non gauge-invariant primitive ideals. Our results generalize the Hong and Szyma$ acute{ mathrm { n } } $ski's description of the primitive ideal space of a graph $ C ^ * $-algebra by a simpler method.

The regular completion of a C*-algebra with large projections

Abstract When a unital C*-algebra A is prime and has very large projections, it is shown that the regular completion A^ of the algebra A is a simple, wild type III AW*-factor that has no non-zero σ-finite projections. For example, the Weaver algebra, the Crabb algebra and the Katsura algebra are prime C*-algebras that have very large projections. As a corollary, such algebras have no non-zero abelian elements, that is, they are not of type I.

Green Tensor and Solution of the Boussinesq Problem in the Generalized Theory of Elasticity

The fundamental spatial problems of the theory of elasticity such as the problem of constructing Green tensor and the Boussinesq problem of the action of a concentrated force on a half-space are considered. According to the classical theory of elasticity, these problems are singular. It is shown that ananalytical solution of such problems can be constructed by the Papkovich—Neuber representation without invoking symmetry conditions. This makes it possible to present the solution of the problems under consideration in a single form and allows us to write an explicit solution of half-space loaded by a concentrated vector-force having non-zero projections onto the normal to the plane bounding the half-space and onto the plane itself. This paper deals with the generalized regular solutions of the considered fundamental problems of the elasticity. The solutions are limited at a singular point and damp at infinity.

Beams of photons with nonzero projections of orbital angular momenta: new results

Solving Maxwell’s equations in cylindrical coordinates yields states in quantum theory with definite values of energy ℏω, longitudinal momentum ℏkz, and total angular momentum projection ℏm on the z-axis (where ℏ is the Planck constant). Experimentally, values of up to ℏm Ȉ 104 ℏ have been obtained for the last quantity. The wave front of such states is like a meat grinder screw, with the lines of force of the Poynting vector representing the screw line. Such states differ from plane waves by the nonzero orbital angular momentum projection on the direction of motion, and from spherical waves, by the definite direction of motion. For brevity, these states are referred to as ‘twisted photons’. In this paper, recent experimental and theoretical results on twisted photons are reviewed, to which the present authors actively contributed. Detailed discussion is given to recent experiments on the production of high-intensity beams of terahertz (wavelength: 140 μm) twisted photons performed on the Novosibirsk free electron laser at the Budker Institute of Nuclear Physics, SB RAS. Recent theoretical work on the interaction of twisted photons with atoms is summarized. Due to their extra degree of freedom — the projection of the total angular momentum on the direction of motion — twisted photons represent a novel research tool of potentially wide application in physics.

The normal modes of vibration of benzene from the trajectories of stress tensor eigenvector projection space

A QTAIM and Stress tensor eigenvector projection Uσ space formalism has been used to provide a detailed description of the participation of each of the bonds of the four infrared active normal modes of benzene. Analysis of the maximum Uσ space trajectory projections revealed a mixture of CC and CH bonding characteristics. The four infrared active Uσ space trajectories were found to be unique. The normal mode with the highest infrared intensity was the only mode with non-zero maximum Uσ space trajectory projections in the most preferred and least preferred directions for the CC and CH bond critical points respectively.

Generalized tricircular projections on minimal norm ideals in [formula omitted

A nonzero projection P on a complex Banach space X is said to be a generalized tricircular projection if there exist distinct modulus one complex numbers λ and μ, not equal to 1, and nonzero projections Q and R on X such that P⊕Q⊕R=I and P+λQ+μR is an isometry. We determine the structure of generalized tricircular projections on minimal norm ideals in B(H), different from the Hilbert–Schmidt class.

Amenable traces and Følner [formula omitted]-algebras

In the present article we review an approximation procedure for amenable traces on unital and separable C∗-algebras acting on a Hilbert space in terms of Følner sequences of non-zero finite rank projections. We apply this method to improve spectral approximation results due to Arveson and Bédos. We also present an abstract characterization in terms of unital completely positive maps of unital separable C∗-algebras admitting a non-degenerate representation which has a Følner sequence or, equivalently, an amenable trace. This is analogous to Voiculescu’s abstract characterization of quasidiagonal C∗-algebras. We define Følner C∗-algebras as those unital separable C∗-algebras that satisfy these equivalent conditions. Finally we also mention some permanence properties related to these algebras.

Extended rotation algebras: Adjoining spectral projections to rotation algebras

Denote by Aθ the rotation algebra corresponding to the rotation 2πθ. The C*-algebra 𝔹θ generated by Aθ together with certain spectral projections of the canonical unitary generators is studied. The C*-algebra 𝔹θ is shown to have a unique tracial state and to be nuclear provided that θ is irrational. Moreover, we study the ideal structure of the C*-algebra 𝔹θ. In particular, it is shown that 𝔹θ is simple if neither the commutative sub-C*-algebra generated by the spectral projections of u in question (assumed to be a set invariant under Ad v) nor the corresponding commutative sub-C*-algebra associated to v contains non-zero minimal projections. In the second part of the paper, we study the extended rotation algebra 𝔹θ generated by the spectral projections (one for each unitary) corresponding to the half-open interval from 0 to θ. (The spectral projections for each half-open interval from nθ to (n + 1)θ are then included for each integer n.) Using simplicity of 𝔹θ for θ irrational, the natural field of C*-algebras on the unit circle with fibres 𝔹θ is shown to be continuous at irrational points. This field is lower semicontinuous on the whole circle. Much more useful is an upper semicontinuous field which is obtained by desingularizing this field at rational points on the circle. The fibres of the desingularized field at rational points are certain (computable) type I C*-algebras. Using this new field, we are able to show that 𝔹θ is an AF algebra with K0(𝔹θ) ≅ ℤ + θℤ for generic θ, in the sense of Baire category, with the class of the unit being 1 ∈ ℤ.

