Normal State Space Research Articles

Normal states are determined by their facial distances II

AbstractLet be a von Neumann algebra with separable predual. For a normal semi‐finite weight on , denote by the von Neumann subalgebra generated by Let be the center of , and be the set of normal semi‐finite weights on . When has no type part (but could have a non‐trivial type part), for every faithful weights with being strictly semi‐finite, if , then there is a positive self‐adjoint operator affiliated with such that . This does not hold for the hyper‐finite type factor. When has no type part, we verify that for strictly semi‐finite weights with and , one has . This is not true for the hyper‐finite type factor. Denote by the subset of consisting of weights with being semi‐finite. When is the direct sum of a semi‐finite algebra and a type algebra, we show that for , if and , then . This fails for any type factor when . Using the above, we establish that when has no type part, the distances of a normal state on to closed faces of the normal state space of uniquely determine this state.

Intelligent fault detection of reciprocating compressor using a novel discrete state space

Aiming to the intelligent fault detection of reciprocating compressors (RCs), a novel method based on discrete state space construction and state space transformation is proposed in this paper. Conventional fault detection methods were conducted with continuous data, which tend to cause overfitting on establishing a model. This paper proposes a novel framework of fault detection on the basis of discrete data. High-dimensional features are extracted from raw signals and discretized into labels representing potential states of operating conditions. These discrete features of normal data are constructed into raw state space, and then transformed into a new state space by latent Dirichlet allocation subsequently. The transformed state space and its distribution can reflect operating conditions with more representative information. Then the distributions of real-time data under the transformed normal state space are calculated. The divergences of real-time state space distribution and normal state space are determined to identify the operating condition of reciprocating compressors, including normal condition and abnormal condition. This proposed method was validated on four cases of RC faults. The result demonstrates the effectiveness of the proposed method on detecting abnormal conditions of RCs.

Tingley's problem through the facial structure of operator algebras

Tingley's problem asks whether every surjective isometry between the unit spheres of two Banach spaces admits an extension to a real linear surjective isometry between the whole spaces. In this paper, we give an affirmative answer to Tingley's problem when both spaces are preduals of von Neumann algebras, the spaces of self-adjoint operators in von Neumann algebras or the spaces of self-adjoint normal functionals on von Neumann algebras. We also show that every surjective isometry between the unit spheres of unital C⁎-algebras restricts to a bijection between their unitary groups. In addition, we show that every surjective isometry between the normal state spaces or the normal quasi-state spaces of two von Neumann algebras extends to a linear surjective isometry.

Skew Normal State Space Modeling of RC Electrical Circuit and Parameters Estimation based on Particle Markov Chain Monte Carlo

Skew Normal State Space Modeling of RC Electrical Circuit and Parameters Estimation based on Particle Markov Chain Monte Carlo

Transition probabilities of normal states determine the Jordan structure of a quantum system

Let Φ : 𝔖(M1) → 𝔖(M2) be a bijection (not assumed affine nor continuous) between the sets of normal states of two quantum systems, modelled on the self-adjoint parts of von Neumann algebras M1 and M2, respectively. This paper concerns with the situation when Φ preserves (or partially preserves) one of the following three notions of “transition probability” on the normal state spaces: the transition probability PU introduced by Uhlmann [Rep. Math. Phys. 9, 273-279 (1976)], the transition probability PR introduced by Raggio [Lett. Math. Phys. 6, 233-236 (1982)], and an “asymmetric transition probability” P0 (as introduced in this article). It is shown that the two systems are isomorphic, i.e., M1 and M2 are Jordan ∗-isomorphic, if Φ preserves all pairs with zero Uhlmann (respectively, Raggio or asymmetric) transition probability, in the sense that for any normal states μ and ν, we have PΦ(μ),Φ(ν)=0 if and only if P(μ, ν) = 0, where P stands for PU (respectively, PR or P0). Furthermore, as an extension of Wigner’s theorem, it is shown that there is a Jordan ∗-isomorphism Θ : M2 → M1 satisfying Φ = Θ∗|𝔖(M1) if and only if Φ preserves the “asymmetric transition probability.” This is also equivalent to Φ preserving the Raggio transition probability. Consequently, if Φ preserves the Raggio transition probability, it will preserve the Uhlmann transition probability as well. As another application, the sets of normal states equipped with either the usual metric, the Bures metric or “the metric induced by the self-dual cone,” are complete Jordan ∗-invariants for the underlying von Neumann algebras.

