Narrow Results

In this paper, we extend a Central Limit Theorem, recently established for the Thompson group $F=F_2$ by Krishnan, to the Brown–Thompson groups $F_p$, where $p$ is any integer greater than or equal to $2$. The non-commutative probability space considered is the group algebra $ℂ[F_p]$, equipped with the canonical trace. The random variables in question are $a_n:=(x_n+x_n^{-1})/\sqrt{2}$, where ${\{x_i\}}_{i\ge 0}$ represents the standard family of infinite generators. Analogously to the case of $F=F_2$, it is established that the limit distribution of $s_n=(a_0+\cdots +a_{n-1})/\sqrt{n}$ converges to the standard normal distribution. Furthermore, it is demonstrated that for a state corresponding to Jones’s oriented subgroup is denoted by $\overrightarrow{F}$, such a Central Limit Theorem does not hold.

A philosophically consistent axiomatic approach to classical and quantum mechanics is given. The approach realizes a strong formal implementation of Bohr's correspondence principle. In all instances, classical and quantum concepts are fully parallel: the same general theory has a classical realization and a quantum realization. Extending the ''probability via expectation'' approach of Whittle to noncommuting quantities, this paper defines quantities, ensembles, and experiments as mathematical concepts and shows how to model complementarity, uncertainty, probability, nonlocality and dynamics in these terms. The approach carries no connotation of unlimited repeatability; hence it can be applied to unique systems such as the universe. Consistent experiments provide an elegant solution to the reality problem, confirming the insistence of the orthodox Copenhagen interpretation on that there is nothing but ensembles, while avoiding its elusive reality picture. The weak law of large numbers explains the emergence of classical properties for macroscopic systems.

A probability density of fractional (or fractal) order is defined by the probability increment pr{x < X ≤ x+dx} = p_α(x)(dx)^α, 0 < α < 1, and appears to be quite suitable to deal with random variables defined in a fractal space. Combining this definition with the fractional Taylor's series denotes the Mittag–Leffler function) provided by the modified Riemann–Liouville derivative, one can expand a probability calculus parallel to the standard one. This approach could be considered as a framework for the derivation of some space fractional partial differential diffusion equations in coarse-grained spaces. It is shown firstly that there is some relation between fractional probability and signed measure of probability, and secondly that when α = 1/2, there is some identity between this fractal probability and quantum probability. Shortly, a wavefunction could be thought of as a fractal probability density of order 1/2. One exhibits further relations with possibility theory and relative information. Lastly, one arrives at a new informational entropy based on the inverse of the Mittag–Leffler function.

Stimulated by the quantum generalization of the canonical representation theory for Gaussian processes in Ref. 1, we first give the representations (not necessarily canonical) of two stationary Gaussian processes X and Y by means of white noises q_t and p_t with no assumptions on their commutator. We then assume that q_t + ip_t annihilates the vacuum state and prove that the representations are the joint Boson–Fock ones if and only if X and Y have a scalar commutator.

Cabana-Duvillard and lonescu¹¹ have proved that any symmetric probability measure with moments of any order can be obtained as central limit theorem of self-adjoint, weakly independent and symmetrically distributed (in a quantum souse) random variables. Results of this type will be called "universal central limit theorem".

Using Interacting Fock Space (IFS) techniques we extend this result in two directions: (i) we prove that the random variables can be taken to be generalized Gaussian in the sense of Accardi and Bożejko³ and we give a realization of such random variables as sums of creation, annihilation and preservation operators acting on an appropriate IFS; (ii) we extend the above-mentioned result to the nonsymmetric case. The nontrivial difference between the symmetric and the nonsymmetric case is explained at the end of the introduction below.

We present a short and elegant proof of the complete theory of strict representations of the algebra of all adjointable operators on a Hilbert -module E by operators on a Hilbert -module F. An analogue for W*-modules and normal representations is also proved. As an application we furnish a new proof of Blecher's Eilenberg–Watts theorem.

For quite some time there were only two rather difficult and involved proofs available, the original one by Arveson and a more recent one by Liebscher, of the fact that for every Arveson system there exists an E₀-semigroup. We put together two recent short proofs, one by Skeide and one by Arveson, to obtain a simpler one, which unifies the advantages of each proof and discards their disadvantages.

Almost every paper about Arveson systems (i.e. product systems of Hilbert spaces) starts by recalling their basic classification assigning to every Arveson system a type and an index. So it is natural to ask in how far an analogue classification can also be proposed for product systems of Hilbert modules. However, while the definition of type is plain, there are obstacles for the definition of index. But all obstacles can be removed when restricting to the category which we introduce here as spatial product systems and that matches the usual definition of spatial in the case of Arveson systems. This is not really a loss because the definition of index for nonspatial Arveson systems is rather formal and does not reflect the information the index carries for spatial Arveson systems.

E₀-semigroups give rise to product systems. Our definition of spatial product system, namely, existence of a unital unit that is central, matches Powers' definition of spatial in the sense that the E₀-semigroup from which the product system is derived admits a semigroup of intertwining isometries. We show that every spatial product system contains a unique maximal completely spatial subsystem (generated by all units) that is isomorphic to a product system of time ordered Fock modules. (There exist nonspatial product systems that are generated by their units. Consequently, these cannot be Fock modules.) The index of a spatial product system we define as the (unique) Hilbert bimodule that determines the Fock module. In order to show that the index merits the name index we provide a product of product systems under which the index is additive (direct sum). While for Arveson systems there is the tensor product, for general product systems the tensor product does not make sense as a product system. Even for Arveson systems our product is, in general, only a subsystem of the tensor product. Moreover, its construction depends explicitly on the choice of the central reference units of its factors.

