Nonzero Projections Research Articles

Coherent Population Transfer among Three States: Full Algebraic Solutions and the Relevance of Non Adiabatic Processes to Transfer by Delayed Pulses

Ongoing work aimed at developing highly efficient methods of populating a chosen sublevel of an energy level highlights the need to understand off-resonant effects in coherent excitation. This motivated us to re-examine some aspects of the theory of coherent excitation in a three-state system with a view to obtaining algebraic expressions for off-resonant eigenvalues and eigenvectors. Earlier work gives simple closed-form expressions for the eigenvalues this system, expressions applying even when the system is not on two-photon resonance. We present here expressions of similar simplicity for the components of the normalised eigenvectors. The analytic properties of these components explain the observed sensitivity of the stimulated-Raman-adiabatic-passage process (STIRAP) to the condition of two-photon resonance. None of the eigenstates is ‘trapped’ or ‘dark’ unless the system is on two-photon resonance; off resonance, all states have nonzero projections on the unperturbed intermediate state. A simple argument shows that no dressed state can be adiabatically connected to both the unperturbed initial and final states when the system is off two-photon resonance. That is, adiabatic transfer from initial to final state requires that these be degenerate before and after the STIRAP pulse sequence, and this implies zero two-photon detuning. However, substantial transfer probabilities are observed experimentally for very small two-photon detunings. We show that such systems are characterised by very sharp avoided crossings of two eigenvalues, and that the observed population transfer can be understood as an effect of non adiabatic transitions occurring at the avoided crossings.

Distance From Projections to Nilpotents

AbstractThe distance from an arbitrary rank-one projection to the set of nilpotent operators, in the space of k × k matrices with the usual operator norm, is shown to be sec(π/(k:+2))/2. This gives improved bounds for the distance between the set of all non-zero projections and the set of nilpotents in the space of k × k matrices. Another result of note is that the shortest distance between the set of non-zero projections and the set of nilpotents in the space of k × k matrices is .

Compact open sets in duals and projections inL1-algebras of certain semi-direct product groups

The main purpose of this paper is to study projections, that is, self-adjoint idempotents, inL1-algebras of semi-direct productsG= ℝ ⋉ ℝd,d≥ 2. We establish necessary and sufficient conditions for the existence of non-zero projections in terms of the action of ℝ on ℝd. In the cases where such projections exist, we describe minimal ones in detail.

Re-examination of eigenvector derivatives

Analytical expressions for eigenvector derivatives for general non-self-adjoint systems using a modal expansion approach have not been correctly derived in several papers and books that address this problem. A common mistake in several developments on deriving eigenvector derivatives (or its perturbation forms) has been to ignore the eigenvector in question, in the expansion of its derivative, although the remaining eigenvectors forming the basis are generally non-orthogonal. This assumption is based upon explicit or implicit heuristic arguments that only directional changes contribute to eigenvector sensitivity. It is shown in this paper that the above assumption and the resulting equations are incorrect, except for the classical and most commonly encountered case of a self-adjoint problem having orthogonal eigenvectors. For the non-self-adjoint case, certain basis coefficients in the eigenvector derivative expansion have not been resolved correctly in the literature. A careful re-examination of the eigenvalue problem reveals that the two independent sets of normalizations (required for uniquely specifying the right and left eigenvectors) can be used to uniquely determine the basis coefficients. The solution derived herein for the eigenvector derivatives is shown to have generally nonzero projections onto all eigenvectors. It is also shown that basis coefficients for left and right eigenvector derivatives are related by a simple expression. A numerical example is included to demonstrate the present formulation for eigenvector derivatives with respect to a scalar parameter. We also extend Nelson's algebraic approach (for self-adjoint eigenvalue problems) to the general non-self-adjoint problem and the modal truncation approach to approximate eigenvector derivatives for the non-self-adjoint case.