Local facial structure and norm-exposed faces of the unit ball in a [formula omitted]-triple

Each tripotent u in the bidual A⁎⁎ of a JB⁎-triple A, compact relative to A, gives rise to a complex Banach space P2(u)A with bidual the JBW⁎-algebra A2⁎⁎(u) that is a weak⁎-closed subtriple of A⁎⁎. Using the fact that the family of tripotents in the JBW⁎-algebra A2⁎⁎(u), compact relative to P2(u)A, coincides with the family of tripotents in A2⁎⁎(u), compact relative to A, the facial structure of the unit balls in P2(u)A and its dual A2⁎⁎(u)⁎, the predual of A2⁎⁎(u), is revealed. The JBW-algebra Bu⁎⁎, that is the self-adjoint part of the JBW⁎-algebra A2⁎⁎(u), is the bidual of the unital GM-space Bu that consists of the space of all real-valued weak⁎-continuous affine functions on the normal state space of A2⁎⁎(u). It is therefore possible to investigate the ‘real’ local facial structure of the unit balls in Bu, Bu⁎, and Bu⁎⁎. Some of the results obtained are applied to the problem of identifying the norm-exposed faces of the unit ball A1 in A. It is shown that both locally and globally the norm-exposed faces of A1 are those corresponding to compact support tripotents of elements of norm one in the dual space A⁎ of A.

Particle filters and Bayesian inference in financial econometrics

AbstractIn this paper we reviewsequential Monte Carlo(SMC) methods, or particle filters (PF), with special emphasis on its potential applications in financial time series analysis and econometrics. We start with the well‐known normal dynamic linear model, also known as the normal linear state space model, for which sequential state learning is available in closed form via standard Kalman filter and Kalman smoother recursions. Particle filters are then introduced as a set of Monte Carlo schemes that enable Kalman‐type recursions when normality or linearity or both are abandoned. The seminalbootstrap filter(BF) of Gordon, Salmond and Smith (1993) is used to introduce the SMC jargon, potentials and limitations. We also review the literature on parameter learning, an area that started to attract much attention from the particle filter community in recent years. We give particular attention to the Liu–West filter (2001), Storvik filter (2002) andparticle learning(PL) of Carvalho, Johannes, Lopes and Polson (2010). We argue that the BF and theauxiliary particle filter(APF) of Pitt and Shephard (1999) define two fundamentally distinct directions within the particle filter literature. We also show that the distinction is more pronounced with parameter learning and argue that PL, which follows the APF direction, is an attractive extension. One of our contributions is to sort out the research from BF to APF (during the 1990s), from APF to now (the 2000s) and from Liu–West filter to Storvik filter to PL. To this end, we provide code in R for all the examples of the paper. Readers are invited to find their own way into this dynamic and active research arena. Copyright © 2010 John Wiley & Sons, Ltd.

C*-Independence andW*-Independence of von Neumann Algebras

It is shown that in any von Neumann algebra with infinite-dimensional non-abelian central part there are always finite-dimensional subalgebras which are C*-independent but not W*-independent. On the other hand, it is proved that C*-independence is very close to W*-independence in the following sense: If A and B are C*-independent subalgebras of a von Neumann algebra M then the pairs (φ1,φ2), where φ1 is a normal state on A and φ2 is a normal state on B, which do admit a common normal extension over M form a norm dense set in the product of normal state spaces. Some other independence conditions (strict locality, logical independence) are studied.