Maps preserving operator pairs whose products are projections

Let B ( H ) be the algebra of all bounded linear operators on a complex Hilbert space H with dim H ⩾ 2 . It is proved that a surjective map φ on B ( H ) preserves operator pairs whose products are nonzero projections in both directions if and only if there is a unitary or an anti-unitary operator U on H such that φ ( A ) = λ U ∗ AU for all A in B ( H ) for some constants λ with λ 2 = 1 . Related results for surjective maps preserving operator pairs whose triple Jordan products are nonzero projections in both directions are also obtained. These show that the operator pairs whose products or triple Jordan products are nonzero projections are isometric invariants of B ( H ) .

Mode processing of transient signals using a deficient vertical receiving array, with application to LOAPEX measurements

Mode processing of transient signals in a deep ocean environment using a deficient vertical receiving array is considered. Here, ‘‘deficient’’ implies that the vertical array is too sparse to resolve modes and/or has gaps. It is well known that mode processing of cw signals using a deficient array leads to significant modal cross‐talk, i.e., nonzero projections of mode n energy onto mode m, m≠n. It is shown, using simulated wave fields, that when phase‐coherent processing is employed, transient signals are less sensitive to modal cross‐talk than their cw counterparts. An explanation of this behavior is provided. Mode processing of a subset of the measurements made during the LOAPEX experiment at approximately 1 Mm range with f0=75 Hz, Δf=35 Hz is shown to give robust estimates of modal group time spreads for mode numbers 0≤m≤70. [Work supported by ONR.]

Decomposability of von Neumann Algebras and the Mazur Property of Higher Level

AbstractThe decomposability number of a von Neumann algebra ℳ (denoted by dec(ℳ)) is the greatest cardinality of a family of pairwise orthogonal non-zero projections in ℳ. In this paper, we explore the close connection between dec(ℳ) and the cardinal level of the Mazur property for the predual ℳ* of ℳ, the study of which was initiated by the second author. Here, our main focus is on those von Neumann algebras whose preduals constitute such important Banach algebras on a locally compact group G as the group algebra L1(G), the Fourier algebra A(G), the measure algebra M(G), the algebra LUC(G)*, etc. We show that for any of these von Neumann algebras, say ℳ, the cardinal number dec(ℳ) and a certain cardinal level of the Mazur property of ℳ* are completely encoded in the underlying group structure. In fact, they can be expressed precisely by two dual cardinal invariants of G: the compact covering number κ(G) of G and the least cardinality ᙭(G) of an open basis at the identity of G. We also present an application of the Mazur property of higher level to the topological centre problem for the Banach algebra A(G)**.

Phonon dispersion of oriented DNA by inelastic x-ray scattering

Films of oriented deoxyribonucleic acid (DNA), prepared by the wet spinning method, have been studied using inelastic x-ray scattering. Spectra were recorded within the range of energy transfers -30< homega <30 meV at momentum transfers homegaQ ranging from 2.5 to 30 nm(-1) whereby the direction of Q essentially coincided with the helical axis. Measurements at ambient temperature cover samples in the A, B, C, and D conformations of DNA. Within the limits of the instrumental resolution, the spectra were analyzed by the response of a damped harmonic oscillator delivering dispersion and damping of modes having displacements with nonzero projections onto Q, i.e., essentially the compression waves traveling along the helical axis. The longitudinal speed of sound resulting from the sinusoidal dispersion varies only weakly with conformation. Our sound speed values are compared to results from Brillouin spectroscopy. The dispersion curves exhibit a minimum at about the inverse rise per residue, which -- together with strong elastic scattering -- reflect the large degree of disorder. Overdamping of the modes is observed for Q>5 nm(-1). The possibility that the observed large damping parameters are due to several contributing modes is discussed in terms of a simple model calculation for an idealized double helix. Whereas the quasicrystalline approximation for an effective disordered chain could well describe the sinusoidal dispersion, it fails to reproduce the observed damping by one order of magnitude. Our results indicate that the high-frequency dynamics of DNA is liquidlike and is most appropriately described by instantaneous normal modes of short correlation length.

Complementarity of representations in quantum mechanics

We show that Bohr's principle of complementarity between position and momentum descriptions can be formulated rigorously as a claim about the existence of representations of the canonical commutation relations. In particular, in any representation where the position operator has eigenstates, there is no momentum operator, and vice versa. Equivalently, if there are nonzero projections corresponding to sharp position values, all spectral projections of the momentum operator map onto the zero element.

LOCAL CHARACTERIZATION OF APPROXIMATELY FINITE OPERATOR ALGEBRAS

We show that the family of nest algebras with r non-zero nest projections is stable, in the sense that an approximate containment of one such algebra within another is close to an exact containment. We use this result to give a local characterization of limits formed from this family. We then consider quite general regular limit algebras and characterize these algebras using a local condition which reflects the assumed regularity of the system.

Wigner-Type Theorem on Symmetry Transformations in Type II Factors

Wigner's theorem on symmetry transformations can be formulated in the followingway. If φ is a bijective map on the set of all nonzero minimal projections in atype I factor A which preserves transition probabilities with respect to a faithfulnormal semifinite trace, then it can be extended to a linear *-automorphism orto a linear *-antiautomorphism of A. In this paper we prove a natural analogueof this statement for type II factors.