Spatiality of a product system means that it may be derived from an E₀-semigroup with an invariant vector expectation, i.e. from a noise. We extend our product of spatial product systems to a product of noises and study its properties.

Finally, we apply our techniques to show the module analogue of Fowler's result that free flows are comletely spatial, and we compute their indices.

We study white noise Heisenberg equations giving rise to flows which are *-automorphisms of the observable algebra, but not necessarily inner automorphisms. We prove that the causally normally ordered form of these white noise Heisenberg equations are equivalent to Evans–Hudson flows. This gives in particular, the microscopic structure of the maps defining these flows, in terms of the original white noise derivations.

We show that every continuous product system of correspondences over a unitalC*-algebra occurs as the product system of a strictly continuousE₀-semigroup.

We solve, mainly by counterexamples, many natural questions regarding maximal commutative subalgebras invariant under CP-maps or semigroups of CP-maps on a von Neumann algebra. In particular, we discuss the structure of the generators of norm continuous semigroups on leaving a maximal commutative subalgebra invariant and show that there exist Markov CP-semigroups on M_d without invariant maximal commutative subalgebras for any d > 2.

This note is devoted to extension of quantum probability calculus to generalizations of complex Hilbert space. Starting with Hilbert space over complex hyperbolic numbers, we derive general hyper-trigonometric interference of probabilities.

We construct a growing graph such that the Jacobi coefficient of the asymptotic spectral distribution is . The distribution is a χ² distribution with one degree of freedom normalized to have mean 0 and variance 1.

We consider a family of infinite dimensional Laplace operators which contains the classical Lévy–Laplacian. We prove a representation of these operators as a quadratic functions of quantum stochastic processes. Particularly, for the classical Lévy–Laplacian, the following formula is proved: Δ_L = lim_ε→0 ∫_‖s-t‖<ε b_sb_tdsdt, where b_t is the annihilation process.

Multi-photon models are theoretically and experimentally important because in them quantum properly phenomena are verified; as well as squeezed light and quantum entanglement also play a relevant role in quantum information and quantum communication (see Refs. 18–20).

In this paper we study a generic model of a multi-photon system with an arbitrary number of pumping and subharmonics fields. This model includes measurement on the system, as could be direct or homodyne detection and we demonstrate the existence of dynamics in the context of Continuous Measurement Theory of Open Quantum Systems (see Refs. 1–11) using Quantum Stochastic Differential Equations with unbounded coefficients (see Refs. 10–15).

We consider Glauber dynamics on finite cycles. By introducing a vacuum state we consider an algebraic probability space for the generator of the dynamics. We obtain a quantum decomposition of the generator and construct an interacting Fock space. As a result we obtain a distribution of the generator in the vacuum state. We also discuss the monotonicity of the moments of spectral measure as the couplings increase. In particular, when the couplings are assumed to be uniform, as the cycle grows to an infinite chain, we show that the distribution (under suitable dilation and translation) converges to a Kesten distribution.

It is well known that freeness appears in the high-dimensional limit of independence for matrices. Thus, for instance, the additive free Brownian motion can be seen as the limit of the Brownian motion on hermitian matrices. More generally, it is quite natural to try to build free Lévy processes as high-dimensional limits of classical matricial Lévy processes.

We will focus here on one specific such construction, discussing and generalizing the work done previously by Biane in Ref.2, who has shown that the (classical) Brownian motion on the Unitary group U(d) converges to the free multiplicative Brownian motion when d goes to infinity. We shall first recall that result and give an alternative proof for it. We shall then see how this proof can be adapted in a more general context in order to get a free Lévy process on the dual group (in the sense of Voiculescu) U〈n〉. This result will actually amount to a truly noncommutative limit theorem for classical random variables, of which Biane's result constitutes the case n = 1.

In a fundamental lemma we characterize “generating functions” of certain functors on the category of algebraic non-commutative probability spaces. Special families of such generating functions correspond to “unital, associative universal products” on this category, which again define a notion of non-commutative stochastic independence. Using the fundamental lemma, we prove the existence of cumulants and of “cumulant Lie algebras” for all independences coming from a unital, associative universal product. These include the five independences (tensor, free, Boolean, monotone, anti-monotone) appearing in Muraki’s classification, c-free independence of Bożejko and Speicher, the indented product of Hasebe and the bi-free independence of Voiculescu. We show how the non-commutative independence can be reconstructed from its cumulants and cumulant Lie algebras.

The subject of the present paper is an application of quantum probability to $p$-adic objects. We give a quantum-probabilistic interpretation of the spherical Hecke algebra for ${\mathrm{\text{PGL}}}_2(F)$, where $F$ is a $p$-adic field. As a byproduct, we obtain a new proof of the Fourier inversion formula for ${\mathrm{\text{PGL}}}_2(F)$.

This is a brief account of Prof. K. R. Parthasarathy’s life and author’s interactions with him.