Recursive residuals and model diagnostics for normal and non-normal state space models

Model diagnostics for normal and non-normal state space models are based on recursive residuals which are defined from the one-step ahead predictive distribution. Routine calculation of these residuals is discussed in detail. Various diagnostic tools are suggested to check, for example, for wrong observation distributions and for autocorrelation. The paper also discusses such topics as model diagnostics for discrete time series and model discrimination via Bayes factors. The case studies cover environmental applications such as analysing a time series of the number of daily rainfall occurrences and a time series of daily sulfur dioxide emissions.

Lattice of tripotents in a JBW*-triple

The complete lattice of tripotents in a JBW*-triple and the unit ball in its predual are respectively proposed as models for the complete lattice of propositions and for the generalized normal state space of a nonassociative, noncommutative physical system. A subsystem of such a system may be defined in terms of either principal ideals in the complete lattice of propositions or norm-closed faces of the generalized state space. It is shown that the two definitions are equivalent and that each subsystem is associative.

GEOMETRY OF THE STATE SPACE OF MODULAR JORDAN ALGEBRAS

In the class of compact convex sets the normal state spaces of modular Jordan algebras are characterized. To wit,the state spaces are precisely those sets that are modular, spectral, and symmetric.

Classes of conditional expectations over von Neumann algebras

Let N ⊂ M be von Neumann algebras and E ω : M → N an ω-conditional expectation mapping. For a state ψ of N an extension \\ ̃ gy E ω of ψ with respect to E ω is described. The relation E ω ~ E ϑ defined to hold if \\ ̃ gy E ω = \\ ̃ gy E ϑ for every ψ is an equivalence relation. The family of equivalence classes possesses an affine structure and shows analogy with the normal state space of a von Neumann algebra.

Convexity, unitary invariance and monotonicity under completely positive maps over injective vN-algebras

The action of dynamical maps over the normal state space of a properly infinite, injective vN-algebra is analyzed and shown to be equivalent to convec unitary mixing with respect to some suitably chosen C *-subalgebra. As an application, it is shown that the conditions usually imposed on (convex) relative state functionals (like the relative entropy etc.) necessarily imply their decrease under completely positive maps.

Normal state spaces of Jordan and von Neumann algebras

Normal state spaces of Jordan and von Neumann algebras are characterized among convex sets. The normal state spaces of JBW-algebras are precisely those which are spectral and elliptic; along these the normal state spaces of von Neumann algebras are distinguished by the global 3-ball property.

State spaces of Jordan algebras

In this chapter we will discuss properties of the normal state space of JBW-algebras. Since every JB-algebra state space is also the normal state space of a JBW-algebra (Corollary 2.61), these properties also apply to JB-algebra state spaces.

Matrix methods for the study of a regulated synchronous machine

A powerful but simple matrix method for the digital computer manipulation of large sets of differential and algebraic equations is introduced to the field of power systems. The method is first illustrated with reference to the linearized equations representing a general-purpose turboalternator model. Subsequently, the form of the system equations obtained as a result of the matrix method is seen to provide a unified approach to the determination of system stability limits using Routh, Nyquist, or eigenvalue methods. The normal state space form of the system equations is also shown to facilitate control studies. A previously suggested performance criterion for an excitation system is generalized, and a systematic method for the simultaneous optimization of governor and exciter control loops is suggested. This method of optimization does not require the evaluation of system transient responses. The manner in which the normal state space form of the system equations leads naturally to the synthesis of controllers which are optimal with respect to a chosen scalar quadratic performance index is indicated. With this form of control, the structure is not defined a priori but emerges from the computational algorithm and it becomes unnecessary to specify the source or the magnitude of the stabilizing signals required to provide an improved dynamic performance. Finally, the matrix method is shown to be effective for the manipulation of the nonlinear machine equations and these are obtained in a new form which is directly amenable to digital or analog computer solution.