A Simple C ∗ -Algebra with no Nontrivial Projections

A C*-algebra is constructed which is separable, simple, nuclear, nonunital, and contains no nonzero projections. Some results on automorphisms of AF algebras are also obtained. A C*-algebra is said to be projectionless if it contains no projections other than 1 (if present) and 0. It has long been an open question whether there exists a projectionless simple C*-algebra (see [13, p. 18], [6, 1.9.6], [8, p. 81], [14, p. 242]). In this paper we construct a projectionless simple separable nuclear nonunital C*-algebra. It is quite possible that the methods of this paper can be modified to yield a projectionless simple unital C*-algebra. It is conjectured that the C*-algebra generated by the regular representation of the free group on two generators (known to be simple and unital) is projectionless. 1. Outline of construction. The general method of construction is motivated by the construction of the Bunce-Deddens weighted shift algebras [5] as described by Green [12, p. 248]. The algebra A is constructed as an inductive limit of C*-algebra An, each of which is a continuous field algebra on a circle T with a constant simple fiber B with the embedding j,,: A, -->A,,+1 inducing the twice around map z -> z2of T onto T. The algebra B will be the (unique) simple unital AF algebra whose ordered group KO(B) is isomorphic to the additive group of real algebraic numbers [10, 2.2]. B has the following properties: (1) B has a unique normalized trace , which is faithful. (2) If p and q are projections in B, then p q if and only if (p) = T(q). (3) If X is any algebraic number with 0 B such that f(l) = a(f(O)). Prim(A(a)) = {J,: 0 e PROPOSITION 1.1. A(a) is projectionless if (and only if) a(1) #6 1. Received by the editors December 13, 1978. AMS (MOS) subject classifications (1970). Primary 46L05. ? 1980 American Mathematical Society 0002-9939/80/0000-01 60/$02.25

A simple 𝐶*-algebra with no nontrivial projections

A C ∗ {C^\ast } -algebra is constructed which is separable, simple, nuclear, non-unital, and contains no nonzero projections. Some results on automorphisms of AF algebras are also obtained.

Essential spectrum Γ(β) of a dual action on a von Neumann algebra

For a dual action β of a locally compact group G on a von Neumann algebra N we define the essential spectrum Γ(β) as the intersection of all spectrum sp βp of the restriction βp of β to Np when p runs over all nonzero projections in Nβ. T(β) is then an algebraic invariant for a covariant dual system {N, β}. Γ(β) is a closed subgroup of G (Theorem 3.7). We introduce three kinds of concept for β such as integrable, regular and dominant (§§4, 5). The former concepts are weaker than the dominance, ϊf β is regular, Γ(/3) coincides with the kerne! of the action β on the center of the crossed dual product N ζξ)dβG (Theorem 6.1). If β is regular, Γ(β) is normal and Γ(β) — Γ(β). If β is ergodic on the center Z(N) and Γ(β) = G, then N(£)dβG is a factor and vice versa (Theorem 6.4). If β is regular, Γ(β) = G is equivalent to Z(N β )CZ(N) (Proposition 6.3). If β is integrable on a factor N and if Γ(β)= G, then there is a lattice isomorphism between the closed subgroups of G and the von Neumann subalgebras of N containing Nβ (Theorem 8.4). Moreover, by N§ξ)dβ(H\G) we mean the von Neumann algebra generated by β(N)and 1 (g)(Lx(G)Π λ'(H)'), where H is a closed subgroup of G and Λ' is the right regular representation of G. N§ξ>dβ(H\G) coincides with the set of x Ξ N(g)^G such that βt(x) = x for all t E H (Theorem 7.2). 0. Introduction. In our previous paper [17, 16, 21] we have generalized Takesaki's duality to a general locally compact group in terms of a dual action and a crossed dual product as the following: In this paper we continue our study on dual actions and Takesaki's duality obtained in the above from the view point of covariant systems {M, a} and covariant dual systems {JV, β}. Then we naturally raise some questions: a. What is an invariant of equivalent covariant dual systems? b. When does Takesaki's duality hold as a covariant (dual) system? Using the spectrum of β given in [17, §5], we can define the essential spectrum Γ(j8) of β in §3 by the same manner as S set. Then Γ(β) is a closed subgroup of G and an algebraic invariant of dual actions on a

A reduction of the Kemmer equation and the singular behaviour of integer-spin relativistic Hamiltonians

A reduced form of the Kemmer equation for spin one particles is derived, wherein the wave function, which has no redundant components, transforms locally (unlike in earlier reductions). The reduced Hamiltonian is identical to the spin one specialisation of the general integer-spin Hamiltonian obtained from recent investigations on second quantizable relativistic wave equations having the Hamiltonian form. It is an observation from the present analysis that while the reduced equation is well defined for the non-zero helicity projections of the wave function, the first time derivative of the zero helicity part of the reduced wave function is essentially undefined (much as if this part described a scalar particle). This helps to explain the “improper” nature of the general Hamiltonian for integer spin.

On Products of Symmetries

In (2) Halmos and Kakutani proved that any unitary operator on an infinite-dimensional Hilbert space is a product of at most four symmetries (self-adjoint unitaries). It is the purpose of this paper to show that if the unitary is an element of a properly infinite von Neumann algebraA(i.e., one with no finite non-zero central projections), then the symmetries may be chosen fromA.A principal tool used in establishing this result is Theorem 1, which was proved by Murray and von Neumann (6, 3.2.3) for type II1factors; see also (3, Lemma 5). The author would like to thank David Topping for raising the question, and for several stimulating conversations on the subject. He is also indebted to the referee for several helpful suggestions.